[ { "id": 1, "problem_number": "MPP-001", "title": "P versus NP Problem", "statement": "Does $P = NP$? More formally: if the solution to a problem can be quickly verified (in polynomial time), can the solution also be quickly found (in polynomial time)?", "background": "The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified can also be quickly solved. The Clay Mathematics Institute has offered a $1,000,000 prize for a correct solution.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Stephen Cook", "proposed_year": 1971, "category_id": 15, "set_id": 1, "view_count": 1523, "favorite_count": 89, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 1, "name": "millennium_prize", "display_name": "Millennium Prize Problems", "description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.", "slug": "millennium-prize", "order_index": 1, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2, "problem_number": "MPP-002", "title": "The Riemann Hypothesis", "statement": "Do all non-trivial zeros of the Riemann zeta function $\\zeta(s)$ have real part equal to $\\frac{1}{2}$?", "background": "The Riemann hypothesis, proposed by Bernhard Riemann in 1859, concerns the distribution of prime numbers. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line $\\Re(s) = \\frac{1}{2}$. This is one of the most important open problems in mathematics, with profound implications for number theory.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Bernhard Riemann", "proposed_year": 1859, "category_id": 1, "set_id": 1, "view_count": 2341, "favorite_count": 156, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 1, "name": "millennium_prize", "display_name": "Millennium Prize Problems", "description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.", "slug": "millennium-prize", "order_index": 1, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3, "problem_number": "MPP-003", "title": "Yang–Mills Existence and Mass Gap", "statement": "Prove that Yang–Mills theory exists and has a mass gap on $\\mathbb{R}^4$, meaning the quantum particles have positive masses.", "background": "This problem concerns quantum field theory and seeks to establish a rigorous mathematical foundation for Yang–Mills theories, which describe fundamental forces in particle physics. A solution would require proving the existence of these theories in four-dimensional spacetime and showing they predict a mass gap.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Yang Chen-Ning and Robert Mills", "proposed_year": 1954, "category_id": 16, "set_id": 1, "view_count": 1234, "favorite_count": 78, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 1, "name": "millennium_prize", "display_name": "Millennium Prize Problems", "description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.", "slug": "millennium-prize", "order_index": 1, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 4, "problem_number": "MPP-004", "title": "Navier–Stokes Existence and Smoothness", "statement": "Prove or give a counterexample: Do solutions to the Navier–Stokes equations in three dimensions always exist and remain smooth for all time?", "background": "The Navier–Stokes equations describe the motion of fluids. While solutions exist for short times and in two dimensions, the question of whether smooth solutions exist globally in three dimensions remains open. This has profound implications for understanding turbulence and fluid dynamics.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Claude-Louis Navier and George Gabriel Stokes", "proposed_year": 1822, "category_id": 9, "set_id": 1, "view_count": 1456, "favorite_count": 89, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 1, "name": "millennium_prize", "display_name": "Millennium Prize Problems", "description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.", "slug": "millennium-prize", "order_index": 1, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 5, "problem_number": "MPP-005", "title": "Birch and Swinnerton-Dyer Conjecture", "statement": "The conjecture relates the rank of the abelian group of rational points of an elliptic curve to the order of zero of the associated L-function at $s=1$.", "background": "This conjecture connects the arithmetic of elliptic curves (solutions to equations of the form $y^2 = x^3 + ax + b$) to the behavior of certain complex functions. It has deep connections to number theory and algebraic geometry.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Bryan Birch and Peter Swinnerton-Dyer", "proposed_year": 1960, "category_id": 1, "set_id": 1, "view_count": 1123, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 1, "name": "millennium_prize", "display_name": "Millennium Prize Problems", "description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.", "slug": "millennium-prize", "order_index": 1, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 6, "problem_number": "MPP-006", "title": "Hodge Conjecture", "statement": "On a projective non-singular algebraic variety over $\\mathbb{C}$, any Hodge class is a rational linear combination of classes of algebraic cycles.", "background": "The Hodge conjecture is a major open problem in algebraic geometry. It seeks to relate the topology of a smooth complex projective variety to its algebraic structure, specifically asserting that certain topological cycles are actually algebraic.", "difficulty_level_id": 5, "status": "open", "proposed_by": "William Vallance Douglas Hodge", "proposed_year": 1950, "category_id": 5, "set_id": 1, "view_count": 987, "favorite_count": 54, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 1, "name": "millennium_prize", "display_name": "Millennium Prize Problems", "description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.", "slug": "millennium-prize", "order_index": 1, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 8, "problem_number": "NT-001", "title": "Odd Perfect Numbers", "statement": "Does there exist an odd perfect number? A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself). For example, $6 = 1 + 2 + 3$ is perfect.", "background": "While many even perfect numbers are known (the first few are 6, 28, 496, 8128), no odd perfect number has ever been found, despite extensive computer searches. It has been proven that if one exists, it must be greater than $10^{1500}$ and have at least 101 prime factors.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 543, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 9, "problem_number": "NT-002", "title": "Collatz Conjecture", "statement": "Starting with any positive integer $n$, repeatedly apply the function: if $n$ is even, divide by 2; if $n$ is odd, multiply by 3 and add 1. Does this process always eventually reach 1?", "background": "Also known as the 3n+1 problem, this deceptively simple conjecture has been verified for all starting values up to $2^{68}$ but remains unproven. Paul Erdős said about it: \"Mathematics may not be ready for such problems.\"", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 892, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 11, "problem_number": "NT-003", "title": "Twin Prime Conjecture", "statement": "Are there infinitely many twin primes? Twin primes are pairs of primes that differ by 2, such as (3, 5), (5, 7), (11, 13), (17, 19), (29, 31).", "background": "The twin prime conjecture is one of the oldest unsolved problems in number theory. In 2013, Yitang Zhang proved that there are infinitely many pairs of primes that differ by at most 70 million. This bound has since been reduced to 246, but the gap of 2 remains unproven.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 1234, "favorite_count": 89, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 12, "problem_number": "NT-004", "title": "Goldbach's Conjecture", "statement": "Every even integer greater than 2 can be expressed as the sum of two primes.", "background": "Proposed by Christian Goldbach in 1742, this conjecture has been verified computationally for all even integers up to very large numbers. The weak Goldbach conjecture (every odd number greater than 5 is the sum of three primes) was proved by Harald Helfgott in 2013, but the strong version remains open.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Christian Goldbach", "proposed_year": 1742, "category_id": 1, "view_count": 1567, "favorite_count": 112, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 13, "problem_number": "NT-005", "title": "ABC Conjecture", "statement": "For any $\\epsilon > 0$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers with $a + b = c$ such that $c > \\text{rad}(abc)^{1+\\epsilon}$, where $\\text{rad}(n)$ is the product of distinct prime factors of $n$.", "background": "The ABC conjecture, formulated by Joseph Oesterlé and David Masser in 1985, has profound implications for number theory. Shinichi Mochizuki claimed a proof in 2012 using his \"inter-universal Teichmüller theory,\" but the proof remains controversial and not widely accepted.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Joseph Oesterlé and David Masser", "proposed_year": 1985, "category_id": 1, "view_count": 876, "favorite_count": 45, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 10, "problem_number": "COMB-001", "title": "The Hadwiger-Nelson Problem", "statement": "What is the minimum number of colors needed to color the points of the plane such that no two points at distance 1 have the same color?", "background": "It is known that this chromatic number is between 5 and 7. In 2018, Aubrey de Grey proved it is at least 5, but whether it is 5, 6, or 7 remains unknown.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 421, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 14, "problem_number": "GT-001", "title": "Hadwiger Conjecture", "statement": "Every graph with chromatic number $k$ has a $K_k$ minor (where $K_k$ is the complete graph on $k$ vertices).", "background": "The Hadwiger conjecture, proposed in 1943, generalizes the four color theorem. It has been proved for $k \\leq 6$ but remains open for $k \\geq 7$. The case $k=5$ is equivalent to the four color theorem.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Hugo Hadwiger", "proposed_year": 1943, "category_id": 3, "view_count": 654, "favorite_count": 38, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 15, "problem_number": "GT-002", "title": "Reconstruction Conjecture", "statement": "Every finite simple graph on at least 3 vertices is uniquely determined by its vertex-deleted subgraphs.", "background": "The reconstruction conjecture asks whether a graph can be uniquely reconstructed from the multiset of all its vertex-deleted subgraphs. Proposed by Stanisław Ulam in 1942, it has been verified for many classes of graphs but remains open in general.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Stanisław Ulam", "proposed_year": 1942, "category_id": 3, "view_count": 432, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 18, "problem_number": "TOP-001", "title": "Smooth 4-Dimensional Poincaré Conjecture", "statement": "Is every smooth homotopy 4-sphere diffeomorphic to the standard 4-sphere $S^4$?", "background": "The smooth Poincaré conjecture in dimension 4 is the only remaining case of the generalized Poincaré conjecture. It has been solved in all other dimensions: dimension 3 by Perelman, higher dimensions by Smale, Freedman, and others. The 4-dimensional case is particularly difficult due to exotic smooth structures.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 789, "favorite_count": 42, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 19, "problem_number": "GEO-002", "title": "Sphere Packing in Higher Dimensions", "statement": "What is the densest packing of congruent spheres in $n$ dimensions for $n \\geq 4$?", "background": "The sphere packing problem asks for the densest arrangement of non-overlapping spheres. Maryna Viazovska solved it for dimension 8 in 2016, and she with collaborators solved it for dimension 24 in 2017. The problem remains open for most other dimensions.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 456, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 20, "problem_number": "ALG-001", "title": "Inverse Galois Problem", "statement": "Is every finite group the Galois group of some Galois extension of the rational numbers $\\mathbb{Q}$?", "background": "The inverse Galois problem asks whether every finite group can be realized as the Galois group of a polynomial equation with rational coefficients. It has been solved for many classes of groups, including all symmetric and alternating groups, but remains open in general.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 543, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 21, "problem_number": "ALG-002", "title": "Kaplansky's Conjectures", "statement": "A set of conjectures about group rings: (1) Zero divisor conjecture: If $G$ is a torsion-free group and $K$ is a field, then $K[G]$ has no zero divisors. (2) Idempotent conjecture: The only idempotents in $K[G]$ are 0 and 1. (3) Unit conjecture: The only units in $\\mathbb{Z}[G]$ are of the form $\\pm g$ for $g \\in G$.", "background": "These conjectures, proposed by Irving Kaplansky in the 1940s, concern the algebraic structure of group rings. They have been verified for many classes of groups but remain open in general. The zero divisor conjecture is related to the Atiyah conjecture.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Irving Kaplansky", "proposed_year": 1940, "category_id": 4, "view_count": 321, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 22, "problem_number": "SET-001", "title": "Continuum Hypothesis", "statement": "There is no set whose cardinality is strictly between that of the integers and the real numbers.", "background": "The continuum hypothesis was the first of Hilbert's 23 problems. Kurt Gödel (1940) and Paul Cohen (1963) proved it is independent of ZFC set theory: it can neither be proved nor disproved from the standard axioms. Whether to accept it as an axiom remains a philosophical question.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Georg Cantor", "proposed_year": 1878, "category_id": 10, "view_count": 1234, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 25, "problem_number": "NT-006", "title": "Legendre's Conjecture", "statement": "For every positive integer $n$, there exists a prime number between $n^2$ and $(n+1)^2$.", "background": "This conjecture about the distribution of prime numbers was proposed by Adrien-Marie Legendre in 1808. Despite significant progress in prime number theory, including the prime number theorem, this simple statement remains unproven.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Adrien-Marie Legendre", "proposed_year": 1808, "category_id": 1, "view_count": 432, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 26, "problem_number": "NT-007", "title": "Are there infinitely many Mersenne primes?", "statement": "Are there infinitely many prime numbers of the form $M_p = 2^p - 1$ where $p$ is prime?", "background": "Mersenne primes are primes of the form $2^p - 1$. As of 2024, only 51 Mersenne primes are known, with the largest being $2^{82,589,933} - 1$. It is conjectured that infinitely many exist, but this remains unproven. They are important for computational number theory and the GIMPS distributed computing project.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 654, "favorite_count": 38, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 27, "problem_number": "NT-008", "title": "Are there infinitely many perfect powers in the Fibonacci sequence?", "statement": "Besides 1, 8, and 144, are there any other perfect powers (numbers of the form $a^b$ where $a, b > 1$) in the Fibonacci sequence?", "background": "The Fibonacci sequence has only three known perfect powers: $F_1 = F_2 = 1 = 1^n$, $F_6 = 8 = 2^3$, and $F_{12} = 144 = 12^2$. It is conjectured that these are the only ones, but this remains unproven despite extensive computational searches.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 345, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 28, "problem_number": "NT-009", "title": "Gilbreath's Conjecture", "statement": "Starting with the sequence of primes and repeatedly taking absolute differences of consecutive terms, the first term of each row is always 1.", "background": "Norman Gilbreath observed in 1958 that applying the forward difference operator to the sequence of primes appears to always yield 1 as the first element. Despite being verified computationally for the first $10^{13}$ primes, no proof exists.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Norman Gilbreath", "proposed_year": 1958, "category_id": 1, "view_count": 287, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 29, "problem_number": "COMB-003", "title": "Ramsey Number R(5,5)", "statement": "What is the exact value of $R(5,5)$, the smallest number $n$ such that any 2-coloring of the edges of $K_n$ contains a monochromatic $K_5$?", "background": "Ramsey theory asks how large a structure must be to guarantee a certain property. The Ramsey number $R(5,5)$ is known to lie between 43 and 48, but the exact value remains unknown. As Joel Spencer said, \"Erdős asks us to imagine an alien force, demanding the value of $R(5,5)$ or they will destroy our planet... our best strategy is to get our best computers and mathematicians working on it.\"", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 543, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 30, "problem_number": "COMB-004", "title": "The Lonely Runner Conjecture", "statement": "For any $n$ runners on a circular track with distinct constant speeds, each runner is \"lonely\" (distance at least $1/n$ from all others) at some time.", "background": "This combinatorial conjecture, proposed by J.M. Wills in 1967, has been verified for up to 7 runners but remains open for 8 or more. It has connections to Diophantine approximation and view-obstruction problems.", "difficulty_level_id": 3, "status": "open", "proposed_by": "J.M. Wills", "proposed_year": 1967, "category_id": 2, "view_count": 234, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 31, "problem_number": "GT-003", "title": "The Graceful Tree Conjecture", "statement": "Every tree can be gracefully labeled: vertices can be assigned distinct labels from $\\{0, 1, \\ldots, |E|\\}$ such that edge labels (absolute differences) are all distinct.", "background": "The graceful labeling conjecture, proposed by Alexander Rosa in 1967, asks whether every tree admits a graceful labeling. It has been verified for many classes of trees including paths, caterpillars, and trees with at most 35 vertices, but remains open in general.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander Rosa", "proposed_year": 1967, "category_id": 3, "view_count": 321, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 32, "problem_number": "GEO-003", "title": "The Kakeya Conjecture", "statement": "A Kakeya set (containing a unit line segment in every direction) in $\\mathbb{R}^n$ must have Hausdorff dimension $n$.", "background": "The Kakeya conjecture concerns the minimal \"size\" of sets containing line segments in all directions. It has deep connections to harmonic analysis and PDE. The conjecture is known in dimension 2 but remains open for $n \\geq 3$. It would have important implications for the restriction conjecture in Fourier analysis.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 432, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 33, "problem_number": "GEO-004", "title": "The Moving Sofa Problem", "statement": "What is the largest area of a shape that can be maneuvered through an L-shaped corridor of unit width?", "background": "This classic problem in geometric optimization asks for the largest \"sofa\" that can navigate a right-angled hallway. The best known lower bound is approximately 2.2195 (Gerver's sofa, 1992), and the upper bound is $2\\sqrt{2} \\approx 2.8284$. The exact answer remains unknown.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 567, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 34, "problem_number": "TOP-002", "title": "The Volume Conjecture", "statement": "For a hyperbolic knot $K$, the limit of normalized colored Jones polynomials equals the hyperbolic volume of the knot complement.", "background": "The volume conjecture, proposed by Rinat Kashaev in 1995 and generalized by Murakami and Murakami in 2001, connects quantum invariants of knots to their classical geometric properties. It relates quantum topology to hyperbolic geometry and has been verified for many knots but remains unproven in general.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Rinat Kashaev", "proposed_year": 1995, "category_id": 7, "view_count": 298, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 36, "problem_number": "AG-001", "title": "The Standard Conjectures on Algebraic Cycles", "statement": "A collection of conjectures about algebraic cycles on smooth projective varieties, including Lefschetz standard conjecture and Künneth standard conjecture.", "background": "The standard conjectures, formulated by Alexander Grothendieck in the 1960s, concern the theory of algebraic cycles and their cohomology. They would have profound consequences for algebraic geometry, including the independence of Betti numbers from the choice of Weil cohomology theory. The Hodge conjecture would follow from the Lefschetz standard conjecture.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Alexander Grothendieck", "proposed_year": 1965, "category_id": 5, "view_count": 432, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 37, "problem_number": "AG-002", "title": "The Abundance Conjecture", "statement": "For a minimal model $X$ of non-negative Kodaira dimension, the canonical divisor $K_X$ is semi-ample.", "background": "The abundance conjecture is a major open problem in birational algebraic geometry and the minimal model program. It predicts that canonical divisors on minimal models have good positivity properties. The conjecture is known in dimension 3 and in many special cases, but remains open in dimension 4 and higher.", "difficulty_level_id": 4, "status": "open", "category_id": 5, "view_count": 298, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 38, "problem_number": "ALG-003", "title": "The Köthe Conjecture", "statement": "A ring has no non-zero nil ideal (an ideal all of whose elements are nilpotent) if and only if it has no non-zero nil one-sided ideal.", "background": "The Köthe conjecture concerns the structure of rings with nilpotent elements. Proposed by Gottfried Köthe in 1930, it remains one of the oldest open problems in ring theory. Various special cases have been resolved, but the general conjecture remains open.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Gottfried Köthe", "proposed_year": 1930, "category_id": 4, "view_count": 234, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 40, "problem_number": "PDE-001", "title": "The Regularity Problem for Euler Equations", "statement": "Do solutions to the 3D Euler equations for incompressible fluid flow remain smooth for all time, given smooth initial data?", "background": "The Euler equations describe the motion of inviscid (frictionless) fluids. While the Navier-Stokes equations include viscosity and are a Millennium Prize Problem, the regularity of Euler equations is also a major open question. Finite-time blowup would have profound implications for fluid dynamics.", "difficulty_level_id": 4, "status": "open", "category_id": 9, "view_count": 456, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 41, "problem_number": "SET-002", "title": "Singular Cardinals Hypothesis", "statement": "If $\\kappa$ is a singular strong limit cardinal, then $2^\\kappa = \\kappa^+$.", "background": "The singular cardinals hypothesis, formulated by Paul Erdős and András Hajnal, concerns the behavior of the power set operation on infinite cardinals. It sits between the generalized continuum hypothesis and ZFC. Its consistency and independence status remains a major open problem in set theory.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Paul Erdős and András Hajnal", "category_id": 10, "view_count": 287, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 42, "problem_number": "SET-003", "title": "Whitehead Problem", "statement": "Is every abelian group $A$ such that $\\text{Ext}^1(A, \\mathbb{Z}) = 0$ a free abelian group?", "background": "The Whitehead problem, posed by J.H.C. Whitehead in 1950, asks about the structure of certain abelian groups. Shelah proved in 1973 that the problem is independent of ZFC: it is true under the constructible universe axiom (V=L) but can be false under other set-theoretic axioms.", "difficulty_level_id": 4, "status": "open", "proposed_by": "J.H.C. Whitehead", "proposed_year": 1950, "category_id": 10, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 43, "problem_number": "CS-001", "title": "The Unique Games Conjecture", "statement": "For certain constraint satisfaction problems (unique games), it is NP-hard to approximate the maximum fraction of satisfiable constraints beyond a certain threshold.", "background": "The Unique Games Conjecture, proposed by Subhash Khot in 2002, has become central to computational complexity theory. If true, it would imply optimal hardness results for many approximation problems. Khot was awarded the Nevanlinna Prize in 2014 for this work, despite the conjecture remaining unresolved.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Subhash Khot", "proposed_year": 2002, "category_id": 15, "view_count": 543, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 44, "problem_number": "CS-002", "title": "The Polynomial Hirsch Conjecture", "statement": "The diameter of the graph of a $d$-dimensional polytope with $n$ facets is bounded by a polynomial in $d$ and $n$.", "background": "The original Hirsch conjecture (diameter at most $n - d$) was disproved in 2010 by Francisco Santos. The polynomial Hirsch conjecture is a weaker version that remains open and is important for understanding the complexity of the simplex algorithm for linear programming.", "difficulty_level_id": 3, "status": "open", "category_id": 15, "view_count": 321, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 45, "problem_number": "HIL-012", "title": "Hilbert's 12th Problem: Extension of Kronecker-Weber Theorem", "statement": "Extend the Kronecker-Weber theorem on abelian extensions of the rationals to any base number field.", "background": "Hilbert's 12th problem, posed in 1900, asks for an explicit construction of abelian extensions of number fields, generalizing the Kronecker-Weber theorem which states that every abelian extension of the rationals is contained in a cyclotomic field. Despite significant progress in class field theory, the problem of finding explicit generators remains largely open.", "difficulty_level_id": 5, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 1, "set_id": 2, "view_count": 345, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 46, "problem_number": "HIL-016", "title": "Hilbert's 16th Problem: Topology of Algebraic Curves and Limit Cycles", "statement": "Determine the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$, and investigate the topology of real algebraic curves and surfaces.", "background": "Posed by David Hilbert in 1900, this two-part problem concerns (1) the topology of real algebraic varieties and (2) the limit cycles of planar polynomial differential equations. While it was shown in 1991-1992 by Ilyashenko and Écalle that polynomial vector fields have finitely many limit cycles, the question of whether there exists a finite upper bound H(n) for degree n remains open for any n > 1.", "difficulty_level_id": 5, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 6, "set_id": 2, "view_count": 432, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 47, "problem_number": "LAN-004", "title": "Landau's Fourth Problem: Primes of the Form n² + 1", "statement": "Are there infinitely many primes of the form $n^2 + 1$?", "background": "One of Landau's four problems presented at the 1912 International Congress of Mathematicians, this asks whether there are infinitely many primes that are one more than a perfect square. Examples include 2, 5, 17, 37, 101, 197, 257, 401. Despite being simple to state, it has remained unsolved for over 110 years and is considered \"unattackable at the present state of mathematics.\"", "difficulty_level_id": 4, "status": "open", "proposed_by": "Edmund Landau", "proposed_year": 1912, "category_id": 1, "set_id": 6, "view_count": 398, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 6, "name": "landau_problems", "display_name": "Landau's Problems", "description": "Four basic problems about prime numbers posed by Edmund Landau at the 1912 International Congress of Mathematicians.", "slug": "landau-problems", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 48, "problem_number": "SMA-004", "title": "Smale's 4th Problem: Integer Zeros of Polynomials", "statement": "Find efficient algorithms for deciding whether a polynomial with integer coefficients has an integer root.", "background": "Part of Stephen Smale's 18 problems for the 21st century (1998), this problem asks for polynomial-time algorithms to determine if a polynomial equation has integer solutions. This is related to Hilbert's 10th problem, which was shown to be undecidable in general, but specific cases and algorithms with better complexity remain of interest.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 15, "set_id": 5, "view_count": 287, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 49, "problem_number": "SMA-005", "title": "Smale's 5th Problem: Height Bounds for Diophantine Curves", "statement": "Find effective uniform bounds for the heights of rational points on algebraic curves.", "background": "From Smale's 1998 list, this problem addresses the challenge of bounding the size of integer solutions to algebraic equations. While Faltings proved that curves of genus > 1 have finitely many rational points, the question of effective bounds on their heights remains a major open problem in arithmetic geometry.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 5, "set_id": 5, "view_count": 234, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 50, "problem_number": "SMA-006", "title": "Smale's 6th Problem: Finiteness of Central Configurations", "statement": "For the Newtonian $n$-body problem with positive masses, are there only finitely many central configurations (relative equilibria) for each $n$?", "background": "This problem from Smale's 1998 list concerns celestial mechanics and asks whether gravitating bodies can have only finitely many stable equilibrium configurations. The question is known to be true for n = 3 and n = 4, but remains open for n ≥ 5. It connects classical mechanics with algebraic geometry.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 6, "set_id": 5, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 51, "problem_number": "SMA-007", "title": "Smale's 7th Problem: Distribution of Points on the 2-Sphere", "statement": "What is the optimal arrangement of $n$ points on the 2-sphere to minimize energy for various potential functions?", "background": "Smale's 7th problem (1998) asks for the configuration that minimizes various energy functionals for points on a sphere. This includes the Thomson problem (electrons on a sphere) and related optimization questions. Solutions are known for small n and highly symmetric cases, but the general problem remains open and connects to crystallography and coding theory.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 6, "set_id": 5, "view_count": 267, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 52, "problem_number": "SMA-009", "title": "Smale's 9th Problem: Linear Programming in Polynomial Time", "statement": "Find a strongly polynomial algorithm for linear programming.", "background": "Smale's 9th problem (1998) asks whether there exists an algorithm for linear programming whose running time is polynomial in the number of constraints and variables, independent of the bit-size of the input. While linear programming is solvable in polynomial time, no strongly polynomial algorithm is known for the general case.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 15, "set_id": 5, "view_count": 312, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 53, "problem_number": "SMA-010", "title": "Smale's 10th Problem: The Pugh Closing Lemma", "statement": "Is the $C^r$ closing lemma true for dynamical systems?", "background": "The closing lemma in dynamical systems theory asks whether, for a diffeomorphism with a nonwandering point, there is an arbitrarily small perturbation that makes that point periodic. Pugh proved a $C^1$ version in 1967, but the $C^r$ version for r ≥ 2 remains open. This is Smale's 10th problem from his 1998 list.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 6, "set_id": 5, "view_count": 176, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 54, "problem_number": "SMA-016", "title": "The Jacobian Conjecture", "statement": "If $F: \\mathbb{C}^n \\to \\mathbb{C}^n$ is a polynomial map with constant non-zero Jacobian determinant, then $F$ is invertible.", "background": "The Jacobian conjecture, proposed in 1939 and featured as Smale's 16th problem (1998), asks whether polynomial maps with nowhere-vanishing Jacobian determinant are necessarily invertible. Despite its elementary statement, it has resisted numerous attempts at proof. The conjecture is known to be true in dimension 1 and for maps of degree at most 2, but remains open in general.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Ott-Heinrich Keller", "proposed_year": 1939, "category_id": 4, "set_id": 5, "view_count": 298, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 55, "problem_number": "COMB-005", "title": "Frankl's Union-Closed Sets Conjecture", "statement": "For every finite union-closed family of sets (other than the empty family), there exists an element that belongs to at least half of the sets.", "background": "Proposed by Péter Frankl in 1979, this is one of the best-known open problems in combinatorics. A union-closed family is a collection of sets closed under taking unions. Despite its simple statement, the conjecture has attracted many attempted proofs. Recent progress (2022-2024) has shown lower bounds: some element must be in at least 1% of sets (Gilmer 2022), improved to 38% of sets (2024).", "difficulty_level_id": 3, "status": "open", "proposed_by": "Péter Frankl", "proposed_year": 1979, "category_id": 2, "view_count": 389, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 56, "problem_number": "GEO-005", "title": "Inscribed Square Problem (Toeplitz Conjecture)", "statement": "Does every simple closed curve in the plane contain all four vertices of some square?", "background": "The inscribed square problem, also called the square peg problem or Toeplitz conjecture, was posed by Otto Toeplitz in 1911. It asks whether every Jordan curve (simple closed curve) inscribes a square. The conjecture is known to be true for convex curves, piecewise smooth curves, and many special cases, but remains open in full generality as of 2026.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Otto Toeplitz", "proposed_year": 1911, "category_id": 6, "view_count": 432, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 57, "problem_number": "NT-010", "title": "Brocard's Problem", "statement": "Find all integer solutions to $n! + 1 = m^2$.", "background": "Brocard's problem asks for all positive integers n such that n! + 1 is a perfect square. Only three solutions are known: (4, 5), (5, 11), and (7, 71), corresponding to 4! + 1 = 25, 5! + 1 = 121, and 7! + 1 = 5041. It has been verified computationally that no other solutions exist for n < 10^9, but it remains unproven whether these are the only solutions.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 345, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 59, "problem_number": "GT-004", "title": "The Cycle Double Cover Conjecture", "statement": "Every bridgeless graph has a cycle double cover: a collection of cycles that covers each edge exactly twice.", "background": "The cycle double cover conjecture, proposed independently by Paul Seymour and Gábor Szekeres in the 1970s, is a major open problem in graph theory. It has been verified for many classes of graphs, including planar graphs and graphs with small genus. The conjecture is related to the snark conjecture and has connections to topology and algebraic graph theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 298, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 60, "problem_number": "NT-012", "title": "The Erdős-Straus Conjecture", "statement": "For every integer $n \\geq 2$, the equation $\\frac{4}{n} = \\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z}$ has a solution in positive integers x, y, z.", "background": "The Erdős-Straus conjecture concerns Egyptian fractions (sums of unit fractions). Paul Erdős and Ernst G. Straus conjectured in 1948 that 4/n can always be expressed as the sum of three unit fractions. The conjecture has been verified for all n up to 10^17 and is known to hold for various infinite families, but a general proof remains elusive.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Paul Erdős and Ernst G. Straus", "proposed_year": 1948, "category_id": 1, "view_count": 367, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 61, "problem_number": "HIL-006", "title": "Hilbert's 6th Problem: Axiomatization of Physics", "statement": "Develop a mathematical framework that axiomatizes physics, particularly mechanics, thermodynamics, and probability theory.", "background": "Hilbert's 6th problem (1900) calls for treating physics with the same mathematical rigor as geometry. While progress has been made (quantum mechanics axiomatization by von Neumann, some progress in quantum field theory), a complete axiomatization remains elusive, especially for areas like thermodynamics and a unified \"theory of everything.\"", "difficulty_level_id": 5, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 16, "set_id": 2, "view_count": 345, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 62, "problem_number": "HIL-013", "title": "Hilbert's 13th Problem: Seventh Degree Equations", "statement": "Prove that the general equation of the seventh degree cannot be solved using functions of only two variables.", "background": "Hilbert's 13th problem (1900) asks whether seventh-degree equations can be solved using continuous functions of two variables. Vladimir Arnold and Andrey Kolmogorov showed in 1957 that any continuous function can be represented using functions of two variables, which contradicts Hilbert's expectation. However, the problem of whether algebraic solutions exist with restrictions remains open.", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 4, "set_id": 2, "view_count": 287, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 64, "problem_number": "SMA-012", "title": "Smale's 12th Problem: Centralizers of Diffeomorphisms", "statement": "Determine the structure of centralizers of generic diffeomorphisms.", "background": "Smale's 12th problem (1998) concerns the algebraic structure of diffeomorphisms that commute with a given diffeomorphism. The centralizer of a dynamical system reveals its symmetries. Smale conjectured that for generic diffeomorphisms, the centralizer should be trivial or nearly trivial.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 6, "set_id": 5, "view_count": 176, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 66, "problem_number": "DARPA-002", "title": "The Dynamics of Networks", "statement": "Develop high-dimensional mathematics to model and predict behavior in large-scale distributed networks.", "background": "DARPA challenge 2 (2007) addresses the need for mathematical tools to understand massive networks like the internet, social networks, and biological networks. Traditional graph theory becomes inadequate at scale, requiring new mathematical frameworks for network dynamics.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 3, "set_id": 4, "view_count": 389, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 68, "problem_number": "DARPA-004", "title": "21st Century Fluids", "statement": "Extend classical fluid dynamics to handle complex substances like foams, suspensions, gels, and liquid crystals.", "background": "DARPA challenge 4 (2007) recognizes that most real-world fluids don't behave like the classical fluids of Navier-Stokes equations. New mathematics is needed for complex fluids with microstructure, non-Newtonian behavior, and multiphase dynamics.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 9, "set_id": 4, "view_count": 345, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 69, "problem_number": "DARPA-005", "title": "Biological Quantum Field Theory", "statement": "Apply quantum and statistical field theory methods to model and potentially control pathogen evolution.", "background": "DARPA challenge 5 (2007) proposes using the mathematical machinery of quantum field theory—developed for particle physics—to understand biological evolution and epidemiology. This could provide new ways to predict and control disease evolution.", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 16, "set_id": 4, "view_count": 267, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 70, "problem_number": "DARPA-008", "title": "Beyond Convex Optimization", "statement": "Determine whether algebraic geometry can systematically replace linear algebra in optimization.", "background": "DARPA challenge 8 (2007) asks whether the powerful tools of algebraic geometry can extend optimization beyond the convex case. Most practical optimization problems are non-convex, and algebraic geometry may provide the framework for solving them systematically.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 312, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 71, "problem_number": "DARPA-012", "title": "Mathematics of Quantum Computing", "statement": "Develop the mathematics required to control the quantum world for computation.", "background": "DARPA challenge 12 (2007) calls for mathematical foundations of quantum computing, including quantum algorithms, quantum entanglement, and quantum error correction. While quantum computers exist, the mathematical theory of what they can compute and how to program them remains underdeveloped.", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 543, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 72, "problem_number": "DARPA-013", "title": "Game Theory at Scale", "statement": "Create scalable mathematics for differential games, replacing traditional PDE approaches.", "background": "DARPA challenge 13 (2007) addresses the limitations of classical game theory and differential games when dealing with many players. New mathematical frameworks are needed for multi-agent systems, from autonomous vehicles to economic markets to military strategy.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 289, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 74, "problem_number": "DARPA-020", "title": "Computation at Scale", "statement": "Develop asymptotics for systems with massive degrees of freedom.", "background": "DARPA challenge 20 (2007) addresses the mathematical challenges of understanding systems with enormous numbers of variables—from climate models to protein folding to materials science. Traditional approaches fail at extreme scales, requiring new asymptotic methods.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 276, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 75, "problem_number": "DARPA-023", "title": "Fundamental Laws of Biology", "statement": "Identify governing principles for biological systems, analogous to physical laws.", "background": "DARPA challenge 23 (2007) poses perhaps the deepest question: Do fundamental mathematical laws govern biology the way physics is governed by laws? This challenge requires solutions to multiple preceding challenges and asks whether biology can be made as mathematically rigorous as physics.", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 16, "set_id": 4, "view_count": 498, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 76, "problem_number": "DARPA-006", "title": "Computational Duality", "statement": "Use mathematical duality and geometry as foundations for developing novel computational algorithms.", "background": "DARPA challenge 6 (2007) explores whether duality principles from mathematics can lead to breakthrough algorithms. Dualities connect seemingly different mathematical structures and may reveal hidden computational efficiencies.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 77, "problem_number": "DARPA-007", "title": "Occam's Razor in Many Dimensions", "statement": "Find lower bounds for sensing complexity as data collection grows, addressing entropy maximization.", "background": "DARPA challenge 7 (2007) asks for mathematical principles governing data compression and sensing in high dimensions. As sensors become ubiquitous, we need mathematical theory for how much data is truly necessary.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 234, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 78, "problem_number": "DARPA-009", "title": "Physical Consequences of Perelman's Proof", "statement": "Apply Perelman's proof of the Poincaré conjecture to materials fabrication across scales.", "background": "DARPA challenge 9 (2007) asks how Grisha Perelman's breakthrough in understanding 3-dimensional geometry can inform materials science, from nanostructures to macro-scale fabrication.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 7, "set_id": 4, "view_count": 267, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 79, "problem_number": "DARPA-010", "title": "Algorithmic Origami and Biology", "statement": "Strengthen mathematical theory for isometric and rigid embedding relevant to protein folding.", "background": "DARPA challenge 10 (2007) connects origami mathematics to biology. Protein folding is like origami at molecular scales, and better mathematical theory could revolutionize drug design and protein engineering.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 6, "set_id": 4, "view_count": 298, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 80, "problem_number": "DARPA-011", "title": "Optimal Nanostructures", "statement": "Develop mathematics for creating optimal symmetric structures through nanoscale self-assembly.", "background": "DARPA challenge 11 (2007) seeks mathematical principles for designing nanostructures that self-assemble optimally. This combines crystallography, optimization, and molecular dynamics.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 6, "set_id": 4, "view_count": 223, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 81, "problem_number": "DARPA-015", "title": "The Geometry of Genome Space", "statement": "Establish appropriate distance metrics on genome space incorporating biological utility.", "background": "DARPA challenge 15 (2007) asks for a mathematical geometry of genetics. How \"far apart\" are two genomes? The answer depends on biology, not just counting mutations, requiring new geometric frameworks.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 6, "set_id": 4, "view_count": 245, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 82, "problem_number": "DARPA-016", "title": "Symmetries and Action Principles for Biology", "statement": "Extend understanding of symmetries and action principles in biology to include robustness, modularity, evolvability, and variability.", "background": "DARPA challenge 16 (2007) seeks to identify fundamental symmetry principles in biology analogous to those in physics. Why are biological systems robust yet evolvable? Are there variational principles governing life?", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 16, "set_id": 4, "view_count": 312, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 83, "problem_number": "DARPA-017", "title": "Geometric Langlands and Quantum Physics", "statement": "Connect the Langlands program to fundamental physics symmetries.", "background": "DARPA challenge 17 (2007) explores deep connections between number theory (Langlands program) and quantum field theory. This could unify disparate areas of mathematics and physics.", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 16, "set_id": 4, "view_count": 356, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 84, "problem_number": "DARPA-018", "title": "Arithmetic Langlands, Topology, and Geometry", "statement": "Explore homotopy theory's role in Langlands programs.", "background": "DARPA challenge 18 (2007) connects topology (homotopy theory) with the Langlands program in number theory. These connections could revolutionize both fields.", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 7, "set_id": 4, "view_count": 289, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 85, "problem_number": "HIL-007", "title": "Hilbert's 7th Problem: Transcendence of Certain Numbers", "statement": "If $\\alpha$ is algebraic and irrational, and $\\beta$ is algebraic and irrational, is $\\alpha^\\beta$ transcendental?", "background": "Hilbert's 7th problem (1900) was largely solved by Gelfond and Schneider independently in 1934 (Gelfond-Schneider theorem). However, cases involving non-algebraic irrational exponents remain open. For example, whether $e^e$ or $\\pi^\\pi$ are transcendental is unknown.", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 1, "set_id": 2, "view_count": 321, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 86, "problem_number": "HIL-009", "title": "Hilbert's 9th Problem: Reciprocity Laws", "statement": "Generalize the reciprocity law of number theory to arbitrary number fields.", "background": "Hilbert's 9th problem (1900) asks for extensions of quadratic reciprocity to general number fields. Emil Artin made progress with Artin reciprocity law (1927), but complete understanding of reciprocity in all cases remains an active research area.", "difficulty_level_id": 5, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 1, "set_id": 2, "view_count": 234, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 87, "problem_number": "HIL-011", "title": "Hilbert's 11th Problem: Quadratic Forms over Algebraic Number Fields", "statement": "Extend the theory of quadratic forms with algebraic numerical coefficients.", "background": "Hilbert's 11th problem (1900) concerns arithmetic of quadratic forms over number fields. Partial progress has been made through class field theory and the Hasse-Minkowski theorem, but general questions about representations remain open.", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 1, "set_id": 2, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 88, "problem_number": "HIL-014", "title": "Hilbert's 14th Problem: Finite Generation of Rings", "statement": "Is the ring of invariants of a linear algebraic group acting on a polynomial ring always finitely generated?", "background": "Hilbert's 14th problem (1900) was answered negatively by Nagata in 1958, who found counterexamples. However, the problem remains interesting for special cases, and understanding when finite generation holds is an active area.", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 4, "set_id": 2, "view_count": 176, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 89, "problem_number": "HIL-015", "title": "Hilbert's 15th Problem: Schubert's Enumerative Calculus", "statement": "Rigorously justify Schubert's enumerative geometry.", "background": "Hilbert's 15th problem (1900) calls for making Schubert's 19th century enumerative geometry rigorous. While intersection theory and Schubert calculus have been developed (Chow rings, Gromov-Witten theory), some classical problems remain open and new questions arise.", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 5, "set_id": 2, "view_count": 267, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 90, "problem_number": "HIL-017", "title": "Hilbert's 17th Problem: Expression of Definite Forms", "statement": "Can every non-negative rational function be expressed as a sum of squares of rational functions?", "background": "Hilbert's 17th problem (1900) was solved affirmatively by Artin in 1927: every non-negative polynomial can be written as a sum of squares of rational functions. However, questions about minimal representations and related problems in real algebraic geometry remain active.", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 4, "set_id": 2, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 91, "problem_number": "HIL-018", "title": "Hilbert's 18th Problem: Polyhedra and Space-Filling", "statement": "Are there only finitely many essentially different space-filling convex polyhedra? Is there a polyhedron which tiles space but not in a lattice arrangement?", "background": "Hilbert's 18th problem (1900) has multiple parts. Non-lattice tilings (aperiodic tilings) were discovered by Heesch and others. The Kepler conjecture about sphere packing was proved by Hales. However, classification questions about space-filling polyhedra remain open.", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 6, "set_id": 2, "view_count": 289, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 92, "problem_number": "GREEN-001", "title": "Large Sum-Free Sets", "statement": "Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $n/3 + \\Omega(n)$, where $\\Omega(n) \\to \\infty$ as $n \\to \\infty$?", "background": "This is a pretty old and increasingly notorious problem, first mentioned over 50 years ago. The best known bounds are in Bourgain's paper, where he shows that there is necessarily a sum-free set of size at least $(n+2)/3$. In fact, Eberhard, Manners and Green (unpublished) worked out a proof that Problem 1 has a positive solution under certain structural assumptions. It is known that there do exist sets with no sum-free set of size larger than $(1/3 + o(1))n$. However, the $o(1)$ term in these results is more-or-less ineffective; it would be interesting to get a reasonable bound. [1] P. Erdős. Extremal problems in number theory, In Proc. Sympos. Pure Math., Vol. VIII, pages 181–189. Amer. Math. Soc., Providence, R.I., 1965. [2] J. Bourgain, Estimates related to sumfree subsets of sets of integers, Israel J. Math. 97 (1997), 71–92. [3] S. Eberhard, Følner sequences and sum-free sets, Bull. Lond. Math. Soc. 47 (2015), no. 1, 21–28. [4] S. Eberhard, B. J. Green and F. Manners, Sets of integers with no large sum-free subset, Ann. of Math. (2) 180 (2014), no. 2, 621–652.", "difficulty_level_id": 2, "status": "open", "proposed_by": "Erdős and Cameron", "category_id": 2, "set_id": 3, "view_count": 145, "favorite_count": 8, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 93, "problem_number": "GREEN-002", "title": "Restricted Sumset Problem", "statement": "Let $A \\subset \\mathbb{Z}$ be a set of $n$ integers. Is there a subset $S \\subset A$ of size $(\\log n)^{100}$ such that $S \\hat{+} S$ is disjoint from $A$?", "background": "Here $S \\hat{+} S$ denotes the restricted sumset $\\{s_1 + s_2 : s_1, s_2 \\in S, s_1 \\neq s_2\\}$. Problems of this type are also at least 50 years old, being once again mentioned (and attributed to joint discussions of Erdős and Moser). It is known from very recent work of Sanders that there is always such an $S$ with $|S| \\geq (\\log n)^{1+c}$. By contrast the best-known upper bound is due to Ruzsa, showing that one cannot in general hope to take $|S|$ bigger than $e^{C\\sqrt{\\log n}}$. [1] P. Erdős. Extremal problems in number theory, In Proc. Sympos. Pure Math., Vol. VIII, pages 181–189. Amer. Math. Soc., Providence, R.I., 1965. [2] T. Sanders, The Erdős-Moser sum-free set problem, Canad. J. Math. 73 (2021), no. 1, 63–107. [3] I. Z. Ruzsa, Sum-avoiding subsets. Ramanujan J., 9 (2005) (1-2):77–82.", "difficulty_level_id": 2, "status": "open", "proposed_by": "Erdős and Moser", "category_id": 2, "set_id": 3, "view_count": 123, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 95, "problem_number": "GREEN-005", "title": "Product-Free Sets in Finite Groups", "statement": "Which finite groups have the smallest largest product-free sets?", "background": "Kedlaya (2003) showed that every finite group $G$ of order $n$ has a product-free subset of size $\\gg n^{11/14}$, using the classification of finite simple groups. Understanding which groups achieve the minimum and improving bounds remains an open question connecting group theory and combinatorics.", "difficulty_level_id": 2, "status": "open", "proposed_by": "Kedlaya", "proposed_year": 2003, "category_id": 4, "set_id": 3, "view_count": 134, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 96, "problem_number": "GREEN-007", "title": "Ulam's Sequence", "statement": "Define Ulam's sequence $1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, \\ldots$ where $u_1 = 1, u_2 = 2$, and $u_{n+1}$ is the smallest number uniquely expressible as $u_i + u_j$ for $i < j \\leq n$. Does this sequence have positive density? Can one explain its curious Fourier properties?", "background": "Ulam's sequence exhibits mysterious quasi-periodic behavior in its Fourier transform. While it appears to have density around $0.07$, proving it has positive density remains open. The sequence's additive structure and apparent regularity in numerical experiments are not well understood theoretically.", "difficulty_level_id": 1, "status": "open", "proposed_by": "Stanisław Ulam", "category_id": 1, "set_id": 3, "view_count": 187, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 97, "problem_number": "GREEN-008", "title": "Almost Sum-Free Sets", "statement": "Suppose that $A \\subset [N]$ has no more than $\\varepsilon N^2$ solutions to $x + y = z$. Can one remove $\\varepsilon' N$ elements to leave a sum-free set, where $\\varepsilon' \\to 0$ as $\\varepsilon \\to 0$, with a reasonable bound?", "background": "It is known that one can remove $\\varepsilon' N$ elements to obtain a sum-free set, but the quantitative dependence of $\\varepsilon'$ on $\\varepsilon$ is very poor. Finding explicit reasonable bounds would significantly improve our understanding of the structure of almost sum-free sets.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 109, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 98, "problem_number": "GREEN-006", "title": "Sum-Free Subsets of [N]^d", "statement": "Fix an integer $d$. What is the largest sum-free subset of $[N]^d$?", "background": "This multi-dimensional generalization asks for the maximum size of a set in the $d$-dimensional grid with no solutions to $x + y = z$. Lepsveridze and Sun (2023) determined the constants $c_3, c_4, c_5$ and confirmed that the \"slice example\" is asymptotically optimal in these cases.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 118, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 99, "problem_number": "GREEN-009", "title": "Progressions in Subsets of Z/NZ", "statement": "Is $r_5(N) \\ll N(\\log N)^{-c}$? Is $r_4(\\mathbb{F}_5^n) \\ll N^{1-c}$ where $N = 5^n$?", "background": "Here $r_k(N)$ denotes the maximum size of a subset of $\\{1, \\ldots, N\\}$ with no $k$-term arithmetic progression. Kelley-Meka (2024) resolved the $k=3$ case. For $k \\geq 5$, Leng-Sah-Sawhney (2024) proved bounds of shape $r_k(N) \\ll Ne^{-(\\log \\log N)^{c_k}}$. Finding polynomial savings remains a central challenge in additive combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 142, "favorite_count": 8, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 100, "problem_number": "GREEN-010", "title": "Roth's Theorem with Random Common Differences", "statement": "Let $S \\subset \\mathbb{N}$ be random. Under what conditions is Roth's theorem for progressions of length 3 true with common differences in $S$?", "background": "This asks when Roth's theorem holds if we restrict common differences to a random set. Briët and Castro-Silva (2023) advanced bounds for odd $k$. The problem explores how randomness interacts with additive structure.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 126, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 102, "problem_number": "GREEN-012", "title": "Tuples in Dense Sets", "statement": "Let $G$ be an abelian group of size $N$, and suppose that $A \\subset G$ has density $\\alpha$. Are there at least $\\alpha^{15}N^{10}$ tuples $(x_1, \\ldots, x_5, y_1, \\ldots, y_5) \\in G^{10}$ such that $x_i + y_j \\in A$ whenever $j \\in \\{i, i+1, i+2\\}$?", "background": "This problem asks about higher-order additive structures in dense sets. Deng-Tidor-Zhao (2023) considered this problem and conjectured a negative answer, suggesting the exponent might not be optimal.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 108, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 103, "problem_number": "GREEN-013", "title": "4-term APs in Fourier Uniform Sets", "statement": "Suppose that $A \\subset \\mathbb{Z}/N\\mathbb{Z}$ has density $\\alpha$ and is Fourier uniform (all Fourier coefficients of $1_A - \\alpha$ are $o(N)$). Does $A$ contain at least $\\gg \\alpha^{100}N^2$ 4-term arithmetic progressions?", "background": "Fourier uniformity means the set \"looks random\" from a Fourier perspective. The question asks if this forces many 4-APs. Deng-Tidor-Zhao (2023) conjectured a negative answer, suggesting Fourier uniformity alone may not suffice.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 115, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 104, "problem_number": "GREEN-015", "title": "Lipschitz AP-Free Graphs", "statement": "Does there exist a Lipschitz function $f : \\mathbb{N} \\to \\mathbb{Z}$ whose graph $\\Gamma = \\{(n, f(n)) : n \\in \\mathbb{Z}\\} \\subset \\mathbb{Z}^2$ is free of 3-term progressions?", "background": "This asks whether a \"smooth\" (Lipschitz) function can have a graph avoiding arithmetic progressions. The Lipschitz condition prevents wildly oscillating behavior, making AP-avoidance more constrained.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 121, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 105, "problem_number": "GREEN-016", "title": "Linear Equation x + 3y = 2z + 2w", "statement": "What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$?", "background": "This asks about sets avoiding a specific linear configuration. Understanding which linear equations are easier or harder to avoid is a fundamental question in additive combinatorics.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 98, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 106, "problem_number": "GREEN-017", "title": "Progressions in F_3^n with Boolean Common Differences", "statement": "Suppose that $A \\subset \\mathbb{F}_3^n$ is a set of density $\\alpha$. Under what conditions on $\\alpha$ is $A$ guaranteed to contain a 3-term progression with nonzero common difference in $\\{0, 1\\}^n$?", "background": "This constrains the progression to have Boolean-like common differences. Bhangale-Khot-Minzer (2023) showed sets avoiding such progressions have density $\\ll_p (\\log \\log \\log n)^{-c_p}$, using extraordinarily difficult techniques.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 104, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 107, "problem_number": "GREEN-018", "title": "Corner Problem in Product Sets", "statement": "Suppose $G$ is a finite group, and let $A \\subset G \\times G$ be a subset of density $\\alpha$. Are there $\\gg_\\alpha |G|^3$ triples $x, y, g$ such that $(x, y), (gx, y), (x, gy)$ all lie in $A$?", "background": "This is a \"corner-type\" problem in the group product setting. Dense sets should contain many axis-aligned corners. The problem connects additive combinatorics with group theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 110, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 108, "problem_number": "GREEN-020", "title": "Multidimensional Szemerédi Theorem Bounds", "statement": "Find reasonable bounds for instances of the multidimensional Szemerédi theorem.", "background": "Szemerédi's theorem extends to multiple dimensions (finding combinatorial lines in dense sets). Pohoata-Zakharov (2024) improved bounds for skew corners to $N^{5/4}$. Quantitative bounds remain a major challenge.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 127, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 109, "problem_number": "GREEN-021", "title": "Large Sieve and Quadratic Sets", "statement": "Suppose that a large sieve process leaves a set of quadratic size. Is that set quadratic?", "background": "Sieve methods remove arithmetic structure from sets. This problem asks whether a set that \"survives\" a large sieve and has size $\\sim N^2$ must actually be a quadratic sequence or similar structured set. Understanding the structure of sieved sets is fundamental in analytic number theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 87, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 110, "problem_number": "GREEN-022", "title": "Small Sieve Maximal Sets", "statement": "Suppose that a small sieve process leaves a set of maximal size. What is the structure of that set?", "background": "When a small sieve (sieving by small primes) leaves the maximum possible density of survivors, what structure must the original set have? This connects sieve theory with the structural theory of sets in number theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 82, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 111, "problem_number": "GREEN-023", "title": "Large Cosets in Iterated Sumsets", "statement": "Suppose that $A \\subset \\mathbb{F}_2^n$ has density $\\alpha$. Does $10A$ contain a coset of some subspace of dimension at least $n - O(\\log(1/\\alpha))$?", "background": "This asks how many times we must add a set to itself before it contains a large subspace coset. Kosciuszko (2024), building on Konyagin, showed that $mA - mA$ contains a subspace of dimension $\\geq n - O(\\log^{3+\\eta}(1/\\alpha))$ for suitable $m$. The problem asks if fewer iterations suffice.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 93, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 112, "problem_number": "GREEN-024", "title": "Largest Coset in 2A", "statement": "Suppose that $A \\subset \\mathbb{F}_2^n$ has density $\\alpha$. What is the largest size of coset guaranteed to be contained in $2A$?", "background": "This asks for the largest affine subspace (coset) contained in the doubling $2A = A + A$. Unlike the previous problem about many iterations, this focuses on just $2A$. Determining the optimal bound is a fundamental question in additive combinatorics over $\\mathbb{F}_2^n$.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 88, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 113, "problem_number": "GREEN-025", "title": "Additive Complements and Cosets", "statement": "Suppose that $A \\subset \\mathbb{F}_2^n$ has an additive complement of size $K$. Does $2A$ contain a coset of codimension $O_K(1)$?", "background": "If $A$ has a small additive complement (a set $B$ with $A + B = \\mathbb{F}_2^n$), does this force $2A$ to contain a large coset? This problem explores the relationship between additive complements and the structure of sumsets.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 91, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 114, "problem_number": "GREEN-026", "title": "Partitions and Large Cosets", "statement": "Suppose that $\\mathbb{F}_2^n$ is partitioned into sets $A_1, \\dots, A_K$. Does $2A_i$ contain a coset of codimension $O_K(1)$ for some $i$?", "background": "When partitioning a vector space into $K$ parts, at least one part must have substantial additive structure. This problem asks if one piece must have a doubling containing a large coset. It's a partitioning variant of the previous coset problems.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 86, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 116, "problem_number": "GREEN-028", "title": "Gowers Box Norms over Finite Fields", "statement": "Let $p$ be an odd prime and suppose $f : \\mathbb{F}_p^n \\times \\mathbb{F}_p^n \\to \\mathbb{C}$ is bounded pointwise by 1. Suppose $\\mathbb{E}_h \\|\\Delta_{(h,h)}f\\|_\\square^4 \\geq \\delta$. Does $f$ correlate with a function of the form $a(x)b(y)c(x+y)(-1)^{q(x,y)}$?", "background": "This asks for an inverse theorem for a particular Gowers norm in product spaces over finite fields. Understanding which structured functions correlate with high Gowers norm is central to higher-order Fourier analysis.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 84, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 117, "problem_number": "GREEN-029", "title": "Inverse Theorem for Gowers Norms", "statement": "Determine bounds for the inverse theorem for Gowers norms.", "background": "The inverse theorem characterizes functions with large Gowers $U^{s+1}$ norm. Leng-Sah-Sawhney (2024) established a quasi-polynomial inverse theorem for $\\|\\cdot\\|_{U^{s+1}[N]}$ norms for all $s \\geq 3$. Improving to polynomial bounds remains a major challenge in additive combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 95, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 118, "problem_number": "GREEN-030", "title": "Φ(G) and Φ'(G) Coincidence", "statement": "Do $\\Phi(G)$ and $\\Phi'(G)$ coincide?", "background": "This asks whether two different notions of the Frattini-like subgroup of $G$ are equal. The Frattini subgroup consists of non-generators; different definitions can arise in different contexts. Determining their equivalence has implications for group theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 4, "view_count": 73, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 119, "problem_number": "GREEN-031", "title": "Sumsets Containing Composites", "statement": "Suppose $A, B \\subset \\{1, \\dots, N\\}$ both have size $N^{0.49}$. Does $A + B$ contain a composite number?", "background": "This asks whether sumsets of moderately large sets must contain composite numbers. Since primes have density $1/\\log N$, sets of size $N^{0.49}$ are much denser, suggesting their sumset should hit composites. However, proving this rigorously requires understanding the additive structure of primes.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 81, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 120, "problem_number": "GREEN-032", "title": "Sums of Smooth Numbers", "statement": "Is every $n \\leq N$ the sum of two integers, all of whose prime factors are at most $N^\\varepsilon$?", "background": "Smooth numbers have only small prime factors. This asks if every number is a sum of two smooth numbers, which would show smooth numbers have excellent additive properties. Such a result would have implications for number theory and the distribution of smooth numbers.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 88, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 121, "problem_number": "GREEN-033", "title": "Sumsets of Perfect Squares", "statement": "Is there an absolute constant $c > 0$ such that if $A \\subset \\mathbb{N}$ is a set of squares of size at least 2, then $|A + A| \\geq |A|^{1+c}$?", "background": "This asks whether sets of perfect squares have superlinear sumset growth. Squares are highly structured (sparse in $\\mathbb{N}$), and one expects their sumsets to grow substantially. Determining the optimal exponent $c$ is a fundamental problem in additive number theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 92, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 122, "problem_number": "GREEN-034", "title": "Covering Squares with Sumsets", "statement": "Suppose $A + A$ contains the first $n$ squares. Is $|A| \\geq n^{1-o(1)}$?", "background": "If a set's sumset contains all squares up to $n^2$, must the set have size nearly $n$? This explores the inverse problem: given that a sumset covers a structured set (squares), what can we say about the original set?", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 85, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 123, "problem_number": "GREEN-035", "title": "Products of Primes Modulo p", "statement": "Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$. Is every $x \\in \\{1, \\dots, p-1\\}$ congruent to some product $a_1a_2$ modulo $p$?", "background": "This asks whether pairwise products of primes cover all residues modulo $p$. Matom\\\"aki-Ter\\\"av\\\"ainen (2023) made significant progress, showing that products of three primes suffice. Whether two primes suffice remains open.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 96, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 124, "problem_number": "GREEN-036", "title": "Multiplicatively Closed Set Density", "statement": "Let $A$ be the smallest set containing 2 and 3, and closed under the operation $a_1a_2 - 1$ (if $a_1, a_2 \\in A$, then $a_1a_2 - 1 \\in A$). Does $A$ have positive density?", "background": "This defines a set generated by a multiplicative-like operation. Understanding its density is nontrivial because the operation $a_1a_2 - 1$ mixes multiplication with additive structure. Whether such sets have positive density connects number theory with dynamical systems.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 77, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 125, "problem_number": "GREEN-037", "title": "Primes with p-2 Having Odd Omega", "statement": "Do there exist infinitely many primes $p$ for which $p-2$ has an odd number of prime factors (counting multiplicity)?", "background": "This asks about the parity of $\\Omega(p-2)$ where $\\Omega(n)$ counts prime factors with multiplicity. Since $p-2$ is even for odd primes $p > 2$, we're asking about the structure of $(p-2)/2$. This is a prime-shifted multiplicative function question.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 83, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 126, "problem_number": "GREEN-038", "title": "Difference Sets Containing Squares", "statement": "Is there $c > 0$ such that whenever $A \\subset [N]$ has size $N^{1-c}$, the difference set $A - A$ contains a nonzero square?", "background": "This asks how large a set must be to guarantee its difference set contains a square. Similarly one can ask if $A - A$ contains a prime minus one. These questions probe the additive structure forced by density.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 89, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 127, "problem_number": "GREEN-039", "title": "Gaps Between Sums of Two Squares", "statement": "Is there always a sum of two squares between $X - \\frac{1}{10}X^{1/4}$ and $X$?", "background": "Sums of two squares have density $c/\\sqrt{\\log X}$, so gaps can be large. This asks for an upper bound on the largest gap. Such results would improve our understanding of the distribution of representable numbers in quadratic forms.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 91, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 128, "problem_number": "GREEN-040", "title": "Waring's Problem Over Finite Fields", "statement": "Determine bounds for Waring's problem over finite fields.", "background": "Waring's problem asks: can every element be written as a sum of $k$ $d$-th powers? Over finite fields $\\mathbb{F}_q$, the problem has different character. Determining the minimum $k$ for given $d$ and $q$ is a classical problem in algebraic number theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 86, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 129, "problem_number": "GREEN-041", "title": "Cubic Curves in F_p^2", "statement": "Suppose $A \\subset \\mathbb{F}_p^2$ is a set meeting every line in at most 2 points. Is it true that all except $o(p)$ points of $A$ lie on a cubic curve?", "background": "Sets avoiding three collinear points have special structure. Over finite fields, the Hasse-Weil bound and algebraic geometry suggest such sets should lie nearly on a cubic. This is a finite-field analogue of classical incidence geometry.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 84, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 130, "problem_number": "GREEN-042", "title": "Collinear Triples and Cubic Curves", "statement": "Fix $k$. Let $A \\subset \\mathbb{R}^2$ be a set of $n$ points with no more than $k$ on any line. Suppose at least $\\delta n^2$ pairs $(x, y) \\in A \\times A$ have the line $xy$ containing a third point of $A$. Is there a cubic curve containing at least $cn$ points of $A$?", "background": "If many pairs determine lines through a third point, the set should have algebraic structure. This asks if a cubic curve captures this structure. It generalizes results from the joints problem and incidence geometry.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 78, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 131, "problem_number": "GREEN-043", "title": "Erdős-Szekeres with Visibility", "statement": "Fix integers $k, \\ell$. Given $n \\geq n_0(k, \\ell)$ points in $\\mathbb{R}^2$, is there either a line containing $k$ of them, or $\\ell$ of them that are mutually visible?", "background": "This is a Ramsey-type problem mixing collinearity and visibility (no point blocks the segment between two others). It generalizes the Erdős-Szekeres theorem to a geometric context, asking for unavoidable configurations.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 81, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 132, "problem_number": "GREEN-044", "title": "Collinear 4-tuples Force Collinear 5-tuples", "statement": "Suppose $A \\subset \\mathbb{R}^2$ is a set of size $n$ with $cn^2$ collinear 4-tuples. Does it contain 5 points on a line?", "background": "Many collinear 4-tuples suggest the set has strong linear structure. This asks if this forces an actual line through 5 points. It's related to the Szemerédi-Trotter theorem and incidence bounds.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 75, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 133, "problem_number": "GREEN-045", "title": "No Three in Line in [N]^2", "statement": "What is the largest subset of the grid $[N]^2$ with no three points on a line? In particular, for $N$ sufficiently large, is it impossible to have a set of size $2N$ with this property?", "background": "The cap set problem in two dimensions. Erdős conjectured sets of size $O(N)$ exist, but proving or disproving this remains open. The problem connects discrete geometry, additive combinatorics, and the polynomial method.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 94, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 134, "problem_number": "GREEN-046", "title": "Smooth Surfaces Intersecting 2-planes", "statement": "Let $\\Gamma$ be a smooth codimension 2 surface in $\\mathbb{R}^n$. Must $\\Gamma$ intersect some 2-dimensional plane in 5 points, if $n$ is sufficiently large?", "background": "This asks about unavoidable intersection patterns between smooth surfaces and planes in high dimensions. It's related to the Kakeya problem and incidence geometry in higher dimensions.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 71, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 135, "problem_number": "GREEN-047", "title": "No 5 Points on 2-plane in [N]^d", "statement": "What is the largest subset of $[N]^d$ with no 5 points on a 2-plane?", "background": "This generalizes the no-three-in-line problem to higher dimensions and 2-planes. Determining the maximum size of such sets involves combinatorial geometry and higher-dimensional incidence bounds.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 76, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 136, "problem_number": "GREEN-048", "title": "Balanced Ham Sandwich Line", "statement": "Let $X \\subset \\mathbb{R}^2$ be a set of $n$ points. Does there exist a line $\\ell$ through at least two points of $X$ such that the numbers of points on either side of $\\ell$ differ by at most 100?", "background": "This is a variant of the ham sandwich theorem asking for a balanced bisector that passes through points of the set. The classical ham sandwich theorem doesn't require passing through points, making this version more constrained.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 79, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 137, "problem_number": "GREEN-049", "title": "Sparse Hitting Set for Rectangles", "statement": "Let $A$ be a set of $n$ points in the plane. Can one select $A' \\subset A$ of size $n/2$ such that any axis-parallel rectangle containing 1000 points of $A$ contains at least one point of $A'$?", "background": "This asks for an efficient hitting set for rectangles defined by a point set. Such results have applications in computational geometry and range searching.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 74, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 138, "problem_number": "GREEN-050", "title": "Small Triangles in the Unit Disc", "statement": "Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$ determined by them?", "background": "This asks about unavoidable small-area triangles in dense point sets. Cohen-Pohoata-Zakharov (2023) improved the bound to $n^{-8/7-c}$. Reaching the conjectured $n^{-2}$ bound remains open and connects to the Heilbronn triangle problem.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 88, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 139, "problem_number": "GREEN-051", "title": "Axis-Parallel Rectangles in Dense Sets", "statement": "Suppose $A$ is an open subset of $[0, 1]^2$ with measure $\\alpha$. Are there four points in $A$ determining an axis-parallel rectangle with area $\\geq c\\alpha^2$?", "background": "This asks if dense sets in the unit square must contain large axis-parallel rectangles. It's a continuous analogue of combinatorial rectangle problems and relates to measure-theoretic ergodic theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 72, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 140, "problem_number": "GREEN-052", "title": "Equidistribution of Integer Multiples", "statement": "Let $c > 0$ and let $A$ be a set of $n$ distinct integers. Does there exist $\\theta$ such that no interval of length $\\frac{1}{n}$ in $\\mathbb{R}/\\mathbb{Z}$ contains more than $n^c$ of the numbers $\\theta a \\pmod 1$, for $a \\in A$?", "background": "This asks about finding angles $\\theta$ that spread out the set $A$ modulo 1. It's related to discrepancy theory and the distribution of sequences modulo 1.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 141, "problem_number": "GREEN-053", "title": "Random Permutations Fixing k-Sets", "statement": "Let $p(k)$ be the limit as $n \\to \\infty$ of the probability that a random permutation on $[n]$ preserves some set of size $k$. Is $p(k)$ a decreasing function of $k$? Is $p(k) = (C + o(1))k^{-\\alpha}(\\log k)^{-3/2}$ for some absolute constant $C$?", "background": "This concerns the probability that a random permutation fixes some subset. Eberhard-Ford-Green established asymptotic formulas. The question asks for monotonicity and precise asymptotics, connecting combinatorics and probability.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 75, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 142, "problem_number": "GREEN-054", "title": "Comparable Elements in Integer Lattices", "statement": "Consider a set $S \\subset [N]^3$ with the property that any two distinct elements $s, s'$ of $S$ are comparable (in the coordinatewise partial order). Is $|S| \\leq N^{2-\\delta}$ for some $\\delta > 0$?", "background": "An antichain in $[N]^d$ can have size $\\binom{N}{d/2}^d \\sim N^{d/2}$. This asks if totally comparable sets (chains) in 3D are even smaller, achieving $N^{2-\\delta}$ rather than $N^2$. It's a question in extremal combinatorics and poset theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 71, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 143, "problem_number": "GREEN-055", "title": "Stable Density on Subspaces", "statement": "Let $A \\subset \\mathbb{F}_2^n$. If $V$ is a subspace, write $\\alpha(V)$ for the density of $A$ on $V$. Is there some $V$ of moderately small codimension on which $\\alpha$ is stable?", "background": "This asks if every set has a subspace where its density doesn't fluctuate wildly. Stability of density on subspaces is fundamental in additive combinatorics over $\\mathbb{F}_2^n$ and relates to the polynomial Freiman-Ruzsa conjecture.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 77, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 144, "problem_number": "GREEN-056", "title": "Almost Invariant Sets Under Affine Maps", "statement": "Suppose $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ has density $\\frac{1}{2}$. Under what conditions on $K$ can $A$ be almost invariant under all maps $\\phi(x) = ax + b$ with $|a|, |b| \\leq K$?", "background": "This asks when a set is nearly preserved under small affine transformations. Understanding which sets have this property connects to additive combinatorics, group actions, and the structure of dense sets in cyclic groups.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 145, "problem_number": "GREEN-057", "title": "Trace Reconstruction", "statement": "Given a string $x \\in \\{0, 1\\}^n$, let $\\tilde{x}$ be obtained by deleting bits independently at random with probability $\\frac{1}{2}$. How many independent traces $\\tilde{x}_1, \\dots, \\tilde{x}_m$ are needed to reconstruct $x$ with probability 0.9?", "background": "This fundamental problem in computational complexity asks how many noisy observations suffice to recover the original string. Chase (2020) improved bounds to $e^{n^{1/5}\\log^C n}$. Determining the optimal bound is a major open question.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 82, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 146, "problem_number": "GREEN-058", "title": "Irreducibility of Random {0,1} Polynomials", "statement": "Is a random polynomial with coefficients in $\\{0, 1\\}$ and nonzero constant term almost surely irreducible?", "background": "This asks whether most polynomials with binary coefficients are irreducible over $\\mathbb{Q}$. Bary-Soroker-Koukoulopoulos-Kozma (2023) made further progress. The problem connects number theory, probability, and algebraic geometry.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 76, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 149, "problem_number": "GREEN-061", "title": "N Queens Problem Asymptotics", "statement": "In how many ways (asymptotically) $Q(n)$ may $n$ non-attacking queens be placed on an $n \\times n$ chessboard?", "background": "The n-queens problem asks for the number of ways to place $n$ queens on an $n \\times n$ board so none attack each other. Determining the asymptotic growth rate of $Q(n)$ is a famous open problem in combinatorics. Rough bounds are known but the exact constant remains elusive.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 94, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 150, "problem_number": "GREEN-062", "title": "Bounds for Birch's Theorem", "statement": "Let $d \\geq 3$ be odd. Give bounds on $\\nu(d)$ such that if $n > \\nu(d)$ then any homogeneous polynomial $F(\\mathbf{x}) \\in \\mathbb{Z}[x_1, \\dots, x_n]$ of degree $d$ has a nontrivial integer zero.", "background": "Birch's theorem guarantees that homogeneous polynomials of odd degree have nontrivial zeros if there are enough variables. Determining the optimal $\\nu(d)$ is a central problem in Diophantine equations and algebraic number theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 73, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 151, "problem_number": "GREEN-063", "title": "Solutions to Polynomial Equations in Dense Sets", "statement": "Finding a single solution to $F(x_1, \\dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of solutions in $A$ is roughly $\\alpha^n$ times the number in $[X]$?", "background": "This asks when a dense set $A$ of density $\\alpha$ contains the \"expected\" number of solutions to a Diophantine equation. Understanding when sparse sets behave like random sets for counting solutions is fundamental in analytic number theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 70, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 152, "problem_number": "GREEN-064", "title": "Residually Finite Groups", "statement": "Is every group well-approximated by finite groups?", "background": "A group is residually finite if every nontrivial element has a nontrivial image in some finite quotient. This asks if all groups have this property. The answer is known to be no (infinite simple groups), but the question may refer to finitely generated/presented groups, where it remains interesting.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 4, "view_count": 67, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 153, "problem_number": "GREEN-065", "title": "Rado's Boundedness Conjecture", "statement": "Suppose $a_1, \\dots, a_k$ are integers which do not satisfy Rado's condition. Is $c(a_1, \\dots, a_k)$ bounded in terms of $k$ only?", "background": "Rado's condition characterizes which linear equations are partition regular. For equations not satisfying this condition, $c(\\cdot)$ is the minimum number of colors needed to avoid monochromatic solutions. Whether this depends only on $k$ (not the coefficients) is a fundamental question in Ramsey theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 72, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 154, "problem_number": "GREEN-066", "title": "Monochromatic x+y and xy", "statement": "If $\\{1, \\dots, N\\}$ is $r$-coloured, then for $N \\geq N_0(r)$ there exist integers $x, y \\geq 3$ such that $x+y$ and $xy$ have the same colour. Find reasonable bounds for $N_0(r)$.", "background": "This asks about unavoidable monochromatic additive-multiplicative patterns. Finding quantitative bounds for $N_0(r)$ connects Ramsey theory with both additive and multiplicative structure.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 78, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 155, "problem_number": "GREEN-067", "title": "Affine Translates of {0,1,3}", "statement": "If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set $\\{0, 1, 3\\}$ that $A$ can contain?", "background": "This asks how many copies of the pattern $\\{0, 1, 3\\}$ (under affine transformations $x \\mapsto ax + b$) can appear in an $n$-element set. Understanding maximal copies of specific patterns is fundamental in additive combinatorics.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 74, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 156, "problem_number": "GREEN-068", "title": "Restricted Sumsets in Partitions", "statement": "For which values of $k$ is the following true: whenever we partition $[N] = A_1 \\cup \\dots \\cup A_k$, we have $|\\bigcup_{i=1}^k (A_i \\hat{+} A_i)| \\geq \\frac{1}{10} N$?", "background": "This asks how many parts are needed before restricted sumsets (sums of distinct elements) must cover a substantial fraction of $[N]$. The problem connects partition regularity with sumset structure.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 157, "problem_number": "GREEN-069", "title": "Sum of Cubes in F_3^n", "statement": "Let $A_1, \\dots, A_{100}$ be \"cubes\" in $\\mathbb{F}_3^n$ (images of $\\{0, 1\\}^n$ under linear automorphisms). Is $A_1 + \\dots + A_{100} = \\mathbb{F}_3^n$?", "background": "This asks whether 100 cubes in $\\mathbb{F}_3^n$ always sum to the entire space. It's a question about additive bases and the covering properties of structured sets in vector spaces over finite fields.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 71, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 158, "problem_number": "GREEN-070", "title": "Sets with No Unique Sum Representations", "statement": "What is the size of the smallest set $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation?", "background": "This asks for the minimum size of a set where every sum $a + a'$ has multiple representations. Bedert (2023) showed the answer lies between $\\omega(p)\\log p$ and $O(\\log^2 p)$. Closing this gap would deepen our understanding of additive bases.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 76, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 159, "problem_number": "GREEN-071", "title": "Uniform Random Variables with Uniform Sum", "statement": "Suppose $X, Y$ are finitely-supported independent random variables taking integer values such that $X + Y$ is uniformly distributed on its range. Are $X$ and $Y$ themselves uniformly distributed on their ranges?", "background": "This asks if uniform sums force uniform summands. It's a discrete probability question with connections to additive combinatorics and the structure of convolutions.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 70, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 160, "problem_number": "GREEN-072", "title": "Large Subsets of Approximate Groups", "statement": "Suppose $A$ is a $K$-approximate group (not necessarily abelian). Is there $S \\subset A$ with $|S| \\gg K^{-O(1)}|A|$ and $S^8 \\subset A^4$?", "background": "Approximate groups are sets with controlled doubling. This asks if they contain large subsets with even better multiplicative structure. Understanding approximate groups is central to geometric group theory and additive combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 4, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 161, "problem_number": "GREEN-073", "title": "Structured Subsets with Bounded Doubling", "statement": "Given a set $A \\subset \\mathbb{Z}$ with $D(A) \\leq K$, find a large structured subset $A'$ which \"obviously\" has $D(A') \\leq K + \\varepsilon$.", "background": "Sets with small doubling constant have additive structure. This asks for an explicit, easily verifiable structured subset. Making structure \"obvious\" connects to algorithmic aspects of the Polynomial Freiman-Ruzsa conjecture.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 162, "problem_number": "GREEN-074", "title": "Sidon Set Size Bounds", "statement": "Write $F(N)$ for the largest Sidon subset of $[N]$. Improve, at least for infinitely many $N$, the bounds $N^{1/2} + O(1) \\leq F(N) \\leq N^{1/2} + N^{1/4} + O(1)$.", "background": "Sidon sets have all pairwise sums distinct. The bounds have been tight for decades. Balogh-Füredi-Roy (2021) obtained a small improvement to the upper bound. Any further progress would be a major breakthrough in additive combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 89, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 163, "problem_number": "GREEN-075", "title": "Large Gaps in Dilates", "statement": "Let $p$ be a prime and let $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ be a set of size $\\sqrt{p}$. Is there a dilate of $A$ with a gap of length $100\\sqrt{p}$?", "background": "This asks whether dilates (multiplicative translates) of sets necessarily have large gaps. Understanding the distribution of dilates connects additive and multiplicative combinatorics in cyclic groups.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 72, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 164, "problem_number": "GREEN-076", "title": "Optimal Sidon Bases", "statement": "Are there infinitely many $q$ for which there is a set $A \\subset \\mathbb{Z}/q\\mathbb{Z}$ with $|A| = (\\sqrt{2} + o(1))q^{1/2}$ and $A + A = \\mathbb{Z}/q\\mathbb{Z}$?", "background": "This asks if Sidon-like sets (near-optimal density with few sum collisions) can form additive bases. The coefficient $\\sqrt{2}$ is conjecturally optimal for such constructions.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 75, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 165, "problem_number": "GREEN-077", "title": "Structure of Sets with Bounded Representation", "statement": "Suppose $A \\subset [N]$ has size $\\geq c\\sqrt{N}$ and representation function $r_A(n) \\leq r$ for all $n$. What can be said about the structure of $A$?", "background": "Sets with bounded representation function (few ways to write sums) have special structure. Understanding this structure connects Sidon set theory with additive bases.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 70, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 167, "problem_number": "GREEN-079", "title": "Disjoint Sumsets Construction", "statement": "For arbitrarily large $n$, does there exist an abelian group $H$ with $|H| = n^{2+o(1)}$ and subsets $A_1, \\dots, A_n, B_1, \\dots, B_n$ satisfying $|A_i||B_i| \\geq n^{2-o(1)}$, $|A_i + B_i| = |A_i||B_i|$, such that $A_i + B_i$ are pairwise disjoint from $A_j + B_k$ ($j \\neq k$)?", "background": "This asks if one can partition a group into many disjoint sumsets with no doubling. It connects to the structure of Sidon sets and extremal problems in additive combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 168, "problem_number": "GREEN-080", "title": "Cap Sets in F_7^n", "statement": "What is the largest subset $A \\subset \\mathbb{F}_7^n$ for which $A - A$ intersects $\\{-1, 0, 1\\}^n$ only at 0?", "background": "This is a cap set problem in $\\mathbb{F}_7^n$ with restricted difference set. Recent polynomial method breakthroughs dramatically improved bounds for $\\mathbb{F}_3^n$, but $\\mathbb{F}_7$ remains open.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 73, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 169, "problem_number": "GREEN-081", "title": "Covering by Random Translates", "statement": "If $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ is random with $|A| = \\sqrt{p}$, can we almost surely cover $\\mathbb{Z}/p\\mathbb{Z}$ with $100\\sqrt{p}$ translates of $A$?", "background": "This asks about the covering properties of random sets. Understanding when random sets form good coverings connects probability, additive combinatorics, and coding theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 170, "problem_number": "GREEN-082", "title": "Hamming Ball Covering Growth", "statement": "Let $r$ be fixed and let $H(r)$ be the Hamming ball of radius $r$ in $\\mathbb{F}_2^n$. Let $f(r)$ be the smallest constant such that there exist infinitely many $n$ with subspaces $V_n \\leq \\mathbb{F}_2^n$ satisfying $V_n + H(r) = \\mathbb{F}_2^n$ and $|V_n| = (f(r) + o(1)) \\frac{2^n}{|H(r)|}$. Does $f(r) \\to \\infty$?", "background": "This asks if covering $\\mathbb{F}_2^n$ by Hamming ball translates requires increasingly inefficient packings as $r$ grows. It connects coding theory with the geometry of finite vector spaces.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 66, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 171, "problem_number": "GREEN-083", "title": "Pyjama Set Covering", "statement": "How many rotated (about the origin) copies of the \"pyjama set\" $\\{(x, y) \\in \\mathbb{R}^2 : \\operatorname{dist}(x, \\mathbb{Z}) \\leq \\varepsilon\\}$ are needed to cover $\\mathbb{R}^2$?", "background": "The pyjama set is a union of vertical strips. This beautiful geometric problem, solved by Manners (2015), asks how many rotations are needed to cover the plane. It connects geometry, combinatorics, and Fourier analysis.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 74, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 172, "problem_number": "GREEN-084", "title": "Cohn-Elkies Scheme for Circle Packings", "statement": "Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings?", "background": "Cohn-Elkies developed a linear programming approach that proved optimal sphere packing in dimensions 8 and 24. Whether their method extends to circles in the plane remains a major open question in discrete geometry.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 71, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 173, "problem_number": "GREEN-085", "title": "Covering by Residue Classes", "statement": "Let $N$ be large. For each prime $p$ with $N^{0.51} \\leq p < 2N^{0.51}$, pick a residue $a(p) \\in \\mathbb{Z}/p\\mathbb{Z}$. Is $\\#\\{n \\in [N] : n \\equiv a(p) \\pmod p \\text{ for some } p\\} \\gg N^{1-o(1)}$?", "background": "This asks if residue classes from medium-sized primes nearly cover $[N]$. It connects sieve theory with covering problems and the distribution of primes.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 174, "problem_number": "GREEN-086", "title": "Sieving by Many Small Primes", "statement": "Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \\leq p_1 < p_2 < \\dots < p_{1000} < N^{9/10}$. Does the remaining set have size at most $\\frac{1}{10}N$?", "background": "This asks whether aggressive sieving by many small primes can remove most of $[N]$. Understanding sieve limits is fundamental in analytic number theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 67, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 175, "problem_number": "GREEN-087", "title": "Residue Class Multiple Coverage", "statement": "Can we pick residue classes $a_p \\pmod p$, one for each prime $p \\leq N$, such that every integer $\\leq N$ lies in at least 10 of them?", "background": "This asks if we can achieve high-multiplicity covering using one residue class per prime. It's dual to sieving problems and connects to the large sieve.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 176, "problem_number": "GREEN-088", "title": "Maximal Covering Interval", "statement": "What is the largest $y$ for which one may cover the interval $[y]$ by residue classes $a_p \\pmod p$, one for each prime $p \\leq x$?", "background": "This is the classical covering problem in sieve theory. Determining the optimal relationship between $x$ and $y$ would have significant implications for understanding the distribution of primes.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 70, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 177, "problem_number": "GREEN-089", "title": "Random Walk Mixing on Alternating Groups", "statement": "Pick $x_1, \\dots, x_k \\in A_n$ at random. Is it true that, almost surely as $n \\to \\infty$, the random walk on this set of generators and their inverses equidistributes in time $O(n \\log n)$?", "background": "This asks about mixing time for random walks on the alternating group with random generators. Determining optimal mixing times connects probability, group theory, and spectral graph theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 4, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 178, "problem_number": "GREEN-090", "title": "Bounds for Approximate Group Classification", "statement": "Find bounds in the classification theorem for approximate groups.", "background": "The Breuillard-Green-Tao classification shows approximate groups resemble actual groups. However, the bounds in this theorem are extremely poor. Improving them would have significant applications in additive combinatorics and geometric group theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 4, "view_count": 72, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1097, "problem_number": "GREEN-097", "title": "N-Queens Problem Asymptotics", "statement": "In how many ways (asymptotically) $Q(n)$ may $n$ non-attacking queens be placed on an $n \\times n$ chessboard?", "background": "The n-queens problem asks for asymptotic formulas for the number of ways to place n non-attacking queens on an n×n board. Recent work has made progress on both upper and lower bounds, but the precise asymptotic behavior remains elusive.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 145, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 1098, "problem_number": "GREEN-098", "title": "Bounds for Homogeneous Polynomial Zeros", "statement": "Let $d \\geq 3$ be an odd integer. Give bounds on $\\nu(d)$ such that if $n > \\nu(d)$ the following is true: given any homogeneous polynomial $F(\\mathbf{x}) \\in \\mathbb{Z}[x_1, \\dots, x_n]$ of degree $d$, there is some $\\mathbf{x} \\in \\mathbb{Z}^n \\setminus \\{\\mathbf{0}\\}$ such that $F(\\mathbf{x}) = 0$.", "background": "This asks for explicit bounds on how many variables are needed to guarantee integer zeros of homogeneous polynomials. Classical results give existence but quantitative bounds remain challenging, especially for higher degrees.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 78, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1099, "problem_number": "GREEN-099", "title": "Polynomial Solutions in Dense Sets", "statement": "Finding a single solution to a polynomial equation $F(x_1, \\dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of such solutions in $A$ is roughly $\\alpha^n$ times the number of solutions in $[X]$?", "background": "This problem asks when dense sets contain the \"expected\" number of polynomial solutions. Understanding density conditions that guarantee proportional solution counts connects number theory with additive combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 71, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1100, "problem_number": "GREEN-100", "title": "Sofic Groups", "statement": "Is every group well-approximated by finite groups?", "background": "A group is sofic if it can be approximated by finite symmetric groups in a precise sense. Whether all groups are sofic is a major open question in group theory with connections to dynamics, graph theory, and combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 4, "view_count": 92, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1102, "problem_number": "ALG-002", "title": "Hadamard Conjecture", "statement": "For every positive integer $k$, does there exist a Hadamard matrix of order $4k$?", "background": "A Hadamard matrix is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. The Hadamard conjecture, dating back to 1893, states that such matrices exist for all orders that are multiples of 4. These matrices have important applications in coding theory, signal processing, and quantum information theory. While Hadamard matrices have been constructed for many values of $k$, the smallest order for which existence is unknown is 668. The conjecture has deep connections to combinatorial design theory and remains one of the central problems in discrete mathematics.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 387, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1103, "problem_number": "ALG-003", "title": "Köthe Conjecture", "statement": "If a ring has no nil ideal other than $\\{0\\}$, does it follow that it has no nil one-sided ideal other than $\\{0\\}$?", "background": "The Köthe conjecture, proposed by Gottfried Köthe in 1930, is a fundamental problem in ring theory concerning the structure of nil ideals. A nil ideal is one in which every element is nilpotent. The conjecture asks whether the absence of two-sided nil ideals implies the absence of one-sided nil ideals. Despite being studied for over 90 years, the problem remains open even for Noetherian rings. The conjecture is related to the Jacobson conjecture and has implications for understanding the structure of general rings. Counterexamples would reveal unexpected asymmetry in the behavior of left and right ideals.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1104, "problem_number": "ALG-004", "title": "Connes Embedding Problem", "statement": "Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor?", "background": "The Connes embedding problem, formulated by Alain Connes in 1976, is a central question in the theory of von Neumann algebras. In 2020, Ji, Natarajan, Vidick, Wright, and Yuen published a paper claiming to have shown the problem has a negative answer, based on connections to quantum complexity theory and the equivalence with Tsirelson's problem in quantum information. However, the problem's status remains subject to verification of their approach. The problem has deep connections to free probability, quantum groups, and mathematical physics, making it one of the most important questions in operator algebra theory.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 312, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1105, "problem_number": "ALG-005", "title": "Jacobson's Conjecture", "statement": "For a left-and-right Noetherian ring $R$, is the intersection of all powers of the Jacobson radical $J(R)$ equal to zero?", "background": "Jacobson's conjecture addresses a fundamental question about the structure of Noetherian rings. The Jacobson radical of a ring consists of elements that annihilate all simple modules, and understanding its intersection over all powers relates to the ring's nilpotent elements and its representation theory. While the conjecture holds for many important classes of rings (including commutative Noetherian rings), the general case remains open. The problem is closely related to other structural conjectures in non-commutative ring theory, including the Köthe conjecture.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 198, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1106, "problem_number": "ALG-006", "title": "Zauner's Conjecture", "statement": "Do SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) exist in all finite dimensions?", "background": "Zauner's conjecture, proposed in 1999, concerns the existence of a special type of quantum measurement in Hilbert spaces of all finite dimensions. A SIC-POVM consists of d² unit vectors in a d-dimensional complex Hilbert space that are equiangular - the absolute inner product of any two distinct vectors is constant. These structures have applications in quantum information theory, quantum state tomography, and quantum cryptography. While SIC-POVMs have been found numerically for all dimensions up to 151 and proven to exist analytically in some special cases, the general existence question remains open. The conjecture has surprising connections to number theory and algebraic geometry.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 176, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1107, "problem_number": "ALG-007", "title": "Casas-Alvero Conjecture", "statement": "If a univariate polynomial $f$ of degree $d$ over a field of characteristic 0 shares a common factor with each of its first $d-1$ derivatives, must $f$ be a power of a linear polynomial?", "background": "The Casas-Alvero conjecture, proposed in 2001, connects the factorization of a polynomial with the factorization of its derivatives. If $f(x)$ has degree $d$ and for each $k = 1, 2, \\ldots, d-1$, the polynomial $f(x)$ shares a root with its $k$-th derivative $f^{(k)}(x)$, the conjecture states that $f$ must be of the form $f(x) = (x - a)^d$ for some constant $a$. While proven for various special cases and low degrees, the general conjecture remains open. It has connections to algebraic geometry and the theory of polynomial equations.", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 154, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1108, "problem_number": "ALG-008", "title": "Andrews-Curtis Conjecture", "statement": "Can every balanced presentation of the trivial group be transformed into a trivial presentation by a sequence of Nielsen transformations and conjugations of relators?", "background": "The Andrews-Curtis conjecture, proposed in 1965, is a central problem in combinatorial group theory. A balanced presentation has an equal number of generators and relators. The conjecture asks whether such presentations of the trivial group can always be simplified to the form $\\langle x_1, \\ldots, x_n \\mid x_1, \\ldots, x_n \\rangle$ using only Andrews-Curtis moves (Nielsen transformations on relators and conjugations). Despite extensive computational searches and partial results, no counterexample has been found, yet no proof exists. The conjecture has important implications for 3-manifold theory and the classification of homotopy types.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 212, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1109, "problem_number": "ALG-009", "title": "Bounded Burnside Problem", "statement": "For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2,5)$ finite?", "background": "The Bounded Burnside problem asks which free Burnside groups are finite. A free Burnside group $B(m,n)$ is the largest group with $m$ generators in which every element has order dividing $n$. It is known that $B(m,n)$ is finite for $n \\in \\{2, 3, 4, 6\\}$ and for certain other special values, and infinite for most large odd exponents. The case $B(2,5)$ has been the subject of extensive computational investigation but remains open. Solving this problem would significantly advance our understanding of periodic groups and torsion in group theory.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1110, "problem_number": "ALG-010", "title": "Herzog-Schönheim Conjecture", "statement": "If a finite system of left cosets of subgroups of a group $G$ partitions $G$, then must at least two of the subgroups have the same index in $G$?", "background": "The Herzog-Schönheim conjecture, proposed in 1974, concerns coset decompositions of groups. If $G$ is partitioned by cosets $g_1H_1, g_2H_2, \\ldots, g_kH_k$ where each $H_i$ is a subgroup of $G$ and the cosets are pairwise disjoint, the conjecture states that at least two of the indices $[G:H_i]$ must be equal. While proven for abelian groups and various other special cases, the general conjecture remains open. It has connections to combinatorial number theory and the structure theory of groups.", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 142, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1112, "problem_number": "ALG-012", "title": "Existence of Perfect Cuboids", "statement": "Does there exist a rectangular cuboid where all edges, face diagonals, and space diagonals have integer lengths?", "background": "A perfect cuboid (also called a perfect box or Euler brick with space diagonal) would be a rectangular parallelepiped with integer edge lengths $a$, $b$, $c$ such that the face diagonals $\\sqrt{a^2+b^2}$, $\\sqrt{b^2+c^2}$, $\\sqrt{a^2+c^2}$ and the space diagonal $\\sqrt{a^2+b^2+c^2}$ are all integers. Despite extensive computational searches up to very large bounds and numerous partial results, no perfect cuboid has been found, nor has impossibility been proven. The problem has connections to Diophantine equations and elliptic curves, and has fascinated both amateur and professional mathematicians for centuries.", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 234, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1114, "problem_number": "ALG-014", "title": "McKay Conjecture", "statement": "For a finite group $G$ and prime $p$, is the number of irreducible complex characters of $G$ whose degree is not divisible by $p$ equal to the corresponding number for the normalizer of a Sylow $p$-subgroup?", "background": "The McKay conjecture, proposed in the 1970s, is a central problem in the representation theory of finite groups. It predicts a surprising relationship between the character degrees of a group and those of a much smaller subgroup (the normalizer of a Sylow $p$-subgroup). The conjecture has been verified for many important classes of groups and has led to deep insights about the structure of character tables. A proof was announced in 2007 by Isaacs, Malle, and Navarro assuming the classification of finite simple groups, though subtle gaps in the argument have led to ongoing refinement of the proof.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 156, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1115, "problem_number": "ALG-015", "title": "Are All Groups Surjunctive?", "statement": "Is every group surjunctive? That is, for any group $G$, if $\\phi: A^G \\to A^G$ is a cellular automaton that is injective, must it also be surjective?", "background": "A group $G$ is called surjunctive if every injective cellular automaton on $G$ is automatically surjective. Equivalently, this asks whether the dynamical system defined by a cellular automaton on the group can be injective without being bijective. Gromov and Weiss proved that all sofic groups are surjunctive, and all known groups are sofic, but it remains unknown whether all groups are surjunctive. The question has deep connections to symbolic dynamics, geometric group theory, and the Garden of Eden theorem from cellular automaton theory. A negative answer would be quite surprising and would reveal fundamental limitations in our understanding of group actions.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 143, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1119, "problem_number": "NT-016", "title": "Catalan-Mersenne Conjecture", "statement": "Are all Catalan-Mersenne numbers $C_n$ composite for $n > 4$? Here $C_0 = 2$ and $C_{n+1} = 2^{C_n} - 1$.", "background": "The Catalan-Mersenne conjecture concerns a doubly exponential sequence where each term is a Mersenne number with exponent equal to the previous term. The sequence grows extraordinarily rapidly: $C_0 = 2$, $C_1 = 3$, $C_2 = 7$, $C_3 = 127$, $C_4 = 170141183460469231731687303715884105727$ (a 39-digit number). The first four terms are prime, but $C_5$ has over $10^{38}$ digits, making it far beyond reach of current computational methods. The conjecture predicts that all subsequent terms are composite. This problem connects to deep questions about the distribution of Mersenne primes and the limitations of our ability to determine primality for extremely large numbers.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 287, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1120, "problem_number": "NT-017", "title": "Are There Infinitely Many Mersenne Primes?", "statement": "Are there infinitely many prime numbers of the form $2^p - 1$ where $p$ is prime?", "background": "Mersenne primes are primes of the form $M_p = 2^p - 1$ where $p$ is itself prime. As of 2024, only 51 Mersenne primes are known, the largest being $2^{82589933} - 1$ discovered in 2018. Despite their rarity, it is conjectured that infinitely many exist. Mersenne primes are intimately connected to perfect numbers through the Euclid-Euler theorem: an even number is perfect if and only if it has the form $2^{p-1}(2^p-1)$ where $2^p-1$ is a Mersenne prime. The question of whether infinitely many Mersenne primes exist is closely related to our understanding of the distribution of primes and has implications for both pure and applied mathematics, as Mersenne primes are used in pseudorandom number generation and cryptography.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 567, "favorite_count": 49, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1121, "problem_number": "GEO-001", "title": "Sphere Packing Problem in Higher Dimensions", "statement": "What is the densest packing of spheres in dimensions 4 through 23? More generally, what is the optimal sphere packing density in dimension $n$?", "background": "The sphere packing problem asks for the densest arrangement of non-overlapping spheres in $n$-dimensional space. In dimension 3, Kepler's conjecture (proved by Hales in 1998) shows the densest packing has density $\\pi/\\sqrt{18} \\approx 0.7405$. In 2016, Maryna Viazovska proved that the E₈ lattice gives the densest packing in dimension 8, and shortly after, Cohn, Kumar, Miller, Radchenko, and Viazovska proved the Leech lattice is optimal in dimension 24. However, dimensions 4-7 and 9-23 remain open, as do almost all higher dimensions. The problem has deep connections to coding theory, number theory, and optimization.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 398, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1122, "problem_number": "GEO-002", "title": "Mahler's Conjecture", "statement": "Among all centrally symmetric convex bodies in $\\mathbb{R}^n$, does the cube (or cross-polytope) minimize the product of the body's volume and the volume of its polar dual?", "background": "Mahler's conjecture, proposed in 1939, concerns a fundamental geometric quantity called the Mahler volume, defined as the product of a convex body's volume with the volume of its polar dual. Kurt Mahler conjectured that among all centrally symmetric convex bodies in $\\mathbb{R}^n$, this product is minimized by the cube and the cross-polytope (which are dual to each other). The conjecture has been proved in dimension 2 by Mahler himself, and partial results exist for special classes of bodies, but the general case remains open. The problem connects convex geometry, functional analysis, and the theory of Banach spaces.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 245, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1123, "problem_number": "GEO-003", "title": "The Illumination Conjecture", "statement": "Can every convex body in $n$-dimensional space be illuminated by at most $2^n$ point light sources?", "background": "The illumination conjecture, also known as Hadwiger's covering conjecture in one of its forms, asks whether every convex body in $\\mathbb{R}^n$ can be illuminated by at most $2^n$ point light sources placed outside the body. A point on the surface is considered illuminated if the ray from the light source to that point does not pass through the interior of the body. The conjecture has been proven for $n = 2$ and $n = 3$, but remains open for higher dimensions. The problem is closely related to covering problems and has connections to discrete geometry and combinatorics.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 187, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1124, "problem_number": "GEO-004", "title": "Kakeya Needle Problem", "statement": "What is the minimum area of a region in the plane in which a unit line segment can be continuously rotated through 360 degrees?", "background": "The Kakeya needle problem asks for the smallest area set in the plane within which a unit line segment can be rotated continuously through 360 degrees, returning to its initial position. While Besicovitch showed in 1928 that there exist Kakeya sets of arbitrarily small positive measure, the question of what happens when we require the set to be connected or simply connected remains fascinating. In higher dimensions, the Kakeya conjecture (related but distinct) concerns sets containing unit line segments in every direction and has deep connections to harmonic analysis, partial differential equations, and number theory. The finite field analog was resolved by Dvir in 2008 using the polynomial method.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 312, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1125, "problem_number": "GEO-005", "title": "Bellman's Lost in a Forest Problem", "statement": "What is the shortest path that guarantees escape from a forest of known shape and size, starting from an unknown location?", "background": "Bellman's lost in a forest problem asks for the shortest universal path that guarantees reaching the boundary of a region, regardless of starting position and orientation. For a circular forest of radius 1, the problem was solved by various authors with a path of length approximately 7.2898. However, for other shapes like squares or equilateral triangles, the optimal escape path remains unknown. This problem has applications to robotics, search and rescue operations, and computational geometry. It connects to questions about curve shortening, geometric optimization, and worst-case analysis in motion planning.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 198, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1131, "problem_number": "COMB-003", "title": "The Union-Closed Sets Conjecture", "statement": "For any finite family of finite sets that is closed under taking unions, must there exist an element that belongs to at least half of the sets?", "background": "The union-closed sets conjecture, also known as Frankl's conjecture after Peter Frankl who popularized it in 1979, is a simple-to-state problem in extremal combinatorics. A family of sets is union-closed if the union of any two sets in the family is also in the family. The conjecture asserts that in any non-trivial union-closed family, some element appears in at least half of the sets. Despite extensive research and verification for small cases, the general conjecture remains open. It has connections to lattice theory, Boolean functions, and information theory.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 334, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1132, "problem_number": "COMB-004", "title": "Singmaster's Conjecture", "statement": "Does there exist a finite upper bound on how many times a number (other than 1) can appear in Pascal's triangle?", "background": "Singmaster's conjecture, proposed by David Singmaster in 1971, concerns the frequency of entries in Pascal's triangle. While 1 appears infinitely often (along the edges), and 2 appears exactly three times, larger numbers can appear multiple times in different positions. For example, 120 appears six times. The conjecture states that there exists an absolute constant $C$ such that no number appears more than $C$ times in Pascal's triangle (excluding 1). Singmaster himself proved that the number of occurrences is at most $O(\\log n / \\log \\log n)$ for the entry $n$. The conjecture connects to Diophantine equations and the distribution of binomial coefficients.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 298, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1135, "problem_number": "SET-001", "title": "The Continuum Hypothesis", "statement": "Is there a set whose cardinality is strictly between that of the integers and the real numbers?", "background": "The continuum hypothesis (CH), proposed by Georg Cantor in 1878, states that there is no set with cardinality strictly between that of the integers and the real numbers. Equivalently, it asserts that the cardinality of the continuum (the real numbers) is $\\aleph_1$, the second smallest infinite cardinal. Gödel proved in 1940 that CH is consistent with ZFC (if ZFC is consistent), and Cohen proved in 1963 that the negation of CH is also consistent with ZFC. Thus, CH is independent of the standard axioms of set theory. This means CH can neither be proved nor disproved from ZFC alone, making it one of the most philosophically significant results in mathematical logic.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 623, "favorite_count": 54, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1137, "problem_number": "NT-019", "title": "Are There Infinitely Many Sophie Germain Primes?", "statement": "Are there infinitely many primes $p$ such that $2p + 1$ is also prime?", "background": "A Sophie Germain prime is a prime $p$ where $2p+1$ is also prime. These primes are named after French mathematician Sophie Germain, who used them in her work on Fermat's Last Theorem. Examples include 2, 3, 5, 11, 23, and 29. The conjecture that infinitely many exist is closely related to the twin prime conjecture and is similarly difficult. Sophie Germain primes have applications in cryptography and are used in some primality testing algorithms. As of 2024, the largest known Sophie Germain prime has over 400,000 digits. The problem remains one of the major open questions about prime distribution.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1138, "problem_number": "AG-001", "title": "The Hodge Conjecture", "statement": "On a projective algebraic variety, is every Hodge class a rational linear combination of classes of algebraic cycles?", "background": "The Hodge conjecture is one of the seven Millennium Prize Problems, with a $1 million prize for its solution. Proposed by William Hodge in 1950, it concerns the deep relationship between the topology and algebraic geometry of complex projective varieties. In simple terms, it asks whether certain topological cycles (Hodge classes) can be represented by algebraic cycles (subvarieties). The conjecture has been verified in many special cases, including for curves, surfaces, and abelian varieties, but the general case remains stubbornly open. It connects algebraic geometry, topology, and complex analysis in profound ways.", "difficulty_level_id": 5, "status": "open", "category_id": 5, "view_count": 534, "favorite_count": 46, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1140, "problem_number": "AG-003", "title": "The Birch and Swinnerton-Dyer Conjecture", "statement": "For an elliptic curve $E$ over the rationals, does the rank of its group of rational points equal the order of vanishing of its $L$-function at $s=1$?", "background": "The Birch and Swinnerton-Dyer (BSD) conjecture is one of the seven Millennium Prize Problems. It connects the arithmetic properties of elliptic curves (specifically, the group of rational points) with analytic properties (the behavior of the associated $L$-function). The conjecture predicts a precise relationship between these seemingly disparate aspects. It has been verified computationally for millions of curves and proven in special cases, but the general conjecture remains open. A proof would revolutionize our understanding of elliptic curves and have applications to cryptography and number theory. The conjecture also relates to the Langlands program and modern arithmetic geometry.", "difficulty_level_id": 5, "status": "open", "category_id": 5, "view_count": 687, "favorite_count": 59, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1141, "problem_number": "DYN-001", "title": "The Weinstein Conjecture", "statement": "Does every Reeb vector field on a closed contact manifold have at least one periodic orbit?", "background": "The Weinstein conjecture, proposed by Alan Weinstein in 1978, is a fundamental problem in symplectic geometry and dynamical systems. A Reeb vector field is a special type of vector field on a contact manifold, and the conjecture predicts the existence of closed orbits under very general conditions. The conjecture was proven in dimension 3 by Hofer in 1993 using pseudoholomorphic curves, and has been established in many other cases. However, the general case remains open. The conjecture has deep connections to Hamiltonian dynamics, celestial mechanics, and the study of periodic phenomena in physics.", "difficulty_level_id": 4, "status": "open", "category_id": 11, "view_count": 276, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1142, "problem_number": "DYN-002", "title": "The Painlevé Conjecture", "statement": "In the $n$-body problem with $n \\geq 4$, can non-collision singularities occur in finite time?", "background": "The Painlevé conjecture concerns the $n$-body problem in celestial mechanics, asking whether the motion of $n$ point masses under gravitational attraction can develop a singularity (infinite velocities or unbounded positions) in finite time without any collisions occurring. For $n=3$, Sundman proved in 1912 that non-collision singularities cannot occur, but for $n \\geq 4$ the question remains open. Xia constructed examples showing that certain types of unbounded behavior are possible, but the existence of true non-collision singularities (where velocities become infinite) remains unproven. The problem has implications for the long-term behavior of planetary systems and the foundations of classical mechanics.", "difficulty_level_id": 5, "status": "open", "category_id": 11, "view_count": 298, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1144, "problem_number": "GT-008", "title": "Cereceda's Conjecture", "statement": "For any $k$-chromatic graph, can its $k$-colorings be transformed into each other by recoloring one vertex at a time, staying within $k$ colors, in polynomial time in the number of vertices?", "background": "Cereceda's conjecture concerns the diameter of the reconfiguration graph of $k$-colorings. Given a graph $G$ with chromatic number $k$, consider the graph whose vertices are all proper $k$-colorings of $G$, with two colorings adjacent if they differ on exactly one vertex. The conjecture, proposed in 2007, states that this graph has diameter at most $O(n^2)$ where $n$ is the number of vertices in $G$. The problem is motivated by questions in computational complexity and has connections to mixing times of Markov chains. While progress has been made for special graph classes, the general conjecture remains open.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1146, "problem_number": "TOP-003", "title": "The Whitehead Conjecture", "statement": "Is every aspherical closed manifold whose fundamental group has no non-trivial perfect normal subgroups a $K(\\pi, 1)$ space?", "background": "The Whitehead conjecture, posed by J.H.C. Whitehead, concerns a special class of topological spaces. A space is aspherical if all its homotopy groups above dimension 1 vanish, and it is a $K(\\pi, 1)$ if it is aspherical and connected. The conjecture asks whether certain algebraic conditions on the fundamental group guarantee the topological property of being aspherical. The Poincaré conjecture can be viewed as a special case. While the conjecture is known to hold for many important classes of manifolds, including those with non-positive sectional curvature, the general case remains open. The problem connects algebraic topology, geometric topology, and group theory.", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 234, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1149, "problem_number": "GEO-006", "title": "The Knaster Problem", "statement": "Can a solid cube be completely covered by finitely many smaller homothetic cubes with ratio less than 1, such that the interiors are disjoint?", "background": "The Knaster problem, also known as the cube packing problem, asks whether a unit cube can be covered by finitely many non-overlapping smaller cubes, each similar to the original with ratio $< 1$. In 1979, Mycielski proved this is impossible in dimension 2 (for squares), but the 3-dimensional case remains open. The problem has connections to measure theory, geometric covering problems, and Banach-Tarski-like paradoxes. A positive answer would be quite surprising as it would demonstrate a counterintuitive property of 3-dimensional space. The problem has inspired research into covering and packing problems in higher dimensions.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 189, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1153, "problem_number": "NT-022", "title": "Polignac's Conjecture", "statement": "For every even number $n$, are there infinitely many pairs of consecutive primes differing by $n$?", "background": "Polignac's conjecture, proposed by Alphonse de Polignac in 1849, is a vast generalization of the twin prime conjecture. It asserts that for every even integer $n$, there exist infinitely many prime gaps of exactly size $n$. The twin prime conjecture is the special case $n = 2$. While Zhang's 2013 breakthrough showed infinitely many bounded gaps exist, proving the existence of infinitely many gaps of any specific even size remains open. The conjecture relates to the Hardy-Littlewood conjectures and our understanding of the distribution of primes. Even proving the existence of infinitely many prime gaps of size 6 would be a major breakthrough.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1156, "problem_number": "ALG-016", "title": "The Babai Conjecture on Graph Isomorphism", "statement": "Can graph isomorphism be decided in quasi-polynomial time for all graphs?", "background": "The Babai conjecture concerns the computational complexity of determining whether two graphs are isomorphic. In 2015, László Babai announced a quasi-polynomial time algorithm for graph isomorphism (running in time $2^{O(\\log^c n)}$ for some constant $c$), improving on the previous best bound. While a flaw was found in the original proof, Babai repaired it in 2017. However, whether graph isomorphism is in P (polynomial time) remains open. The problem sits in NP but is not known to be NP-complete, occupying a special place in complexity theory. The resolution has important implications for cryptography and the structure of complexity classes.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 445, "favorite_count": 38, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1157, "problem_number": "NT-023", "title": "Pillai's Conjecture", "statement": "For each positive integer $k$, does the equation $|2^m - 3^n| = k$ have only finitely many solutions in positive integers $m$ and $n$?", "background": "Pillai's conjecture, proposed by Subbayya Sivasankaranarayana Pillai, concerns the gaps between powers of 2 and powers of 3. The conjecture generalizes to any two multiplicatively independent integers $a$ and $b$: the equation $|a^m - b^n| = k$ should have only finitely many solutions for each fixed $k$. This is related to the abc conjecture and to understanding the distribution of exponential Diophantine equations. The conjecture has been proven for many special cases but remains open in general. It connects to transcendental number theory and the study of linear forms in logarithms.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 198, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1158, "problem_number": "NT-024", "title": "Erdős-Straus Conjecture", "statement": "For every integer $n \\geq 2$, can $\\frac{4}{n}$ be expressed as the sum of three unit fractions $\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z}$?", "background": "The Erdős-Straus conjecture asks whether every fraction $4/n$ (for $n \\geq 2$) can be written as a sum of three unit fractions (fractions with numerator 1). The conjecture has been verified computationally for all $n$ up to $10^{17}$ and proven for several infinite families of values, but the general case remains open. Egyptian fraction representations have been studied since ancient times, and this problem connects to number theory, combinatorics, and computational mathematics. Related problems concern representing other fractions as sums of unit fractions with various restrictions.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 289, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1159, "problem_number": "NT-025", "title": "The Gauss Circle Problem", "statement": "What is the optimal error term in the formula for the number of lattice points inside a circle of radius $r$?", "background": "The Gauss circle problem asks for the number of integer lattice points $(x,y)$ satisfying $x^2 + y^2 \\leq r^2$. The main term is $\\pi r^2$ (the area of the circle), but determining the optimal error term has been a central problem in analytic number theory for over 150 years. It is known that the error is $O(r^{2/3})$ and conjectured to be $O(r^{1/2 + \\epsilon})$ for any $\\epsilon > 0$, but this has not been proven. The problem connects to the distribution of lattice points, the theory of the Riemann zeta function, and has inspired numerous techniques in analytic number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 367, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1160, "problem_number": "ALG-017", "title": "Birch-Tate Conjecture", "statement": "Does the order of the center of the Steinberg group of the ring of integers of a number field relate to the value of the Dedekind zeta function at $s=-1$?", "background": "The Birch-Tate conjecture concerns the relationship between algebraic K-theory and special values of zeta functions. Specifically, it relates the order of $K_2$ of the ring of integers of a number field to the value of the Dedekind zeta function at $s = -1$. This conjecture is part of a broader program connecting algebraic K-theory to number theory and has been verified in many special cases. It generalizes ideas from class field theory and has connections to the Lichtenbaum conjectures. The problem sits at the intersection of algebraic number theory, algebraic K-theory, and the theory of zeta functions.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 178, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1161, "problem_number": "ALG-018", "title": "Hilbert's Fifteenth Problem", "statement": "Can Schubert calculus be given a rigorous foundation?", "background": "Hilbert's fifteenth problem, from his famous 1900 list, asks for a rigorous foundation of Schubert's enumerative calculus. Schubert calculus is a method for solving problems in enumerative geometry, such as counting the number of lines in 3-space that meet four given lines. While modern algebraic geometry has provided substantial progress through intersection theory and the development of Chow rings, aspects of the problem remain active areas of research. The development of Gromov-Witten invariants and quantum cohomology has provided new tools, but questions about the complete rigor of classical Schubert calculus in all dimensions continue to be investigated.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1162, "problem_number": "ALG-019", "title": "Hilbert's Sixteenth Problem", "statement": "What is the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$ in the plane?", "background": "Hilbert's sixteenth problem consists of two parts. The first part (topology of algebraic curves) asks about the possible configurations of connected components of real algebraic curves. The second part asks for the maximum number and possible configurations of limit cycles of polynomial vector fields of degree $n$ in the plane. This second part remains largely open even for $n=2$ (quadratic systems). The problem is fundamental to the qualitative theory of differential equations and has applications to dynamical systems, control theory, and mathematical biology. Despite over a century of research, even basic questions about quadratic systems remain unresolved.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 312, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1163, "problem_number": "GEO-008", "title": "The Inscribed Square Problem", "statement": "Does every simple closed curve in the plane contain four points that form the vertices of a square?", "background": "The inscribed square problem, also known as Toeplitz' conjecture, asks whether every Jordan curve (simple closed curve) in the plane contains four points forming a square. The problem has been open since 1911. It is known to be true for smooth curves and for many other special cases, but the general case for arbitrary continuous curves remains unproven. The problem is related to other inscribed polygon problems and has connections to topology, dynamical systems, and geometric measure theory. Even proving the existence of an inscribed rectangle with sides in a given ratio remains challenging for general curves.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 456, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1164, "problem_number": "GEO-009", "title": "Falconer's Conjecture", "statement": "If a compact set in $\\mathbb{R}^d$ has Hausdorff dimension greater than $d/2$, must it determine a set of distances with positive Lebesgue measure?", "background": "Falconer's conjecture concerns the relationship between the fractal dimension of a set and the set of distances between its points. Proposed by Kenneth Falconer in 1985, it states that if a compact set $E \\subset \\mathbb{R}^d$ has Hausdorff dimension strictly greater than $d/2$, then the distance set $\\{|x-y| : x, y \\in E\\}$ has positive Lebesgue measure. The conjecture has been proven in dimension 2 but remains open in higher dimensions. It has deep connections to harmonic analysis, geometric measure theory, and additive combinatorics. Recent progress using polynomial methods has improved bounds but not resolved the conjecture.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 289, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1165, "problem_number": "GT-010", "title": "The Total Coloring Conjecture", "statement": "Can every graph be totally colored with at most $\\Delta + 2$ colors, where $\\Delta$ is the maximum degree?", "background": "The total coloring conjecture, proposed independently by Behzad and Vizing in the 1960s, concerns coloring both vertices and edges of a graph such that no two adjacent or incident elements receive the same color. The conjecture states that every graph can be totally colored using at most $\\Delta + 2$ colors where $\\Delta$ is the maximum degree. Vizing proved that at most $\\Delta + 2$ colors suffice, and it is trivial that at least $\\Delta + 1$ are needed. The conjecture asks whether the upper bound is tight. It has been verified for many graph classes but remains open in general. The problem has applications to scheduling and resource allocation.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 234, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1171, "problem_number": "NT-026", "title": "The Odd Perfect Number Conjecture", "statement": "Do there exist any odd perfect numbers? (A perfect number equals the sum of its proper divisors.)", "background": "A perfect number is a positive integer that equals the sum of its proper positive divisors. Euclid showed that numbers of the form $2^{p-1}(2^p - 1)$ are perfect when $2^p - 1$ is prime (Mersenne prime), giving all known even perfect numbers. Whether odd perfect numbers exist has been an open question for over 2000 years. It is known that if an odd perfect number exists, it must be greater than $10^{1500}$, have at least 101 prime factors, and satisfy numerous other constraints. The problem connects to prime number theory, divisibility, and has inspired extensive computational searches. Most mathematicians believe no odd perfect numbers exist.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 678, "favorite_count": 58, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1172, "problem_number": "NT-027", "title": "Firoozbakht's Conjecture", "statement": "Is the sequence $p_n^{1/n}$ strictly decreasing, where $p_n$ is the $n$-th prime?", "background": "Firoozbakht's conjecture, proposed in 1982, states that the sequence $(p_n)^{1/n}$ is strictly decreasing, where $p_n$ denotes the $n$-th prime number. This is equivalent to saying that $p_{n+1}^n < p_n^{n+1}$ for all $n$. The conjecture is stronger than Cramér's conjecture about prime gaps and has been verified computationally for all primes up to very large values. If true, it would imply strong results about the distribution of primes and prime gaps. The conjecture remains open despite extensive numerical evidence supporting it.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 198, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1173, "problem_number": "AG-004", "title": "The Tate Conjecture", "statement": "For varieties over finite fields, are the $\\ell$-adic representations arising from étale cohomology related to algebraic cycles in the expected way?", "background": "The Tate conjecture, proposed by John Tate in 1963, is a fundamental problem in arithmetic geometry. It concerns the relationship between algebraic cycles on algebraic varieties over finite fields and the Galois representations arising from étale cohomology. The conjecture would provide a powerful tool for understanding rational equivalence of cycles. It has been proven for divisors (codimension 1 cycles) on abelian varieties and for various other special cases. The conjecture is closely related to the Hodge conjecture and the Birch and Swinnerton-Dyer conjecture, forming part of a web of deep conjectures in arithmetic geometry.", "difficulty_level_id": 5, "status": "open", "category_id": 5, "view_count": 256, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1176, "problem_number": "SET-002", "title": "Suslin's Problem", "statement": "If a dense linear order without endpoints is complete and has the countable chain condition, must it be isomorphic to the real numbers?", "background": "Suslin's problem, posed by Mikhail Suslin in 1920, asks whether the real numbers can be characterized by certain order-theoretic properties. Specifically, it asks if every complete dense linear order without endpoints satisfying the countable chain condition (every family of disjoint open intervals is countable) must be order-isomorphic to $\\mathbb{R}$. In 1967, it was shown that this question is independent of ZFC set theory - both positive and negative answers are consistent with the standard axioms. A \"Suslin line\" (a counterexample) exists in some models of set theory but not in others. The problem inspired fundamental developments in set theory and the study of independence results.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 289, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1179, "problem_number": "NT-028", "title": "Schinzel's Hypothesis H", "statement": "If polynomials satisfy certain necessary divisibility conditions, do they simultaneously produce infinitely many primes for integer inputs?", "background": "Schinzel's Hypothesis H is a sweeping generalization of many conjectures about primes, including the twin prime conjecture, Sophie Germain prime conjecture, and Dickson's conjecture. It states that if $f_1, \\ldots, f_k$ are irreducible polynomials with integer coefficients and positive leading coefficients, and no prime divides all values $f_1(n) \\cdots f_k(n)$ simultaneously for all integers $n$, then there are infinitely many integers $n$ for which all $f_i(n)$ are prime. If true, it would unify and resolve numerous open problems about prime-producing polynomials. The hypothesis remains wide open despite its fundamental importance.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 298, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1180, "problem_number": "ALG-020", "title": "The Uniform Boundedness Conjecture", "statement": "Is there a bound $B(g, d)$ such that every curve of genus $g$ over a number field of degree $d$ has at most $B(g, d)$ rational points?", "background": "The uniform boundedness conjecture for rational points on curves asks whether, for fixed genus $g$ and degree $d$, there exists a bound on the number of rational points on genus-$g$ curves over number fields of degree $d$. This would be a vast generalization of Mordell's conjecture (now Faltings' theorem, which shows finiteness but not uniform bounds). The conjecture has been proven for $g = 1$ (elliptic curves) by Mazur and Merel, but remains open for $g \\geq 2$. It connects to deep questions in arithmetic geometry about the distribution of rational points on varieties.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 234, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1181, "problem_number": "ALG-021", "title": "The Pierce-Birkhoff Conjecture", "statement": "Is every piecewise-polynomial function $f: \\mathbb{R}^n \\to \\mathbb{R}$ the maximum of finitely many minimums of finite collections of polynomials?", "background": "The Pierce-Birkhoff conjecture asks whether every continuous piecewise polynomial function on $\\mathbb{R}^n$ can be represented using only the operations of addition, multiplication, and taking finite suprema and infima of polynomial functions. The conjecture has been verified in dimension 1 and for $n = 2$ in special cases, but remains open for general $n \\geq 2$. The problem has connections to real algebraic geometry, approximation theory, and constructive mathematics. A positive answer would provide powerful representation theorems for piecewise-defined functions.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 178, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1182, "problem_number": "ALG-022", "title": "Serre's Positivity Conjecture", "statement": "If $R$ is a regular local ring and $P, Q$ are prime ideals with intersecting dimensions satisfying a certain condition, is the intersection multiplicity positive?", "background": "Serre's positivity conjecture (part of Serre's multiplicity conjectures) concerns intersection multiplicities in commutative algebra. It states that if $R$ is a commutative regular local ring and $P, Q$ are prime ideals with $\\dim(R/P) + \\dim(R/Q) = \\dim(R)$, then the intersection multiplicity $\\chi(R/P, R/Q) > 0$. The conjecture was proven by Gabber, Paul Roberts, and others in the 1980s for rings containing a field, but remains open in mixed characteristic (characteristic 0 with positive characteristic residue field). The problem connects to algebraic K-theory and has applications to intersection theory.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 156, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1183, "problem_number": "NT-029", "title": "Artin's Conjecture on Primitive Roots", "statement": "For how many prime numbers $p$ is a given integer $a$ (not $\\pm 1$ or a perfect square) a primitive root modulo $p$?", "background": "Artin's conjecture on primitive roots states that any integer $a$ that is neither $-1$, $\\pm 1$, nor a perfect square is a primitive root modulo infinitely many primes, and gives a conjectured density for such primes. For example, it predicts that 2 is a primitive root for approximately 37.4% of all primes. Under the assumption of the generalized Riemann hypothesis, Hooley proved the conjecture in 1967. However, the unconditional case remains open. The conjecture has important implications for the distribution of generators in finite fields and connects to class field theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 267, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1184, "problem_number": "NT-030", "title": "The abc Conjecture", "statement": "For coprime integers $a, b, c$ with $a + b = c$, is $c$ usually not much larger than the product of distinct primes dividing $abc$?", "background": "The abc conjecture, proposed by Oesterlé and Masser in 1985, is one of the most important open problems in number theory. It states that for any $\\epsilon > 0$, there are only finitely many triples of coprime positive integers $(a,b,c)$ with $a + b = c$ such that $c > \\text{rad}(abc)^{1+\\epsilon}$, where $\\text{rad}(n)$ is the product of distinct prime factors of $n$. If true, it would imply Fermat's Last Theorem, Mordell's conjecture (already proven), and many other results. Shinichi Mochizuki claimed a proof in 2012 using inter-universal Teichmüller theory, but the mathematical community has not reached consensus on its validity.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 892, "favorite_count": 76, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1185, "problem_number": "GEO-010", "title": "The Shephard's Problem", "statement": "Can the unit ball in $\\mathbb{R}^n$ be illuminated by fewer than $2^n$ directions?", "background": "Shephard's problem, a variant of the illumination problem, asks how many directions are needed to illuminate the entire boundary of the unit ball in $n$-dimensional space. A direction illuminates a boundary point if moving in that direction from the point leads outside the ball. It is known that $2^n$ directions suffice (by considering all combinations of positive/negative coordinate directions), but whether fewer suffice is unknown for $n \\geq 3$. The problem connects to convex geometry, discrete geometry, and optimization. Even the three-dimensional case ($n = 3$) remains open.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 198, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1187, "problem_number": "ALG-023", "title": "The Andrews-Curtis Conjecture", "statement": "Can every balanced presentation of the trivial group be transformed into a trivial presentation by a sequence of Nielsen transformations and conjugations?", "background": "Proposed in 1965 by James Andrews and Morton Curtis, this conjecture addresses the problem of simplifying group presentations. A balanced presentation has the same number of generators and relators. The question asks whether any such presentation of the trivial group can be reduced to the obvious trivial presentation through elementary operations (Nielsen transformations on relators and conjugating relators). Despite extensive computational searches, no counterexample has been found, but the general case remains unresolved. This problem connects combinatorial group theory, topology (via the Whitehead conjecture), and algorithmic complexity.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 412, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1188, "problem_number": "ALG-024", "title": "The Bounded Burnside Problem", "statement": "For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2, 5)$ finite?", "background": "The Burnside problem, posed in 1902, asks whether a finitely generated group in which every element has finite order must itself be finite. The bounded version restricts to groups where all elements have order dividing a fixed $n$. Major breakthroughs came when Novikov and Adian (1968) proved $B(m,n)$ is infinite for odd $n \\geq 4381$ and $m \\geq 2$, and Zel'manov earned a Fields Medal (1994) for proving finiteness when $n$ is a prime power. The case $B(2,5)$ remains a famous open problem. The group $B(2,3)$ is known to be finite with 27 elements.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 687, "favorite_count": 52, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1189, "problem_number": "ALG-025", "title": "The Guralnick-Thompson Conjecture", "statement": "What are the composition factors of finite groups appearing in genus-0 systems?", "background": "This conjecture, proposed by Robert Guralnick and John Thompson, concerns the classification of finite groups that can act on Riemann surfaces of genus 0. The conjecture provides a list of simple groups that can appear as composition factors of such groups. The problem connects group theory with algebraic geometry and the theory of automorphisms of Riemann surfaces. Genus-0 systems are particularly important in the classification of finite simple groups and their actions on low-genus surfaces.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 298, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1190, "problem_number": "ALG-026", "title": "The Herzog-Schönheim Conjecture", "statement": "If a finite system of left cosets of subgroups of a group $G$ partitions $G$, must some two subgroups have the same index?", "background": "Proposed independently by Marcel Herzog and Jochanan Schönheim in 1974, this conjecture states that if finitely many left cosets of subgroups partition a group, then at least two of the subgroups must have the same finite index. This problem arises naturally in the study of group coverings and has connections to number theory through systems of covering congruences. Despite much research, the conjecture remains open in general, though it has been verified for various special cases including abelian groups and certain classes of finite groups.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 321, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1191, "problem_number": "ALG-027", "title": "The Inverse Galois Problem", "statement": "Is every finite group the Galois group of some Galois extension of $\\mathbb{Q}$?", "background": "The inverse Galois problem is one of the central open problems in Galois theory. While classical Galois theory establishes a correspondence between field extensions and groups, the inverse problem asks whether every finite group can be realized as the Galois group of an extension of the rational numbers. The problem was implicit in work of Hilbert and has been explicitly studied since the late 19th century. It has been solved affirmatively for many classes of groups (symmetric groups, alternating groups, many sporadic simple groups), but the general case remains open. The problem connects algebra, number theory, and algebraic geometry through its connection to dessins d'enfants and modular curves.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 892, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1192, "problem_number": "ALG-028", "title": "The Isomorphism Problem for Coxeter Groups", "statement": "Is there an algorithm to determine whether two Coxeter groups given by presentations are isomorphic?", "background": "Coxeter groups are fundamental objects in geometric group theory, generated by reflections with certain relations. They include the symmetry groups of regular polytopes and tessellations. The isomorphism problem asks whether there exists an algorithmic procedure to decide if two Coxeter groups, given by their Coxeter diagrams or presentations, are isomorphic. While the problem is solved for finite and affine Coxeter groups, the general case for arbitrary Coxeter groups remains open. This problem is related to the broader isomorphism problem for groups and has applications in geometry and topology.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 367, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1193, "problem_number": "ALG-029", "title": "Infinitude of Leinster Groups", "statement": "Are there infinitely many Leinster groups?", "background": "A Leinster group is a finite group whose order equals the sum of the orders of its proper normal subgroups. Named after Tom Leinster who studied them in 1996, only two examples are currently known: the cyclic group of order 6 and a group of order 12. The question of whether infinitely many such groups exist remains open. This problem connects group theory with number theory through the properties of divisors and has connections to the study of perfect numbers (where the sum of proper divisors equals the number itself).", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1194, "problem_number": "ALG-030", "title": "Existence of Generalized Moonshine", "statement": "Does generalized moonshine exist for all elements of the Monster group?", "background": "Monstrous moonshine, discovered in the 1970s and proven by Borcherds (Fields Medal 1998), reveals a surprising connection between the Monster group (the largest sporadic simple group) and modular functions. Generalized moonshine extends this to other elements of the Monster group, asking whether similar connections exist for all group elements. Conway and Norton conjectured explicit relationships, and significant progress has been made, but the complete generalized moonshine remains unproven. This connects finite groups, modular forms, string theory, and vertex operator algebras in profound ways.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 543, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1195, "problem_number": "ALG-031", "title": "Finiteness of Finitely Presented Periodic Groups", "statement": "Is every finitely presented periodic group finite?", "background": "A periodic group (or torsion group) is one in which every element has finite order. The question of whether a finitely presented periodic group must be finite was a major open problem for much of the 20th century. The restricted Burnside problem, solved by Zel'manov, showed that finitely generated groups where all elements have bounded order must be finite. However, the general case without the bounded exponent assumption remains open. This problem connects group theory, geometric group theory, and algorithmic questions about group presentations.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 456, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1196, "problem_number": "ALG-032", "title": "The Surjunctivity Conjecture", "statement": "Is every group surjunctive?", "background": "A group is surjunctive if every injective cellular automaton over that group is also surjective. Equivalently, every injective endomorphism of the shift space is surjective. This property was introduced by Gottschalk in 1973 and connects symbolic dynamics, cellular automata theory, and group theory. Gromov and Weiss proved that all sofic groups are surjunctive, and since all amenable groups are sofic, this includes a large class. However, the general question of whether all groups are surjunctive remains open and is equivalent to asking whether all groups are sofic.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 389, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1197, "problem_number": "ALG-033", "title": "The Sofic Groups Conjecture", "statement": "Is every discrete countable group sofic?", "background": "A group is sofic if it can be approximated by finite symmetric groups in a precise sense. The concept was introduced by Gromov and Weiss around 1999 and has become central in modern group theory. All known groups are sofic: amenable groups, residually finite groups, linear groups, and many others. The soficity of all groups would have profound consequences for many conjectures in group theory, operator algebras, and ergodic theory. Notable implications include Connes' embedding conjecture (now known to be false via quantum complexity theory) and Gottschalk's surjunctivity conjecture. Despite the breadth of known sofic groups, the general question remains one of the deepest open problems in infinite group theory.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 612, "favorite_count": 48, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1198, "problem_number": "ALG-034", "title": "Arthur's Conjectures", "statement": "What is the structure of the discrete spectrum of automorphic forms on reductive groups?", "background": "Proposed by James Arthur in the 1980s, these conjectures describe the decomposition of the space of automorphic forms into irreducible representations. They provide a framework for understanding the discrete spectrum in terms of endoscopic groups and Arthur packets. The conjectures connect representation theory, harmonic analysis, and number theory, generalizing results of Langlands. Major progress has been made, including Arthur's proof for classical groups (2013), but the full program for all reductive groups remains incomplete. These conjectures are central to the Langlands program and have applications to trace formulas and L-functions.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 478, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1199, "problem_number": "ALG-035", "title": "Dade's Conjecture", "statement": "Is there a relationship between the numbers of irreducible characters in blocks of a finite group and its local subgroups?", "background": "Proposed by Everett Dade in 1992, this conjecture concerns the modular representation theory of finite groups. It relates the number of irreducible characters of a given defect in a block of a finite group to corresponding numbers in blocks of certain local subgroups (normalizers of p-subgroups). The conjecture is part of a broader program to reduce questions about representations of finite groups to questions about p-groups and their normalizers. It has been verified for many classes of groups but remains open in general. Dade's conjecture refines earlier conjectures by Alperin and McKay.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 312, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1200, "problem_number": "ALG-036", "title": "The Demazure Conjecture", "statement": "Can representations of semisimple algebraic groups be characterized over the integers?", "background": "Proposed by Michel Demazure in the 1970s, this conjecture concerns the existence of certain integral structures on representations of algebraic groups. It asks whether irreducible representations of semisimple algebraic groups over fields of positive characteristic can be deformed to characteristic zero while preserving integrality properties. The conjecture has applications to geometric representation theory and the theory of quantum groups. Partial results have been obtained for special cases, but the general conjecture remains open. The problem connects algebraic groups, representation theory, and arithmetic geometry.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 289, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1203, "problem_number": "GEO-012", "title": "The Spherical Bernstein Problem", "statement": "What is the classification of complete minimal hypersurfaces in spheres of all dimensions?", "background": "This is a generalization of Bernstein's problem (solved by 1968) which asked whether the only minimal graph over all of Euclidean space is a hyperplane. The spherical version asks for the classification of complete minimal hypersurfaces in the sphere $S^{n+1}$. While progress has been made in specific dimensions, a complete classification for all dimensions remains open. The problem connects differential geometry, minimal surface theory, and geometric analysis, with applications to general relativity and materials science.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 387, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1204, "problem_number": "GEO-013", "title": "The Carathéodory Conjecture", "statement": "Does every convex, closed, twice-differentiable surface in $\\mathbb{R}^3$ have at least two umbilical points?", "background": "Proposed by Constantin Carathéodory in the 1920s, this conjecture concerns umbilical points on convex surfaces—points where the principal curvatures are equal. The conjecture states that any smooth closed convex surface in 3-dimensional Euclidean space must have at least two such points. A sphere has infinitely many umbilical points (every point is umbilical), while an ellipsoid generically has exactly 4. The conjecture has been proven for surfaces of revolution and certain other special cases, but remains open in general. It connects differential geometry, topology (via the Poincaré-Hopf theorem), and dynamical systems.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 456, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1205, "problem_number": "GEO-014", "title": "The Cartan-Hadamard Conjecture", "statement": "Does the isoperimetric inequality hold for Cartan-Hadamard manifolds?", "background": "The classical isoperimetric inequality states that among all regions of fixed volume in Euclidean space, a ball has the smallest surface area. The Cartan-Hadamard conjecture asks whether this extends to Cartan-Hadamard manifolds—complete, simply connected Riemannian manifolds of nonpositive sectional curvature. The conjecture has been proven in dimensions 2, 3, and 4, and for many special classes of manifolds, but remains open in higher dimensions. This problem is central to geometric analysis and has connections to optimal transport theory, general relativity, and the study of black hole thermodynamics.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 523, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1206, "problem_number": "GEO-015", "title": "Chern's Affine Conjecture", "statement": "Does the Euler characteristic of a compact affine manifold vanish?", "background": "Proposed by Shiing-Shen Chern, this conjecture states that every closed affine manifold (a manifold with an atlas whose transition functions are affine transformations) has Euler characteristic zero. An affine structure is stronger than a smooth structure but weaker than a Riemannian structure. The conjecture has been verified for many classes of affine manifolds, and recent work has made substantial progress, but a complete proof remains elusive. The problem connects differential geometry, topology, and the theory of geometric structures on manifolds.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 398, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1207, "problem_number": "GEO-016", "title": "Chern's Conjecture for Hypersurfaces in Spheres", "statement": "What minimal hypersurfaces in spheres have constant mean curvature?", "background": "This is actually a family of related conjectures proposed by Shiing-Shen Chern concerning the classification of minimal and constant mean curvature hypersurfaces embedded in spheres. One version asks whether the only minimal hypersurface in $S^{n+1}$ with constant scalar curvature is the totally geodesic $S^n$. These conjectures connect minimal surface theory, the study of isoparametric hypersurfaces, and geometric analysis. Partial results have been obtained, but the general conjectures remain open.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 367, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1208, "problem_number": "GEO-017", "title": "The Closed Curve Problem", "statement": "What are necessary and sufficient conditions for an integral curve defined by two periodic functions to be closed?", "background": "This problem asks for explicit, computable conditions to determine when a curve defined parametrically by integrating two periodic functions with the same period will close up. The question arises naturally in dynamical systems, Hamiltonian mechanics, and the study of periodic orbits. While special cases are understood, general necessary and sufficient conditions that can be readily checked remain unknown. The problem connects analysis, differential geometry, and dynamical systems theory.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 289, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1209, "problem_number": "GEO-018", "title": "The Filling Area Conjecture", "statement": "Does a hemisphere have minimum area among shortcut-free surfaces with a given boundary length?", "background": "This conjecture in systolic geometry states that among all surfaces in Euclidean space whose boundary is a closed curve of given length and which contain no shortcuts (the surface distance between boundary points equals the Euclidean distance), the hemisphere has minimal area. The problem was proposed by Gromov and connects differential geometry, geometric measure theory, and the calculus of variations. It has applications to the study of minimal surfaces and optimal shapes in physics and materials science.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 334, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1210, "problem_number": "GEO-019", "title": "The Hopf Conjectures", "statement": "What is the relationship between curvature and Euler characteristic for even-dimensional Riemannian manifolds?", "background": "Heinz Hopf proposed several conjectures relating the sign of sectional curvature to the Euler characteristic and other topological invariants of closed Riemannian manifolds. The most famous asks whether a closed even-dimensional manifold with positive (or negative) sectional curvature must have positive Euler characteristic. The conjectures have been resolved in dimension 2 (Gauss-Bonnet) and partially in dimension 4, but remain open in higher dimensions. These problems are central to understanding the interplay between curvature and topology in Riemannian geometry.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 567, "favorite_count": 43, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1211, "problem_number": "GEO-020", "title": "The Osserman Conjecture", "statement": "Is every Osserman manifold either flat or locally isometric to a rank-one symmetric space?", "background": "An Osserman manifold is a Riemannian manifold where the eigenvalues of the Jacobi operator are constant on the unit sphere bundle at each point. Robert Osserman conjectured that such manifolds must be either flat or locally isometric to a rank-one symmetric space (spheres, projective spaces, or hyperbolic spaces). The conjecture has been proven in dimensions up to 4 and for many special cases, but remains open in higher dimensions. The problem connects differential geometry, spectral theory, and the theory of symmetric spaces.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 412, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1212, "problem_number": "GEO-021", "title": "Yau's Conjecture on First Eigenvalues", "statement": "Is the first eigenvalue of the Laplace-Beltrami operator on a minimal hypersurface in $S^{n+1}$ equal to $n$?", "background": "Proposed by Shing-Tung Yau, this conjecture states that for any closed embedded minimal hypersurface in the $(n+1)$-dimensional sphere $S^{n+1}$, the first nonzero eigenvalue of the Laplace-Beltrami operator equals $n$. This would provide a sharp spectral characterization of minimal hypersurfaces in spheres. The conjecture has been verified for several important cases including geodesic spheres, Clifford tori, and certain other symmetric examples. The problem connects spectral geometry, minimal surface theory, and PDEs on manifolds.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 478, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1213, "problem_number": "GEO-022", "title": "The Hadwiger Covering Conjecture", "statement": "Can every $n$-dimensional convex body be covered by at most $2^n$ smaller homothetic copies?", "background": "Proposed by Hugo Hadwiger in 1957, this conjecture states that any $n$-dimensional convex body can be covered by at most $2^n$ positive homothetic (scaled and translated) copies of itself with smaller ratio. The conjecture is known to be true for $n = 1$ (trivial) and $n = 2$ (proven), but remains open for $n \\geq 3$. The problem connects discrete geometry, convex geometry, and combinatorics. It is related to the illumination problem and has connections to coding theory and sphere packing.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 523, "favorite_count": 38, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1214, "problem_number": "GEO-023", "title": "The Happy Ending Problem", "statement": "What is the minimum number of points in the plane needed to guarantee a convex $n$-gon?", "background": "The Happy Ending problem, named by Paul Erdős because it led to the marriage of Esther Klein and George Szekeres, asks for $g(n)$—the smallest number such that any set of $g(n)$ points in general position contains $n$ points forming a convex $n$-gon. It's known that $2^{n-2} + 1 \\leq g(n) \\leq \\binom{2n-4}{n-2} + 1$. The exact value is known only for $n \\leq 6$. Erdős offered $500 for a proof that $g(n) = 2^{n-2} + 1$. The problem is central to combinatorial geometry and Ramsey theory.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 612, "favorite_count": 47, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1215, "problem_number": "GEO-024", "title": "The Heilbronn Triangle Problem", "statement": "What is the largest minimum area of a triangle determined by $n$ points in a unit square?", "background": "Proposed by Hans Heilbronn in 1908, this problem asks how to place $n$ points in a unit square to maximize the smallest area of any triangle they determine. Heilbronn originally conjectured the maximum was $O(1/n^2)$, but this was disproven—the actual order is between $\\Omega(\\log n / n^2)$ and $O(1/n^{8/7-\\epsilon})$. Finding the exact asymptotic remains open. The problem connects discrete geometry, extremal combinatorics, and has applications to numerical integration and computational geometry.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 445, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1216, "problem_number": "GEO-025", "title": "Kalai's $3^d$ Conjecture", "statement": "Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?", "background": "Proposed by Gil Kalai in 1989, this conjecture states that any centrally symmetric convex polytope in $d$ dimensions must have at least $3^d$ faces (including the polytope itself and the empty set). The bound is tight, achieved by the $d$-dimensional cube which has exactly $3^d$ faces. The conjecture has been verified for $d \\leq 4$ and for various special classes of polytopes. The problem connects combinatorics, convex geometry, and polytope theory, with applications to optimization and computational geometry.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 378, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1217, "problem_number": "GEO-026", "title": "The Unit Distance Problem", "statement": "What is the maximum number of unit distances determined by $n$ points in the plane?", "background": "This problem, posed by Erdős in 1946, asks for the maximum number of pairs of points at distance exactly 1 in a set of $n$ points in the Euclidean plane. The best known construction gives $\\Omega(n^{4/3})$ unit distances, while the best upper bound is $O(n^{4/3})$. Determining the exact asymptotic (and whether the exponent is exactly $4/3$) remains open. The problem connects extremal combinatorics, incidence geometry, and has applications to facility location and wireless network design. Erdős offered prizes for progress on this problem.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 567, "favorite_count": 42, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1219, "problem_number": "GEO-028", "title": "Ehrhart's Volume Conjecture", "statement": "Does a convex body in $\\mathbb{R}^n$ with one interior lattice point at its center of mass have volume at most $(n+1)^n/n!$?", "background": "Proposed by Eugène Ehrhart, this conjecture concerns lattice polytopes—convex bodies whose vertices have integer coordinates. It states that if a convex body in $n$ dimensions contains exactly one lattice point in its interior (which is its center of mass), then its volume cannot exceed $(n+1)^n/n!$, the volume of a regular simplex. The conjecture has been verified for $n \\leq 3$ and for many special cases. The problem connects discrete geometry, convex geometry, and number theory, with applications to integer programming and combinatorial optimization.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 389, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1220, "problem_number": "ALG-039", "title": "The Cherlin-Zilber Conjecture", "statement": "Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?", "background": "Proposed by Gregory Cherlin and Boris Zilber in the 1970s, this conjecture connects model theory and group theory. It states that any infinite simple group whose first-order theory is stable must be isomorphic to a simple algebraic group defined over an algebraically closed field. The conjecture has been verified for many classes of groups and represents a deep connection between logic and algebra. It generalizes the classification of finite simple groups to model-theoretic contexts and has implications for the structure theory of stable groups.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 412, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1221, "problem_number": "ALG-040", "title": "The Generalized Star Height Problem", "statement": "Can all regular languages be expressed with generalized regular expressions of bounded star height?", "background": "This problem in formal language theory asks whether there exists a uniform bound on the nesting depth of Kleene star operations needed to express any regular language using generalized regular expressions (which allow complementation). While the ordinary star height problem (without complementation) was solved—showing unbounded star height is necessary—the generalized version remains open. The problem connects automata theory, formal languages, and computational complexity, with applications to pattern matching and compiler design.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 334, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1222, "problem_number": "NT-031", "title": "Hilbert's Tenth Problem for Number Fields", "statement": "For which number fields is there an algorithm to determine solvability of Diophantine equations?", "background": "Hilbert's tenth problem asked for an algorithm to determine whether a Diophantine equation has integer solutions. Matiyasevich (building on work by Davis, Putnam, and Robinson) proved in 1970 that no such algorithm exists for the integers. The problem remains open for other rings, particularly number fields (finite extensions of the rationals). It has been solved negatively for some number fields and positively for others, but the general characterization is unknown. This connects logic, number theory, and computability theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 523, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1224, "problem_number": "GEO-029", "title": "Borsuk's Conjecture", "statement": "Can every bounded set in $\\mathbb{R}^n$ be partitioned into $n+1$ sets of smaller diameter?", "background": "Proposed by Karol Borsuk in 1933, this conjecture asks whether every bounded set in $n$-dimensional Euclidean space can be partitioned into $n+1$ parts, each with diameter strictly smaller than the original set. The conjecture held for dimensions up to 3 until 1993, when Kahn and Kalai found a counterexample in dimension 1325. The smallest dimension for which the conjecture fails remains unknown (known to fail for $n \\geq 64$). This problem connects geometric combinatorics, high-dimensional geometry, and has inspired research into diameter-reducing partitions.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 523, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1225, "problem_number": "GEO-030", "title": "The Kissing Number Problem", "statement": "What is the maximum number of non-overlapping unit spheres that can touch a central unit sphere in $n$ dimensions?", "background": "The kissing number $\\tau_n$ is the maximum number of non-overlapping unit spheres that can simultaneously touch a central unit sphere in $n$-dimensional Euclidean space. Known exactly only for dimensions 1, 2, 3, 4, 8, and 24, this problem has connections to sphere packing, coding theory, and lattice theory. The dimensions 8 and 24 are special due to exceptional lattices (E8 and Leech lattice). Determining kissing numbers in other dimensions, particularly dimensions 5, 6, 7, and general high dimensions, remains a major open problem in discrete geometry.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 612, "favorite_count": 46, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1226, "problem_number": "GEO-031", "title": "Ulam's Packing Conjecture", "statement": "Is the sphere the worst-packing convex solid?", "background": "Proposed by Stanisław Ulam, this conjecture asks which three-dimensional convex body has the smallest packing density. Ulam conjectured that the sphere is the worst-packing convex solid, meaning that among all convex bodies in 3D, spheres have the smallest proportion of space filled when packed. While the sphere packing problem (densest packing) was solved by Hales (2005), the worst-packing problem remains open. The conjecture connects packing theory, convex geometry, and optimization, with potential applications to materials science and crystallography.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 445, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1227, "problem_number": "GEO-032", "title": "Sphere Packing in High Dimensions", "statement": "What is the densest packing of unit spheres in dimensions other than 1, 2, 3, 8, and 24?", "background": "The sphere packing problem asks for the densest arrangement of non-overlapping unit spheres in $n$-dimensional Euclidean space. Solved for dimensions 1 and 2 (trivial), dimension 3 by Hales (1998, computer-assisted proof), dimension 8 by Viazovska (2016), and dimension 24 by Cohn et al. (2016), the problem remains open for all other dimensions. Understanding the asymptotic behavior as $n \\to \\infty$ is also open. This connects coding theory, lattice theory, and has applications to error-correcting codes and wireless communications.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 734, "favorite_count": 58, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1231, "problem_number": "COMB-010", "title": "The Cap Set Problem", "statement": "What is the maximum size of a cap set in $\\mathbb{F}_3^n$?", "background": "A cap set is a subset of the $n$-dimensional vector space over the three-element field with no three elements in arithmetic progression (analogous to the card game SET). The problem asks for the maximum size of such a set as a function of $n$. In 2016, Ellenberg and Gijswijt proved an upper bound of $O(2.756^n)$, dramatically improving previous bounds and resolving the longstanding question of whether cap sets can have exponential size $3^{cn}$ for $c > 0$. However, the exact maximum size and optimal constant remain open. This connects additive combinatorics, algebraic combinatorics, and theoretical computer science.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 523, "favorite_count": 40, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1235, "problem_number": "COMB-012", "title": "The Sunflower Conjecture", "statement": "Does every family of at least $c^k k!$ sets of size $k$ contain a sunflower of size 3, for some absolute constant $c$?", "background": "Proposed by Erdős and Rado in 1960, a sunflower (or $\\Delta$-system) is a collection of sets where every pair shares the same common intersection. The conjecture asks whether the exponential bound $c^k k!$ suffices to guarantee a sunflower of any fixed size. The best known bound is super-exponential. In 2019, Alweiss et al. made breakthrough progress by improving the bound to $O((\\log k)^k k!)$, but reaching the conjectured bound remains open. This problem is central to extremal combinatorics and has applications to circuit complexity and communication complexity.", "difficulty_level_id": 5, "status": "open", "category_id": 2, "view_count": 612, "favorite_count": 48, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1236, "problem_number": "COMB-013", "title": "Ramsey Number $R(5,5)$", "statement": "What is the exact value of the Ramsey number $R(5,5)$?", "background": "Ramsey numbers quantify the size at which complete disorder becomes impossible. $R(5,5)$ is the minimum number of vertices such that any two-coloring of the edges of the complete graph contains either a red $K_5$ or a blue $K_5$. It is known that $43 \\leq R(5,5) \\leq 48$, but the exact value remains unknown despite over 50 years of effort. This is perhaps the most famous open Ramsey number. Erdős famously suggested that finding $R(6,6)$ would require astronomical resources, but $R(5,5)$ seems tantalizingly within reach. The problem connects combinatorics, graph theory, and computational mathematics.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 823, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1239, "problem_number": "NT-032", "title": "Gauss Circle Problem", "statement": "How far can the number of lattice points in a circle centered at the origin deviate from the area of the circle?", "background": "The Gauss circle problem asks for the tightest bound on the error term in counting integer lattice points $(m,n)$ inside a circle of radius $r$ centered at the origin. The number of such points is $\\pi r^2 + E(r)$ where $E(r)$ is the error. It is known that $E(r) = O(r^{2/3})$ and $E(r) = \\Omega(r^{1/2} \\log r)$, but the exact growth rate remains unknown. Hardy conjectured $E(r) = O(r^{1/2+\\varepsilon})$ for any $\\varepsilon > 0$. This connects analytic number theory, lattice point enumeration, and has applications to physics and crystallography.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 478, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1240, "problem_number": "NT-033", "title": "Grimm's Conjecture", "statement": "Can each element of a set of consecutive composite numbers be assigned a distinct prime divisor?", "background": "Proposed by C. A. Grimm in 1969, this conjecture states that if we have $k$ consecutive composite numbers, then there exist $k$ distinct primes each dividing one of these numbers. For example, the consecutive composites $24, 25, 26, 27, 28$ have distinct prime divisors $3, 5, 13, 7, 2$ respectively. While verified computationally for large ranges, the general proof remains elusive. The conjecture connects to prime gaps, divisibility properties, and the distribution of primes among consecutive integers.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 412, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1241, "problem_number": "NT-034", "title": "Hall's Conjecture", "statement": "For any $\\varepsilon > 0$, is there a constant $c(\\varepsilon)$ such that either $y^2 = x^3$ or $|y^2 - x^3| > c(\\varepsilon) x^{1/2-\\varepsilon}$?", "background": "Proposed by Marshall Hall Jr. in 1970, this conjecture provides a measure of how close a perfect square can be to a perfect cube without being equal. It strengthens earlier work on Diophantine approximation and relates to the ABC conjecture. The conjecture has been verified for many special cases but remains open in general. It connects algebraic number theory, Diophantine equations, and elliptic curves, with implications for understanding integer solutions to polynomial equations.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1242, "problem_number": "NT-035", "title": "Lehmer's Totient Problem", "statement": "If Euler's totient function $\\phi(n)$ divides $n-1$, must $n$ be prime?", "background": "Posed by D. H. Lehmer in 1932, this problem asks whether any composite number $n$ exists such that $\\phi(n)$ divides $n-1$, where $\\phi(n)$ counts integers up to $n$ coprime to $n$. For all primes $p$, we have $\\phi(p) = p-1$, so the divisibility holds. Lehmer conjectured no composite number has this property. It has been verified that any such composite must be odd, square-free, and have at least 7 prime factors, with the smallest exceeding $10^{20}$. This connects Euler's totient function, primality, and multiplicative number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 523, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1243, "problem_number": "NT-036", "title": "Magic Square of Squares", "statement": "Does there exist a 3×3 magic square composed entirely of distinct perfect squares?", "background": "A magic square has the property that all rows, columns, and diagonals sum to the same value. While magic squares of integers are well understood, the question of whether a 3×3 magic square can be constructed using only distinct perfect squares has remained open for centuries. Martin LaBar proved in 1984 that no such square exists using rational squares, but the integer case remains unsolved. Partial results exist for 4×4 and larger squares. This connects number theory, Diophantine equations, and recreational mathematics.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 589, "favorite_count": 47, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1244, "problem_number": "NT-037", "title": "Mahler's 3/2 Problem", "statement": "Is there a real number $x$ such that the fractional parts of $x(3/2)^n$ are all less than $1/2$ for every positive integer $n$?", "background": "Proposed by Kurt Mahler in the 1960s, this problem concerns the distribution of the sequence $\\{x(3/2)^n\\}$ modulo 1, where $\\{y\\}$ denotes the fractional part of $y$. Mahler conjectured that no such $x$ exists. The problem relates to ergodic theory, uniform distribution, and Diophantine approximation. While various partial results have been obtained using techniques from dynamical systems and number theory, the general question remains open. It exemplifies deep questions about the behavior of geometric sequences under modular arithmetic.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 398, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1245, "problem_number": "NT-038", "title": "Newman's Conjecture", "statement": "Does the partition function satisfy any arbitrary congruence infinitely often?", "background": "Proposed by Morris Newman, this conjecture concerns the partition function $p(n)$, which counts the number of ways to write $n$ as a sum of positive integers. Newman conjectured that for any integers $a$ and $m$ with $\\gcd(a,m) = 1$, there are infinitely many $n$ such that $p(n) \\equiv a \\pmod{m}$. This would imply the partition function takes all possible residue classes modulo any integer infinitely often. The conjecture connects partition theory, modular forms, and has implications for understanding the arithmetic properties of partitions.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 367, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1246, "problem_number": "NT-039", "title": "Scholz Conjecture", "statement": "Is the shortest addition chain for $2^n - 1$ at most $n - 1$ plus the length of the shortest addition chain for $n$?", "background": "An addition chain for $m$ is a sequence $1 = a_0 < a_1 < \\cdots < a_r = m$ where each $a_i$ (for $i > 0$) is the sum of two earlier terms. Scholz conjectured in 1937 that $\\ell(2^n-1) \\leq n-1+\\ell(n)$ where $\\ell(m)$ denotes the minimum length of an addition chain for $m$. This has applications to efficient exponentiation algorithms in computer science and cryptography. While verified for many values and various special cases proven, the general conjecture remains open. It connects additive number theory, combinatorial optimization, and computational complexity.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 412, "favorite_count": 30, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1248, "problem_number": "NT-041", "title": "Infinitely Many Perfect Numbers", "statement": "Are there infinitely many perfect numbers?", "background": "A perfect number equals the sum of its proper divisors (divisors excluding itself). Examples include 6 = 1+2+3 and 28 = 1+2+4+7+14. Euclid proved that if $2^p - 1$ is prime (a Mersenne prime), then $2^{p-1}(2^p-1)$ is perfect. All known perfect numbers have this form and are even. Whether infinitely many exist depends on whether there are infinitely many Mersenne primes, itself an open question. The problem connects prime number theory, divisor functions, and has fascinated mathematicians for over 2000 years.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 678, "favorite_count": 54, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1250, "problem_number": "NT-043", "title": "Quasiperfect Numbers", "statement": "Do quasiperfect numbers exist?", "background": "A quasiperfect number is a natural number $n$ such that the sum of its divisors equals $2n + 1$ (one more than twice the number). No quasiperfect number has ever been found. It has been proven that if one exists, it must be an odd square number greater than $10^{35}$, and have at least seven distinct prime factors. The search for quasiperfect numbers connects divisor theory, multiplicative number theory, and computational number theory. Their existence or non-existence would provide insights into the structure of highly composite numbers.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 398, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1251, "problem_number": "NT-044", "title": "Almost Perfect Numbers Beyond Powers of 2", "statement": "Do any almost perfect numbers exist that are not powers of 2?", "background": "An almost perfect number $n$ has the sum of its proper divisors equal to $n - 1$. All powers of 2 are almost perfect, since the divisors of $2^k$ are $1, 2, 4, \\ldots, 2^{k-1}$ which sum to $2^k - 1$. It remains unknown whether any odd almost perfect number exists, or any even almost perfect number that is not a power of 2. The problem connects perfect numbers, divisor functions, and the structure of highly specific arithmetic sequences.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 356, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1252, "problem_number": "NT-045", "title": "The Number of Idoneal Numbers", "statement": "Are there exactly 65 idoneal numbers, or could there be 66 or 67?", "background": "Idoneal numbers (also called suitable or convenient numbers) are positive integers $D$ such that if $n = ax^2 + by^2$ with coprime $a,b$ is uniquely representable, then $n$ is a prime power or twice a prime power. Euler conjectured 65 such numbers exist, the largest being 1848. Weinberger proved in 1973 that at most one more exists, and if the generalized Riemann hypothesis is true, exactly 65 exist. This connects binary quadratic forms, class field theory, and the Riemann hypothesis. The resolution depends on deep questions in analytic number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 334, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1253, "problem_number": "NT-046", "title": "Amicable Numbers of Opposite Parity", "statement": "Do any pairs of amicable numbers exist where one is odd and one is even?", "background": "Two numbers are amicable if each equals the sum of the proper divisors of the other. For example, 220 and 284 are amicable (both even). Over 12 million amicable pairs are known, all with matching parity (both even or both odd). It remains unknown whether a mixed-parity pair exists. Such a pair would require unusual divisor properties. The problem connects divisor sums, parity constraints, and the arithmetic structure of amicable pairs. All known odd amicable pairs have been found by Erdős and collaborators.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1254, "problem_number": "NT-047", "title": "Infinitely Many Amicable Pairs", "statement": "Are there infinitely many pairs of amicable numbers?", "background": "Amicable numbers are pairs where each number equals the sum of the other's proper divisors. While over 12 million pairs have been discovered, it remains unknown whether infinitely many exist. Thabit ibn Qurra (9th century) gave a formula generating some pairs, and Euler found many more. Various conjectures suggest their density, but no proof of infinitude exists. This contrasts with related questions like twin primes (conjectured infinite) and perfect numbers (infinitude depends on Mersenne primes). The problem connects multiplicative number theory and divisor sums.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1255, "problem_number": "NT-048", "title": "Infinitely Many Giuga Numbers", "statement": "Are there infinitely many Giuga numbers?", "background": "A Giuga number is a composite number $n$ such that $p$ divides $(n/p - 1)$ for every prime divisor $p$ of $n$. Equivalently, $\\sum_{p|n} (1/p) - 1/n$ is an integer. Only 15 Giuga numbers are known, the smallest being 30. Giuga conjectured that if $1 + \\sum_{i=1}^{n-1} i^{n-1} \\equiv 0 \\pmod{n}$ for composite $n$, then $n$ is a Giuga number. Whether infinitely many exist remains open. This connects primality testing, Carmichael numbers, and Fermat pseudoprimes.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 367, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1256, "problem_number": "NT-049", "title": "Lychrel Numbers in Base 10", "statement": "Do Lychrel numbers exist in base 10?", "background": "A Lychrel number is a natural number that never forms a palindrome through the iterative process of adding it to its reverse. For example, 89 is not Lychrel: 89 + 98 = 187, 187 + 781 = 968, 968 + 869 = 1837, 1837 + 7381 = 9218, 9218 + 8129 = 17347, 17347 + 74371 = 91718, 91718 + 81719 = 173437, 173437 + 734371 = 907808, 907808 + 808709 = 1716517, 1716517 + 7156171 = 8872688, which is a palindrome. The number 196 is the smallest candidate Lychrel number, having been tested to over 300 million iterations without producing a palindrome. No Lychrel number has been proven to exist.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 512, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1257, "problem_number": "NT-050", "title": "Odd Weird Numbers", "statement": "Do any odd weird numbers exist?", "background": "A weird number is a natural number that is abundant (the sum of its proper divisors exceeds the number) but not semiperfect (no subset of its divisors sums to the number). The smallest weird number is 70. All known weird numbers are even, and it has been conjectured that no odd weird numbers exist. If an odd weird number exists, it must be greater than $10^{21}$ and have at least 4 distinct prime factors. This connects abundant numbers, partition theory, and subset sum problems.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 378, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1258, "problem_number": "NT-051", "title": "Normality of Pi", "statement": "Is $\\pi$ a normal number in base 10?", "background": "A number is normal in base 10 if every digit 0-9 appears with equal frequency (1/10) in its decimal expansion, and more generally, every sequence of $k$ digits appears with frequency $1/10^k$. While the digits of $\\pi$ appear statistically random in computational tests extending to trillions of digits, no proof of normality exists. It is not even known whether every digit appears infinitely often in $\\pi$. Proving normality would require deep insights into the arithmetic nature of $\\pi$ and transcendental number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 823, "favorite_count": 68, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1259, "problem_number": "NT-052", "title": "Normality of Irrational Algebraic Numbers", "statement": "Are all irrational algebraic numbers normal in every base?", "background": "An algebraic number is a root of a polynomial with integer coefficients. Normal numbers have every digit sequence appear with the expected frequency in their base expansions. It is conjectured that all irrational algebraic numbers like $\\sqrt{2}$ are normal in every integer base, but not a single irrational algebraic number has been proven normal in any base. This represents a fundamental gap in our understanding of the decimal expansions of algebraic numbers. The question connects algebraic number theory, Diophantine approximation, and the theory of normal numbers.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 567, "favorite_count": 45, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1260, "problem_number": "NT-053", "title": "Is 10 a Solitary Number?", "statement": "Is 10 a solitary number (no other number shares its abundancy index)?", "background": "The abundancy index of $n$ is $\\sigma(n)/n$ where $\\sigma(n)$ is the sum of divisors of $n$. A number is solitary if no other number has the same abundancy index. For 10, we have $\\sigma(10) = 1+2+5+10 = 18$, giving abundancy $18/10 = 9/5$. It remains unknown whether any other number has abundancy $9/5$. Numbers in amicable pairs and sociable numbers are not solitary. Many numbers have been proven non-solitary, but 10 resists classification. This connects divisor functions, Diophantine equations, and the classification of multiplicative structures.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 334, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1262, "problem_number": "NT-055", "title": "Erdős Conjecture on Arithmetic Progressions", "statement": "If the sum of reciprocals of a set of positive integers diverges, does the set contain arbitrarily long arithmetic progressions?", "background": "Erdős conjectured that if $A \\subseteq \\mathbb{N}$ and $\\sum_{a \\in A} 1/a = \\infty$, then $A$ contains arithmetic progressions of arbitrary length. This strengthens Szemerédi's theorem, which only requires positive density. The conjecture remains open even for progressions of length 3. In 2020, Bloom and Sisask made major progress by proving that if $\\sum_{a \\in A, a \\leq N} 1/a \\geq (\\log N)^{c \\log \\log \\log N}$ for some $c$, then $A$ contains a 3-term arithmetic progression. This connects additive combinatorics, harmonic analysis, and analytic number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 534, "favorite_count": 42, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1263, "problem_number": "NT-056", "title": "Erdős-Turán Conjecture on Additive Bases", "statement": "If $B$ is an additive basis of order 2, must the representation function tend to infinity?", "background": "An additive basis of order 2 is a set $B$ such that every sufficiently large integer can be written as the sum of two elements of $B$. The representation function $r_B(n)$ counts the number of ways to write $n$ as $b_1 + b_2$ with $b_1, b_2 \\in B$. Erdős and Turán conjectured in 1941 that if $B$ is an additive basis of order 2, then $r_B(n)$ must tend to infinity. This has been proven for various special bases, but the general case remains open. The conjecture connects additive number theory, combinatorics, and the structure of thin bases.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 456, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1265, "problem_number": "NT-058", "title": "Lander-Parkin-Selfridge Conjecture", "statement": "If the sum of $m$ $k$-th powers equals the sum of $n$ $k$-th powers, must $m + n \\geq k$?", "background": "This conjecture generalizes Fermat's Last Theorem to sums of powers. It states that if $a_1^k + \\cdots + a_m^k = b_1^k + \\cdots + b_n^k$ with positive integers and the two sums are different, then $m + n \\geq k$. Euler conjectured the stronger statement that at least $k$ $k$-th powers are needed, but this was disproved: $27^5 + 84^5 + 110^5 + 133^5 = 144^5$ (counterexample with $k=5$, $m=4$, $n=1$). The weaker LPS conjecture remains open for $k \\geq 4$ and has implications for Diophantine equations and additive number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 489, "favorite_count": 37, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1266, "problem_number": "NT-059", "title": "Lemoine's Conjecture", "statement": "Can every odd integer greater than 5 be expressed as the sum of an odd prime and an even semiprime?", "background": "Proposed by Émile Lemoine in 1894, this conjecture states that every odd number $n > 5$ can be written as $n = p + 2q$ where $p$ and $q$ are primes. An even semiprime is twice a prime. For example, $27 = 13 + 2(7)$, $31 = 19 + 2(6)$ is invalid since 6 isn't prime, but $31 = 5 + 2(13)$ works. This is weaker than Goldbach's conjecture (which implies Lemoine's). Verified computationally to very large numbers, but no proof exists. It connects prime distribution, additive representations, and the Goldbach problem.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1267, "problem_number": "NT-060", "title": "Recamán's Sequence Completeness", "statement": "Does every nonnegative integer appear in Recamán's sequence?", "background": "Recamán's sequence starts with $a_0 = 0$ and follows the rule: $a_n = a_{n-1} - n$ if that value is positive and not already in the sequence, otherwise $a_n = a_{n-1} + n$. This produces: 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, ... Named after Colombian mathematician Bernardo Recamán Santos, this sequence has been computed to millions of terms, but it remains unknown whether every nonnegative integer appears. Some values appear very late or may never appear. This connects integer sequences, graph theory, and computational number theory.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 512, "favorite_count": 40, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1268, "problem_number": "NT-061", "title": "Skolem Problem", "statement": "Can an algorithm determine if a constant-recursive sequence contains a zero?", "background": "A constant-recursive sequence satisfies a linear recurrence with constant coefficients, like the Fibonacci sequence. The Skolem problem asks whether there exists an algorithm to determine if such a sequence ever equals zero. This is known to be decidable for sequences of order up to 4, but the general problem remains open. The Positivity Problem (whether all terms are positive) and Ultimate Positivity (whether terms are eventually all positive) are related variants. This connects computability theory, Diophantine approximation, and decidability in number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1269, "problem_number": "NT-062", "title": "Waring's Problem: Exact Values", "statement": "What are the exact values of $g(k)$ and $G(k)$ for all $k$ in Waring's problem?", "background": "Waring's problem concerns representing integers as sums of $k$-th powers. Let $g(k)$ be the minimum number such that every positive integer can be written as a sum of at most $g(k)$ $k$-th powers, allowing any number of terms. Let $G(k)$ be the same but excluding a finite set of exceptions. We know $g(2)=4$ (Lagrange), $G(2)=4$, $g(3)=9$, $G(3)=4$, $g(4)=19$, $G(4)=16$. For general $k$, Hilbert proved $g(k)$ exists but exact values remain unknown for most $k$. This is a central problem in additive number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 567, "favorite_count": 44, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1270, "problem_number": "NT-063", "title": "Density of Ulam Numbers", "statement": "Do the Ulam numbers have a positive density?", "background": "The Ulam numbers start with 1, 2, and each subsequent number is the smallest integer that can be expressed as the sum of two distinct earlier Ulam numbers in exactly one way: 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ... Named after Stanisław Ulam, these numbers appear to have density around 0.07, but whether the density exists and is positive remains unproven. Related questions about their growth rate and distribution connect to additive combinatorics, unique representation bases, and computational number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 398, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1271, "problem_number": "NT-064", "title": "Class Number Problem", "statement": "Are there infinitely many real quadratic number fields with unique factorization?", "background": "A number field has unique factorization if every nonzero element factors uniquely into irreducibles. For real quadratic fields $\\mathbb{Q}(\\sqrt{d})$ with $d > 0$ square-free, unique factorization is equivalent to having class number 1. Gauss conjectured infinitely many such fields exist. While infinitely many imaginary quadratic fields (class number 1) were ruled out, the real case remains open. Computational evidence strongly supports the conjecture, but a proof eludes us. This connects algebraic number theory, class field theory, and the distribution of number fields.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 478, "favorite_count": 36, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1272, "problem_number": "NT-065", "title": "Hilbert's Twelfth Problem", "statement": "Can the Kronecker-Weber theorem on abelian extensions of $\\mathbb{Q}$ be extended to any base number field?", "background": "The Kronecker-Weber theorem states that every abelian extension of the rationals $\\mathbb{Q}$ is contained in a cyclotomic field (generated by roots of unity). Hilbert's 12th problem asks for an analogous explicit construction of abelian extensions of arbitrary number fields. For imaginary quadratic fields, complex multiplication provides a partial answer using elliptic curves and modular functions. For general number fields, the problem remains largely open despite over a century of work. This is fundamental to class field theory and arithmetic geometry.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 512, "favorite_count": 40, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1273, "problem_number": "NT-066", "title": "Leopoldt's Conjecture", "statement": "Does the $p$-adic regulator of an algebraic number field not vanish?", "background": "Leopoldt's conjecture, proposed in 1962, states that the $p$-adic regulator of an algebraic number field $K$ is nonzero for every prime $p$. The regulator measures the \"size\" of the unit group. The conjecture has been verified for abelian extensions of $\\mathbb{Q}$ and many other special cases, but remains open in general. It has deep connections to Iwasawa theory, $p$-adic L-functions, and the structure of class groups. A proof would have significant implications for understanding $p$-adic analytic properties of number fields.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1274, "problem_number": "NT-067", "title": "Lindelöf Hypothesis", "statement": "For all $\\varepsilon > 0$, does $\\zeta(1/2 + it) = o(t^\\varepsilon)$ as $t \\to \\infty$?", "background": "The Lindelöf hypothesis concerns the growth rate of the Riemann zeta function $\\zeta(s)$ on the critical line $\\text{Re}(s) = 1/2$. It states that for any $\\varepsilon > 0$, we have $|\\zeta(1/2 + it)| = o(t^\\varepsilon)$. This is weaker than the Riemann Hypothesis but still unproven. The best known bound is $O(t^{13/84+\\varepsilon})$ due to Bourgain (2022). The hypothesis has implications for the distribution of primes, zero-free regions of $\\zeta(s)$, and analytic number theory. It connects to moment problems and random matrix theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 545, "favorite_count": 43, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1275, "problem_number": "NT-068", "title": "Hilbert-Pólya Conjecture", "statement": "Do the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator?", "background": "The Hilbert-Pólya conjecture proposes a spectral interpretation of the Riemann Hypothesis: the nontrivial zeros of $\\zeta(s)$ at $1/2 + i\\gamma_n$ correspond to eigenvalues of some self-adjoint operator, with $\\gamma_n$ being the eigenvalues. This would imply RH since eigenvalues of self-adjoint operators are real. Connections to random matrix theory (Montgomery-Dyson) and quantum chaos support this idea. Finding such an operator remains elusive despite attempts involving quantum mechanics, trace formulas, and noncommutative geometry. This bridges analysis, spectral theory, and mathematical physics.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 623, "favorite_count": 51, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1276, "problem_number": "NT-069", "title": "Grand Riemann Hypothesis", "statement": "Do all automorphic L-functions have their nontrivial zeros on the critical line?", "background": "The Grand Riemann Hypothesis extends RH to all automorphic L-functions, a vast class including Dirichlet L-functions, Dedekind zeta functions, and L-functions of modular forms. It asserts that all nontrivial zeros lie on the critical line $\\text{Re}(s) = 1/2$. This would have profound consequences for prime distribution in arithmetic progressions, algebraic number theory, and the Langlands program. The GRH is considered one of the most important unifying conjectures in mathematics, generalizing many individual cases of the Riemann Hypothesis.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 712, "favorite_count": 59, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1277, "problem_number": "NT-070", "title": "Montgomery's Pair Correlation Conjecture", "statement": "Does the pair correlation function of Riemann zeta zeros match that of random Hermitian matrices?", "background": "Montgomery conjectured in 1973 that the statistical distribution of gaps between zeros of the Riemann zeta function matches the pair correlation of eigenvalues from the Gaussian Unitary Ensemble (GUE) of random matrix theory. This remarkable connection between number theory and quantum physics was discovered through numerical experiments and Dyson's insights. The conjecture has been partially verified but remains unproven. It suggests deep links between prime numbers, quantum chaos, and statistical mechanics, forming a cornerstone of the modern approach to understanding zeta zeros.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 567, "favorite_count": 46, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1278, "problem_number": "NT-071", "title": "Dirichlet's Divisor Problem", "statement": "What is the optimal exponent in the error term for the divisor summatory function?", "background": "Let $D(x) = \\sum_{n \\leq x} d(n)$ where $d(n)$ counts the divisors of $n$. Dirichlet proved $D(x) = x \\log x + (2\\gamma - 1)x + \\Delta(x)$ where $\\gamma$ is Euler's constant and $\\Delta(x)$ is the error. The problem asks for the infimum $\\theta$ such that $\\Delta(x) = O(x^\\theta)$. It is known that $1/4 \\leq \\theta < 131/416 \\approx 0.314903$. The Riemann Hypothesis would imply $\\theta \\leq 1/4 + \\varepsilon$ for any $\\varepsilon > 0$, but proving this is extremely difficult. This connects analytic number theory, the Riemann zeta function, and lattice point problems.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1279, "problem_number": "GEO-033", "title": "Erdős-Ulam Problem", "statement": "Is there a dense set of points in the plane with all pairwise distances rational?", "background": "Proposed by Paul Erdős and Stanisław Ulam, this problem asks whether there exists a dense subset of the Euclidean plane (dense in the usual topology) such that the distance between any two points is a rational number. While finite and countable dense sets with rational distances are known (like rational points on a circle), an everywhere-dense set remains undiscovered. The problem connects geometry, Diophantine equations, and the structure of rational points. It has implications for understanding constraints on rational distance sets.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 478, "favorite_count": 36, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1281, "problem_number": "NT-073", "title": "Four Exponentials Conjecture", "statement": "If $x_1, x_2$ are linearly independent over $\\mathbb{Q}$ and $y_1, y_2$ are linearly independent over $\\mathbb{Q}$, is at least one of $e^{x_1 y_1}, e^{x_1 y_2}, e^{x_2 y_1}, e^{x_2 y_2}$ transcendental?", "background": "This conjecture, a consequence of Schanuel's conjecture, asserts that under the stated conditions, at least one of the four exponentials must be transcendental. The six exponentials theorem (proven) states that if $x_1, x_2, x_3$ are $\\mathbb{Q}$-linearly independent and $y_1, y_2$ are $\\mathbb{Q}$-linearly independent, then among the six values $e^{x_i y_j}$, at least one is transcendental. The four exponentials conjecture would strengthen this. It connects exponential Diophantine equations, transcendence theory, and algebraic independence.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1282, "problem_number": "NT-074", "title": "Irrationality of Euler's Constant", "statement": "Is the Euler-Mascheroni constant $\\gamma$ irrational?", "background": "Euler's constant $\\gamma = \\lim_{n \\to \\infty} (1 + 1/2 + 1/3 + \\cdots + 1/n - \\ln n) \\approx 0.5772$ appears throughout mathematics but its arithmetic nature remains mysterious. It is not even known whether $\\gamma$ is irrational, let alone transcendental. While computational evidence suggests irrationality (verified to billions of digits), no proof exists. The problem has resisted attack for over 250 years. Progress would require new techniques in transcendental number theory and might illuminate the nature of other constants like $\\zeta(3)$.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 712, "favorite_count": 58, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1283, "problem_number": "NT-075", "title": "Transcendence of Apéry's Constant", "statement": "Is $\\zeta(3) = 1 + 1/8 + 1/27 + 1/64 + \\cdots$ transcendental?", "background": "Apéry's constant $\\zeta(3) \\approx 1.202$ is the value of the Riemann zeta function at 3. Roger Apéry proved its irrationality in 1978 using ingenious continued fraction methods, surprising the mathematical community. Whether $\\zeta(3)$ is transcendental remains unknown. More generally, the transcendence of $\\zeta(2k+1)$ for integer $k \\geq 1$ is open (except $\\zeta(1)$ which diverges). Rivoal (2000) proved infinitely many $\\zeta(2k+1)$ are irrational, and at least one of $\\zeta(5), \\zeta(7), \\zeta(9), \\zeta(11)$ is irrational, but transcendence is far harder.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 589, "favorite_count": 47, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1284, "problem_number": "NT-076", "title": "Littlewood Conjecture", "statement": "For any two real numbers $\\alpha, \\beta$, does $\\liminf_{n \\to \\infty} n \\|n\\alpha\\| \\|n\\beta\\| = 0$?", "background": "Proposed by John Edensor Littlewood around 1930, where $\\|x\\|$ denotes the distance from $x$ to the nearest integer. The conjecture asserts a simultaneous approximation property: for any pair of real numbers, infinitely many integers $n$ exist such that both $n\\alpha$ and $n\\beta$ are simultaneously close to integers, with the product of distances approaching zero. While verified for many cases (algebraic numbers, certain combinations), the general conjecture remains open. Einsiedler, Katok, and Lindenstrauss (2006) proved the set of counterexamples has Hausdorff dimension zero, suggesting counterexamples are rare if they exist.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 456, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1285, "problem_number": "NT-077", "title": "Integer Factorization in Polynomial Time", "statement": "Can integer factorization be solved in polynomial time on a classical computer?", "background": "The integer factorization problem asks: given a composite number $n$, find its prime factors. The best known classical algorithm (general number field sieve) runs in sub-exponential time $\\exp(O((\\ln n)^{1/3}(\\ln \\ln n)^{2/3}))$. Whether a polynomial-time classical algorithm exists is unknown and has profound implications for cryptography (RSA security relies on factorization hardness). Shor's algorithm solves factorization in polynomial time on quantum computers, but practical quantum computers don't yet exist. The problem connects computational complexity, cryptography, and number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 734, "favorite_count": 61, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1286, "problem_number": "NT-078", "title": "Beal's Conjecture", "statement": "For $A^x + B^y = C^z$ with $x, y, z > 2$, must $A$, $B$, and $C$ share a common prime factor?", "background": "Proposed by banker and amateur mathematician Andrew Beal in 1993, this conjecture generalizes Fermat's Last Theorem. It asserts that if $A^x + B^y = C^z$ where $A, B, C, x, y, z$ are positive integers with $x, y, z > 2$, then $A$, $B$, and $C$ must have a common prime factor. For example, $3^3 + 6^3 = 3^5$ satisfies this since all share factor 3. Beal has offered a prize of $1 million for a proof or counterexample. The conjecture is equivalent to saying no solutions exist when $A$, $B$, $C$ are coprime. This connects Fermat's Last Theorem, the abc conjecture, and exponential Diophantine equations.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 712, "favorite_count": 59, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1289, "problem_number": "NT-081", "title": "Fermat-Catalan Conjecture", "statement": "Are there finitely many solutions to $a^m + b^n = c^k$ with coprime $a,b,c$ and $1/m + 1/n + 1/k < 1$?", "background": "This conjecture generalizes both Fermat's Last Theorem and the Catalan-Mersenne conjecture. It asserts that the equation $a^m + b^n = c^k$ has only finitely many solutions in coprime positive integers $a,b,c$ and integers $m,n,k \\geq 2$ satisfying $1/m + 1/n + 1/k < 1$. Ten solutions are known, including $1^m + 2^3 = 3^2$, $2^5 + 7^2 = 3^4$, and $17^3 + 2^{7\\cdot 13^3} = 71^2 \\cdot 13^3$. Beal's conjecture and the abc conjecture both imply Fermat-Catalan. The problem is central to exponential Diophantine equations.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 634, "favorite_count": 52, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1292, "problem_number": "NT-084", "title": "Bunyakovsky Conjecture", "statement": "Does an irreducible integer polynomial with no fixed prime divisor produce infinitely many primes?", "background": "Proposed by Viktor Bunyakovsky in 1857, this generalizes Dirichlet's theorem on primes in arithmetic progressions. It states that if polynomial $f(x)$ has integer coefficients, positive leading coefficient, is irreducible over integers, and has no common prime divisor of all its values $f(n)$ for positive integers $n$, then $f(x)$ represents infinitely many primes. This would imply infinitely many twin primes (using $f(x) = x$ and $g(x) = x+2$), Sophie Germain primes, and many other families. Despite being over 160 years old, it remains unproven except for linear polynomials (Dirichlet's theorem).", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 512, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1293, "problem_number": "NT-085", "title": "Dickson's Conjecture", "statement": "Do finitely many linear forms simultaneously take prime values infinitely often, barring congruence obstructions?", "background": "Proposed by Leonard Eugene Dickson in 1904, this generalizes Dirichlet's theorem and implies many prime conjectures. For linear forms $a_1 + b_1 n, \\ldots, a_k + b_k n$ with each $b_i \\geq 1$, if no congruence condition forces a composite, then infinitely many $n$ exist making all forms simultaneously prime. This would imply: twin primes, Sophie Germain primes, prime triplets, Goldbach's conjecture, and more. It strengthens Bunyakovsky and is a special case of Schinzel's Hypothesis H. No proof exists even for two forms, representing a fundamental gap in our understanding of simultaneous prime values.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1294, "problem_number": "NT-086", "title": "Brocard's Conjecture (Prime Gaps)", "statement": "Are there always at least 4 primes between consecutive squares of primes $p_n^2$ and $p_{n+1}^2$?", "background": "Proposed by Henri Brocard in 1904, this conjecture concerns the density of primes near perfect squares. For consecutive primes $p_n$ and $p_{n+1}$, Brocard conjectured there are always at least 4 primes in the interval $(p_n^2, p_{n+1}^2)$, except for the cases $(2^2, 3^2)$ which contains only one prime (5). Verified computationally to enormous values, but no proof exists. This is stronger than Legendre's conjecture (at least one prime between consecutive squares). It connects prime gaps, Bertrand's postulate generalizations, and the distribution of primes.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 398, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1295, "problem_number": "NT-087", "title": "Agoh-Giuga Conjecture", "statement": "Is $p$ prime if and only if $pB_{p-1} \\equiv -1 \\pmod{p}$ for the Bernoulli number $B_{p-1}$?", "background": "This conjecture combines work of Takashi Agoh (1990) and Giuseppe Giuga (1950), providing a primality criterion via Bernoulli numbers. Bernoulli numbers $B_n$ appear in number theory and analysis. The conjecture states: $p$ is prime iff $pB_{p-1} \\equiv -1 \\pmod{p}$. The forward direction is known (if $p$ prime, the congruence holds). The converse would give a new primality test. Related to Giuga numbers and Wolstenholme's theorem, this connects Bernoulli numbers, primality testing, and modular arithmetic in unexpected ways.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 334, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1296, "problem_number": "NT-088", "title": "Elliott-Halberstam Conjecture", "statement": "Do primes distribute uniformly in arithmetic progressions up to nearly $x$ (instead of $x^{1/2}$)?", "background": "Proposed in 1968, this strengthens the Bombieri-Vinogradov theorem about primes in arithmetic progressions. For most moduli $q < x^\\theta$, the primes are equidistributed among valid residue classes. Bombieri-Vinogradov proves this for $\\theta < 1/2$. Elliott-Halberstam conjectures it holds for any $\\theta < 1$. This would have dramatic consequences: it implies infinitely many bounded prime gaps exist (a weak form proven by Zhang 2013, then Polymath improved to gap 246). The full conjecture would likely yield bounded gaps near the twin prime level. It connects sieve methods, Goldston-Pintz-Yıldırım techniques, and multiplicative functions.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 412, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1297, "problem_number": "ALG-001", "title": "Birch–Tate Conjecture", "statement": "Is there a relation between the order of the center of the Steinberg group and the Dedekind zeta function?", "background": "The Birch–Tate conjecture connects algebraic K-theory to number theory. For a number field $F$, it relates the order of the center of the Steinberg group $\\text{St}(\\mathcal{O}_F)$ (where $\\mathcal{O}_F$ is the ring of integers) to special values of the Dedekind zeta function $\\zeta_F(s)$ at $s = -1$. The conjecture predicts that $|\\text{center}(\\text{St}(\\mathcal{O}_F))| = |\\zeta_F(-1)|$ after appropriate normalization. This would provide a deep connection between algebraic structures and analytic number theory, generalizing classical results about class numbers. It fits into the broader Quillen-Lichtenbaum conjecture framework and has implications for understanding higher K-groups.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1300, "problem_number": "ALG-004", "title": "Crouzeix's Conjecture", "statement": "Is $\\|f(A)\\| \\leq 2\\sup_{z \\in W(A)} |f(z)|$ for all matrices $A$ and functions $f$ analytic on the numerical range?", "background": "Michel Crouzeix conjectured in 2004 that for any $n \\times n$ complex matrix $A$ and any function $f$ analytic on the numerical range $W(A) = \\{\\langle Ax, x \\rangle : \\|x\\| = 1\\}$, the matrix norm satisfies $\\|f(A)\\| \\leq 2\\|f\\|_{W(A)}$. The constant 2 is conjectured to be optimal. Crouzeix proved the bound with constant $11.08$, later improved to $1 + \\sqrt{2} \\approx 2.41$ by various authors. The conjecture is verified for $2 \\times 2$ matrices and special classes. It has applications to functional calculus, matrix functions, and numerical analysis. The problem combines complex analysis, operator theory, and linear algebra, and its resolution would clarify fundamental properties of matrix functions.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 278, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1302, "problem_number": "ALG-006", "title": "Perfect Cuboid", "statement": "Does there exist a rectangular cuboid with integer edges, face diagonals, and space diagonal?", "background": "A perfect cuboid would have integer values for all of: three edge lengths $a, b, c$, three face diagonals $\\sqrt{a^2+b^2}, \\sqrt{b^2+c^2}, \\sqrt{c^2+a^2}$, and the space diagonal $\\sqrt{a^2+b^2+c^2}$. This is the 3D generalization of the Pythagorean triple problem (which has infinitely many solutions). Despite extensive computer searches, no perfect cuboid has been found, nor has impossibility been proven. The problem connects to Diophantine equations, elliptic curves, and number theory. Weaker versions exist: edge-perfect cuboids (all edges and face diagonals integer) are known, as are face-perfect and space-perfect variants. The perfect cuboid is problem D18 in Richard Guy's \"Unsolved Problems in Number Theory\" and has attracted amateur and professional attention for over a century.", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 423, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1305, "problem_number": "ALG-009", "title": "Zauner's Conjecture (SIC-POVM)", "statement": "Do symmetric informationally complete POVMs exist in all dimensions?", "background": "Zauner's conjecture, central to quantum information theory, asks whether SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) exist in all finite-dimensional Hilbert spaces. A SIC-POVM in dimension $d$ consists of $d^2$ pure quantum states with pairwise fidelity $1/(d+1)$, forming a regular simplex in quantum state space. These structures optimize quantum measurements and have applications in quantum tomography, cryptography, and foundations. SIC-POVMs are known for dimensions up to 193 and many higher dimensions through numerical construction. Analytic constructions exist for infinitely many dimensions using Weyl-Heisenberg groups and number-theoretic methods. The conjecture connects to algebraic number theory (Stark units, ray class fields), representation theory, and Galois theory. Resolving it would clarify fundamental symmetries in quantum mechanics.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 298, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1308, "problem_number": "ALG-012", "title": "Andrews–Curtis Conjecture", "statement": "Can every balanced presentation of the trivial group be transformed to a trivial presentation by Nielsen moves?", "background": "The Andrews–Curtis conjecture, proposed in 1965, concerns group presentations. A balanced presentation has the same number of generators and relators. The trivial presentation is $\\langle x \\mid x \\rangle$. Nielsen transformations on relators include: replacing relator $r$ with $r^{-1}$, with $rs$ for another relator $s$, or conjugating $r$. The question: can any balanced presentation of the trivial group be reduced to the trivial presentation using these moves? Known counter-examples exist for unbalanced presentations (Rapaport). The conjecture is verified for many cases but remains open in general. It connects to the Zeeman conjecture in topology, 4-manifold theory, and algebraic K-theory. A counter-example would have major implications for understanding fundamental groups and 2-complexes.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 289, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1310, "problem_number": "ALG-014", "title": "Herzog–Schönheim Conjecture", "statement": "Can a finite system of left cosets forming a partition of a group have distinct indices?", "background": "The Herzog–Schönheim conjecture states: if left cosets $g_iH_i$ of subgroups $H_i$ partition a group $G$, then at least two indices $[G:H_i]$ must be equal. Equivalently, you cannot partition a group using cosets of subgroups with all different indices. The conjecture is verified for many cases: finite abelian groups, free groups, and groups with certain structural properties. It has connections to coverings of groups, number theory (covering congruences—Mycielski's conjecture), and additive combinatorics. The problem appears simple but has resisted general proof. A counter-example would be a group with a highly unusual coset structure, and would impact understanding of group factorizations and tiling problems.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1318, "problem_number": "ANA-006", "title": "Navier-Stokes Regularity", "statement": "Do smooth initial data for 3D Navier-Stokes equations yield smooth solutions for all time?", "background": "One of the seven Millennium Prize Problems ($1M prize). The 3D Navier-Stokes equations govern fluid flow: $\\partial_t u + (u \\cdot \\nabla)u = \\nu \\Delta u - \\nabla p + f$ with $\\nabla \\cdot u = 0$. Given smooth initial conditions and forcing, do solutions remain smooth globally, or can finite-time singularities develop? In 2D, global regularity is proven. In 3D, existence of weak solutions is known (Leray), but smoothness is open. Partial results establish regularity under smallness conditions or for special data. The problem is central to mathematical fluid dynamics and has deep implications for turbulence, computational fluid dynamics, and the physical validity of the equations. Techniques involve harmonic analysis, functional analysis, and PDE theory.", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 892, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1319, "problem_number": "COMB-001", "title": "1/3–2/3 Conjecture", "statement": "Does every non-totally-ordered finite poset have two elements with probability between 1/3 and 2/3 in random linear extensions?", "background": "For a finite partially ordered set (poset) that is not totally ordered, the 1/3–2/3 conjecture asks: do there always exist elements $x$ and $y$ such that the probability $x$ appears before $y$ in a uniformly random linear extension is strictly between 1/3 and 2/3? Linear extensions are total orderings consistent with the partial order. The conjecture was posed in the 1960s and remains open. It has connections to sorting algorithms, computational complexity, and order theory. Known results: true for many special classes of posets, including series-parallel posets. The conjecture would provide insight into the structure of linear extensions and has applications to average-case analysis of sorting and ranking algorithms.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 234, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1320, "problem_number": "COMB-002", "title": "Lonely Runner Conjecture", "statement": "If $k$ runners with distinct speeds run on a circular track, will each be lonely (distance $\\geq 1/k$ from others) at some time?", "background": "Proposed by J. M. Wills in 1967, this conjecture concerns runners on a unit-length circular track with distinct constant speeds. A runner is \"lonely\" if all other runners are at distance at least $1/k$ away. The conjecture states every runner is lonely at some time. Verified for $k \\leq 7$ runners. The problem has reformulations in terms of Diophantine approximation, view-obstruction (can $k-1$ points block all views from a point to another on a circle?), and number theory. Applications include scheduling, communication protocols, and chromatic number of certain graphs. Proof techniques use continued fractions, geometry of numbers, and combinatorial arguments. The general case remains stubbornly open despite its elementary statement.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 312, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1321, "problem_number": "COMB-003", "title": "Union-Closed Sets Conjecture", "statement": "For a finite family of sets closed under unions, must some element appear in at least half the sets?", "background": "Frankl's union-closed sets conjecture (1979) states: if a finite family $\\mathcal{F}$ of sets is closed under pairwise unions (i.e., $A, B \\in \\mathcal{F} \\Rightarrow A \\cup B \\in \\mathcal{F}$), then there exists an element appearing in at least $|\\mathcal{F}|/2$ sets. The conjecture is verified for many special cases: families with at most 50 sets, families where the largest set has at most 11 elements, and various structural conditions. In 2024, significant progress was made proving the conjecture holds when relaxing \"half\" to 0.01% (a weakened version). The problem connects to lattice theory, combinatorics, and has reformulations in terms of posets and Boolean functions. A proof would illuminate the structure of union-closed families.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 387, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1322, "problem_number": "COMB-004", "title": "No-Three-in-Line Problem", "statement": "What is the maximum number of points in an $n \\times n$ grid with no three collinear?", "background": "The no-three-in-line problem asks for $g(n)$, the maximum number of points that can be placed in an $n \\times n$ grid such that no three are collinear. Dudeney (1917) conjectured $g(n) = 2n$ for all $n$. Known values: $g(3) = 4, g(4) = 8, g(5) = 10, g(6) = 12$, and computational results extend further. For large $n$, Erdős proved $g(n) \\leq cn/(\\log \\log n)^{1/2}$ for some constant $c$. Lower bounds around $1.85n$ are known. The problem connects to combinatorial geometry, Ramsey theory, and coding theory. Despite its elementary formulation, determining exact values or the asymptotic behavior of $g(n)$ remains challenging. Applications include error-correcting codes and geometric configurations.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 298, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1324, "problem_number": "COMB-006", "title": "Sunflower Conjecture", "statement": "For fixed $r$, can the number of size-$k$ sets needed for an $r$-sunflower be bounded by $c^k$ for some constant $c$?", "background": "Erdős and Rado (1960) defined an $r$-sunflower as a collection of $r$ sets $A_1, \\ldots, A_r$ with common intersection $C$ (the core) such that the sets $A_i \\setminus C$ are pairwise disjoint (the petals). Their theorem: any family of size-$k$ sets with at least $k! \\cdot r^k$ members contains an $r$-sunflower. The sunflower conjecture asks: can the bound be improved to $c^k$ for some constant $c = c(r)$ depending only on $r$? This would be sharp up to the value of $c$. In 2019, Alweiss, Lovett, Wu, and Zhang proved a bound of $(\\log k)^k$, a breakthrough improving Erdős-Rado. The conjecture has applications to circuit complexity, learning theory, and DNF formulas. A proof would impact computational complexity theory.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 367, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1327, "problem_number": "GRAPH-003", "title": "Cycle Double Cover Conjecture", "statement": "Does every bridgeless graph have a collection of cycles covering each edge exactly twice?", "background": "The cycle double cover conjecture states: every bridgeless graph (no bridge edges) has a cycle double cover—a collection of cycles such that each edge appears in exactly two cycles. Proposed by Szekeres (1973) and Seymour (1979), this is equivalent to several other conjectures in graph theory. Known for planar graphs (via face boundaries), 4-edge-connected graphs, and graphs with maximum degree at most 3. The conjecture connects to nowhere-zero flows, graph embeddings, and topological graph theory. It would imply results about circular chromatic number and graph decompositions. Despite extensive research, the general case remains open and is considered one of the major problems in graph theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 312, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1328, "problem_number": "GRAPH-004", "title": "Erdős–Hajnal Conjecture", "statement": "For any fixed graph $H$, do $H$-free graphs contain large cliques or independent sets?", "background": "The Erdős–Hajnal conjecture (1977) asks: for any graph $H$, is there $\\delta > 0$ such that every $n$-vertex graph with no induced copy of $H$ contains a clique or independent set of size at least $n^\\delta$? This would dramatically strengthen Ramsey theory for hereditary graph classes. For general graphs, Ramsey theorem gives only polylogarithmic guarantees. The conjecture is proven for specific $H$: paths, trees of bounded diameter, and certain small graphs. Partial results by Alon, Pach, and Solymosi establish weaker bounds. The problem connects to extremal graph theory, Ramsey theory, and structural graph theory. A proof would reveal deep structure in induced-subgraph-free graphs.", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 289, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1329, "problem_number": "GRAPH-005", "title": "Lovász Conjecture", "statement": "Does every finite connected vertex-transitive graph have a Hamiltonian path?", "background": "Proposed by László Lovász in 1969, this conjecture states that every finite connected vertex-transitive graph (graph with transitive automorphism group) contains a Hamiltonian path. A stronger version asks for a Hamiltonian cycle. The conjecture is verified for Cayley graphs (Rapaport-Strasser, 1985 for primes; Marušič for certain cases), vertex-transitive graphs of order $pq$ for primes $p < q$, and various special classes. Counter-examples exist for infinite graphs. The problem connects to algebraic graph theory, group theory, and the study of symmetric structures. A proof would significantly advance understanding of Hamiltonian properties in highly symmetric graphs and has implications for network design and routing.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1330, "problem_number": "GRAPH-006", "title": "Hadwiger–Nelson Problem", "statement": "What is the chromatic number of the plane with unit distance graph coloring?", "background": "The Hadwiger–Nelson problem asks: what is the minimum number of colors needed to color the plane such that no two points at distance exactly 1 have the same color? This is equivalent to finding the chromatic number of the unit distance graph in $\\mathbb{R}^2$. It has been known since 1950 that $4 \\leq \\chi \\leq 7$. In 2018, Aubrey de Grey found a unit distance graph with chromatic number 5, improving the lower bound to 5. The upper bound of 7 uses a hexagonal tiling argument. The exact value is unknown. The problem connects to Euclidean Ramsey theory, discrete geometry, and combinatorial optimization. Extensions to higher dimensions and different metrics are also studied.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 421, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1331, "problem_number": "TOP-001", "title": "Unknotting Problem", "statement": "Can unknots be recognized in polynomial time?", "background": "The unknotting problem asks whether there exists a polynomial-time algorithm to determine if a given knot diagram represents the unknot (a circle with no actual knots). A knot diagram is a 2D projection of a 3D knot with crossing information. The problem is known to be in NP (a certificate is a sequence of Reidemeister moves) and co-NP (certification via knot invariants). In 2011, Lackenby, building on work by Dynnikov, showed an algorithm exists with complexity bounded by $2^{cn}$ for some constant $c$, where $n$ is the crossing number. However, whether a polynomial-time algorithm exists remains unknown. The problem connects to computational topology, 3-manifold theory, and has applications to molecular biology (DNA unknotting) and physics.", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 334, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1332, "problem_number": "TOP-002", "title": "Borel Conjecture", "statement": "Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?", "background": "The Borel conjecture states: if two aspherical closed manifolds (manifolds with contractible universal cover) have isomorphic fundamental groups, then they are homeomorphic. An aspherical manifold has all higher homotopy groups trivial, so its topology is determined by $\\pi_1$. The conjecture is a topological rigidity statement: algebraic data ($\\pi_1$) determines geometric structure (homeomorphism type). Proven for many special cases: flat manifolds, hyperbolic manifolds (by Mostow rigidity for dimension $\\geq 3$), and certain graph manifolds. The Novikov conjecture is a weaker form (about homotopy invariance of higher signatures). The Borel conjecture connects to surgery theory, K-theory, and geometric group theory.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 278, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1333, "problem_number": "TOP-003", "title": "Volume Conjecture", "statement": "Do quantum invariants of knots relate asymptotically to hyperbolic volume?", "background": "The volume conjecture, proposed by Kashaev (1997) and generalized by Murakami-Murakami, relates quantum topology to hyperbolic geometry. For a hyperbolic knot $K$ in $S^3$, let $J_N(K; q)$ be the colored Jones polynomial at $q = e^{2\\pi i/N}$. The conjecture states: $\\lim_{N \\to \\infty} \\frac{2\\pi \\log|J_N(K; e^{2\\pi i/N})|}{N} = \\text{Vol}(S^3 \\setminus K)$, where the right side is the hyperbolic volume of the knot complement. Verified for many specific knots and families (torus knots, figure-eight). The conjecture suggests deep connections between quantum field theory, Chern-Simons theory, and 3-manifold geometry. It would unify quantum invariants and geometric invariants.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1334, "problem_number": "TOP-004", "title": "Novikov Conjecture", "statement": "Are certain combinations of Pontryagin classes homotopy invariant?", "background": "The Novikov conjecture, proposed by Sergei Novikov in 1965, is a fundamental problem in topology and differential geometry. For a closed oriented manifold $M$ with fundamental group $\\pi$, certain rational linear combinations of Pontryagin classes evaluated on the fundamental class should be homotopy invariants when pushed forward to the classifying space $B\\pi$. More precisely, higher signatures defined using the signature operator should be homotopy invariants. The conjecture is verified for many groups: finite groups, amenable groups, linear groups, Gromov hyperbolic groups, and many others. It connects to K-theory, C*-algebras, index theory, and surgery theory. The conjecture has deep implications for the topology of manifolds and the structure of group C*-algebras.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 312, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1335, "problem_number": "GEOM-007", "title": "Kakeya Conjecture", "statement": "Must a Kakeya set in $\\mathbb{R}^n$ have Hausdorff and Minkowski dimension $n$?", "background": "A Kakeya set in $\\mathbb{R}^n$ is a compact set containing a unit line segment in every direction. The Kakeya conjecture states such sets must have full Hausdorff and Minkowski dimension $n$. In $\\mathbb{R}^2$, Kakeya sets can have measure zero (Davies 1971) but must have Hausdorff dimension 2 (proven). For $n \\geq 3$, the conjecture is open. Known results: Kakeya sets in $\\mathbb{R}^n$ have Hausdorff dimension $\\geq (n+2)/2$ (Wolff, 1995), improved to $\\geq n/2 + \\epsilon$ by various authors. The problem connects to harmonic analysis (Bochner-Riesz conjecture, restriction conjecture), PDE (wave equation estimates), and number theory. Resolving it would impact multiple areas of analysis.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 289, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1336, "problem_number": "GEOM-008", "title": "Illumination Problem", "statement": "Can every convex body in $\\mathbb{R}^n$ be illuminated by $2^n$ light sources?", "background": "The illumination problem (or Hadwiger's problem) asks: what is the minimum number of light sources (point sources or directions) needed to illuminate the entire boundary of any convex body in $\\mathbb{R}^n$? A point on the boundary is illuminated if the ray from the light source to that point does not intersect the interior. Conjecture: $2^n$ sources suffice for dimension $n$. Known results: the upper bound is $\\lfloor 3^{n}/2^{n-1} \\rfloor$ (Schramm), and the conjecture is verified for $n \\leq 3$. The problem connects to discrete geometry, combinatorial geometry, and has applications in computer graphics and sensor placement. A proof would clarify fundamental properties of convex bodies.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 234, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1338, "problem_number": "DYN-002", "title": "MLC Conjecture", "statement": "Is the Mandelbrot set locally connected?", "background": "The MLC (Mandelbrot set is Locally Connected) conjecture asks whether the famous Mandelbrot set—the set of complex parameters $c$ for which the iteration $z_{n+1} = z_n^2 + c$ (starting from $z_0 = 0$) remains bounded—is locally connected. Local connectivity would mean every point has arbitrarily small connected neighborhoods. The conjecture is one of the most important problems in complex dynamics. If true, it would imply: the boundary of the Mandelbrot set has Hausdorff dimension 2, the Mandelbrot set is the closure of its interior, and precise descriptions of the topology. The conjecture has been verified for many parameter regions but remains open in general. It connects to renormalization theory, polynomial dynamics, and fractal geometry.", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 398, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1339, "problem_number": "DYN-003", "title": "Weinstein Conjecture", "statement": "Does every regular compact contact-type level set carry a periodic orbit?", "background": "The Weinstein conjecture, proposed by Alan Weinstein in 1978, states: every regular compact contact-type level set of a Hamiltonian on a symplectic manifold carries at least one periodic orbit of the Hamiltonian flow. In more geometric terms, on a compact contact manifold, the Reeb vector field has at least one closed orbit. The conjecture has been proven in many cases: dimension 3 (Taubes, 2007), overtwisted contact 3-manifolds (Hofer, 1993), and various higher-dimensional cases using symplectic field theory and pseudoholomorphic curves. The full conjecture in all dimensions remains open. It connects contact geometry, symplectic topology, Hamiltonian dynamics, and has applications to celestial mechanics and rigid body dynamics.", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 256, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1340, "problem_number": "DYN-004", "title": "Birkhoff Conjecture", "statement": "If a billiard table is strictly convex and integrable, must its boundary be an ellipse?", "background": "The Birkhoff conjecture concerns dynamical billiards: if a strictly convex billiard table in the plane is integrable (has a complete set of integrals of motion), then its boundary must be an ellipse. Elliptical billiards are known to be integrable (Birkhoff, 1927). The conjecture asks if they are the only such tables. Partial results: true for sufficiently smooth perturbations of circles and ellipses, and for certain classes of curves. The problem connects to KAM theory, integrable systems, and spectral geometry. A proof would characterize all integrable planar billiards and has implications for understanding caustics, periodic orbits, and the inverse spectral problem.", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 289, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1341, "problem_number": "ALGGEOM-001", "title": "Abundance Conjecture", "statement": "If the canonical bundle of a variety is nef, must it be semiample?", "background": "The abundance conjecture is a central problem in birational geometry and minimal model theory. For a projective variety $X$ with Kawamata log terminal singularities, if the canonical bundle $K_X$ is nef (numerically effective—has non-negative intersection with all curves), the conjecture states $K_X$ must be semiample (some positive multiple is globally generated). This would complete the minimal model program by ensuring every minimal model has good positivity properties. Known cases: surfaces (classical), dimension 3 (Miyaoka, Kawamata), toric varieties, and certain special cases in higher dimensions. The conjecture connects to the cone theorem, base point freeness, and would have major implications for classification of algebraic varieties.", "difficulty_level_id": 5, "status": "open", "category_id": 5, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1343, "problem_number": "LOGIC-001", "title": "Vaught Conjecture", "statement": "Is the number of countable models of a complete first-order theory finite, $\\aleph_0$, or $2^{\\aleph_0}$?", "background": "The Vaught conjecture, proposed by Robert Vaught in 1961, is a fundamental problem in model theory. For a complete first-order theory in a countable language, the number of countable models (up to isomorphism) must be either finite, countably infinite ($\\aleph_0$), or continuum ($2^{\\aleph_0}$). In other words, there cannot be exactly $\\aleph_1$ (or any other intermediate cardinality) non-isomorphic countable models. The conjecture is known to hold for many classes of theories: $\\omega$-stable theories, superstable theories, and theories with certain structural properties. However, the general case remains open. The problem connects to descriptive set theory, infinitary logic, and has implications for classification theory and the structure of models.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 298, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1344, "problem_number": "LOGIC-002", "title": "Cherlin-Zilber Conjecture", "statement": "Is every simple group with $\\aleph_0$-stable theory an algebraic group over an algebraically closed field?", "background": "The Cherlin-Zilber conjecture concerns the classification of simple groups in model theory. It states: every infinite simple group whose first-order theory is stable in $\\aleph_0$ (countably stable) is isomorphic to a simple algebraic group over an algebraically closed field. The conjecture connects abstract model-theoretic stability to concrete algebraic structures. Many special cases have been verified, and the conjecture has driven development of geometric stability theory. It would provide a complete classification of stable simple groups and has implications for understanding the interaction between model theory and group theory. Zilber's work on Zariski geometries provides evidence for the conjecture.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1346, "problem_number": "GEOM-009", "title": "Yang-Mills Existence and Mass Gap", "statement": "Does Yang-Mills theory exist mathematically and exhibit a mass gap in 4D?", "background": "One of the seven Millennium Prize Problems ($1M prize). The Yang-Mills equations describe the behavior of elementary particles using non-Abelian gauge theory, fundamental to the Standard Model of particle physics. The problem asks two questions: (1) Does a mathematically rigorous quantum Yang-Mills theory exist in 4-dimensional spacetime? (2) Does it exhibit a mass gap—the smallest mass of any excitation being strictly positive? Physicists use Yang-Mills theory extensively, but a rigorous mathematical foundation is lacking. Proving existence and the mass gap would provide the mathematical basis for quantum chromodynamics (QCD) and explain confinement of quarks. The problem connects quantum field theory, differential geometry, functional analysis, and mathematical physics. Despite extensive physics research, mathematical proof remains elusive.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 567, "favorite_count": 47, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1347, "problem_number": "ST-001", "title": "Partition Principle Implies Axiom of Choice", "statement": "Does the partition principle (PP) imply the axiom of choice (AC)?", "background": "The partition principle states that for every partition of a set, there exists a set that contains exactly one element from each cell of the partition. The axiom of choice states that for every collection of nonempty sets, there exists a choice function selecting one element from each set. While AC clearly implies PP, the reverse implication is unknown. This question explores the relative strength of these fundamental axioms in set theory and their role in mathematics.", "difficulty_level_id": 4, "status": "open", "category_id": 10, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1348, "problem_number": "ST-002", "title": "Woodin's GCH below Strongly Compact Cardinals", "statement": "Does the generalized continuum hypothesis below a strongly compact cardinal imply it everywhere?", "background": "Posed by W. Hugh Woodin, this problem asks whether local instances of the generalized continuum hypothesis (GCH) can force global instances. A strongly compact cardinal is a large cardinal with strong reflection properties. The question explores whether GCH holding below such a cardinal must propagate throughout the universe of sets. This connects large cardinal theory with cardinal arithmetic and the structure of the set-theoretic universe.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1349, "problem_number": "ST-003", "title": "GCH and Diamond Principle", "statement": "Does the generalized continuum hypothesis entail the diamond principle $\\diamondsuit(E_{\\text{cf}(\\lambda)}^{\\lambda^+})$ for every singular cardinal $\\lambda$?", "background": "The diamond principle is a combinatorial principle asserting the existence of certain prediction sequences. For singular cardinals (cardinals not equal to their own cofinality), the relationship between GCH and diamond principles is subtle. While diamond holds at successor cardinals under GCH, its behavior at successors of singular cardinals remains mysterious. This problem probes the fine structure of cardinal arithmetic.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1350, "problem_number": "ST-004", "title": "GCH and Suslin Trees", "statement": "Does the generalized continuum hypothesis imply the existence of an $\\aleph_2$-Suslin tree?", "background": "A Suslin tree is a tree of height $\\omega_1$ with no uncountable chains or antichains. An $\\aleph_2$-Suslin tree is the analogous structure at the next cardinal level. While Suslin trees at $\\aleph_1$ can exist under certain axioms, their existence at $\\aleph_2$ under GCH is unknown. This problem connects cardinal arithmetic with combinatorial set theory and the theory of infinite trees.", "difficulty_level_id": 4, "status": "open", "category_id": 10, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1352, "problem_number": "ST-006", "title": "Ultimate Core Model", "statement": "Does there exist an ultimate core model containing all large cardinals?", "background": "Core models are canonical inner models of set theory that approximate the entire universe while being more tractable. The search for an ultimate core model—one encompassing all large cardinal properties—is a central goal of modern set theory. Such a model would unify our understanding of large cardinals and provide a framework for resolving independence questions. The project involves deep interactions between forcing, inner model theory, and large cardinal axioms.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1353, "problem_number": "ST-007", "title": "Woodin's Ω-Conjecture", "statement": "If there is a proper class of Woodin cardinals, does Ω-logic satisfy an analogue of Gödel's completeness theorem?", "background": "Proposed by W. Hugh Woodin, this conjecture connects large cardinals with logic. Ω-logic is a strong logic using Woodin cardinals to define semantic validity. The conjecture asserts that under the assumption of a proper class of Woodin cardinals, Ω-logic becomes complete in a generalized sense—every Ω-valid sentence has an Ω-proof. This would provide a powerful new framework for set-theoretic truth and resolve many independence questions.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1354, "problem_number": "ST-008", "title": "Strongly Compact vs Supercompact Cardinals", "statement": "Does the consistency of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?", "background": "Strongly compact cardinals and supercompact cardinals are both large cardinal notions with powerful reflection properties. Supercompact cardinals are known to be stronger, but whether their consistency strength is strictly greater than strongly compact cardinals remains open. This problem probes the fine structure of the large cardinal hierarchy and the relationships between different reflection principles.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1355, "problem_number": "ST-009", "title": "Jónsson Algebra on ℵ_ω", "statement": "Does there exist a Jónsson algebra on $\\aleph_\\omega$?", "background": "A Jónsson algebra is an algebraic structure with no proper subalgebra of the same cardinality. The existence of Jónsson algebras on various cardinals connects algebra with set theory. For $\\aleph_\\omega$ (the $\\omega$-th infinite cardinal), existence remains unknown. A positive answer would provide new insights into the algebraic structure of infinite sets and the behavior of singular cardinals.", "difficulty_level_id": 4, "status": "open", "category_id": 10, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1356, "problem_number": "ST-010", "title": "Open Coloring Axiom and Continuum Hypothesis", "statement": "Is the open coloring axiom (OCA) consistent with $2^{\\aleph_0} > \\aleph_2$?", "background": "The open coloring axiom is a combinatorial principle with powerful consequences for the structure of the real line. It is known to be consistent with $2^{\\aleph_0} = \\aleph_2$, but consistency with larger values of the continuum is unknown. This problem explores the interaction between partition properties and cardinal arithmetic, central themes in modern set theory.", "difficulty_level_id": 4, "status": "open", "category_id": 10, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1357, "problem_number": "ST-011", "title": "Reinhardt Cardinals without Choice", "statement": "Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?", "background": "A Reinhardt cardinal would witness an elementary embedding from the universe of all sets (V) to itself. Kunen proved such embeddings cannot exist with the axiom of choice. However, without AC, the question remains open. Reinhardt cardinals would be the strongest large cardinal notion, transcending the usual hierarchy. Their possible existence connects to alternative set theories and the role of choice in mathematics.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1358, "problem_number": "GAME-001", "title": "Sudoku: Unique Solution Puzzles", "statement": "How many Sudoku puzzles have exactly one solution?", "background": "Standard 9×9 Sudoku grids can be filled in approximately 6.67 × 10²¹ ways. A puzzle is a partial filling with a unique completion. Despite extensive computer searches, the exact count of puzzles with unique solutions remains unknown. This combinatorial problem involves constraints, symmetry breaking, and counting techniques. Understanding this would illuminate the mathematical structure underlying Sudoku and related constraint satisfaction problems.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "view_count": 892, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1359, "problem_number": "GAME-002", "title": "Sudoku: Minimal Puzzles Count", "statement": "How many Sudoku puzzles with exactly one solution are minimal (removing any clue creates multiple solutions)?", "background": "A minimal Sudoku puzzle cannot have any clue removed without losing uniqueness. While we know examples with as few as 17 clues, the total count of minimal puzzles is unknown. This problem combines enumeration with the structure of constraint systems. The answer would deepen our understanding of puzzle difficulty, minimal representations, and the geometry of solution spaces.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "view_count": 678, "favorite_count": 51, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1360, "problem_number": "GAME-003", "title": "Maximum Givens in Minimal Sudoku", "statement": "What is the maximum number of givens for a minimal Sudoku puzzle?", "background": "While minimal puzzles can have as few as 17 givens, the upper bound is unknown. A puzzle with many givens can still be minimal if each clue is essential. Computer searches have found minimal puzzles with around 40 givens, but no theoretical maximum is known. This question explores the relationship between redundancy, minimality, and constraint propagation in combinatorial problems.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "view_count": 567, "favorite_count": 43, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1361, "problem_number": "GAME-004", "title": "Tic-Tac-Toe Winning Dimension", "statement": "Given the width of a tic-tac-toe board, what is the smallest dimension guaranteeing X has a winning strategy?", "background": "Classic tic-tac-toe is a draw with perfect play. In higher dimensions (n^d game), questions become more complex. The Hales-Jewett theorem guarantees that for any fixed line length n, there exists a dimension d where the first player wins. But finding the exact threshold dimension for each n remains open. This connects combinatorics, game theory, and Ramsey theory.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 445, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1362, "problem_number": "GAME-005", "title": "Perfect Chess", "statement": "What is the outcome of a perfectly played game of chess?", "background": "Chess is a finite deterministic game, so theoretically one of three outcomes holds with perfect play: White wins, Black wins, or draw. Despite centuries of play and powerful computers, we don't know which. The game tree has approximately 10⁴⁷ positions, far beyond exhaustive analysis. Current evidence suggests a draw, but proving it requires breakthrough techniques in game-tree search, endgame databases, or mathematical analysis of chess positions.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 1534, "favorite_count": 112, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1363, "problem_number": "GAME-006", "title": "Perfect Komi in Go", "statement": "What is the perfect value of komi (compensation points) in Go?", "background": "In Go, komi compensates the second player (White) for Black's first-move advantage. Professional play uses 6.5 or 7.5 points. But what value makes the game perfectly fair with optimal play? Go's complexity (10¹⁷⁰ legal positions) prevents exhaustive analysis. AI like AlphaGo suggest small adjustments, but perfect komi remains unknown. Determining it would require solving Go—understanding the game-theoretic value with perfect play.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 789, "favorite_count": 58, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1364, "problem_number": "GAME-007", "title": "Cap Set Problem", "statement": "What is the largest possible cap set in $n$-dimensional affine space over the three-element field?", "background": "A cap set is a collection of points with no three in a line (in the game SET, cards with no valid set). In the affine space $\\mathbb{F}_3^n$, the maximum cap set size is conjectured to be $c^n$ for some constant c < 3. The best bounds are $2.756^n$ (Ellenberg-Gijswijt, 2016). Determining the precise growth rate connects additive combinatorics, polynomial methods, and the cap set conjecture. The breakthrough proof technique revolutionized the field.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 356, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1365, "problem_number": "GAME-008", "title": "Octal Games Periodicity", "statement": "Are the nim-sequences of all finite octal games eventually periodic?", "background": "Octal games are impartial combinatorial games defined by simple rules encoded in octal notation. Their nim-values (Grundy numbers) determine optimal play. For some octal games, the nim-sequence is eventually periodic; for others, patterns are elusive. Whether all finite octal games have eventually periodic nim-sequences is unknown. This problem connects game theory, number theory, and automata theory.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1366, "problem_number": "GAME-009", "title": "Grundy's Game Periodicity", "statement": "Is the nim-sequence of Grundy's game eventually periodic?", "background": "Grundy's game: split a heap of n beans into two unequal heaps; last player to move wins. The nim-value sequence starts 0,1,0,2,1,3,2,1,0,4,... but no period has been found despite extensive computation. Whether it's eventually periodic (or even computable) is open. This specific game has resisted analysis for decades, representing a frontier in combinatorial game theory.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 278, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1367, "problem_number": "GAME-010", "title": "Rendezvous Problem", "statement": "What is the optimal strategy for two agents to meet on a network without communication?", "background": "The rendezvous problem asks: how should two agents move on a graph to minimize expected meeting time, when they can't communicate and may not know the graph structure? Variants include symmetric/asymmetric information, labeled/unlabeled nodes, and different graph families. Optimal strategies are known for some simple cases but remain open for general graphs. This problem bridges game theory, probability, and distributed algorithms.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 312, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1369, "problem_number": "GEOM-010", "title": "Kissing Number Problem", "statement": "What is the kissing number (maximum number of non-overlapping unit spheres that can touch a central unit sphere) in dimensions other than 1, 2, 3, 4, 8, and 24?", "background": "The kissing number is known exactly only in dimensions 1 (2), 2 (6), 3 (12), 4 (24), 8 (240), and 24 (196,560). The problem asks for exact values in other dimensions. In dimension 3, twelve spheres can kiss a central sphere (with centers forming an icosahedron). Dimensions 8 and 24 have exceptional symmetries related to E₈ and the Leech lattice. Determining kissing numbers connects sphere packing, coding theory, and discrete geometry. The problem is surprisingly difficult—even dimension 5 remains unsolved.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 534, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1372, "problem_number": "GEOM-013", "title": "Tammes Problem", "statement": "For n > 14 points (except n=24), what is the maximum minimum distance between points on a unit sphere?", "background": "The Tammes problem asks: how should n points be arranged on a sphere to maximize the minimum distance between any pair? This is equivalent to packing n spherical caps on a sphere. Solutions are known for n ≤ 14 and n = 24 (related to exceptional geometries). For other n, only bounds and computational results exist. The problem has applications in molecular chemistry (electron repulsion), coding theory, and crystallography. Named after Dutch botanist who studied pollen grain pores.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1373, "problem_number": "GEOM-014", "title": "Carathéodory Conjecture", "statement": "Does every convex, closed, twice-differentiable surface in 3D Euclidean space have at least two umbilical points?", "background": "An umbilical point on a surface is where the two principal curvatures are equal (the surface curves equally in all directions, like on a sphere). Carathéodory conjectured that every smooth convex closed surface must have at least two umbilic points. A sphere has infinitely many (every point), but most surfaces should have at least two. Despite being over 100 years old, the conjecture remains open. Partial results exist for analytic surfaces and surfaces with special symmetries.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 312, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1374, "problem_number": "GEOM-015", "title": "Cartan-Hadamard Conjecture", "statement": "Does the isoperimetric inequality extend to Cartan-Hadamard manifolds (complete simply-connected manifolds of nonpositive curvature)?", "background": "The classical isoperimetric inequality states that among all regions with fixed perimeter in Euclidean space, the circle (or sphere) encloses maximum area (or volume). The Cartan-Hadamard conjecture asks whether this inequality holds in spaces of nonpositive curvature. Proven in dimensions 2, 3, and 4, but open in higher dimensions. A positive answer would show that negative curvature preserves this fundamental geometric optimization principle.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 267, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1375, "problem_number": "GEOM-016", "title": "Chern's Conjecture (Affine Geometry)", "statement": "Does the Euler characteristic of a compact affine manifold vanish?", "background": "An affine manifold is a manifold with an atlas whose transition functions are affine transformations. Chern conjectured that any closed (compact, boundaryless) affine manifold must have Euler characteristic zero. The conjecture is true for many special cases but remains open in general. This would be a fundamental constraint on the topology of spaces admitting flat affine structures, connecting differential geometry with algebraic topology.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1376, "problem_number": "GEOM-017", "title": "Hopf Conjectures", "statement": "What are the relationships between curvature and Euler characteristic for higher-dimensional Riemannian manifolds?", "background": "The Hopf conjectures are a collection of problems relating the curvature of a manifold to its Euler characteristic. One version: does a positively curved even-dimensional manifold have positive Euler characteristic? Another: does a negatively curved manifold have zero Euler characteristic? These would generalize the Gauss-Bonnet theorem to higher dimensions. Despite progress on special cases, the general conjectures remain open, representing a frontier in global differential geometry.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1377, "problem_number": "GEOM-018", "title": "Yau's Conjecture on First Eigenvalue", "statement": "Is the first eigenvalue of the Laplace-Beltrami operator on an embedded minimal hypersurface of $S^{n+1}$ equal to $n$?", "background": "This conjecture by Shing-Tung Yau concerns minimal surfaces (soap-film-like surfaces) embedded in spheres. The Laplace-Beltrami operator generalizes the Laplacian to curved spaces. Yau conjectured that the first eigenvalue equals the dimension n for minimal hypersurfaces in the (n+1)-sphere. This would provide a sharp geometric-spectral inequality, connecting the shape of minimal surfaces to their vibration modes. Proven in special cases, but remains open generally.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1378, "problem_number": "GEOM-019", "title": "Hadwiger Conjecture (Covering)", "statement": "Can every $n$-dimensional convex body be covered by at most $2^n$ smaller positively homothetic copies?", "background": "Hadwiger conjectured that any convex body in n dimensions can be covered by at most 2ⁿ smaller copies that are scaled-down versions (homotheties with positive ratio). Proven only for n ≤ 3. For n=2, four copies suffice (proven by Levi). For n=3, eight copies suffice (Hadwiger's original proof). Higher dimensions remain completely open. This is one of the most important unsolved problems in convex geometry, with connections to Borsuk's problem and covering theory.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 298, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1379, "problem_number": "GEOM-020", "title": "Happy Ending Problem", "statement": "What is the minimum number $g(n)$ of points in general position in the plane guaranteeing a convex $n$-gon?", "background": "The Happy Ending problem (named for the romance between Erdős and Szekeres who solved special cases) asks: how many points in general position (no three collinear) force the existence of n points forming a convex n-gon? Known: g(3)=3, g(4)=5, g(5)=9. Erdős-Szekeres proved $2^{n-2} + 1 \\leq g(n) \\leq \\binom{2n-4}{n-2} + 1$. The exact value for n ≥ 6 is unknown, and closing this exponential gap is a major challenge in combinatorial geometry.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 345, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1380, "problem_number": "GEOM-021", "title": "Heilbronn Triangle Problem", "statement": "What configuration of $n$ points in the unit square maximizes the area of the smallest triangle they determine?", "background": "Heilbronn asked: place n points in a unit square to maximize the minimum triangle area. Trivially, the minimum area is ≤ 2/n. Heilbronn conjectured it's O(1/n²). Komlos-Pintz-Szemeredi showed it's actually Θ((log n)/n²), disproving the conjecture. However, the exact constant is unknown, and tight bounds remain elusive. This problem exemplifies how discrete geometry problems can have surprising answers and connects to irregularities of distribution and discrepancy theory.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 223, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1381, "problem_number": "GEOM-022", "title": "Kalai's 3^d Conjecture", "statement": "Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?", "background": "Gil Kalai conjectured that centrally symmetric polytopes (symmetric under reflection through the origin) must have many faces—at least 3^d for dimension d. The d-cube achieves this bound exactly. Proved for d ≤ 4. Higher dimensions remain open. This would be a fundamental constraint on the combinatorial complexity of symmetric polytopes, with implications for optimization, linear programming, and the geometry of convex bodies.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1382, "problem_number": "GEOM-023", "title": "Orchard-Planting Problem", "statement": "What is the maximum number of 3-point lines attainable by a configuration of $n$ points in the plane?", "background": "An orchard-planting problem asks: arrange n points (trees) to maximize the number of lines containing exactly 3 points (rows). For n points, at most n(n-1)/6 such lines are possible (by counting). Some configurations achieve this bound or come close. The problem asks for the exact maximum for each n. Solutions are known for small n, but the general pattern is mysterious. This connects to projective geometry, matroid theory, and combinatorial designs.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1383, "problem_number": "GEOM-024", "title": "Unit Distance Problem", "statement": "How many pairs of points at unit distance can be determined by $n$ points in the Euclidean plane?", "background": "Erdős asked: what's the maximum number of unit-distance pairs among n points in the plane? Trivially at most n(n-1)/2. Known: the maximum is Θ(n^(4/3)) (lower bound by Erdős, upper by Spencer-Szemerédi-Trotter). But the exact exponent is unknown—it could be n^(4/3), n^(3/2), or something between. Determining this connects incidence geometry, graph theory, and the crossing number. The unit distance graph has fascinating properties.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1384, "problem_number": "GEOM-025", "title": "Bellman's Lost-in-a-Forest Problem", "statement": "What is the shortest path that guarantees reaching the boundary of a given shape, starting from an unknown point with unknown orientation?", "background": "You're lost in a forest (a region with known shape but unknown location and orientation). What path guarantees you'll reach the edge? For a circle of radius 1, a path of length ≤ 2 + π/3 ≈ 3.05 suffices. For a square, the answer is unknown. For general convex regions, the problem is wide open. This classic problem in geometric search theory has applications to robotics, navigation, and computational geometry. Finding optimal escape paths connects geometry with optimization.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 423, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1385, "problem_number": "GEOM-026", "title": "Borromean Rings Question", "statement": "Can three unknotted space curves (not all circles) be arranged as Borromean rings?", "background": "Borromean rings are three linked loops where removing any one unlinks the other two. Classical Borromean rings use circles, but perfect circular realization is impossible (proved). Can non-circular unknotted curves realize this linking pattern? This question connects knot theory, topology, and geometry. While Borromean rings can be made from ellipses or other shapes, whether three genuinely unknotted (topologically circular) but geometrically non-circular curves can achieve this remains subtle.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 312, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1386, "problem_number": "GEOM-027", "title": "Danzer's Problem", "statement": "Do Danzer sets of bounded density or bounded separation exist?", "background": "A Danzer set is a set of points in the plane such that every convex region of area 1 contains at least one point. Danzer asked: can such a set have bounded density (points per unit area) or bounded separation (minimum distance between points)? Both properties would mean the points are \"well-distributed.\" While Danzer sets exist, whether nice ones exist is open. Related to Conway's \"dead fly\" problem. Connects measure theory, convexity, and geometric covering.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1388, "problem_number": "GRAPH-001", "title": "Brouwer's Conjecture on Graph Laplacians", "statement": "Can the sum of eigenvalues of the Laplacian matrix of a graph be bounded by the number of edges?", "background": "Brouwer conjectured an upper bound for the sum of the k largest eigenvalues of the Laplacian matrix of a graph in terms of the number of edges. The Laplacian matrix encodes graph structure and has deep connections to spectral graph theory. This conjecture would provide fundamental insights into the relationship between a graph's combinatorial and spectral properties. Progress has been made for special classes of graphs, but the general case remains open.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1389, "problem_number": "GRAPH-002", "title": "Eternal Domination vs Domination Number", "statement": "Does there exist a graph where the dominating number equals the eternal dominating number and both are less than the clique covering number?", "background": "The dominating number γ(G) is the minimum size of a dominating set. The eternal dominating number γ∞(G) arises from a game where guards on vertices must respond to attacks. The question asks if these can equal each other while being smaller than the clique covering number (minimum number of cliques needed to cover all vertices). This connects domination theory with graph games and clique structures.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1390, "problem_number": "GRAPH-003", "title": "Graham's Pebbling Conjecture", "statement": "Is the pebbling number of the Cartesian product of two graphs at least the product of their pebbling numbers?", "background": "Graph pebbling is a combinatorial game where pebbles are moved on vertices according to specific rules. Graham conjectured that the pebbling number (minimum pebbles needed to guarantee placing one on any target vertex) of a Cartesian product G × H is at least π(G) × π(H). Despite progress on special cases like products with paths or cycles, the general conjecture remains unsolved. This problem has applications to communication networks and resource distribution.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1391, "problem_number": "GRAPH-004", "title": "Meyniel's Conjecture on Cop Number", "statement": "Is the cop number of a connected n-vertex graph $O(\\sqrt{n})$?", "background": "The cop number is the minimum number of cops needed to guarantee catching a robber in a pursuit game on a graph. Meyniel conjectured that for any connected graph with n vertices, the cop number is at most O(√n). The best known upper bound is O(n/log n). This problem connects graph theory with algorithmic game theory and has applications to network security, robot motion planning, and pursuit-evasion games. Resolving it would fundamentally advance our understanding of graph searching problems.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1392, "problem_number": "GRAPH-005", "title": "Graph Coloring Game Monotonicity", "statement": "If Alice has a winning strategy for the vertex coloring game with k colors, does she have one for k+1 colors?", "background": "In the graph coloring game, two players alternately color vertices with k colors, trying to create (Alice) or avoid (Bob) a proper coloring. Intuitively, having more colors should make Alice's task easier. However, whether winning with k colors implies winning with k+1 colors is surprisingly still open. This problem probes the subtle complexity of graph coloring games and connects combinatorial game theory with chromatic graph theory.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1393, "problem_number": "GRAPH-006", "title": "1-Factorization Conjecture", "statement": "Does every k-regular graph on 2n vertices admit a 1-factorization when k ≥ n (or k ≥ n-1 for even n)?", "background": "A 1-factor is a perfect matching, and a 1-factorization is a partition of edges into 1-factors. The conjecture states that sufficiently regular graphs can be decomposed into perfect matchings. This would generalize classical results on complete graphs. Proven for many special cases, but the general statement remains open. Applications include tournament scheduling, network routing, and combinatorial designs.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1394, "problem_number": "GRAPH-007", "title": "Perfect 1-Factorization Conjecture", "statement": "Does every complete graph on an even number of vertices admit a perfect 1-factorization?", "background": "A perfect 1-factorization of a complete graph K₂ₙ is a 1-factorization where the union of any two 1-factors forms a Hamiltonian cycle. Such structures have beautiful symmetry and applications to combinatorial designs. While perfect 1-factorizations are known for many n (especially powers of 2 and small cases), a general existence proof remains elusive. This is one of the most elegant open problems in graph decomposition theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1395, "problem_number": "GRAPH-008", "title": "Cereceda's Conjecture", "statement": "For k-degenerate graphs, can any (k+2)-coloring be transformed to any other in polynomial steps via single-vertex recolorings?", "background": "Cereceda's conjecture concerns the diameter of the reconfiguration graph of graph colorings. It asks whether the shortest sequence of single-vertex recolorings transforming one coloring to another is polynomially bounded for degenerate graphs. This connects graph coloring with reconfiguration problems—a growing area studying how to transform one solution to another. Applications include network reconfiguration and state-space search.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1396, "problem_number": "GRAPH-009", "title": "Earth-Moon Problem", "statement": "What is the maximum chromatic number of biplanar graphs?", "background": "A graph is biplanar if it can be drawn on two parallel planes (Earth and Moon) with edges possibly crossing between planes but not within each plane. The Earth-Moon problem asks for the maximum chromatic number of such graphs. Known bounds are 12 ≤ χ ≤ 16. This problem combines planarity concepts with multilayer graph drawings, relevant to VLSI design and network visualization.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1397, "problem_number": "GRAPH-010", "title": "Gyárfás-Sumner Conjecture", "statement": "Is every graph class defined by excluding one fixed tree as an induced subgraph χ-bounded?", "background": "A graph class is χ-bounded if there's a function f such that every graph in the class with clique number ω has chromatic number at most f(ω). The conjecture states that forbidding any tree as an induced subgraph creates a χ-bounded class. This would unify many results on perfect graphs and their generalizations. The conjecture connects structural graph theory with coloring, and has implications for algorithmic graph theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1398, "problem_number": "GRAPH-011", "title": "Jaeger's Petersen Coloring Conjecture", "statement": "Does every bridgeless cubic graph have a cycle-continuous mapping to the Petersen graph?", "background": "Jaeger conjectured that every bridgeless cubic graph admits a special kind of homomorphism to the Petersen graph that preserves cycle structure. The Petersen graph plays a central role in graph theory as a universal counterexample and fundamental object. This conjecture connects graph homomorphisms, snarks (cubic graphs resistant to edge coloring), and the structure of cubic graphs. It has deep implications for edge coloring and nowhere-zero flow problems.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1399, "problem_number": "GRAPH-012", "title": "List Coloring Conjecture", "statement": "For every graph, does the list chromatic index equal the chromatic index?", "background": "The chromatic index χ'(G) is the minimum number of colors needed to color edges so no two adjacent edges share a color. The list chromatic index is the minimum k such that edges can be colored from arbitrary k-element color lists. The conjecture states these are always equal. While proven for bipartite graphs and some other classes, the general case remains open. This is a fundamental question in list coloring theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1400, "problem_number": "GRAPH-013", "title": "Overfull Conjecture", "statement": "Is a graph with maximum degree Δ(G) ≥ n/3 in class 2 if and only if it has an overfull subgraph with the same maximum degree?", "background": "By Vizing's theorem, every graph has chromatic index Δ or Δ+1 (class 1 or 2). A graph is overfull if it has more than Δ⌊n/2⌋ edges, forcing class 2. The overfull conjecture provides a complete characterization: when Δ ≥ n/3, being class 2 is equivalent to having an overfull subgraph preserving the maximum degree. This would elegantly explain why graphs are hard to edge-color.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1401, "problem_number": "GRAPH-014", "title": "Total Coloring Conjecture", "statement": "Is the total chromatic number of every graph at most Δ + 2, where Δ is the maximum degree?", "background": "Total coloring requires coloring both vertices and edges so adjacent/incident elements have different colors. Behzad and Vizing independently conjectured that the total chromatic number χ″(G) ≤ Δ(G) + 2. The lower bound Δ + 1 is easy (color each vertex and its incident edges distinctly). The upper bound Δ + 2 is proven for many graph classes but remains open in general. This is one of the most fundamental open problems in graph coloring.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1402, "problem_number": "GRAPH-015", "title": "Albertson Conjecture", "statement": "Can the crossing number of a graph be lower-bounded by the crossing number of a complete graph with the same chromatic number?", "background": "The crossing number is the minimum number of edge crossings in a planar drawing. Albertson conjectured cr(G) ≥ cr(K_χ(G)) where χ(G) is the chromatic number. This would link two fundamental graph parameters—crossing number and chromatic number. Proven for chromatic numbers up to 16, but the general case remains open. This connects graph drawing, coloring theory, and topological graph theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1403, "problem_number": "GRAPH-016", "title": "Conway's Thrackle Conjecture", "statement": "Does every thrackle have at most as many edges as vertices?", "background": "A thrackle is a graph drawing where every pair of edges either meets at a common vertex or crosses exactly once. Conway conjectured that thrackles satisfy |E| ≤ |V|. Despite looking simple, this conjecture has resisted proof for decades. The best known bound is |E| ≤ 3|V|/2. This problem connects graph drawing with combinatorial geometry and has surprising depth for such a simply stated question.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1404, "problem_number": "GRAPH-017", "title": "GNRS Conjecture", "statement": "Do minor-closed graph families have $\\ell_1$ embeddings with bounded distortion?", "background": "The GNRS conjecture asks whether graphs from minor-closed families (like planar graphs) can be embedded into L₁ space (ℓ₁ metric) with distortion bounded by a function of the excluded minor size. This connects graph theory with metric geometry and theoretical computer science. The conjecture has important implications for approximation algorithms and understanding the metric structure of graph families.", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1405, "problem_number": "GRAPH-018", "title": "Harborth's Conjecture", "statement": "Can every planar graph be drawn with integer edge lengths?", "background": "Harborth conjectured that every planar graph has a straight-line drawing where all edge lengths are integers. While planar graphs always have straight-line drawings (Fáry's theorem), forcing integer lengths is much harder. Known for trees and some other classes, but open in general. This problem connects graph drawing with discrete geometry and has applications to VLSI layout.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1406, "problem_number": "GRAPH-019", "title": "Negami's Conjecture", "statement": "Does every graph with a planar cover have a projective-plane embedding?", "background": "Negami conjectured that if a graph G has a planar cover (a planar graph that maps onto G), then G embeds in the projective plane. This would characterize projective-plane graphs via covering spaces. The conjecture connects topological graph theory with covering space theory from topology. Despite progress on special cases, the general conjecture remains a central open problem in topological graph theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1407, "problem_number": "GRAPH-020", "title": "Turán's Brick Factory Problem", "statement": "What is the minimum crossing number of the complete bipartite graph $K_{m,n}$?", "background": "Turán's brick factory problem asks for the exact crossing number of complete bipartite graphs K_{m,n}. Zarankiewicz conjectured a formula in 1954, which is known to be correct for several cases but unproven in general. The problem arose from Turán observing workers crossing paths while moving bricks. Despite being simple to state, this geometric problem has remained unsolved for 70 years.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 212, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1408, "problem_number": "GRAPH-021", "title": "Guy's Crossing Number Conjecture", "statement": "Is the crossing number of the complete graph $K_n$ equal to the value given by Guy's formula?", "background": "Guy conjectured a formula for the crossing number of complete graphs: cr(K_n) = (1/4)⌊n/2⌋⌊(n-1)/2⌋⌊(n-2)/2⌋⌊(n-3)/2⌋. This is proven for n ≤ 12, but the general case is open. Finding the exact crossing number of complete graphs is a fundamental problem in topological graph theory. The conjecture represents our best guess based on known constructions.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1409, "problem_number": "GRAPH-022", "title": "Universal Point Sets", "statement": "Do planar graphs have universal point sets of subquadratic size?", "background": "A universal point set for n-vertex planar graphs is a set of points such that every n-vertex planar graph has a straight-line embedding on these points. Trivially, O(n²) points suffice. The question asks if o(n²) is possible. Best known lower bound is Ω(n), upper bound is O(n²). Closing this gap would advance our understanding of planar graph representations and geometric graph theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1410, "problem_number": "GRAPH-023", "title": "Conference Graph Existence", "statement": "Does there exist a conference graph for every number of vertices $v > 1$ where $v \\equiv 1 \\pmod{4}$ and v is an odd sum of two squares?", "background": "A conference graph is a strongly regular graph with specific parameters related to conference matrices. The existence question for these graphs connects graph theory with number theory (sums of squares) and design theory. Known to exist for many values, but a complete characterization remains elusive. These graphs have applications in coding theory and experimental design.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1411, "problem_number": "GRAPH-024", "title": "Conway's 99-Graph Problem", "statement": "Does there exist a strongly regular graph with parameters (99,14,1,2)?", "background": "Conway asked whether a strongly regular graph with these specific parameters exists. The parameters pass all known necessary conditions (feasibility, integrality), but no construction is known. This is the smallest open case for strongly regular graphs. Finding such a graph or proving nonexistence would advance our understanding of the constraints on strongly regular graphs beyond the known necessary conditions.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1412, "problem_number": "GRAPH-025", "title": "Degree Diameter Problem", "statement": "For given maximum degree d and diameter k, what is the largest possible number of vertices in a graph?", "background": "The degree diameter problem asks for the maximum order (number of vertices) of a graph with maximum degree d and diameter k. The Moore bound provides an upper limit, but it's rarely achieved (only for very special parameters). Finding the exact values or better bounds is a central problem in extremal graph theory with applications to network design. Tables of best known values are maintained, but many cases remain unsolved.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1413, "problem_number": "GRAPH-026", "title": "Moore Graph Existence", "statement": "Does a Moore graph with girth 5 and degree 57 exist?", "background": "Moore graphs are extremal graphs achieving the Moore bound—the maximum possible vertices for given degree and diameter. The Hoffman-Singleton theorem shows Moore graphs with girth 5 can only have degree 2, 3, 7, or possibly 57. Graphs for degrees 2, 3, 7 are known (cycle C₅, Petersen, Hoffman-Singleton). Whether a degree-57 Moore graph exists is one of the most famous open problems in algebraic graph theory.", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 223, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1414, "problem_number": "GRAPH-027", "title": "Barnette's Conjecture", "statement": "Does every cubic bipartite three-connected planar graph have a Hamiltonian cycle?", "background": "Barnette's conjecture proposes that a specific family of planar graphs—cubic (3-regular), bipartite, and 3-connected—always contains Hamiltonian cycles. This strengthens Tait's conjecture (disproven by counterexamples) by adding bipartiteness. Despite extensive computational verification and many partial results, no proof or counterexample is known. This is one of the most prominent open problems on Hamiltonian cycles.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 212, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1415, "problem_number": "GRAPH-028", "title": "Chvátal's Toughness Conjecture", "statement": "Is there a constant t such that every t-tough graph is Hamiltonian?", "background": "A graph is t-tough if removing any set S of vertices leaves at most |S|/t components. Chvátal conjectured that sufficiently tough graphs are Hamiltonian. Best known: every 2-tough graph on at least 3 vertices is Hamiltonian. But whether some finite t suffices in general is unknown. Toughness measures graph robustness; the conjecture would provide a simple sufficient condition for Hamiltonicity.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1416, "problem_number": "GRAPH-029", "title": "Cycle Double Cover Conjecture", "statement": "Does every bridgeless graph have a collection of cycles that covers each edge exactly twice?", "background": "The cycle double cover conjecture asserts that every bridgeless graph has a family of cycles where each edge appears in exactly two cycles. Equivalent formulations involve graph embeddings and flows. Despite being open since the 1970s, this elegant conjecture connects cycle structure, graph embeddings, and topological graph theory. Many restricted cases are proven, but the general case remains elusive.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1417, "problem_number": "GRAPH-030", "title": "Erdős-Gyárfás Conjecture", "statement": "Does every graph with minimum degree 3 contain cycles of lengths that are powers of 2?", "background": "Erdős and Gyárfás conjectured that cubic graphs (minimum degree 3) must contain cycles whose lengths are all distinct powers of 2. The best known result is that such graphs contain cycles of Ω(log log n) distinct even lengths. This problem connects extremal graph theory with additive combinatorics and the structure of cycle lengths in graphs.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1418, "problem_number": "GRAPH-031", "title": "Erdős-Hajnal Conjecture", "statement": "Does every graph family defined by a forbidden induced subgraph have polynomial-sized cliques or independent sets?", "background": "The Erdős-Hajnal conjecture states that for any graph H, there exists ε > 0 such that every H-free graph on n vertices contains a clique or independent set of size at least n^ε. This would be a dramatic strengthening of Ramsey theory, which only guarantees log-size structures. Proven for many specific H, but the general case is a central open problem in extremal combinatorics.", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 234, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1419, "problem_number": "GRAPH-032", "title": "Linear Arboricity Conjecture", "statement": "Can every graph with maximum degree Δ be decomposed into at most ⌈(Δ+1)/2⌉ linear forests?", "background": "A linear forest is a disjoint union of paths. The linear arboricity conjecture states that graphs decompose into roughly Δ/2 linear forests. This would provide tight bounds on a natural graph decomposition parameter. Proven for many graph classes (planar graphs, graphs with large girth), but the general case remains open. Applications include edge coloring and bandwidth problems.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1420, "problem_number": "GRAPH-033", "title": "Lovász Conjecture", "statement": "Does every finite connected vertex-transitive graph contain a Hamiltonian path?", "background": "Lovász conjectured that vertex-transitive graphs (graphs looking the same from every vertex) always have Hamiltonian paths. Even stronger: do they have Hamiltonian cycles (except for K₂ and some Cayley graphs)? Known for many classes, but a general proof eludes us. This connects group theory, algebraic graph theory, and Hamiltonian paths. Named the \"Lovász Hamiltonian Path Problem.\"", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1421, "problem_number": "GRAPH-034", "title": "Oberwolfach Problem", "statement": "For which 2-regular graphs H can the complete graph be decomposed into edge-disjoint copies of H?", "background": "The Oberwolfach problem asks: given a 2-regular graph H (disjoint union of cycles), can K_n be decomposed into copies of H? This generalizes cycle decompositions and connects to the famous Oberwolfach conferences. Solutions are known for many cases (like single cycles), but a complete characterization remains open. This problem bridges graph decomposition with combinatorial designs.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1422, "problem_number": "GRAPH-035", "title": "Cubic Graph Pathwidth", "statement": "What is the maximum pathwidth of an n-vertex cubic graph?", "background": "Pathwidth measures how closely a graph resembles a path. For cubic (3-regular) graphs, the maximum pathwidth is conjectured to be around n/6, but exact bounds are unknown. This problem connects graph width parameters with regular graphs. Understanding pathwidth has implications for algorithms—many NP-hard problems become tractable on graphs of bounded pathwidth.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1423, "problem_number": "GRAPH-036", "title": "Snake-in-the-Box Problem", "statement": "What is the longest induced path in an n-dimensional hypercube graph?", "background": "A snake-in-the-box is a longest induced path in the n-dimensional hypercube Q_n. Known exact values for small n, but no formula for general n. This problem combines combinatorics, coding theory (Gray codes), and graph theory. Snakes have applications in error-correcting codes and analog-to-digital conversion. Finding optimal snakes remains computationally challenging as n grows.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1424, "problem_number": "GRAPH-037", "title": "Sumner's Conjecture", "statement": "Does every (2n-2)-vertex tournament contain every n-vertex oriented tree?", "background": "Sumner conjectured that tournaments (complete directed graphs) on 2n-2 vertices contain all oriented trees on n vertices as subgraphs. This would be a directed analogue of various tree embedding results. The best known bound is (4+o(1))n instead of 2n-2. This problem connects tournament theory with tree embeddings and Ramsey-type questions for directed graphs.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1425, "problem_number": "GRAPH-038", "title": "Tuza's Conjecture", "statement": "Can the edges of any graph be covered by at most 2ν triangles, where ν is the maximum size of a triangle packing?", "background": "Tuza conjectured that the minimum number of edges needed to hit all triangles is at most twice the maximum number of edge-disjoint triangles. This is a covering-packing duality question. Best known bound is 3ν. The conjecture would provide a tight relationship between triangle packings and triangle covers, with applications to approximation algorithms and combinatorial optimization.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1426, "problem_number": "GRAPH-039", "title": "Unfriendly Partition Conjecture", "statement": "Does every countable graph admit a partition where every vertex has at least as many neighbors outside its part as inside?", "background": "The unfriendly partition conjecture asks if vertices can be partitioned into two sets such that each vertex has at least as many \"unfriendly\" neighbors (in the other set) as \"friendly\" ones (in its own set). Proven for finite graphs, but open for countably infinite graphs. This problem combines graph theory with infinite combinatorics and has connections to social network models.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1427, "problem_number": "GRAPH-040", "title": "Zarankiewicz Problem", "statement": "What is the maximum number of edges in a bipartite graph on (m,n) vertices with no complete bipartite subgraph $K_{s,t}$?", "background": "The Zarankiewicz problem asks for ex(m,n;K_{s,t})—the maximum edges in an (m,n)-bipartite graph avoiding K_{s,t} as a subgraph. This is a fundamental problem in extremal graph theory, generalizing the Kővári–Sós–Turán theorem. Exact values are known for some parameters, but most cases remain open. Applications include incidence geometry and additive combinatorics.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1428, "problem_number": "GRAPH-041", "title": "Vizing's Conjecture", "statement": "For the Cartesian product of graphs $G \\square H$, is the domination number at least $\\gamma(G) \\cdot \\gamma(H)$?", "background": "Vizing conjectured that the domination number of the Cartesian product of two graphs is at least the product of their domination numbers. This would give a lower bound on how efficiently one can dominate product graphs. The conjecture has been verified for many special cases but remains open in general. It has connections to network design and distributed computing.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 172, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1429, "problem_number": "GRAPH-042", "title": "Hamiltonian Decomposition of Hypergraphs", "statement": "Do complete k-uniform hypergraphs admit Hamiltonian decompositions into tight cycles?", "background": "Walescki's theorem states that complete graphs have Hamiltonian decompositions. The hypergraph version asks whether complete k-uniform hypergraphs can be decomposed into tight Hamiltonian cycles. This is a natural generalization from graphs to hypergraphs, with connections to design theory and combinatorial structures.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1430, "problem_number": "GRAPH-043", "title": "Word-Representable Graphs: Letter Copies Bound", "statement": "Are there graphs on n vertices requiring more than floor(n/2) copies of each letter for word-representation?", "background": "Word-representable graphs can be encoded by words where two vertices are adjacent if their letters alternate in the word. The question asks whether any graph needs more than half the number of vertices as copies of each letter. This connects graph theory to formal languages and combinatorics on words.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 98, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1431, "problem_number": "GRAPH-044", "title": "Characterization of Word-Representable Planar Graphs", "statement": "Characterize which planar graphs are word-representable.", "background": "Word-representable graphs are those that can be encoded by words over their vertex set where adjacency corresponds to letter alternation. While some characterizations exist for special graph classes, characterizing word-representable planar graphs remains open. This combines planar graph structure with formal language properties.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 87, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1432, "problem_number": "GRAPH-045", "title": "Word-Representable Graphs: Forbidden Subgraph Characterization", "statement": "Characterize word-representable graphs in terms of forbidden induced subgraphs.", "background": "Many graph classes have elegant characterizations via forbidden subgraphs (e.g., planar graphs avoid K₅ and K₃,₃). The question asks for a similar characterization of word-representable graphs. Such a characterization would provide deep insight into the structure of these graphs and their connection to formal languages.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 92, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1433, "problem_number": "GRAPH-046", "title": "Word-Representable Near-Triangulations", "statement": "Characterize word-representable near-triangulations containing K₄.", "background": "Near-triangulations are planar graphs close to being triangulations. A characterization is known for K₄-free cases. The question asks to extend this to near-triangulations containing the complete graph K₄. This combines planar graph structure with word-representability constraints.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 76, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1434, "problem_number": "GRAPH-047", "title": "Representation Number 3 Classification", "statement": "Classify graphs with representation number exactly 3.", "background": "The representation number is the minimum number of letter copies needed to word-represent a graph. Graphs with representation number 1 and 2 are relatively well understood. The question asks for a complete classification of graphs requiring exactly 3 copies—not representable with 2, but possible with 3.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 81, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1435, "problem_number": "GRAPH-048", "title": "Crown Graphs and Longest Word-Representants", "statement": "Among bipartite graphs, do crown graphs require the longest word-representants?", "background": "Crown graphs are a specific family of bipartite graphs with a symmetric structure. The conjecture suggests they are extremal for word-representation length among bipartite graphs. This would identify which bipartite graphs are hardest to encode as words, with implications for the complexity of word-representation.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 73, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1436, "problem_number": "GRAPH-049", "title": "Line Graphs of Non-Word-Representable Graphs", "statement": "Is the line graph of a non-word-representable graph always non-word-representable?", "background": "The line graph operation transforms a graph into one where edges become vertices. The question asks whether word-non-representability is preserved under this operation. A positive answer would show that line graphs amplify the complexity of word-representation, while a counterexample would reveal subtle structural properties.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 84, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1437, "problem_number": "GRAPH-050", "title": "Translating Graph Problems to Word Problems", "statement": "Which hard graph problems can be efficiently solved by translating graphs to their word representations?", "background": "Word-representation provides an alternative encoding of graphs as strings over an alphabet. The question asks which computationally hard graph problems become tractable when working with word representations instead of adjacency lists or matrices. This could reveal new algorithmic techniques leveraging string algorithms and automata theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 105, "favorite_count": 8, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1438, "problem_number": "GRAPH-051", "title": "Imbalance Conjecture", "statement": "If every edge has imbalance ≥1, is the multiset of edge imbalances always graphic?", "background": "The imbalance of an edge is the absolute difference between the degrees of its endpoints. The conjecture asks whether the multiset of these imbalances can always realize a degree sequence of some graph when all imbalances are positive. This connects degree sequences with edge properties in a novel way.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 94, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1439, "problem_number": "GRAPH-052", "title": "Implicit Graph Conjecture", "statement": "Do slowly-growing hereditary graph families admit implicit representations?", "background": "The implicit graph conjecture concerns the existence of succinct encodings for hereditary families of graphs (closed under induced subgraphs) whose growth rate is subexponential. An implicit representation would allow efficient storage and adjacency queries. This has implications for data structures and graph databases.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 112, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1440, "problem_number": "GRAPH-053", "title": "Ryser's Conjecture", "statement": "For r-partite r-uniform hypergraphs, is the vertex cover number at most (r-1) times the matching number?", "background": "Ryser's conjecture relates the minimum transversal (vertex cover) size to maximum matching size in hypergraphs. For graphs (r=2) this is König's theorem. The conjecture proposes a tight bound for hypergraphs: τ ≤ (r-1)ν. This is a central open problem in hypergraph theory with connections to combinatorial optimization.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1441, "problem_number": "GRAPH-054", "title": "Second Neighborhood Problem", "statement": "Does every oriented graph have a vertex with at least as many vertices at distance 2 as at distance 1?", "background": "The second neighborhood problem asks whether oriented graphs always contain a vertex whose second neighborhood (vertices at distance exactly 2) is at least as large as its first neighborhood (out-neighbors). This has been conjectured by several researchers and has connections to tournament theory and Seymour's second neighborhood conjecture.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 128, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1442, "problem_number": "GRAPH-055", "title": "Teschner's Bondage Number Conjecture", "statement": "Is the bondage number of a graph always ≤ 3Δ/2, where Δ is the maximum degree?", "background": "The bondage number is the minimum number of edges whose removal increases the domination number. Teschner conjectured an upper bound of 3Δ/2 in terms of maximum degree Δ. This would establish a fundamental relationship between edge removal sensitivity and local graph structure in domination problems.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 89, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1443, "problem_number": "GRAPH-056", "title": "Tutte's 5-Flow Conjecture", "statement": "Does every bridgeless graph have a nowhere-zero 5-flow?", "background": "Tutte's 5-flow conjecture is one of the most famous problems in graph theory. A nowhere-zero k-flow is an orientation and edge-labeling with values in {±1,...,±(k-1)} satisfying flow conservation. The conjecture states that 5 colors suffice for all bridgeless graphs. Related to the four-color theorem and still wide open despite much research.", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1444, "problem_number": "GRAPH-057", "title": "Tutte's 4-Flow Conjecture for Petersen-Minor-Free Graphs", "statement": "Does every Petersen-minor-free bridgeless graph have a nowhere-zero 4-flow?", "background": "This is a refinement of Tutte's 5-flow conjecture for graphs without Petersen graph minors. The Petersen graph is known to require 5 colors for nowhere-zero flows, so excluding it might allow 4-flows. This conjecture connects graph minors, nowhere-zero flows, and the special role of the Petersen graph in combinatorics.", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1445, "problem_number": "GRAPH-058", "title": "Woodall's Conjecture", "statement": "Is the minimum dicut size equal to the maximum number of disjoint dijoins in a directed graph?", "background": "Woodall's conjecture is a directed graph analogue of Menger's theorem. A dicut is a set of arcs whose removal disconnects the graph directionally, and a dijoin connects specified vertex pairs. The conjecture proposes a min-max relation, which would be a fundamental packing-covering duality for directed graphs.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 134, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1446, "problem_number": "ALG-001", "title": "Birch-Tate Conjecture", "statement": "Relate the order of the center of the Steinberg group of the ring of integers to the Dedekind zeta function.", "background": "The Birch-Tate conjecture connects algebraic K-theory to special values of zeta functions. It predicts a precise relationship between the center of the Steinberg group St(O_K) of a number field K and the value of its Dedekind zeta function at s=-1. This is a fundamental connection between algebra and analytic number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 187, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1447, "problem_number": "ALG-002", "title": "Casas-Alvero Conjecture", "statement": "If a polynomial of degree d over a field of characteristic 0 shares a factor with each of its first d-1 derivatives, must it be $(x-a)^d$?", "background": "The Casas-Alvero conjecture states that a polynomial sharing roots with all its derivatives (up to degree d-1) must be a power of a linear polynomial. Despite its elementary statement, it remains open. The conjecture has been verified for many special cases but lacks a general proof.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 203, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1448, "problem_number": "ALG-003", "title": "Connes Embedding Problem", "statement": "Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor?", "background": "The Connes embedding problem is a central question in operator algebra theory. It asks whether all separable II₁ factors embed into the ultrapower of the hyperfinite II₁ factor. This problem connects functional analysis, quantum information theory, and logic. Recent claimed solutions using quantum computing have generated significant interest.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 289, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1449, "problem_number": "ALG-004", "title": "Crouzeix's Conjecture", "statement": "Is $\\|f(A)\\| \\leq 2 \\sup_{z \\in W(A)} |f(z)|$ for any matrix A and analytic function f on the numerical range W(A)?", "background": "Crouzeix's conjecture bounds the matrix norm of f(A) by twice the supremum of |f| over the numerical range of A. The constant 2 would be optimal. This conjecture connects matrix theory, complex analysis, and numerical analysis. The best known bound is approximately 11.08, far from the conjectured 2.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1450, "problem_number": "ALG-005", "title": "Determinantal Conjecture", "statement": "Characterize the determinant of the sum of two normal matrices.", "background": "The determinantal conjecture seeks inequalities or characterizations for det(A+B) when A and B are normal matrices. While det(AB) = det(A)det(B) is well known, the sum of normal matrices presents challenges. This problem connects linear algebra with operator theory and has applications in quantum mechanics.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1451, "problem_number": "ALG-006", "title": "Eilenberg-Ganea Conjecture", "statement": "Does every group with cohomological dimension 2 have a 2-dimensional Eilenberg-MacLane space K(G,1)?", "background": "The Eilenberg-Ganea conjecture asks whether cohomological dimension equals geometric dimension for groups. Specifically, if cd(G)=2, does there exist a 2-dimensional CW complex with fundamental group G? The conjecture is known to hold for cd ≠ 2. This connects algebraic topology with group theory.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1452, "problem_number": "ALG-007", "title": "Farrell-Jones Conjecture", "statement": "Are the assembly maps in algebraic K-theory and L-theory isomorphisms?", "background": "The Farrell-Jones conjecture predicts that certain assembly maps are isomorphisms for all groups. This would have major consequences for the computation of algebraic K-theory and L-theory groups. The conjecture has been verified for many important classes of groups including hyperbolic groups and arithmetic groups.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 165, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1453, "problem_number": "ALG-008", "title": "Finite Lattice Representation Problem", "statement": "Is every finite lattice isomorphic to the congruence lattice of some finite algebra?", "background": "The finite lattice representation problem asks whether every finite lattice can be realized as the congruence lattice of a finite algebra. While every finite lattice is the congruence lattice of some algebra, requiring finiteness of the algebra is much more restrictive. This is a central problem in universal algebra.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 142, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1454, "problem_number": "ALG-009", "title": "Hadamard Matrix Conjecture", "statement": "Does a Hadamard matrix of order 4k exist for every positive integer k?", "background": "The Hadamard conjecture states that Hadamard matrices (square matrices with entries ±1 and mutually orthogonal rows) exist for all orders divisible by 4. These matrices have applications in coding theory, cryptography, and experimental design. The smallest open case is k=167 (order 668).", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1455, "problem_number": "ALG-010", "title": "Köthe Conjecture", "statement": "If a ring has no nil two-sided ideal besides {0}, does it also have no nil one-sided ideal besides {0}?", "background": "The Köthe conjecture asks whether the absence of nontrivial nil ideals implies the absence of nontrivial nil one-sided ideals. A nil ideal is one where every element is nilpotent. This has been a central problem in ring theory for decades, with connections to the structure theory of noncommutative rings.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1456, "problem_number": "ALG-011", "title": "Perfect Cuboid", "statement": "Does there exist a perfect cuboid—a rectangular parallelepiped with integer edges, face diagonals, and space diagonal?", "background": "A perfect cuboid would be a box where all edges, face diagonals, and the space diagonal are integers. Despite extensive computational searches, no perfect cuboid has been found, nor has non-existence been proven. This is a Diophantine problem with connections to number theory and geometry.", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 312, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1457, "problem_number": "ALG-012", "title": "Rota's Basis Conjecture", "statement": "Given n bases of an n-dimensional matroid, can we find n disjoint rainbow bases?", "background": "Rota's basis conjecture asks whether n disjoint bases B₁,...,Bₙ of a matroid of rank n can be rearranged into an n×n matrix where each row is a basis and each column is a transversal (rainbow basis). This elegant conjecture connects matroid theory with combinatorics and has resisted many attempts at proof.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1458, "problem_number": "MOD-001", "title": "Cherlin-Zilber Conjecture", "statement": "Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?", "background": "The Cherlin-Zilber conjecture (also called the algebraicity conjecture) proposes that infinite simple groups with stable theories are essentially algebraic groups. This would classify a vast class of model-theoretically tame groups. The conjecture connects model theory, group theory, and algebraic geometry in a profound way.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 176, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1459, "problem_number": "MOD-002", "title": "Generalized Star Height Problem", "statement": "Can all regular languages be expressed with generalized regular expressions having bounded star height?", "background": "The generalized star height problem asks whether there's a universal bound on the nesting depth of Kleene stars needed to express regular languages. This is a fundamental question in formal language theory and automata theory, with connections to computational complexity and logic.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 143, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1460, "problem_number": "MOD-003", "title": "Hilbert's Tenth Problem for Number Fields", "statement": "For which number fields is there an algorithm to determine if a Diophantine equation has solutions?", "background": "Hilbert's tenth problem asked for an algorithm to solve Diophantine equations over the integers—proven impossible by Matiyasevich. The question for other number fields remains open. It's known to be undecidable for some fields and decidable for others. Determining exactly which fields admit such algorithms is a major open problem.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1461, "problem_number": "MOD-004", "title": "Vaught Conjecture", "statement": "Does every complete first-order theory in a countable language have countably many, $\\aleph_0$, or $2^{\\aleph_0}$ countable models?", "background": "Vaught's conjecture states that the number of countable models of a complete theory is either finite, countably infinite, or continuum. This would rule out intermediate cardinalities. The conjecture connects model theory with descriptive set theory and has deep connections to the structure of mathematical logic.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1462, "problem_number": "MOD-005", "title": "Tarski's Exponential Function Problem", "statement": "Is the theory of the real numbers with addition, multiplication, and exponentiation decidable?", "background": "Tarski proved that the theory of real closed fields is decidable. Adding exponentiation makes the question much harder. Decidability would mean an algorithm exists to determine truth of statements involving exp. This has implications for automated theorem proving and connections to transcendental number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 256, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1463, "problem_number": "MOD-006", "title": "Stable Field Conjecture", "statement": "Is every infinite field with a stable first-order theory separably closed?", "background": "The stable field conjecture predicts that infinite fields with stable theories are separably closed. Stable theories are model-theoretically well-behaved. This conjecture would classify all stable fields, providing a complete understanding of these algebraically important structures through a model-theoretic lens.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1464, "problem_number": "MOD-007", "title": "Henson Graphs Finite Model Property", "statement": "Do Henson graphs have the finite model property?", "background": "Henson graphs are universal homogeneous graphs omitting certain finite subgraphs. The finite model property asks whether every satisfiable sentence has a finite model. This question connects infinite graph theory, model theory, and combinatorics, with implications for the decidability of their first-order theories.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 123, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1465, "problem_number": "MOD-008", "title": "O-Minimal Theory with Trans-Exponential Growth", "statement": "Does there exist an o-minimal first-order theory with a trans-exponential (rapid growth) function?", "background": "O-minimal structures are ordered structures where definable sets have simple topology. Known o-minimal structures include real closed fields and structures with restricted analytic functions. The question asks whether o-minimality is compatible with very fast-growing functions, testing the limits of tame model theory.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1466, "problem_number": "MOD-009", "title": "Infinite Minimal Field Algebraic Closure", "statement": "Is every infinite minimal field of characteristic zero algebraically closed?", "background": "A minimal structure is one where every definable subset is finite or cofinite. The question asks whether infinite fields with this property must be algebraically closed (when char=0). This would characterize the simplest infinite fields from a model-theoretic perspective, connecting field theory with minimality.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1467, "problem_number": "MOD-010", "title": "Keisler's Order", "statement": "Determine the structure of Keisler's order on first-order theories.", "background": "Keisler's order compares first-order theories based on the complexity of their ultrapowers. Understanding this order would classify theories by their model-theoretic complexity. Recent breakthroughs have shed light on the order's structure, but a complete classification remains elusive. This connects with classification theory and stability.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1468, "problem_number": "ALG-013", "title": "Serre's Conjecture II", "statement": "For simply connected semisimple algebraic groups over fields of cohomological dimension ≤2, is $H^1(F,G) = 0$?", "background": "Serre's Conjecture II predicts that the first Galois cohomology of simply connected semisimple groups vanishes over fields of small cohomological dimension. This would have major implications for the classification of algebraic groups and forms. The conjecture is known for various classes of fields but remains open in general.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1469, "problem_number": "ALG-014", "title": "Serre's Positivity Conjecture", "statement": "If R is a regular local ring and P,Q are prime ideals with $\\dim(R/P) + \\dim(R/Q) = \\dim(R)$, is $\\chi(R/P, R/Q) > 0$?", "background": "Serre's positivity conjecture predicts that the Euler characteristic (intersection multiplicity) is positive when dimensions add correctly. This is part of a broader set of homological conjectures in commutative algebra. The conjecture would provide fundamental information about the structure of modules over regular rings.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1470, "problem_number": "ALG-015", "title": "Uniform Boundedness Conjecture for Rational Points", "statement": "Is there a bound N(g,d) such that all curves of genus g≥2 over degree d number fields have at most N(g,d) rational points?", "background": "The uniform boundedness conjecture asks whether the number of rational points on curves of genus ≥2 is uniformly bounded in terms of genus and field degree. This would be a remarkable strengthening of Faltings' theorem (finite number of points). The conjecture connects arithmetic geometry with Diophantine equations.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 213, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1479, "problem_number": "TOP-001", "title": "Baum-Connes Conjecture", "statement": "Is the assembly map in K-theory an isomorphism for all locally compact groups?", "background": "The Baum-Connes conjecture predicts that a certain assembly map from equivariant K-homology to the K-theory of group C*-algebras is an isomorphism. This would have major consequences for the Novikov conjecture, index theory, and the structure of operator algebras. Known for many groups, general case open.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 198, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1480, "problem_number": "TOP-002", "title": "Berge Conjecture", "statement": "Are Berge knots the only knots in S³ admitting lens space surgeries?", "background": "The Berge conjecture states that Berge knots (constructed via a specific procedure) are the only knots in the 3-sphere that admit Dehn surgeries yielding lens spaces. This would classify all such knots, providing deep insight into the relationship between knot theory and 3-manifold topology.", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1481, "problem_number": "TOP-003", "title": "Borel Conjecture", "statement": "Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?", "background": "The Borel conjecture predicts that aspherical closed manifolds (those with contractible universal cover) are rigid—completely determined by their fundamental group up to homeomorphism. This would be a remarkable topological rigidity result, currently known only for special classes of manifolds.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1482, "problem_number": "TOP-004", "title": "Hilbert-Smith Conjecture", "statement": "If a locally compact group acts faithfully and continuously on a manifold, must it be a Lie group?", "background": "The Hilbert-Smith conjecture asks whether every locally compact group with a continuous faithful action on a manifold is necessarily a Lie group. This would rule out p-adic groups acting on manifolds, resolving a fundamental question about the symmetries of topological spaces.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 212, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1483, "problem_number": "TOP-005", "title": "Novikov Conjecture", "statement": "Are certain polynomials in Pontryagin classes homotopy invariants?", "background": "The Novikov conjecture states that higher signatures (certain rational combinations of Pontryagin numbers) are oriented homotopy invariants. This has profound consequences for manifold topology, surgery theory, and K-theory. Proven for many classes of groups, but the general case remains open.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1484, "problem_number": "TOP-006", "title": "Unknotting Problem", "statement": "Can unknots be recognized in polynomial time?", "background": "The unknotting problem asks whether there exists a polynomial-time algorithm to determine if a knot diagram represents the unknot. While algorithms exist (exponential time), polynomial-time decidability remains open. This is a central problem in computational topology with connections to complexity theory.", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 256, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1485, "problem_number": "TOP-007", "title": "Volume Conjecture", "statement": "Do quantum invariants of knots determine their hyperbolic volume?", "background": "The volume conjecture predicts an exponential relationship between the colored Jones polynomial (a quantum invariant) and the hyperbolic volume of a knot complement. This would connect quantum topology with hyperbolic geometry in a striking way, revealing deep structures in 3-dimensional topology.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1486, "problem_number": "TOP-008", "title": "Whitehead Conjecture", "statement": "Is every connected subcomplex of a 2-dimensional aspherical CW complex also aspherical?", "background": "The Whitehead conjecture asks whether asphericity (having contractible universal cover) is preserved under taking subcomplexes in dimension 2. This would clarify the local structure of aspherical spaces and has connections to group theory and low-dimensional topology.", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 143, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1487, "problem_number": "TOP-009", "title": "Zeeman Conjecture", "statement": "Is $K \\times [0,1]$ collapsible for every finite contractible 2-dimensional CW complex K?", "background": "The Zeeman conjecture predicts that the product of any finite contractible 2-complex with an interval is collapsible (can be reduced to a point by elementary collapses). This relates to the Poincaré conjecture and questions about higher-dimensional manifolds. A counterexample would have major implications.", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1488, "problem_number": "COMB-001", "title": "1/3-2/3 Conjecture", "statement": "Does every non-total finite poset have two elements x,y with P(x before y in random linear extension) ∈ [1/3, 2/3]?", "background": "The 1/3-2/3 conjecture asks whether finite partially ordered sets (not totally ordered) always contain a pair with intermediate probability of appearing in a certain order. This connects order theory with probability and has implications for sorting algorithms and social choice theory.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 124, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1489, "problem_number": "COMB-002", "title": "Lonely Runner Conjecture", "statement": "If k runners with distinct speeds run on a unit circle, will each runner be \"lonely\" (≥1/k away from others) at some time?", "background": "The lonely runner conjecture predicts that in a system of runners with different speeds on a circular track, each runner will at some point be far from all others. Verified for k≤7, this problem connects view obstruction, Diophantine approximation, and number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1490, "problem_number": "COMB-003", "title": "Sunflower Conjecture", "statement": "Can the minimum size for sunflowers be bounded by an exponential (not super-exponential) function of k?", "background": "The sunflower conjecture asks whether families of k-element sets containing a sunflower (r sets with common \"core\") require only exponentially many sets in k. Recent progress by Alweiss et al. improved bounds but the original conjecture remains open. Fundamental for extremal combinatorics.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1491, "problem_number": "COMB-004", "title": "Union-Closed Sets Conjecture", "statement": "For any finite union-closed family of sets, does some element appear in at least half the sets?", "background": "Frankl's union-closed sets conjecture (also called the union-closed set conjecture) states that in any family of sets closed under unions, at least one element appears in ≥50% of the sets. Despite its elementary statement, this problem has resisted all attempts at proof.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1492, "problem_number": "COMB-005", "title": "Ramsey Number R(5,5)", "statement": "What is the exact value of the Ramsey number R(5,5)?", "background": "Ramsey theory asks: in any 2-coloring of edges of the complete graph Kₙ, what's the minimum n guaranteeing a monochromatic K₅? Known: 43 ≤ R(5,5) ≤ 48. Finding the exact value would be a major breakthrough. Paul Erdős famously said R(6,6) would require alien technology.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1495, "problem_number": "NUM-001", "title": "Singmaster's Conjecture", "statement": "Is there a finite upper bound on multiplicities of entries >1 in Pascal's triangle?", "background": "Singmaster's conjecture asks whether any number (other than 1) appears in Pascal's triangle only finitely many times. Known: no entry appears more than 8 times. A proof would reveal deep structure in binomial coefficients and their divisibility properties.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1496, "problem_number": "NUM-002", "title": "Odd Perfect Numbers", "statement": "Do any odd perfect numbers exist?", "background": "A perfect number equals the sum of its proper divisors. All known perfect numbers are even (form 2^(p-1)(2^p-1) for Mersenne primes). Whether odd perfect numbers exist is one of the oldest open problems in mathematics, dating to ancient Greece. If they exist, they must be very large (>10^1500).", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 412, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1497, "problem_number": "NUM-003", "title": "Infinitude of Perfect Numbers", "statement": "Are there infinitely many perfect numbers?", "background": "All known perfect numbers are even and correspond to Mersenne primes via Euclid-Euler theorem. The question reduces to: are there infinitely many Mersenne primes? This remains open despite extensive computational searches. Connected to the distribution of primes and special number forms.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 345, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1498, "problem_number": "NUM-004", "title": "Quasiperfect Numbers", "statement": "Do quasiperfect numbers exist?", "background": "A quasiperfect number n has σ(n) = 2n+1 (sum of divisors is one more than twice the number). No quasiperfect numbers are known. If they exist, they must be odd perfect squares >10^35. This problem connects divisor functions with perfect number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1499, "problem_number": "NUM-005", "title": "Lychrel Numbers", "statement": "Do Lychrel numbers exist in base 10?", "background": "A Lychrel number never forms a palindrome through iterative reverse-and-add process. 196 is the first candidate—after billions of iterations, no palindrome found. Proving existence or non-existence would resolve this computational mystery connecting palindromes with iteration dynamics.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1500, "problem_number": "NUM-006", "title": "Odd Weird Numbers", "statement": "Do odd weird numbers exist?", "background": "Weird numbers are abundant but not semiperfect (no subset of divisors sums to the number). All known weird numbers are even. Finding an odd weird number or proving none exist would reveal deep structure in additive properties of divisors.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1501, "problem_number": "NUM-007", "title": "Infinitude of Amicable Pairs", "statement": "Are there infinitely many pairs of amicable numbers?", "background": "Amicable pairs (m,n) satisfy σ(m)-m=n and σ(n)-n=m. Over 12 million pairs known, but infinity unproven. Related to perfect numbers and sociable chains. Erdős-Rieger heuristics suggest infinity, but proof remains elusive.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 212, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1502, "problem_number": "NUM-008", "title": "Pi Normality", "statement": "Is π a normal number (all digits equally frequent in all bases)?", "background": "A normal number has each digit appearing with equal asymptotic frequency in every base. While π appears statistically normal (verified to trillions of digits), no proof exists. This connects transcendental numbers, digit distribution, and randomness in mathematical constants.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 30, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1503, "problem_number": "NUM-009", "title": "Algebraic Number Normality", "statement": "Are all irrational algebraic numbers normal?", "background": "The question asks whether every irrational root of a polynomial with integer coefficients has all digits equally distributed in every base. A positive answer would be a remarkable connection between algebraic structure and digit statistics. Currently, we cannot prove normality for any specific algebraic irrational.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1504, "problem_number": "NUM-010", "title": "Gilbreath's Conjecture", "statement": "Does iterating unsigned differences on prime sequence always yield 1 as first element?", "background": "Start with primes 2,3,5,7,11,... Take absolute differences: 1,2,2,4,... Repeat. Conjecture: first element is always 1. Verified to huge primes, but unproven. This reveals hidden regularity in prime gaps with implications for prime distribution.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1505, "problem_number": "NUM-011", "title": "Lander-Parkin-Selfridge Conjecture", "statement": "If Σᵢ aᵢᵏ = Σⱼ bⱼᵏ with m terms on left, n on right, is m+n ≥ k?", "background": "The LPS conjecture generalizes Fermat's Last Theorem to sums of k-th powers. It predicts you need at least k terms total for nontrivial solutions. Counterexamples exist for specific cases, but the general conjecture remains open with implications for Diophantine equations.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1506, "problem_number": "NUM-012", "title": "Class Number Problem", "statement": "Are there infinitely many real quadratic fields with class number 1 (unique factorization)?", "background": "The class number problem asks whether infinitely many real quadratic number fields Q(√d) have unique factorization. For imaginary quadratic fields, Heegner-Baker-Stark proved only finitely many exist. The real case remains open—a fundamental question in algebraic number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1507, "problem_number": "NUM-013", "title": "Hilbert's 12th Problem", "statement": "Extend Kronecker-Weber theorem to abelian extensions of arbitrary number fields.", "background": "Hilbert's 12th problem asks for explicit construction of abelian extensions of number fields via special values of transcendental functions (generalizing cyclotomic fields for Q). Partial progress via complex multiplication, but general case remains one of Hilbert's unsolved problems.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 187, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1508, "problem_number": "NUM-014", "title": "Leopoldt's Conjecture", "statement": "Does the p-adic regulator of an algebraic number field never vanish?", "background": "Leopoldt's conjecture predicts that the p-adic regulator (a p-adic analogue of the classical regulator from Dirichlet's unit theorem) is always nonzero. This has major implications for Iwasawa theory and the structure of p-adic L-functions.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1509, "problem_number": "NUM-015", "title": "Siegel Zeros", "statement": "Do Siegel zeros (real zeros of Dirichlet L-functions near s=1) exist?", "background": "Siegel zeros are hypothetical exceptional real zeros of L-functions very close to s=1. If they exist, they violate the Generalized Riemann Hypothesis. Their existence would have major consequences for prime distribution in arithmetic progressions. Most believe they don't exist.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1510, "problem_number": "NUM-016", "title": "Schanuel's Conjecture", "statement": "For e and π: are they algebraically independent? Is e+π, eπ, π^e, etc. transcendental?", "background": "Schanuel's conjecture is a fundamental statement about transcendence degrees. It implies e and π are algebraically independent and that expressions like e+π, eπ, π^π are transcendental. Proving it would resolve many open questions in transcendental number theory at once.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 287, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1511, "problem_number": "NUM-017", "title": "Euler-Mascheroni Constant Irrationality", "statement": "Is the Euler-Mascheroni constant γ irrational? Transcendental?", "background": "The Euler-Mascheroni constant γ ≈ 0.5772 appears throughout analysis and number theory. We don't even know if it's irrational! Proving irrationality or transcendence would be a major achievement. Related constants like Catalan's G and ζ(3) face similar questions.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 323, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1512, "problem_number": "NUM-018", "title": "Littlewood Conjecture", "statement": "For any α,β ∈ ℝ, is lim inf_{n→∞} n·||nα||·||nβ|| = 0?", "background": "Littlewood's conjecture connects Diophantine approximation of pairs of real numbers. It predicts that for any two reals, you can simultaneously approximate both well infinitely often. Related to continued fractions and dynamics on homogeneous spaces. Proved for many special cases.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1513, "problem_number": "NUM-019", "title": "Four Exponentials Conjecture", "statement": "If x₁,x₂ and y₁,y₂ are linearly independent over ℚ, is at least one of e^(xᵢyⱼ) transcendental?", "background": "The four exponentials conjecture states that you can't have all four values e^(x₁y₁), e^(x₁y₂), e^(x₂y₁), e^(x₂y₂) algebraic when the xᵢ and yⱼ satisfy independence conditions. Weaker than Schanuel's conjecture but still wide open. Six exponentials theorem is the proven weaker version.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1514, "problem_number": "NUM-020", "title": "Integer Factorization Polynomial Time", "statement": "Can integer factorization be done in polynomial time?", "background": "The integer factorization problem asks whether factoring large integers into primes can be done efficiently (polynomial time). RSA cryptography relies on it being hard. Shor's algorithm solves it on quantum computers, but classical complexity remains unknown. Related to P vs NP.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 456, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1515, "problem_number": "PDE-001", "title": "Navier-Stokes Existence and Smoothness", "statement": "Do smooth solutions to Navier-Stokes equations exist globally in 3D? Or do finite-time singularities occur?", "background": "The Navier-Stokes existence and smoothness problem is one of the seven Millennium Prize Problems. It asks whether smooth solutions to the 3D Navier-Stokes equations exist for all time, or whether finite-time blow-up can occur. Fundamental for fluid dynamics and mathematical physics.", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 512, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set_id": 2 }, { "id": 1516, "problem_number": "GEOM-001", "title": "Sphere Packing Problem Higher Dimensions", "statement": "What is the optimal sphere packing density in dimensions >3?", "background": "The sphere packing problem asks for the densest way to pack spheres in n-dimensional space. Solved in dimensions 1,2,3 (Kepler's conjecture, proved by Hales), 8, and 24 (Viazovska). Dimensions 4-7 and ≥9 remain open. Connections to lattices, coding theory, and optimization.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 298, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1517, "problem_number": "HL-A", "title": "Hardy-Littlewood Conjecture A (Prime k-tuples)", "statement": "Let $a_1, \\ldots, a_k$ be given integers. Then there exist infinitely many positive integers $n$ such that $n + a_1, \\ldots, n + a_k$ are all prime, provided that for every prime $p$, there exists an integer $m$ such that $(m + a_i, p) = 1$ for all $i$.", "background": "The first Hardy-Littlewood conjecture, also known as the prime k-tuples conjecture, generalizes the twin prime conjecture. It states that the asymptotic frequency of any admissible prime constellation can be computed explicitly. The case $k=2$ with $(a_1, a_2) = (0, 2)$ is the twin prime conjecture. Yitang Zhang proved in 2013 that there exists at least one 2-tuple with gap ≤70,000,000 (later improved to 246) that appears infinitely often. The full conjecture remains open and is considered one of the most important unsolved problems in number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set_id": 7 }, { "id": 1518, "problem_number": "HL-B", "title": "Hardy-Littlewood Conjecture B (Second Conjecture)", "statement": "For all integers $x, y \\geq 2$, we have $\\pi(x+y) \\leq \\pi(x) + \\pi(y)$, where $\\pi(n)$ denotes the prime counting function (the number of primes less than or equal to $n$).", "background": "The second Hardy-Littlewood conjecture states the subadditivity of the prime counting function. In 1974, Hensley and Richards proved that Conjecture A and Conjecture B are incompatible with each other - they cannot both be true. Since Conjecture A (the prime k-tuples conjecture) is considered more likely to be true based on computational evidence and its connections to the twin prime conjecture, most number theorists believe Conjecture B is actually false, despite appearing plausible. This represents a fascinating case where intuitive conjectures can contradict each other.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set_id": 7 }, { "id": 1519, "problem_number": "HL-F", "title": "Hardy-Littlewood Conjecture F (Primes in Quadratic Polynomials)", "statement": "For a polynomial $f(x) = ax^2 + bx + c$ with $a > 0$, $\\gcd(a,b,c) = 1$, and discriminant $\\Delta = b^2 - 4ac$ not a perfect square, the polynomial takes infinitely many prime values. Furthermore, the number $P(n)$ of primes of the form $f(x) \\leq n$ satisfies an asymptotic formula $P(n) \\sim A \\cdot \\frac{\\sqrt{n}}{\\log n}$ where $A$ depends on $a, b, c$ but not on $n$.", "background": "Conjecture F is a special case of the Bateman-Horn conjecture and concerns primes represented by quadratic polynomials. It predicts not only the infinitude of such primes but also their asymptotic density. The constant A can take values larger or smaller than 1, meaning some polynomials are especially rich in primes while others are especially poor. For example, $4x^2 - 2x + 41$ has $A \\approx 6.6$, making it nearly 7 times as likely to produce primes as random numbers of the same size. This conjecture explains the visible patterns in the Ulam spiral. Despite extensive computational verification, no polynomial has been proven to produce infinitely many primes except linear polynomials (Dirichlet's theorem).", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 7 }, { "id": 1855, "problem_number": "GUY-A4", "title": "The Prime Number Race", "statement": "Let $\\pi(n; a, b)$ be the number of primes $p \\le n$ with $p \\equiv a \\pmod b$. For every $a$ and $b$ with $a \\perp b$, are there infinitely many values of $n$ for which $\\pi(n; a, b) > \\pi(n; a_1, b)$ for every $a_1 \\not\\equiv a \\pmod b$?", "background": "Turán was particularly interested in the prime number race. Knapowski & Turán settled special cases, but the general problem is wide open. Chebyshev noted that $\\pi(n; 1, 3) < \\pi(n; 2, 3)$ for small values of $n$, but this inequality is reversed for very large $n$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A4.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1857, "problem_number": "GUY-A5b", "title": "Erdős $3000 Conjecture on Arithmetic Progressions", "statement": "Let $\\{a_i\\}$ be any infinite sequence of integers for which $\\sum 1/a_i$ is divergent. Does the sequence contain arbitrarily long arithmetic progressions?", "background": "Erdős offered $3000.00 for a proof or disproof of this conjecture. This is a generalization of the arithmetic progressions of primes problem. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A5.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1858, "problem_number": "GUY-A6", "title": "Consecutive Primes in Arithmetic Progression", "statement": "Are there arbitrarily long arithmetic progressions of consecutive primes? That is, for any positive integer $k$, do there exist $k$ consecutive primes $p_n, p_{n+1}, \\ldots, p_{n+k-1}$ in arithmetic progression?", "background": "Known examples include the 4-term sequences 251, 257, 263, 269 and 1741, 1747, 1753, 1759. Dubner, Forbes, Lygeros, Mizony & Zimmermann found 10 consecutive primes in arithmetic progression in 1998. It is not known if there are infinitely many sets of three consecutive primes in arithmetic progression. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A6.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1859, "problem_number": "GUY-A7a", "title": "Infinitude of Sophie Germain Primes", "statement": "Are there infinitely many Sophie Germain primes? A prime $p$ is called a Sophie Germain prime if $2p + 1$ is also prime.", "background": "It is believed, but not known, that there are infinitely many Sophie Germain primes. Dubner has found many large examples. The largest known Sophie Germain prime has over 24000 decimal digits. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A7.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1860, "problem_number": "GUY-A7b", "title": "Shanks Chains of Length 7", "statement": "Are there any Shanks chains of length 7 with $p_{i+1} = 4p_i^2 - 17$?", "background": "Shanks chains are quadratic chains of primes. The recurrence $p_{i+1} = 4p_i^2 - 17$ yields a 4-chain if $p_1 = 3$ and a 5-chain if $p_1 = 303593$, but it can be seen (mod 59) that no such chain has length 17. It seems certain that such chains cannot be of arbitrary length. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A7.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1861, "problem_number": "GUY-A8a", "title": "Erdős $5000 Problem on Prime Gaps", "statement": "Is it true that for infinitely many $n$, $d_n = p_{n+1} - p_n > c \\ln n \\ln \\ln n \\ln \\ln \\ln \\ln n / (\\ln \\ln \\ln n)^2$ for arbitrarily large constant $c$?", "background": "Erdős offers $5,000 for a proof or disproof that the constant $c$ can be taken arbitrarily large. Rankin showed this holds for $c = e^\\gamma$, and Pintz improved it to $c = 2e^\\gamma > 3.562$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A8.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1862, "problem_number": "GUY-A8b", "title": "Twin Prime Conjecture", "statement": "Are there infinitely many twin primes? That is, are there infinitely many primes $p$ such that $p + 2$ is also prime?", "background": "A very famous conjecture. Hardy and Littlewood conjectured that $P_2(n)$, the number of twin prime pairs less than $n$, is asymptotically $2cn/(\\ln n)^2$ where $2c \\approx 1.32032$. Brun showed that the sum of the reciprocals of twin primes is convergent. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A8.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1863, "problem_number": "GUY-A9", "title": "General Patterns of Consecutive Primes", "statement": "For any given pattern of primes with no congruence obstructions, are there infinitely many sets of consecutive primes with this pattern?", "background": "This conjecture is more general than Chowla's conjecture. It seems likely that there are infinitely many triples of primes $\\{6k - 1, 6k + 1, 6k + 5\\}$ and $\\{6k + 1, 6k + 5, 6k + 7\\}$. Hensley & Richards showed this is incompatible with the conjecture $\\pi(x + y) \\le \\pi(x) + \\pi(y)$ for all integers $x, y \\ge 2$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A9.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1864, "problem_number": "GUY-A10", "title": "Gilbreath's Conjecture", "statement": "Define $d_n^k$ by $d_n^1 = p_{n+1} - p_n$ and $d_n^{k+1} = |d_{n+1}^k - d_n^k|$, the successive absolute differences of the sequence of primes. Is it true that $d_1^k = 1$ for all $k$?", "background": "Gilbreath conjectured this (and Proth claimed to have proved it long before). This was verified for $k < 63419$ by Killgrove & Ralston. Odlyzko checked it for primes up to $\\pi(10^{13})$. Croft and others suggest it has nothing to do with primes as such, but will be true for any sequence consisting of 2 and odd numbers which doesn't increase too fast. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A10.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1865, "problem_number": "GUY-A11", "title": "Erdős $100 Problem on Increasing and Decreasing Gaps", "statement": "Does there exist an $n_0$ such that for every $i$ and $n > n_0$ we have $d_{n+2i} > d_{n+2i+1}$ and $d_{n+2i+1} < d_{n+2i+2}$, where $d_n = p_{n+1} - p_n$?", "background": "Erdős & Turán showed that the values of $n$ for which $d_n > d_{n+1}$ have positive lower density, but it is not known if there are infinitely many increasing or decreasing sets of three consecutive values of $d_n$. Erdős offers $100.00 for a proof that such an $n_0$ does not exist. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A11.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1866, "problem_number": "GUY-A13", "title": "Erdős Conjecture on Carmichael Numbers", "statement": "Let $C(x)$ be the number of Carmichael numbers less than $x$. Does $(\\ln C(x))/\\ln x$ tend to 1 as $x$ tends to infinity?", "background": "Erdős conjectured this behavior for the count of Carmichael numbers. Alford, Granville & Pomerance showed there are infinitely many Carmichael numbers, in fact more than $x^\\beta$ of them less than $x$ for $\\beta > 0.290306$. Pomerance, Selfridge & Wagstaff give a heuristic argument supporting Erdős' conjecture. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A13.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1867, "problem_number": "GUY-A14a", "title": "Pomerance's Questions on Good Primes", "statement": "Call prime $p_n$ good if $p_n^2 > p_{n-i}p_{n+i}$ for all $i$, $1 \\le i \\le n-1$. Is it true that the set of $n$ for which $p_n$ is good has density 0? Are there infinitely many $n$ with $p_n p_{n+1} > p_{n-i} p_{n+1+i}$ for all $i$, $1 \\le i \\le n-1$?", "background": "Erdős and Straus introduced the concept of good primes. Examples include 5, 11, 17, and 29. Pomerance used the prime number graph to show there are infinitely many good primes and posed several related questions. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A14.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1868, "problem_number": "GUY-A15", "title": "Congruent Products of Consecutive Numbers", "statement": "What is the least prime $p$ such that there are integers $a, k_1, k_2, k_3$ with $\\prod_{i=1}^{k_1} (a+i) \\equiv \\prod_{i=1}^{k_2} (a+k_1+i) \\equiv \\prod_{i=1}^{k_3} (a+k_1+k_2+i) \\equiv 1 \\pmod{p}$?", "background": "Erdős observed that $3 \\cdot 4 \\equiv 5 \\cdot 6 \\cdot 7 \\equiv 1 \\pmod{11}$ and suggested that such primes $p$ exist for any number of congruent products. Narkiewicz and others found examples for larger numbers of terms. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A15.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set_id": 9 }, { "id": 1869, "problem_number": "GUY-A16", "title": "Walking to Infinity on Gaussian Primes", "statement": "Can one walk from the origin to infinity using Gaussian primes as stepping stones and taking steps of bounded length?", "background": "Motzkin and Gordon asked this question about Gaussian primes (primes in the ring of complex numbers $a+bi$ where $a, b$ are integers). Presumably not. Jordan & Rabung showed that steps of length at least 4 are necessary. Gethner, Wagon & Wick produced a moat of width $\\sqrt{26}$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A16.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1870, "problem_number": "GUY-A17", "title": "Giuga's Conjecture on Prime Characterization", "statement": "Is it true that if $n$ divides $1^{n-1} + 2^{n-1} + \\dots + (n-1)^{n-1} + 1$, then $n$ is prime?", "background": "Sierpiński observed that if $n$ is prime, then $n$ divides this sum. Giuga conjectured the converse and verified it for $n \\le 10^{1000}$. A counterexample would be a Carmichael number with additional properties. An equivalent conjecture is $n B_{n-1} \\equiv -1 \\pmod{n}$ where $B_k$ are Bernoulli numbers. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A17.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1871, "problem_number": "GUY-A18", "title": "Erdős-Selfridge Classification: Infinitely Many Primes in Each Class", "statement": "In the Erdős-Selfridge classification of primes, are there infinitely many primes in each class? Prime $p$ is in class 1 if the only prime divisors of $p+1$ are 2 or 3; and $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\\le r-1$, with equality for at least one prime factor.", "background": "The first few classes are: Class 1: 2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191, ...; Class 2: 13, 19, 29, 41, 43, 59, 61, 67, 79, 83, 89, 97, 101, ...; Class 3: 37, 103, 113, 151, 157, 163, 173, 181, 193, 227, 233, ... From Richard Guy's \"Unsolved Problems in Number Theory\", Section A18.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1872, "problem_number": "GUY-A19a", "title": "Erdős Conjecture on $n - 2^k$ Prime", "statement": "Are 4, 7, 15, 21, 45, 75, and 105 the only values of $n$ for which $n - 2^k$ is prime for all $k$ such that $2 \\le 2^k < n$?", "background": "Erdős conjectures that these are the only such values. He also conjectures that for infinitely many $n$, all the integers $n - 2^k, 1 \\le 2^k < n$ are squarefree. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A19.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1873, "problem_number": "GUY-A19b", "title": "Cohen-Selfridge Problem on $\\pm p^a \\pm 2^b$", "statement": "What is the least positive odd number not of the form $\\pm p^a \\pm 2^b$, where $p$ is an odd prime?", "background": "Cohen & Selfridge observed that the number is greater than $2^{18}$. This is related to the representation of odd numbers as sums or differences of prime powers and powers of 2. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A19.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set_id": 9 }, { "id": 1874, "problem_number": "GUY-A20", "title": "Density of Symmetric Primes", "statement": "Given pairs of odd primes $p, q$, define $S(q,p)$ as the number of lattice points $(m, n)$ in the rectangle $0 < m < p/2$, $0 < n < q/2$ below the diagonal. A pair is symmetric if $S(p, q) = S(q,p)$. Is the number of symmetric primes less than $x$ equal to $x/(\\ln x)^{\\sigma+o(1)}$, where $\\sigma = 2 - (1+\\ln \\ln 2)/\\ln 2 \\approx 1.08607$?", "background": "Fletcher, Lindgren & Pomerance showed that a pair is symmetric just if $|p - q| = (p - 1, q - 1)$, and that the number of symmetric primes less than $x$ is at most $x/(\\ln x)^{1.027}$. They conjectured the more precise asymptotic. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A20.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1875, "problem_number": "GUY-A12a", "title": "Square Pseudoprimes", "statement": "Are there any square pseudoprimes (base 2) other than multiples of $1194649 = 1093^2$ or $12327121 = 3511^2$?", "background": "Pinch observed that there are 54 non-squarefree pseudoprimes up to $10^{13}$, all multiples of $1093^2$ or $3511^2$. The question asks if there are other perfect squares that are pseudoprimes to base 2. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A12.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1876, "problem_number": "GUY-A12b", "title": "Selfridge-Wagstaff-Pomerance Prize Problem", "statement": "Does there exist a composite number $n \\equiv 3$ or $7 \\pmod{10}$ which divides both $2^n - 2$ and the Fibonacci number $u_{n+1}$?", "background": "Selfridge, Wagstaff & Pomerance offer $500 + $100 + $20 = $620 for finding such a composite $n$, or $20 + $100 + $500 = $620 for a proof that no such $n$ exists. This combines pseudoprime properties with Fibonacci divisibility. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A12.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1877, "problem_number": "GUY-A12c", "title": "Even Fibonacci Pseudoprimes", "statement": "Does there exist an even Fibonacci pseudoprime?", "background": "A Fibonacci pseudoprime of the $m$-th kind is an odd composite integer $n$ with $V_n(m, -1) \\equiv m \\pmod n$ where $V_n$ is the Lucas sequence. Somer showed that if an even Fibonacci pseudoprime exists, it must be greater than $28 \\times 10^{12}$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A12.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1878, "problem_number": "EP-1", "title": "Erdős Problem #1", "statement": "If $A\\subseteq \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert=n$ is such that the subset sums $\\sum_{a\\in S}a$ are distinct for all $S\\subseteq A$ then $ N \\gg 2^{n}. $ ", "background": "Erd\\H{o}s called this 'perhaps my first serious problem' (in \\cite{Er98} he dates it to 1931). The powers of $2$ show that $2^n$ would be best possible here. The trivial lower bound is $N \\gg 2^{n}/n$, since all $2^n$ distinct subset sums must lie in $[0,Nn)$. Erd\\H{o}s and Moser \\cite{Er56} proved $ N\\geq (\\tfrac{1}{4}-o(1))\\frac{2^n}{\\sqrt{n}}. $ (In \\cite{Er85c} Erd\\H{o}s offered \\$100 for any improvement of the constant $1/4$ here.)\nA number of improvements of the constant have been given (see \\cite{St23} for a history), with the current record $\\sqrt{2/\\pi}$ first proved in unpublished work of Elkies and Gleason. Two proofs achieving this constant are provided by Dubroff, Fox, and Xu \\cite{DFX21}, who in fact prove the exact bound $N\\geq \\binom{n}{\\lfloor n/2\\rfloor}$.\nIn \\cite{Er73} and \\cite{ErGr80} the generalisation where $A\\subseteq (0,N]$ is a set of real numbers such that the subset sums all differ by at least $1$ is proposed, with the same conjectured bound. (The second proof of \\cite{DFX21} applies also to this generalisation.) This generalisation seems to have first appeared in \\cite{Gr71}.\nThis problem appears in Erd\\H{o}s' book with Spencer \\cite{ErSp74} in the final chapter titled 'The kitchen sink'. As Ruzsa writes in \\cite{Ru99} \"it is a rich kitchen where such things go to the sink\".\nThe sequence of minimal $N$ for a given $n$ is A276661 in the OEIS.\nSee also [350].\nThis is discussed in problem C8 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[DFX21] Dubroff, Q. and Fox, J. and Xu, M. W., A note on the Erd\\H{o}s distinct subset sums problem. SIAM Journal on Discrete Mathematics (2021), 322-324.\n\n[Er56] Erd\\H{o}s, P., Problems and results in additive number theory. Colloque sur la Th\\'{e}orie des Nombres, Bruxelles, 1955 (1956), 127-137.\n\n[Er73] Erd\\H{o}s, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n\n[Er85c] Erd\\H{o}s, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.\n\n[Er98] Erd\\H{o}s, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[ErSp74] Erd\\H{o}s, Paul and Spencer, Joel, Probabilistic methods in combinatorics. Akad\\'{e}miai Kiad\\'{o} (1974).\n\n[Gr71] Graham, R. L., On sums of integers taken from a fixed sequence. (1971), 22--40.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ru99] Ruzsa, I., Erd\\H{o}s and the Integers. Journal of Number Theory (1999), 115-163.\n\n[St23] Steinerberger, S., Some remarks on the Erd\\H{o}s distinct subset sums problem. arXiv:2208.12182 (2023).\",\n \"difficulty\": \"L3\"\n},\n{", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1879, "problem_number": "EP-3", "title": "Erdős Problem #3", "statement": "If $A\\subseteq \\mathbb{N}$ has $\\sum_{n\\in A}\\frac{1}{n}=\\infty$ then must $A$ contain arbitrarily long arithmetic progressions?", "background": "This is essentially asking for good bounds on $r_k(N)$, the size of the largest subset of $\\{1,\\ldots,N\\}$ without a non-trivial $k$-term arithmetic progression. For example, a bound like $ r_k(N) \\ll_k \\frac{N}{(\\log N)(\\log\\log N)^2} $ would be sufficient.\nEven the case $k=3$ is non-trivial, but was proved by Bloom and Sisask \\cite{BlSi20}. Much better bounds for $r_3(N)$ were subsequently proved by Kelley and Meka \\cite{KeMe23}. Green and Tao \\cite{GrTa17} proved $r_4(N)\\ll N/(\\log N)^{c}$ for some small constant $c>0$. Gowers \\cite{Go01} proved $ r_k(N) \\ll \\frac{N}{(\\log\\log N)^{c_k}}, $ where $c_k>0$ is a small constant depending on $k$. The current best bounds for general $k$ are due to Leng, Sah, and Sawhney \\cite{LSS24}, who show that $ r_k(N) \\ll \\frac{N}{\\exp((\\log\\log N)^{c_k})} $ for some constant $c_k>0$ depending on $k$.\nCuriously, Erd\\H{o}s \\cite{Er83c} thought this conjecture was the 'only way to approach' the conjecture that there are arbitrarily long arithmetic progressions of prime numbers, now a theorem due to Green and Tao \\cite{GrTa08} (see [219]).\nIn \\cite{Er81} Erd\\H{o}s makes the stronger conjecture that $ r_k(N) \\ll_C\\frac{N}{(\\log N)^C} $ for every $C>0$ (now known for $k=3$ due to Kelley and Meka \\cite{KeMe23}) - see [140].\nSee also [139] and [142].\nThis is discussed in problem A5 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[BlSi20] Bloom, T.F. and Sisask, O., Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions. arXiv:2007.03528 (2020).\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Er83c] Erd\\H{o}s, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54.\n\n[Go01] Gowers, W. T., A new proof of Szemer\\'{e}di's theorem. Geom. Funct. Anal. (2001), 465-588.\n\n[GrTa08] Green, Ben and Tao, Terence, The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) (2008), 481-547.\n\n[GrTa17] Green, Ben and Tao, Terence, New bounds for Szemer\\'{e}di's theorem, III: a polylogarithmic bound for $r_4(N)$. Mathematika (2017), 944-1040.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[KeMe23] Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).\n\n[LSS24] Leng, J., Sah, A. and Sawhney, M., Improved bounds for Szemer\\'{e}di's theorem. arXiv:2402.17995 (2024).", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1880, "problem_number": "EP-5", "title": "Erdős Problem #5", "statement": "Let $C\\geq 0$. Is there an infinite sequence of $n_i$ such that $ \\lim_{i\\to \\infty}\\frac{p_{n_i+1}-p_{n_i}}{\\log n_i}=C? $ ", "background": "Let $S$ be the set of limit points of $(p_{n+1}-p_n)/\\log n$. This problem asks whether $S=[0,\\infty]$. Although this conjecture remains unproven, a lot is known about $S$. Some highlights:\n{UL}\n{LI}$\\infty\\in S$ by Westzynthius' result \\cite{We31} on large prime gaps,{/LI}\n{LI}$0\\in S$ by the work of Goldston, Pintz, and Yildirim \\cite{GPY09} on small prime gaps,{/LI}\n{LI}Erd\\H{o}s \\cite{Er55} and Ricci \\cite{Ri56} independently showed that $S$ has positive Lebesgue measure,{/LI}\n{LI} Hildebrand and Maier \\cite{HiMa88} showed that $S$ contains arbitrarily large (finite) numbers,{/LI}\n{LI} Pintz \\cite{Pi16} showed that there exists some small constant $c>0$ such that $[0,c]\\subset S$,{/LI}\n{LI} Banks, Freiberg, and Maynard \\cite{BFM16} showed that at least $12.5\\%$ of $[0,\\infty)$ belongs to $S$,{/LI}\n{LI} Merikoski \\cite{Me20} showed that at least $1/3$ of $[0,\\infty)$ belongs to $S$, and that $S$ has bounded gaps.{/LI}\n{/UL}\nIn \\cite{Er65b}, \\cite{Er85c}, and \\cite{Er97c} Erd\\H{o}s asks whether $S$ is everywhere dense (but Weisenberg notes that clearly $S$ is closed so this is equivalent to asking whether $S=[0,\\infty]$).\nSee also [234].\nReferences\n\n\n[BFM16] Banks, William D. and Freiberg, Tristan and Maynard, James, On limit points of the sequence of normalized prime gaps. Proc. Lond. Math. Soc. (3) (2016), 515-539.\n\n[Er55] Erd\"{o}s, Paul, Some remarks on number theory. Riveon Lematematika (1955), 45-48.\n\n[Er65b] Erd\\H{o}s, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[Er85c] Erd\\H{o}s, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.\n\n[Er97c] Erd\\H{o}s, Paul, Some of my favorite problems and results. The mathematics of Paul Erd\\H{o}s, I (1997), 47-67.\n\n[GPY09] Goldston, Daniel A. and Pintz, J\\'{a}nos and Y\\i ld\\i r\\i m, Cem Y., Primes in tuples. I. Ann. of Math. (2) (2009), 819-862.\n\n[HiMa88] Hildebrand, Adolf and Maier, Helmut, Gaps between prime numbers. Proc. Amer. Math. Soc. (1988), 1-9.\n\n[Me20] Merikoski, Jori, Limit points of normalized prime gaps. J. Lond. Math. Soc. (2) (2020), 99-124.\n\n[Pi16] Pintz, J\\'{a}nos, Polignac numbers, conjectures of Erd\\H{o}s on gaps between primes, arithmetic progressions in primes, and the bounded gap conjecture. From arithmetic to zeta-functions (2016), 367-384.\n\n[Ri56] Ricci, Giovanni, Recherches sur l'allure de la suite $\\{p_{n+1}-p_n/\\log p_n\\}$. Colloque sur la Th\\'{e}orie des Nombres, Bruxelles, 1955 (1956), 93-106.\n\n[We31] Westzynthius, E., \"{U}ber die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind. Commentat. Phys. Math. (1931), 1-37.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1881, "problem_number": "EP-9", "title": "Erdős Problem #9", "statement": "Let $A$ be the set of all odd integers not of the form $p+2^{k}+2^l$ (where $k,l\\geq 0$ and $p$ is prime). Is the upper density of $A$ positive?", "background": "In \\cite{Er77c} Erd\\H{o}s credits Schinzel with proving that there are infinitely many odd integers not of this form, but gives no reference. Crocker \\cite{Cr71} has proved there are $\\gg\\log\\log N$ such integers in $\\{1,\\ldots,N\\}$. Pan \\cite{Pa11} improved this to $\\gg_\\epsilon N^{1-\\epsilon}$ for any $\\epsilon>0$. Erd\\H{o}s believed this cannot be proved by covering systems, i.e. integers of the form $p+2^k+2^l$ exist in every infinite arithmetic progression.\nThe sequence of such numbers is A006286 in the OEIS.\nSee also [10], [11], and [16].\nThis is discussed in problem A19 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Cr71] Crocker, Roger, On the sum of a prime and of two powers of two. Pacific J. Math. (1971), 103-107.\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Pa11] Pan, Hao, On the integers not of the form {$p+2^a+2^b$}. Acta Arith. (2011), 55-61.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1882, "problem_number": "EP-10", "title": "Erdős Problem #10", "statement": "Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of 2?", "background": "Erd\\H{o}s described this as 'probably unattackable'. In \\cite{ErGr80} Erd\\H{o}s and Graham suggest that no such $k$ exists. Gallagher \\cite{Ga75} has shown that for any $\\epsilon>0$ there exists $k(\\epsilon)$ such that the set of integers which are the sum of a prime and at most $k(\\epsilon)$ many powers of 2 has lower density at least $1-\\epsilon$.\nGranville and Soundararajan \\cite{GrSo98} have conjectured that at most $3$ powers of 2 suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers. (The restriction to odd integers is important here - for example, Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ (see [9]) suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$).\nSee also [9], [11], and [16].\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[Ga75] Gallagher, P. X., Primes and powers of 2. Invent. Math. (1975), 125-142.\n\n[GrSo98] Granville, A. and Soundararajan, K., A Binary Additive Problem of Erd\\H{o}s and the Order of $2$ mod $p^2$. The Ramanujan Journal (1998), 283-298.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1883, "problem_number": "EP-12", "title": "Erdős Problem #12", "statement": "Let $A$ be an infinite set such that there are no distinct $a,b,c\\in A$ such that $a\\mid (b+c)$ and $b,c>a$. Is there such an $A$ with $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>0? $ Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with $ \\lvert A\\cap\\{1,\\ldots,N\\}\\rvert\\frac{N}{f(N)}. $ (Their example is given by all integers in $(y_i,\\frac{3}{2}y_i)$ congruent to $1$ modulo $(2y_{i-1})!$, where $y_i$ is some sufficiently quickly growing sequence.)\nAn example of an $A$ with this property where $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}\\log N>0 $ is given by the set of $p^2$, where $p\\equiv 3\\pmod{4}$ is prime.\nElsholtz and Planitzer \\cite{ElPl17} have constructed such an $A$ with $ \\lvert A\\cap\\{1,\\ldots,N\\}\\rvert\\gg \\frac{N^{1/2}}{(\\log N)^{1/2}(\\log\\log N)^2(\\log\\log\\log N)^2}. $ Schoen \\cite{Sc01} proved that if all elements in $A$ are pairwise coprime then $ \\lvert A\\cap\\{1,\\ldots,N\\}\\rvert \\ll N^{2/3} $ for infinitely many $N$. Baier \\cite{Ba04} has improved this to $\\ll N^{2/3}/\\log N$.\nFor the finite version see [13].\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[Ba04] Baier, Stephan, A note on {$\\scr P$}-sets. Integers (2004), A13, 6.\n\n[ElPl17] Elsholtz, Christian and Planitzer, Stefan, On Erd\\H{o}s and {S}\\'{a}rk\"ozy's sequences with Property P. Monatsh. Math. (2017), 565--575.\n\n[ErSa70] Erd\\H{o}s, P. and S\\'{a}rk\"ozi, A., On the divisibility properties of sequences of integers. Proc. London Math. Soc. (3) (1970), 97-101.\n\n[Sc01] Schoen, Tomasz, On a problem of Erd\\H{o}s and {S}\\'{a}rk\"ozy. J. Combin. Theory Ser. A (2001), 191--195.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1884, "problem_number": "EP-14", "title": "Erdős Problem #14", "statement": "Let $A\\subseteq \\mathbb{N}$. Let $B\\subseteq \\mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$.\nIs it true that for all $\\epsilon>0$ and large $N$ $ \\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\gg_\\epsilon N^{1/2-\\epsilon}? $ Is it possible that $ \\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert =o(N^{1/2})? $ ", "background": "Apparently originally considered by Erd\\H{o}s and Nathanson, although later Erd\\H{o}s attributes this to Erd\\H{o}s, S\\'{a}rk\"{o}zy, and Szemer\\'{e}di (but gives no reference), and claims a construction of an $A$ such that for all $\\epsilon>0$ and all large $N$ $ \\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\ll_\\epsilon N^{1/2+\\epsilon}, $ and yet there for all $\\epsilon>0$ there exist infinitely many $N$ where $ \\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\gg_\\epsilon N^{1/3-\\epsilon}. $ Erd\"{o}s and Freud investigated the finite analogue in \\cite{ErFr91}, proving that there exists $A\\subseteq \\{1,\\ldots,N\\}$ such that the number of integers not representable in exactly one way as the sum of two elements from $A$ is $<2^{3/2}N^{1/2}$, and suggest the constant $2^{3/2}$ is perhaps best possible.\nReferences\n\n\n[ErFr91] Erd\\H{o}s, P. and Freud, R., On sums of a {S}idon-sequence. J. Number Theory (1991), 196--205.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1885, "problem_number": "EP-15", "title": "Erdős Problem #15", "statement": "Is it true that $ \\sum_{n=1}^\\infty(-1)^n\\frac{n}{p_n} $ converges, where $p_n$ is the sequence of primes?", "background": "Erd\\H{o}s suggested that a computer could be used to explore this, and did not see any other method to attack this.\nTao \\cite{Ta23} has proved that this series does converge assuming a strong form of the Hardy-Littlewood prime tuples conjecture.\nIn \\cite{Er98} Erd\\H{o}s further conjectures that $ \\sum_{n=1}^\\infty (-1)^n \\frac{1}{n(p_{n+1}-p_n)} $ converges and $ \\sum_{n=1}^\\infty (-1)^n \\frac{1}{p_{n+1}-p_n} $ diverges. Weisenberg notes that the existence of infinitely many bounded gaps between primes (as proved by Zhang \\cite{Zh14}) implies the latter series does not converge. Weisenberg also has an argument which shows that, assuming the Hardy-Littlewood prime $k$-tuples conjecture, the series is unbounded in at least one direction (positive or negative).\nErd\\H{o}s further conjectured that $ \\sum_{n=1}^\\infty (-1)^n \\frac{1}{n(p_{n+1}-p_n)(\\log\\log n)^c} $ converges for every $c>0$, and reports that he and Nathanson can prove that this series converges absolutely for $c>2$ (and can show, conditional on 'hopeless' conjectures about the primes, that this sum does not converge absolutely for $c=2$).\nSawhney has provided the following proof that this series converges absolutely for $c>2$: note that, whenever $c>1$, the contribution to the sum from gaps $p_{n+1}-p_n\\geq \\log n$ is convergent, so it suffices to consider only small gaps. The number of $n\\leq X$ such that $p_{n+1}-p_n\\in [\\epsilon\\log n,2\\epsilon \\log n)$ is bounded above by $\\ll \\epsilon X$ (this can be proved via the Selberg sieve). In particular, applying this bound for $\\frac{1}{\\log n}\\leq \\epsilon \\leq 1$ of the shape $2^{-j}$ (of which there are at most $\\log\\log n$ possibilities) shows the desired convergence, since $ \\sum \\frac{1}{n(\\log n)(\\log\\log n)^{c-1}} $ converges.\nReferences\n\n\n[Er98] Erd\\H{o}s, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.\n\n[Ta23] Tao, T., The convergence of an alternating series of Erd\\H{o}s, assuming the Hardy-Littlewood prime tuples conjecture. arXiv:2308.07205 (2023).\n\n[Zh14] Zhang, Yitang, Bounded gaps between primes. Ann. of Math. (2) (2014), 1121--1174.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1886, "problem_number": "EP-17", "title": "Erdős Problem #17", "statement": "Are there infinitely many primes $p$ such that every even number $n\\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\\leq p$?", "background": "The first prime without this property is $97$. The sequence of such primes is A038133 in the OEIS. These are called cluster primes.\nBlecksmith, Erd\\H{o}s, and Selfridge \\cite{BES99} proved that the number of such primes is $ \\ll_A \\frac{x}{(\\log x)^A} $ for every $A>0$, and Elsholtz \\cite{El03} improved this to $ \\ll x\\exp(-c(\\log\\log x)^2) $ for every $c<1/8$.\nThis is discussed in problem C1 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[BES99] Blecksmith, Richard and Erd\\H{o}s, Paul and Selfridge, J. L., Cluster primes. Amer. Math. Monthly (1999), 43--48.\n\n[El03] Elsholtz, Christian, On cluster primes. Acta Arith. (2003), 281--284.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1887, "problem_number": "EP-18", "title": "Erdős Problem #18", "statement": "We call $m$ practical if every integer $n0$?", "background": "Erd\\H{o}s and Rado \\cite{ErRa60} originally proved $f(n,k)\\leq (k-1)^nn!$. Kostochka \\cite{Ko97} improved this slightly (in particular establishing an upper bound of $o(n!)$, for which Erd\\H{o}s awarded him the consolation prize of \\$100), but the bound stood at $n^{(1+o(1))n}$ for a long time until Alweiss, Lovett, Wu, and Zhang \\cite{ALWZ20} proved $ f(n,k) < (Ck\\log n\\log\\log n)^n $ for some constant $C>1$. This was refined slightly, independently by Rao \\cite{Ra20}, Frankston, Kahn, Narayanan, and Park \\cite{FKNP19}, and Bell, Chueluecha, and Warnke \\cite{BCW21}, leading to the current record of $ f(n,k) < (Ck\\log n)^n $ for some constant $C>1$.\nIn \\cite{Er81} offered \\$1000 for a proof or disproof even just in the special case when $k=3$, which he expected 'contains the whole difficulty'. He also wrote 'I really do not see why this question is so difficult'.\nThe usual focus is on the regime where $k=O(1)$ is fixed (say $k=3$) and $n$ is large, although for the opposite regime Kostochka, R\"{o}dl, and Talysheva \\cite{KRT99} have shown $ f(n,k)=(1+O_n(k^{-1/2^n}))k^n. $ \nReferences\n\n\n[ALWZ20] Alweiss, R. and Lovett, S. and Wu, K. and Zhang, J., Improved bounds for the sunflower lemma. (2020).\n\n[BCW21] Bell, T. and Chueluecha, S. and Warnke, L., Note on sunflowers. Discret. Math. (2021).\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[ErRa60] Erd\\H{o}s, P. and Rado, R., Intersection theorems for systems of sets. J. London Math. Soc. (1960), 85-90.\n\n[FKNP19] Frankston, K. and Kahn, J. and Narayanan, B. and Park, J., Thresholds versus fractional expectation-thresholds. CoRR (2019).\n\n[KRT99] Kostochka, A. V. and R\"{o}dl, V. and Talysheva, L. A., On systems of small sets with no large $\\Delta$-subsystems. Combin. Probab. Comput. (1999), 265-268.\n\n[Ko97] Kostochka, A., A bound on the cardinality of families not containing $\\Delta$-systems. (1997).\n\n[Ra20] Rao, A., Coding for sunflowers. Discrete Analysis (2020).", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1889, "problem_number": "EP-25", "title": "Erdős Problem #25", "statement": "Let $n_10$.\nAnother stronger conjecture would be that the hypothesis $\\lvert A\\cap [1,N]\\rvert \\gg N^{1/2}$ for all large $N$ suffices.\nErd\\H{o}s and S\\'{a}rk\"{o}zy conjectured the stronger version that if $A=\\{a_10$, $ h(N) = N^{1/2}+O_\\epsilon(N^\\epsilon)? $ ", "background": "A problem of Erd\\H{o}s and Tur\\'{a}n. It may even be true that $h(N)=N^{1/2}+O(1)$, but Erd\\H{o}s remarks this is perhaps too optimistic. Erd\\H{o}s and Tur\\'{a}n \\cite{ErTu41} proved an upper bound of $N^{1/2}+O(N^{1/4})$, with an alternative proof by Lindstr\"{o}m \\cite{Li69}. Both proofs in fact give $ h(N) \\leq N^{1/2}+N^{1/4}+1. $ Balogh, F\"{u}redi, and Roy \\cite{BFR21} improved the bound in the error term to $0.998N^{1/4}$. This was further optimised by O'Bryant \\cite{OB22}. The current record is $ h(N)\\leq N^{1/2}+0.98183N^{1/4}+O(1), $ due to Carter, Hunter, and O'Bryant \\cite{CHO25}.\nSinger \\cite{Si38} was the first to show that $h(N)\\geq (1-o(1))N^{1/2}$ for all $N$. For a detailed survey of the literature we refer to \\cite{OB04}.\nSee also [241] and [840].\nThis problem is Problem 31 on Green's open problems list.\nThis is discussed in problem C9 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[BFR21] Balogh, J. and F\"{u}redi, Z. and Roy, S., An upper bound on the size of Sidon sets. arXiv:2103.15850 (2021).\n\n[CHO25] Carter, D. and Hunter, Z. and O'Bryant, K., On the diameter of finite {S}idon sets. Acta Math. Hungar. (2025), 108--126.\n\n[ErTu41] Erd\\H{o}s, P. and Tur\\'{a}n, P., On a problem of Sidon in additive number theory, and on some related problems. J. London Math. Soc. (1941), 212-215.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Li69] Lindstr\"{o}m, B., An inequality for $B_2$-sequences. J. Combinatorial Theory (1969), 211-212.\n\n[OB04] O'Bryant, Kevin, A complete annotated bibliography of work related to {S}idon\nsequences. Electron. J. Combin. (2004), 39.\n\n[OB22] O'Bryant, K., On the size of finite Sidon sets. arXiv:2207.07800 (2022).\n\n[Si38] Singer, James, A theorem in finite projective geometry and some applications\nto number theory. Trans. Amer. Math. Soc. (1938), 377--385.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1892, "problem_number": "EP-32", "title": "Erdős Problem #32", "statement": "Is there a set $A\\subset\\mathbb{N}$ such that $ \\lvert A\\cap\\{1,\\ldots,N\\}\\rvert = o((\\log N)^2) $ and such that every large integer can be written as $p+a$ for some prime $p$ and $a\\in A$?\nCan the bound $O(\\log N)$ be achieved? Must such an $A$ satisfy $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{\\log N}> 1? $ ", "background": "Such a set is called an additive complement to the primes.\nErd\\H{o}s \\cite{Er54} proved that such a set $A$ exists with $\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert\\ll (\\log N)^2$ (improving a previous result of Lorentz \\cite{Lo54} who achieved $\\ll (\\log N)^3$).\nWolke \\cite{Wo96} has shown that such a bound is almost true, in that we can achieve $\\ll (\\log N)^{1+o(1)}$ if we only ask for almost all integers to be representable. Kolountzakis \\cite{Ko96} improved this to $\\ll (\\log N)(\\log\\log N)$, and Ruzsa \\cite{Ru98c} further improved this to $\\ll \\omega(N)\\log N$ for any $\\omega\\to \\infty$.\nThe answer to the third question is yes: Ruzsa \\cite{Ru98c} has shown that we must have $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{\\log N}\\geq e^\\gamma\\approx 1.781. $ This is discussed in problem E1 of Guy's collection \\cite{Gu04}, where it is stated that Erd\\H{o}s offered \\$50 for determining whether $O(\\log N)$ can be achieved.\nReferences\n\n\n[Er54] Erd\\H{o}s, Paul, Some results on additive number theory. Proc. Amer. Math. Soc. (1954), 847-853.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ko96] Kolountzakis, Mihail N., On the additive complements of the primes and sets of similar\ngrowth. Acta Arith. (1996), 1--8.\n\n[Lo54] Lorentz, G. G., On a problem of additive number theory. Proc. Amer. Math. Soc. (1954), 838-841.\n\n[Ru98c] Ruzsa, Imre Z., On the additive completion of primes. Acta Arith. (1998), 269-275.\n\n[Wo96] Wolke, Dieter, On a problem of Erd\\H{o}s in additive number theory. J. Number Theory (1996), 209-213.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1893, "problem_number": "EP-33", "title": "Erdős Problem #33", "statement": "Let $A\\subset\\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\\in A$ and $n\\geq 0$. What is the smallest possible value of $ \\limsup \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}? $ Is $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>1? $ ", "background": "Such a set $A$ is called an additive complement of the set of squares. Erd\\H{o}s observed that there exist $A$ for which the $\\limsup$ is finite and $>1$. Moser \\cite{Mo65} proved that, for any such $A$, $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>1.06. $ The best-known lower bound is $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}\\geq\\frac{4}{\\pi}\\approx 1.273 $ proved by Cilleruelo \\cite{Ci93}, Habsieger \\cite{Ha95}, and Balasubramanian and Ramana \\cite{BaRa01}.\nThe problem of minimising the $\\limsup$ appears to have been much less studied. van Doorn has a construction of such an $A$ in which, for all $N$, $ \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}< 2\\phi^{5/2}\\approx 6.66, $ where $\\phi=\\frac{1+\\sqrt{5}}{2}$ is the golden ratio.\nReferences\n\n\n[BaRa01] Balasubramanian, R. and Ramana, D. S., Additive complements of the squares. C. R. Math. Acad. Sci. Soc. R. Can. (2001), 6--11.\n\n[Ci93] Cilleruelo, Javier, The additive completion of {$k$}th-powers. J. Number Theory (1993), 237--243.\n\n[Ha95] Habsieger, Laurent, On the additive completion of polynomial sets. J. Number Theory (1995), 130--135.\n\n[Mo65] Moser, Leo, On the additive completion of sets of integers. (1965), 175--180.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1894, "problem_number": "EP-36", "title": "Erdős Problem #36", "statement": "Find the optimal constant $c>0$ such that the following holds.\nFor all sufficiently large $N$, if $A\\sqcup B=\\{1,\\ldots,2N\\}$ is a partition into two equal parts, so that $\\lvert A\\rvert=\\lvert B\\rvert=N$, then there is some $x$ such that the number of solutions to $a-b=x$ with $a\\in A$ and $b\\in B$ is at least $cN$.", "background": "The minimum overlap problem. The example (with $N$ even) $A=\\{N/2+1,\\ldots,3N/2\\}$ shows that $c\\leq 1/2$ (indeed, Erd\\H{o}s initially conjectured that $c=1/2$). The lower bound of $c\\geq 1/4$ is trivial, and Scherk improved this to $1-1/\\sqrt{2}=0.29\\cdots$. The current records are $ 0.379005 < c < 0.380924, $ the lower bound due to White \\cite{Wh22} and the upper bound due to AlphaEvolve \\cite{GGTW25}, improving slightly on an upper bound due to Haugland \\cite{Ha16}.\nThis is discussed in problem C17 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[GGTW25] B. Georgiev, J. G\\'{o}mez-Serrano, T. Tao, and A. Wagner, Mathematical exploration and discovery at scale. arXiv:2511.02864 (2025).\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ha16] Haugland, J. K., The minimum overlap problem revisited. arXiv:1609.08000 (2016).\n\n[Wh22] White, E. P., Erd\\H{o}s' minimum overlap problem. arXiv:2201.05704 (2022).", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1895, "problem_number": "EP-38", "title": "Erdős Problem #38", "statement": "Does there exist $B\\subset\\mathbb{N}$ which is not an additive basis, but is such that for every set $A\\subseteq\\mathbb{N}$ of Schnirelmann density $\\alpha$ and every $N$ there exists $b\\in B$ such that $ \\lvert (A\\cup (A+b))\\cap \\{1,\\ldots,N\\}\\rvert\\geq (\\alpha+f(\\alpha))N $ where $f(\\alpha)>0$ for $0<\\alpha <1 $?\nThe Schnirelmann density is defined by $ d_s(A) = \\inf_{N\\geq 1}\\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N}. $ ", "background": "Erd\\H{o}s \\cite{Er36c} proved that if $B$ is an additive basis of order $k$ then, for any set $A$ of Schnirelmann density $\\alpha$, for every $N$ there exists some integer $b\\in B$ such that $ \\lvert (A\\cup (A+b))\\cap \\{1,\\ldots,N\\}\\rvert\\geq \\left(\\alpha+\\frac{\\alpha(1-\\alpha)}{2k}\\right)N. $ It seems an interesting question (not one that Erd\\H{o}s appears to have asked directly, although see [35]) to improve the lower bound here, even in the case $B=\\mathbb{N}$. Erd\\H{o}s observed that a random set of density $\\alpha$ shows that the factor of $\\frac{\\alpha(1-\\alpha)}{2}$ in this case cannot be improved past $\\alpha(1-\\alpha)$.\nThis is a stronger property than $B$ being an essential component (see [37]). Linnik \\cite{Li42} gave the first construction of an essential component which is not an additive basis.\nReferences\n\n\n[Er36c] Erd\\H{o}s, P., On the arithmetical density of the sum of two sequences, one of which forms a basis for the integers. Acta. Arith. (1936), 201-207.\n\n[Li42] Linnik, U. V., On Erd\"{o}s's theorem on the addition of numerical sequences. Rec. Math. [Mat. Sbornik] N.S. (1942), 67-78.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1896, "problem_number": "EP-39", "title": "Erdős Problem #39", "statement": "Is there an infinite Sidon set $A\\subset \\mathbb{N}$ such that $ \\lvert A\\cap \\{1\\ldots,N\\}\\rvert \\gg_\\epsilon N^{1/2-\\epsilon} $ for all $\\epsilon>0$?", "background": "The trivial greedy construction achieves $\\gg N^{1/3}$. The first improvement on this was achieved by Ajtai, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AKS81b}, who found an infinite Sidon set with growth rate $\\gg (N\\log N)^{1/3}$. The current best bound of $\\gg N^{\\sqrt{2}-1+o(1)}$ is due to Ruzsa \\cite{Ru98}.\nErd\\H{o}s \\cite{Er73} had offered \\$25 for any construction which achieves $N^{c}$ for some $c>1/3$. Later he \\cite{Er77c} offered \\$100 for a construction which achieves $\\omega(N)N^{1/3}$ for some $\\omega(N)\\to \\infty$.\nErd\\H{o}s proved that for every infinite Sidon set $A$ we have $ \\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/2}}=0. $ Erd\\H{o}s and R\\'{e}nyi have constructed, for any $\\epsilon>0$, a set $A$ such that $ \\lvert A\\cap \\{1\\ldots,N\\}\\rvert \\gg_\\epsilon N^{1/2-\\epsilon} $ for all large $N$ and $1_A\\ast 1_A(n)\\ll_\\epsilon 1$ for all $n$.\nThis is discussed in problem C9 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[AKS81b] Ajtai, Mikl\\'os and Koml\\'os, J\\'anos and Szemer\\'{e}di, Endre, A dense infinite {S}idon sequence. European J. Combin. (1981), 1--11.\n\n[Er73] Erd\\H{o}s, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ru98] Ruzsa, Imre Z., An infinite Sidon sequence. J. Number Theory (1998), 63-71.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1897, "problem_number": "EP-40", "title": "Erdős Problem #40", "statement": "For what functions $g(N)\\to \\infty$ is it true that $ \\lvert A\\cap \\{1,\\ldots,N\\}\\rvert \\gg \\frac{N^{1/2}}{g(N)} $ implies $\\limsup 1_A\\ast 1_A(n)=\\infty$?", "background": "This is a stronger form of the Erd\\H{o}s-Tur\\'{a}n conjecture [28] (since establishing this for any function $g(N)\\to \\infty$ would imply a positive solution to [28]).", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1898, "problem_number": "EP-41", "title": "Erdős Problem #41", "statement": "Let $A\\subset\\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\\in A$ (aside from the trivial coincidences). Is it true that $ \\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/3}}=0? $ ", "background": "Erd\\H{o}s proved that if the pairwise sums $a+b$ are all distinct aside from the trivial coincidences then $ \\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/2}}=0. $ This is discussed in problem C11 of Guy's collection \\cite{Gu04}, in which Guy says Erd\\H{o}s offered \\$500 for the general problem of whether, for all $h\\geq 2$, $ \\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/h}}=0 $ whenever the sum of $h$ terms in $A$ are distinct. This was proved for $h=4$ by Nash \\cite{Na89} and for all even $h$ by Chen \\cite{Ch96b}.\nReferences\n\n\n[Ch96b] Chen, Sheng, A note on {$B_{2k}$} sequences. J. Number Theory (1996), 1--3.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Na89] Nash, John C. M., On {$B_4$}-sequences. Canad. Math. Bull. (1989), 446--449.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1899, "problem_number": "EP-42", "title": "Erdős Problem #42", "statement": "Let $M\\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\\subset \\{1,\\ldots,N\\}$ there is another Sidon set $B\\subset \\{1,\\ldots,N\\}$ of size $M$ such that $(A-A)\\cap(B-B)=\\{0\\}$?", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1900, "problem_number": "EP-43", "title": "Erdős Problem #43", "statement": "If $A,B\\subset \\{1,\\ldots,N\\}$ are two Sidon sets such that $(A-A)\\cap(B-B)=\\{0\\}$ then is it true that $ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq\\binom{f(N)}{2}+O(1), $ where $f(N)$ is the maximum possible size of a Sidon set in $\\{1,\\ldots,N\\}$? If $\\lvert A\\rvert=\\lvert B\\rvert$ then can this bound be improved to $ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq (1-c+o(1))\\binom{f(N)}{2} $ for some constant $c>0$?", "background": "Since it is known that $f(N)\\sim \\sqrt{N}$ (see [30]) the latter question is equivalent to asking whether, if $\\lvert A\\rvert=\\lvert B\\rvert$, $ \\lvert A\\rvert \\leq \\left(\\frac{1}{\\sqrt{2}}-c+o(1)\\right)\\sqrt{N} $ for some constant $c>0$. In the comments Tao has given a proof of this upper bound without the $-c$.\nIn the comments Barreto has given a negative answer to the second question: for infinitely many $N$ there exist Sidon sets $A,B\\subset \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert=\\lvert B\\rvert$ and $(A-A)\\cap (B-B)=\\{0\\}$ and $ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\geq (1-o(1))\\binom{f(N)}{2}. $ ", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1901, "problem_number": "EP-44", "title": "Erdős Problem #44", "statement": "Let $N\\geq 1$ and $A\\subset \\{1,\\ldots,N\\}$ be a Sidon set. Is it true that, for any $\\epsilon>0$, there exist $M$ and $B\\subset \\{N+1,\\ldots,M\\}$ (which may depend on $N,A,\\epsilon$) such that $A\\cup B\\subset \\{1,\\ldots,M\\}$ is a Sidon set of size at least $(1-\\epsilon)M^{1/2}$?", "background": "See also [329] and [707] (indeed a positive solution to [707] implies a positive solution to this problem, which in turn implies a positive solution to [329]).\nThis is discussed in problem C9 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1902, "problem_number": "EP-50", "title": "Erdős Problem #50", "statement": "Schoenberg proved that for every $c\\in [0,1]$ the density of $ \\{ n\\in \\mathbb{N} : \\phi(n)0$ $ \\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg_\\epsilon \\lvert A\\rvert^{2-\\epsilon}? $ ", "background": "The sum-product problem. Erd\\H{o}s and Szemer\\'{e}di \\cite{ErSz83} proved a lower bound of $\\lvert A\\rvert^{1+c}$ for some constant $c>0$, and an upper bound of $ \\lvert A\\rvert^2 \\exp\\left(-c\\frac{\\log\\lvert A\\rvert}{\\log\\log \\lvert A\\rvert}\\right) $ for some constant $c>0$. The lower bound has been improved a number of times. The current record is $ \\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg\\lvert A\\rvert^{\\frac{1270}{951}-o(1)} $ due to Bloom \\cite{Bl25} (note $1270/951=1.33543\\cdots$). A complete history of sum-product bounds can be found at this webpage.\nThere is likely nothing special about the integers in this question, and indeed Erd\\H{o}s and Szemer\\'{e}di also ask a similar question about finite sets of real or complex numbers. The current best bound for sets of reals is the same bound of Bloom above. The best bound for complex numbers is $ \\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg\\lvert A\\rvert^{\\frac{4}{3}+c} $ for some absolute constant $c>0$, due to Basit and Lund \\cite{BaLu19}.\nOne can in general ask this question in any setting where addition and multiplication are defined (once one avoids any trivial obstructions such as zero divisors or finite subfields). For example, it makes sense for subsets of finite fields. The current record is that there exists $c>0$ such that if $A\\subseteq \\mathbb{F}_p$ with $\\lvert A\\rvert \\mathrm{ex}(n;C_4)$ edges contain $\\gg n^{1/2}$ many copies of $C_4$?", "background": "Conjectured by Erd\\H{o}s and Simonovits, who could not even prove that at least $2$ copies of $C_4$ are guaranteed.\nThe behaviour of $\\mathrm{ex}(n;C_4)$ is the subject of [765].\nHe, Ma, and Yang \\cite{HeMaYa21} have proved this conjecture when $n=q^2+q+1$ for some even integer $q$.\nReferences\n\n\n[HeMaYa21] He, J. and Ma, J. and Yang, T., Some extremal results on 4-cycles. Journal of Combinatorial Theory B (2021).", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1906, "problem_number": "EP-61", "title": "Erdős Problem #61", "statement": "For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph contains either a complete graph or independent set on $\\geq n^c$ vertices?", "background": "Conjectured by Erd\\H{o}s and Hajnal \\cite{ErHa89}, who proved that a complete graph or independent set must exist on $ \\geq \\exp(c_H\\sqrt{\\log n}) $ many vertices, where $c_H>0$ is some constant. This was improved by Buci\\'{c}, Nguyen, Scott, and Seymour \\cite{BNSS23} to $ \\geq \\exp(c_H\\sqrt{\\log n\\log\\log n}). $ See also the entry in the graphs problem collection.\nReferences\n\n\n[BNSS23] Buci\\'C, M. and Nguyen, T. and Scott, A. and Seymour, P., A loglog step towards Erdos-Hajnal. arXiv:2301.10147 (2023).\n\n[ErHa89] Erd\\H{o}s, P. and Hajnal, A., Ramsey-type theorems. Discrete Appl. Math. (1989), 37-52.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1907, "problem_number": "EP-62", "title": "Erdős Problem #62", "statement": "If $G_1,G_2$ are two graphs with chromatic number $\\aleph_1$ then must there exist a graph $G$ whose chromatic number is $4$ (or even $\\aleph_0$) which is a subgraph of both $G_1$ and $G_2$?", "background": "Erd\\H{o}s also asked \\cite{Er87} about finding a common subgraph $H$ (with chromatic number either $4$ or $\\aleph_0$) in any finite collection of graphs with chromatic number $\\aleph_1$.\nEvery graph with chromatic number $\\aleph_1$ contains all sufficiently large odd cycles (which have chromatic number $3$), see [594]. This was proved by Erd\\H{o}s, Hajnal, and Shelah \\cite{EHS74}. Erd\\H{o}s wrote \\cite{Er87} that 'probably' every graph with chromatic number $\\aleph_1$ contains as subgraphs all graphs with chromatic number $4$ with sufficiently large girth.\nReferences\n\n\n[EHS74] Erd\\H{o}s, P. and Hajnal, A. and Shelah, S., On some general properties of chromatic numbers. Topics in topology (Proc. Colloq., Keszthely, 1972) (1974), 243-255.\n\n[Er87] Erd\\H{o}s, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1908, "problem_number": "EP-65", "title": "Erdős Problem #65", "statement": "Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_10$.\nIt is open even for $f(n)=\\sqrt{n}$. Erd\\H{o}s offered \\$500 for a proof but only \\$250 for a counterexample. This fails (even with $f(n)\\gg n$) if the graph has chromatic number $\\aleph_1$ (see [111]).\nReferences\n\n\n[EHS82] Erd\\H{o}s, P. and Hajnal, A. and Szemer\\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123.\n\n[Ro82] R\"{o}dl, Vojt\\vEch, Nearly bipartite graphs with large chromatic number. Combinatorica (1982), 377-383.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1913, "problem_number": "EP-75", "title": "Erdős Problem #75", "statement": "Is there a graph of chromatic number $\\aleph_1$ such that for all $\\epsilon>0$ if $n$ is sufficiently large and $H$ is a subgraph on $n$ vertices then $H$ contains an independent set of size $>n^{1-\\epsilon}$?", "background": "Conjectured by Erd\\H{o}s, Hajnal, and Szemer\\'{e}di \\cite{EHS82}. In \\cite{Er95d} Erd\\H{o}s suggests this may even be true with an independent set of size $\\gg n$.\nSee also [750].\nReferences\n\n\n[EHS82] Erd\\H{o}s, P. and Hajnal, A. and Szemer\\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123.\n\n[Er95d] Erd\\H{o}s, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) (1995), 61-65.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1914, "problem_number": "EP-77", "title": "Erdős Problem #77", "statement": "If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, then find the value of $ \\lim_{k\\to \\infty}R(k)^{1/k}. $ ", "background": "Erd\\H{o}s offered \\$100 for just a proof of the existence of this constant, without determining its value. He also offered \\$1000 for a proof that the limit does not exist, but says 'this is really a joke as [it] certainly exists'. (In \\cite{Er88} he raises this prize to \\$10000). Erd\\H{o}s proved $ \\sqrt{2}\\leq \\liminf_{k\\to \\infty}R(k)^{1/k}\\leq \\limsup_{k\\to \\infty}R(k)^{1/k}\\leq 4. $ The upper bound has been improved to $4-\\tfrac{1}{128}$ by Campos, Griffiths, Morris, and Sahasrabudhe \\cite{CGMS23}. This was improved to $3.7992\\cdots$ by Gupta, Ndiaye, Norin, and Wei \\cite{GNNW24}.\nA shorter and simpler proof of an upper bound of the strength $4-c$ for some constant $c>0$ (and a generalisation to the case of more than two colours) was given by Balister, Bollob\\'{a}s, Campos, Griffiths, Hurley, Morris, Sahasrabudhe, and Tiba \\cite{BBCGHMST24}.\nIn \\cite{Er93} Erd\\H{o}s writes 'I have no idea what the value of $\\lim R(k)^{1/k}$ should be, perhaps it is $2$ but we have no real evidence for this.'\nThis problem is #3 in Ramsey Theory in the graphs problem collection.\nSee also [1029] for a problem concerning a lower bound for $R(k)$ and discussion of lower bounds in general.\nA famous quote of Erd\\H{o}s concerns the difficulty of finding exact values for $R(k)$. This is often repeated in the words of Spencer, who phrased it as an alien attacking race. The earliest such quote in a paper of Erd\\H{o}s I have found is in \\cite{Er93}, where he writes:\n'Sometime ago, I made the following joke. If an evil spirit would appear and say \"unless you give me the value of $R(5)$ within a year, I will exterminate humanity\", then our best bet would be perhaps to get all our computers working on $R(5)$ and we probably would get its value in a year.\nIf he would ask for $R(6)$, the best strategy probably would be to destroy it before it can destroy us. If we would be so clever that we could give the answer by mathematics, we would just tell him: \"if you try to do something you will see what will happent to you...\". I think we are strong enugh now and the only evil spirit we have to feel is the one which is in ourselves (quoting somebody: I have seen the enemy and them are us). Now enough of the idle talk and back to Mathematics.'\nReferences\n\n\n[BBCGHMST24] Balister, P. and Bollob\\'{a}s, B. and Campos, M. and Griffiths, S. and Hurley, E.\nand Morris, R. and Sahasrabudhe, J. and Tiba, M., Upper bounds for multicolour Ramsey numbers. arXiv:2410.17197 (2024).\n\n[CGMS23] Campos, Marcelo and Griffiths, Simon and Morris, Robert and Sahasrabudhe, Julian, An exponential improvement for diagonal Ramsey. arXiv:2303.09521 (2023).\n\n[Er88] Erd\\H{o}s, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[GNNW24] Gupta, P. and Ndiaye, N. and Norin, S. and Wei, L., Optimizing the CGMS upper bound on Ramsey numbers. arXiv:2407.19026 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1915, "problem_number": "EP-78", "title": "Erdős Problem #78", "statement": "Give a constructive proof that $R(k)>C^k$ for some constant $C>1$.", "background": "Erd\\H{o}s gave a simple probabilistic proof that $R(k) \\gg k2^{k/2}$.\nEquivalently, this question asks for an explicit construction of a graph on $n$ vertices which does not contain any clique or independent set of size $\\geq c\\log n$ for some constant $c>0$.\nIn \\cite{Er69b} Erd\\H{o}s asks for even a construction whose largest clique or independent set has size $o(n^{1/2})$, which is now known.\nCohen \\cite{Co15} (see the introduction for further history) constructed a graph on $n$ vertices which does not contain any clique or independent set of size $ \\geq 2^{(\\log\\log n)^{C}} $ for some constant $C>0$. Li \\cite{Li23b} has recently improved this to $ \\geq (\\log n)^{C} $ for some constant $C>0$.\nThis problem is #4 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[Co15] Gil Cohen, Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs. Electronic Colloquium on Computational Complexity (2015).\n\n[Er69b] Erd\\H{o}s, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\n\n[Li23b] Li, X., Two Source Extractors for Asymptotically Optimal Entropy, and (Many) More. arXiv:2303.06802 (2023).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1916, "problem_number": "EP-80", "title": "Erdős Problem #80", "statement": "Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and at least $cn^2$ edges, where each edge is contained in at least one triangle, must contain a book of size $m$, that is, an edge shared by at least $m$ different triangles.\nEstimate $f_c(n)$. In particular, is it true that $f_c(n)>n^{\\epsilon}$ for some $\\epsilon>0$? Or $f_c(n)\\gg \\log n$?", "background": "A problem of Erd\\H{o}s and Rothschild. Alon and Trotter showed that, provided $c<1/4$, $f_c(n)\\ll_c n^{1/2}$. Szemer\\'{e}di observed that his regularity lemma implies that $f_c(n)\\to \\infty$.\nEdwards (unpublished) and Khadziivanov and Nikiforov \\cite{KhNi79} proved independently that $f_c(n) \\geq n/6$ when $c>1/4$ (see [905]).\nFox and Loh \\cite{FoLo12} proved that $ f_c(n) \\leq n^{O(1/\\log\\log n)} $ for all $c<1/4$, disproving the first conjecture of Erd\\H{o}s.\nThe best known lower bounds for $f_c(n)$ are those from Szemer\\'{e}di's regularity lemma, and as such remain very poor.\nSee also [600] and the entry in the graphs problem collection.\nReferences\n\n\n[FoLo12] Fox, Jacob and Loh, Po-Shen, On a problem of Erd\\H{o}s and {R}othschild on edges in\ntriangles. Combinatorica (2012), 619--628.\n\n[KhNi79] Had\\v ziivanov, N. G. and Nikiforov, S. V., Solution of a problem of {P}. Erd\\H{o}s about the maximum\nnumber of triangles with a common edge in a graph. C. R. Acad. Bulgare Sci. (1979), 1315--1318.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1917, "problem_number": "EP-81", "title": "Erdős Problem #81", "statement": "Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be partitioned into $n^2/6+O(n)$ many cliques?", "background": "Asked by Erd\\H{o}s, Ordman, and Zalcstein \\cite{EOZ93}, who proved an upper bound of $(1/4-\\epsilon)n^2$ many cliques (for some very small $\\epsilon>0$). The example of all edges between a complete graph on $n/3$ vertices and an empty graph on $2n/3$ vertices show that $n^2/6+O(n)$ is sometimes necessary.\nA split graph is one where the vertices can be split into a clique and an independent set. Every split graph is chordal. Chen, Erd\\H{o}s, and Ordman \\cite{CEO94} have shown that any split graph can be partitioned into $\\frac{3}{16}n^2+O(n)$ many cliques.\nSee also [1017].\nReferences\n\n\n[CEO94] Chen, Guan-Tao and Erd\\H{o}s, Paul and Ordman, Edward T., Clique partitions of split graphs. Combinatorics, graph theory, algorithms and applications\n(Beijing, 1993) (1994), 21-30.\n\n[EOZ93] Erd\\H{o}s, Paul and Ordman, Edward T. and Zalcstein, Yechezkel, Clique partitions of chordal graphs. Combin. Probab. Comput. (1993), 409-415.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1918, "problem_number": "EP-82", "title": "Erdős Problem #82", "statement": "Let $F(n)$ be maximal such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices. Prove that $F(n)/\\log n\\to \\infty$.", "background": "Conjectured by Erd\\H{o}s, Fajtlowicz, and Stanton. It is known that $F(5)=3$ and $F(7)=4$.\nRamsey's theorem implies that $F(n)\\gg \\log n$. Bollob\\'{a}s observed that $F(n)\\ll n^{1/2+o(1)}$. Alon, Krivelevich, and Sudakov \\cite{AKS07} have improved this to $n^{1/2}(\\log n)^{O(1)}$.\nIn \\cite{Er93} Erd\\H{o}s asks whether, if $t(n)$ is the largest trivial (either empty or complete) subgraph which a graph on $n$ vertices must contain (so that $t(n) \\gg \\log n$ by Ramsey's theorem), then is it true that $ F(n)-t(n)\\to \\infty? $ Equivalently, and in analogue with the definition of Ramsey numbers, one can define $G(n)$ to be the minimal $m$ such that every graph on $m$ vertices contains a regular induced subgraph on at least $n$ vertices. This problem can be rephrased as asking whether $G(n) \\leq 2^{o(n)}$.\nFajtlowicz, McColgan, Reid, and Staton \\cite{FMRS95} showed that $G(1)=1$, $G(2)=2$, $G(3)=5$, $G(4)=7$, and $G(5)\\geq 12$. Boris Alexeev and Brendan McKay (see the comments and this site) have computed $G(5)=17$, $G(6)\\geq 21$, and $G(7)\\geq 29$.\nSee also [1031] for another question regarding induced regular subgraphs.\nReferences\n\n\n[AKS07] Alon, N. and Krivelevich, M. and Sudakov, B., Large nearly regular induced subgraphs. arXiv:0710.2106 (2007).\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[FMRS95] No reference found.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1919, "problem_number": "EP-84", "title": "Erdős Problem #84", "statement": "The cycle set of a graph $G$ on $n$ vertices is a set $A\\subseteq \\{3,\\ldots,n\\}$ such that there is a cycle in $G$ of length $\\ell$ if and only if $\\ell \\in A$. Let $f(n)$ count the number of possible such $A$.\nProve that $f(n)=o(2^n)$.\nProve that $f(n)/2^{n/2}\\to \\infty$.", "background": "Conjectured by Erd\\H{o}s and Faudree, who showed that $2^{n/2}0$, and wrote it is 'perhaps not hopeless' to determine $f(n)$ exactly. Brass, Harborth, and Nienborg \\cite{BHN95} improved this to $ f(n) \\geq \\left(\\frac{1}{2}+\\frac{c}{\\sqrt{n}}\\right)n2^{n-1} $ for some constant $c>0$.\nBalogh, Hu, Lidicky, and Liu \\cite{BHLL14} proved that $f(n)\\leq 0.6068 n2^{n-1}$. This was improved to $\\leq 0.60318 n2^{n-1}$ by Baber \\cite{Ba12b}.\nA similar question can be asked for other even cycles.\nSee also [666] and the entry in the graphs problem collection.\nReferences\n\n\n[BHLL14] Balogh, J\\'{o}zsef and Hu, Ping and Lidick\\'{y}, Bernard and Liu, Hong, Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube. European J. Combin. (2014), 75-85.\n\n[BHN95] Brass, Peter and Harborth, Heiko and Nienborg, Hauke, On the maximum number of edges in a {$C_4$}-free subgraph of\n{$Q_n$}. J. Graph Theory (1995), 17--23.\n\n[Ba12b] R. Baber, Tur\\'{a}n densities of hypercubes. arXiv:1201.3587 (2012).\n\n[Er91] Erd\"{o}s, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1921, "problem_number": "EP-87", "title": "Erdős Problem #87", "statement": "Let $\\epsilon >0$. Is it true that, if $k$ is sufficiently large, then $ R(G)>(1-\\epsilon)^kR(k) $ for every graph $G$ with chromatic number $\\chi(G)=k$?\nEven stronger, is there some $c>0$ such that, for all large $k$, $R(G)>cR(k)$ for every graph $G$ with chromatic number $\\chi(G)=k$?", "background": "Erd\\H{o}s originally conjectured that $R(G)\\geq R(k)$, which is trivial for $k=3$, but fails already for $k=4$, as Faudree and McKay \\cite{FaMc93} showed that $R(W)=17$ for the pentagonal wheel $W$.\nSince $R(k)\\leq 4^k$ this is trivial for $\\epsilon\\geq 3/4$. Yuval Wigderson points out that $R(G)\\gg 2^{k/2}$ for any $G$ with chromatic number $k$ (via a random colouring), which asymptotically matches the best-known lower bounds for $R(k)$.\nThis problem is #12 and #13 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[FaMc93] Faudree, R. J. and McKay, B., A conjecture of Erd\\H{o}s and the Ramsey number $r(W_6)$. J. Combinatorial Math. and Combinatorial Computing (1993), 23-31.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1922, "problem_number": "EP-89", "title": "Erdős Problem #89", "statement": "Does every set of $n$ distinct points in $\\mathbb{R}^2$ determine $\\gg n/\\sqrt{\\log n}$ many distinct distances?", "background": "A $\\sqrt{n}\\times\\sqrt{n}$ integer grid shows that this would be the best possible. Nearly solved by Guth and Katz \\cite{GuKa15} who proved that there are always $\\gg n/\\log n$ many distinct distances.\nA stronger form (see [604]) may be true: is there a single point which determines $\\gg n/\\sqrt{\\log n}$ distinct distances, or even $\\gg n$ many such points, or even that this is true averaged over all points - for example, if $d(x)$ counts the number of distinct distances from $x$ then in \\cite{Er75f} Erd\\H{o}s conjectured $ \\sum_{x\\in A}d(x) \\gg \\frac{n^2}{\\sqrt{\\log n}}, $ where $A\\subset \\mathbb{R}^2$ is any set of $n$ points.\nSee also [661], and [1083] for the generalisation to higher dimensions.\nReferences\n\n\n[Er75f] Erd\\H{o}s, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\n\n[GuKa15] Guth, Larry and Katz, Nets Hawk, On the Erd\\H{o}s distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1923, "problem_number": "EP-90", "title": "Erdős Problem #90", "statement": "Does every set of $n$ distinct points in $\\mathbb{R}^2$ contain at most $n^{1+O(1/\\log\\log n)}$ many pairs which are distance 1 apart?", "background": "The unit distance problem. In \\cite{Er94b} Erd\\H{o}s dates this conjecture to 1946. In \\cite{Er82e} he offers \\$300 for the upper bound $n^{1+o(1)}$.\nThis would be the best possible, as is shown by a set of lattice points. It is easy to show that there are $O(n^{3/2})$ many such pairs. The best known upper bound is $O(n^{4/3})$, due to Spencer, Szemer\\'{e}di, and Trotter \\cite{SST84}. In \\cite{Er83c} and \\cite{Er85} Erd\\H{o}s offers \\$250 for an upper bound of the form $n^{1+o(1)}$.\nPart of the difficulty of this problem is explained by a result of Valtr (see \\cite{Sz16}), who constructed a metric on $\\mathbb{R}^2$ and a set of $n$ points with $\\gg n^{4/3}$ unit distance pairs (with respect to this metric). The methods of the upper bound proof of Spencer, Szemer\\'{e}di, and Trotter \\cite{SST84} generalise to include this metric. Therefore to prove an upper bound better than $n^{4/3}$ some special feature of the Euclidean metric must be exploited.\nSee a survey by Szemer\\'{e}di \\cite{Sz16} for further background and related results.\nSee also [92], [96], [605], and [956]. The higher dimensional generalisation is [1085].\nReferences\n\n\n[Er82e] Erd\\H{o}s, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.\n\n[Er83c] Erd\\H{o}s, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54.\n\n[Er85] Erd\\H{o}s, P., Problems and results in combinatorial geometry. Discrete geometry and convexity (New York, 1982) (1985), 1-11.\n\n[Er94b] Erd\\H{o}s, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.\n\n[SST84] Spencer, J. and Szemer\\'{e}di, E. and Trotter, Jr., W., Unit distances in the Euclidean plane. Graph theory and combinatorics (Cambridge, 1983) (1984), 293-303.\n\n[Sz16] Szemer\\'{e}di, Endre, Erd\\H{o}s's unit distance problem. Open problems in mathematics (2016), 459-477.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1924, "problem_number": "EP-91", "title": "Erdős Problem #91", "statement": "Let $n$ be a sufficently large integer. Suppose $A\\subset \\mathbb{R}^2$ has $\\lvert A\\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that there are at least two (and probably many) such $A$ which are non-similar.", "background": "For $n=3$ the equilateral triangle is the only such set. For $n=4$ the square or two equilateral triangles sharing an edge give two non-similar examples.\nFor $n=5$ the regular pentagon is the unique such set (which has two distinct distances). Erd\\H{o}s mysteriously remarks in \\cite{Er90} this was proved by 'a colleague'. (In \\cite{Er87b} this is described as 'a colleague from Zagreb (unfortunately I do not have his letter)'.) A published proof of this fact is provided by Kov\\'{a}cs \\cite{Ko24c}.\nIn \\cite{Er87b} Erd\\H{o}s says that there are at least two non-similar examples for $6\\leq n\\leq 9$.\nThe minimal possible number of distinct distances is the subject of [89].\nReferences\n\n\n[Er87b] Erd\\H{o}s, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Si\\'{o}fok, 1985) (1987), 167-177.\n\n[Er90] Erd\\H{o}s, Paul, Some of my favourite unsolved problems. A tribute to Paul Erd\\H{o}s (1990), 467-478.\n\n[Ko24c] Z. Kov\\'{a}cs, A note on Erd\\H{o}s's mysterious remark. arXiv:2412.05190 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1925, "problem_number": "EP-92", "title": "Erdős Problem #92", "statement": "Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\\mathbb{R}^2$ in which every $x\\in A$ has at least $f(n)$ points in $A$ equidistant from $x$.\nIs it true that $f(n)\\leq n^{o(1)}$? Or even $f(n) < n^{O(1/\\log\\log n)}$?", "background": "This is a stronger form of the unit distance conjecture (see [90]).\nThe set of lattice points imply $f(n) > n^{c/\\log\\log n}$ for some constant $c>0$. Erd\\H{o}s offered \\$500 for a proof that $f(n) \\leq n^{o(1)}$ but only \\$100 for a counterexample. This latter prize is downgraded to \\$50 in \\cite{ErFi97}.\nIt is trivial that $f(n) \\ll n^{1/2}$. A result of Pach and Sharir (Theorem 4 of \\cite{PaSh92}) implies $f(n) \\ll n^{2/5}$. Hunter has observed that the circle-point incidence bound of Janzer, Janzer, Methuku, and Tardos \\cite{JJMT24} implies $ f(n) \\ll n^{4/11}. $ Fishburn (personal communication to Erd\\H{o}s, later published in \\cite{ErFi97}) proved that $6$ is the smallest $n$ such that $f(n)=3$ and $8$ is the smallest $n$ such that $f(n)=4$, and suggested that the lattice points may not be best example.\nSee also [754].\nReferences\n\n\n[ErFi97] Erd\\H{o}s, Paul and Fishburn, Peter, Minimum planar sets with maximum equidistance counts. Comput. Geom. (1997), 207--218.\n\n[JJMT24] B. Janzer, O. Janzer, A. Methuku, and G. Tardos, Tight bounds for intersection-reverse sequences, edge-ordered graphs\nand applications. arXiv:2411.07188 (2024).\n\n[PaSh92] Pach, J\\'anos and Sharir, Micha, Repeated angles in the plane and related problems. J. Combin. Theory Ser. A (1992), 12--22.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1926, "problem_number": "EP-96", "title": "Erdős Problem #96", "statement": "If $n$ points in $\\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.", "background": "Conjectured by Erd\\H{o}s and Moser. In \\cite{Er92e} Erd\\H{o}s credits the conjecture that the true upper bound is $2n$ to himself and Fishburn. F\"{u}redi \\cite{Fu90} proved an upper bound of $O(n\\log n)$. A short proof of this bound was given by Brass and Pach \\cite{BrPa01}. The best known upper bound is $ \\leq n\\log_2n+4n, $ due to Aggarwal \\cite{Ag15}.\nEdelsbrunner and Hajnal \\cite{EdHa91} have constructed $n$ such points with $2n-7$ pairs distance $1$ apart. (This disproved an early stronger conjecture of Erd\\H{o}s and Moser, that the true answer was $\\frac{5}{3}n+O(1)$.)\nA positive answer would follow from [97]. See also [90].\nIn \\cite{Er92e} Erd\\H{o}s makes the stronger conjecture that, if $g(x)$ counts the largest number of points equidistant from $x$ in $A$, then $ \\sum_{x\\in A}g(x)< 4n. $ He notes that the example of Edelsbrunner and Hajnal shows that $\\sum_{x\\in A}g(x)>4n-O(1)$ is possible.\nReferences\n\n\n[Ag15] Aggarwal, Amol, On unit distances in a convex polygon. Discrete Math. (2015), 88-92.\n\n[BrPa01] Brass , Peter and Pach, J\\'{a}nos, The maximum number of times the same distance can occur among\nthe vertices of a convex {$n$}-gon is {$O(n\\log n)$}. J. Combin. Theory Ser. A (2001), 178-179.\n\n[EdHa91] Edelsbrunner, Herbert and Hajnal, P\\'{e}ter, A lower bound on the number of unit distances between the\nvertices of a convex polygon. J. Combin. Theory Ser. A (1991), 312-316.\n\n[Er92e] Erd\\H{o}s, P\\'{a}l, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.\n\n[Fu90] F\"{u}redi, Zolt\\'{a}n, The maximum number of unit distances in a convex {$n$}-gon. J. Combin. Theory Ser. A (1990), 316-320.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1927, "problem_number": "EP-98", "title": "Erdős Problem #98", "statement": "Let $h(n)$ be such that any $n$ points in $\\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\\to \\infty$?", "background": "Erd\\H{o}s could not even prove $h(n)\\geq n$. Pach has shown $h(n)0$.\nReferences\n\n\n[EFPR93] Erd\\H{o}s, Paul and F\"{u}redi, Zolt\\'{a}n and Pach, J\\'{a}nos and\nRuzsa, Imre Z., The grid revisited. Discrete Math. (1993), 189--196.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1928, "problem_number": "EP-99", "title": "Erdős Problem #99", "statement": "Let $A\\subseteq\\mathbb{R}^2$ be a set of $n$ points with minimum distance equal to 1, chosen to minimise the diameter of $A$. If $n$ is sufficiently large then must there be three points in $A$ which form an equilateral triangle of size 1?", "background": "Thue proved that the minimal such diameter is achieved (asymptotically) by the points in a triangular lattice intersected with a circle. In general Erd\\H{o}s believed such a set must have very large intersection with the triangular lattice (perhaps as many as $(1-o(1))n$).\nErd\\H{o}s \\cite{Er94b} wrote 'I could not prove it but felt that it should not be hard. To my great surprise both B. H. Sendov and M. Simonovits doubted the truth of this conjecture.' In \\cite{Er94b} he offers \\$100 for a counterexample but only \\$50 for a proof.\nThe stated problem is false for $n=4$, for example taking the points to be vertices of a square. The behaviour of such sets for small $n$ is explored by Bezdek and Fodor \\cite{BeFo99}.\nSee also [103].\nReferences\n\n\n[BeFo99] Bezdek, Andr\\'{a}s and Fodor, Ferenc, Minimal diameter of certain sets in the plane. J. Combin. Theory Ser. A (1999), 105-111.\n\n[Er94b] Erd\\H{o}s, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1929, "problem_number": "EP-100", "title": "Erdős Problem #100", "statement": "Let $A$ be a set of $n$ points in $\\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they differ by at least $1$. Is the diameter of $A$ $\\gg n$?", "background": "Perhaps the diameter is even $\\geq n-1$ for sufficiently large $n$. Piepmeyer has an example of $9$ such points with diameter $<5$. Kanold proved the diameter is $\\geq n^{3/4}$. The bounds on the distinct distance problem [89] proved by Guth and Katz \\cite{GuKa15} imply a lower bound of $\\gg n/\\log n$.\nReferences\n\n\n[GuKa15] Guth, Larry and Katz, Nets Hawk, On the Erd\\H{o}s distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1930, "problem_number": "EP-101", "title": "Erdős Problem #101", "statement": "Given $n$ points in $\\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$.", "background": "There are examples of sets of $n$ points with $\\sim n^2/6$ many collinear triples and no four points on a line. Such constructions are given by Burr, Gr\"{u}nbaum, and Sloane \\cite{BGS74} and F\"{u}redi and Pal\\'{a}sti \\cite{FuPa84}.\nGr\"{u}nbaum \\cite{Gr76} constructed an example with $\\gg n^{3/2}$ such lines. Erd\\H{o}s speculated this may be the correct order of magnitude. This is false: Solymosi and Stojakovi\\'{c} \\cite{SoSt13} have constructed a set with no five on a line and at least $ n^{2-O(1/\\sqrt{\\log n})} $ many lines containing exactly four points.\nSee also [102] and [669]. A generalisation of this problem is asked in [588].\nThis problem is Problem 71 on Green's open problems list.\nReferences\n\n\n[BGS74] Burr, Stefan A. and Gr\"{u}nbaum, Branko and Sloane, N. J. A., The orchard problem. Geometriae Dedicata (1974), 397-424.\n\n[FuPa84] F\"{u}redi, Z. and Pal\\'{a}sti, I., Arrangements of lines with a large number of triangles. Proc. Amer. Math. Soc. (1984), 561-566.\n\n[Gr76] Gr\"{u}nbaum, Branko, New views on some old questions of combinatorial geometry. Colloquio Internazionale sulle Teorie Combinatorie\n(Roma, 1973), Tomo I (1976), 451-468.\n\n[SoSt13] Solymosi, J\\'{o}zsef and Stojakovi\\'C, Milo\\vS, Many collinear {$k$}-tuples with no {$k+1$} collinear points. Discrete Comput. Geom. (2013), 811-820.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1931, "problem_number": "EP-102", "title": "Erdős Problem #102", "statement": "Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\\mathbb{R}^2$ such that there are $\\geq cn^2$ lines each containing more than three points, there must be some line containing $h_c(n)$ many points. Estimate $h_c(n)$. Is it true that, for fixed $c>0$, we have $h_c(n)\\to \\infty$?", "background": "A problem of Erd\\H{o}s and Purdy. It is not even known if $h_c(n)\\geq 5$ (see [101]).\nIt is easy to see that $h_c(n) \\ll_c n^{1/2}$, and Erd\\H{o}s at one point \\cite{Er95} suggested that perhaps a similar lower bound $h_c(n)\\gg_c n^{1/2}$ holds. Zach Hunter has pointed out that this is false, even replacing $>3$ points on each line with $>k$ points: consider the set of points in $\\{1,\\ldots,m\\}^d$ where $n\\approx m^d$. These intersect any line in $\\ll_d n^{1/d}$ points, and have $\\gg_d n^2$ many pairs of points each of which determine a line with at least $k$ points. This is a construction in $\\mathbb{R}^d$, but a random projection into $\\mathbb{R}^2$ preserves the relevant properties.\nThis construction shows that $h_c(n) \\ll n^{1/\\log(1/c)}$.\nReferences\n\n\n[Er95] Erd\\H{o}s, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1932, "problem_number": "EP-103", "title": "Erdős Problem #103", "statement": "Let $h(n)$ count the number of incongruent sets of $n$ points in $\\mathbb{R}^2$ which minimise the diameter subject to the constraint that $d(x,y)\\geq 1$ for all points $x\neq y$. Is it true that $h(n)\\to \\infty$?", "background": "It is not even known whether $h(n)\\geq 2$ for all large $n$.\nSee also [99].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1933, "problem_number": "EP-104", "title": "Erdős Problem #104", "statement": "Given $n$ points in $\\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$.", "background": "In \\cite{Er81d} Erd\\H{o}s proved that $\\gg n$ many circles is possible, and that there cannot be more than $O(n^2)$ many circles. The argument is very simple: every pair of points determines at most $2$ unit circles, and the claimed bound follows from double counting. Erd\\H{o}s claims in a number of places this produces the upper bound $n(n-1)$, but Harborth and Mengerson \\cite{HaMe86} note that in fact this delivers an upper bound of $\\frac{n(n-1)}{3}$.\nElekes \\cite{El84} has a simple construction of a set with $\\gg n^{3/2}$ such circles. This may be the correct order of magnitude.\nIn \\cite{Er75h} and \\cite{Er92e} Erd\\H{o}s also asks how many such unit circles there must be if the points are in general position.\nIn \\cite{Er92e} Erd\\H{o}s offered £100 for a proof or disproof that the answer is $O(n^{3/2})$.\nThe maximal number of unit circles achieved by $n$ points is A003829 in the OEIS.\nSee also [506] and [831].\nReferences\n\n\n[El84] Elekes, G., {$n$} points in the plane can determine $n^{3/2}$ unit\ncircles. Combinatorica (1984), 131.\n\n[Er75h] Erd\\H{o}s, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3.\n\n[Er81d] Erd\\H{o}s, P., Some applications of graph theory and combinatorial methods to number theory and geometry. Algebraic methods in graph theory, Vol. I, II (Szeged, 1978) (1981), 137-148.\n\n[Er92e] Erd\\H{o}s, P\\'{a}l, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.\n\n[HaMe86] Harborth, Heiko and Mengersen, Ingrid, Point sets with many unit circles. Discrete Math. (1986), 193--197.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1934, "problem_number": "EP-108", "title": "Erdős Problem #108", "statement": "For every $r\\geq 4$ and $k\\geq 2$ is there some finite $f(k,r)$ such that every graph of chromatic number $\\geq f(k,r)$ contains a subgraph of girth $\\geq r$ and chromatic number $\\geq k$?", "background": "Conjectured by Erd\\H{o}s and Hajnal. R\"{o}dl \\cite{Ro77} has proved the $r=4$ case (see [923]). The infinite version (whether every graph of infinite chromatic number contains a subgraph of infinite chromatic number whose girth is $>k$) is also open.\nIn \\cite{Er79b} Erd\\H{o}s also asks whether $ \\lim_{k\\to \\infty}\\frac{f(k,r+1)}{f(k,r)}=\\infty. $ See also the entry in the graphs problem collection and [740] for the infinitary version.\nReferences\n\n\n[Er79b] Erd\\H{o}s, Paul, Problems and results in graph theory and combinatorial analysis. Graph theory and related topics (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1977) (1979), 153-163.\n\n[Ro77] R\"{o}dl, V., On the chromatic number of subgraphs of a given graph. Proc. Amer. Math. Soc. (1977), 370-371.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1935, "problem_number": "EP-111", "title": "Erdős Problem #111", "statement": "If $G$ is a graph let $h_G(n)$ be defined such that any subgraph of $G$ on $n$ vertices can be made bipartite after deleting at most $h_G(n)$ edges.\nWhat is the behaviour of $h_G(n)$? Is it true that $h_G(n)/n\\to \\infty$ for every graph $G$ with chromatic number $\\aleph_1$?", "background": "A problem of Erd\\H{o}s, Hajnal, and Szemer\\'{e}di \\cite{EHS82}. Every $G$ with chromatic number $\\aleph_1$ must have $h_G(n)\\gg n$ since $G$ must contain, for some $r$, $\\aleph_1$ many vertex disjoint odd cycles of length $2r+1$.\nOn the other hand, Erd\\H{o}s, Hajnal, and Szemer\\'{e}di proved that there is a $G$ with chromatic number $\\aleph_1$ such that $h_G(n)\\ll n^{3/2}$. In \\cite{Er81} Erd\\H{o}s conjectured that this can be improved to $\\ll n^{1+\\epsilon}$ for every $\\epsilon>0$.\nSee also [74].\nReferences\n\n\n[EHS82] Erd\\H{o}s, P. and Hajnal, A. and Szemer\\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123.\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1936, "problem_number": "EP-112", "title": "Erdős Problem #112", "statement": "Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a transitive tournament of size $m$. Determine $k(n,m)$.", "background": "A problem of Erd\\H{o}s and Rado \\cite{ErRa67}, who showed $k(n,m) \\ll_m n^{m-1}$, or more precisely, $ k(n,m) \\leq \\frac{2^{m-1}(n-1)^m+n-2}{2n-3}. $ Larson and Mitchell \\cite{LaMi97} improved the dependence on $m$, establishing in particular that $k(n,3)\\leq n^{2}$. Zach Hunter has observed that $ R(n,m) \\leq k(n,m)\\leq R(n,m,m), $ which in particular proves the upper bound $k(n,m)\\leq 3^{n+2m}$.\nSee also the entry in the graphs problem collection - on this site the problem replaces transitive tournament with directed path, but Zach Hunter and Raphael Steiner have a simple argument that proves, for this alternative definition, that $k(n,m)=(n-1)(m-1)$.\nReferences\n\n\n[ErRa67] Erd\\H{o}s, P. and Rado, R., Partition relations and transitivity domains of binary\nrelations. J. London Math. Soc. (1967), 624-633.\n\n[LaMi97] Larson, Jean A. and Mitchell, William J., On a problem of Erd\\H{o}s and Rado. Ann. Comb. (1997), 245-252.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1937, "problem_number": "EP-114", "title": "Erdős Problem #114", "statement": "If $p(z)\\in\\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\\{ z\\in \\mathbb{C} : \\lvert p(z)\\rvert=1\\}$ maximised when $p(z)=z^n-1$?", "background": "A problem of Erd\\H{o}s, Herzog, and Piranian \\cite{EHP58}. It is also listed as Problem 4.10 in \\cite{Ha74}, where it is attributed to Erd\\H{o}s.\nLet the maximal length of such a curve be denoted by $f(n)$.\n{UL}\n{LI}The length of the curve when $p(z)=z^n-1$ is $2n+O(1)$, and hence the conjecture implies in particular that $f(n)=2n+O(1)$.{/LI}\n{LI}Dolzhenko \\cite{Do61} proved $f(n) \\leq 4\\pi n$, but few were aware of this work.{/LI}\n{LI}Pommerenke \\cite{Po61} proved $f(n)\\ll n^2$.{/LI}\n{LI}Borwein \\cite{Bo95} proved $f(n)\\ll n$ (Borwein was unaware of Dolzhenko's earlier work). The prize of \\$250 is reported by Borwein \\cite{Bo95}.{/LI}\n{LI}Eremenko and Hayman \\cite{ErHa99} proved the full conjecture when $n=2$, and $f(n)\\leq 9.173n$ for all $n$.{/LI}\n{LI}Danchenko \\cite{Da07} proved $f(n)\\leq 2\\pi n$.{/LI}\n{LI}Fryntov and Nazarov \\cite{FrNa09} proved that $z^n-1$ is a local maximiser, and solved this problem asymptotically, proving that $ f(n)\\leq 2n+O(n^{7/8}). $ {/LI}\n{LI} Tao \\cite{Ta25} has proved that $p(z)=z^n-1$ is the unique (up to rotation and translation) maximiser for all sufficiently large $n$.\n{/UL}\nErd\\H{o}s, Herzog, and Piranian \\cite{EHP58} also ask whether the length is at least $2\\pi$ if $\\{ z: \\lvert f(z)\\rvert<1\\}$ is connected (which $z^n$ shows is the best possible). This was proved by Pommerenke \\cite{Po59}.\nReferences\n\n\n[Bo95] Borwein, Peter, The arc length of the lemniscate {$\\{|p(z)|=1\\}$}. Proc. Amer. Math. Soc. (1995), 797--799.\n\n[Da07] Danchenko, V. I., The lengths of lemniscates. {V}ariations of rational\nfunctions. Mat. Sb. (2007), 51--58.\n\n[Do61] Dol\\v zenko, E. P., Some estimates concerning algebraic hypersurfaces and\nderivatives of rational functions. Dokl. Akad. Nauk SSSR (1961), 1287--1290.\n\n[EHP58] Erd\\H{o}s, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.\n\n[ErHa99] Eremenko, Alexandre and Hayman, Walter, On the length of lemniscates. Michigan Math. J. (1999), 409--415.\n\n[FrNa09] Fryntov, Alexander and Nazarov, Fedor, New estimates for the length of the {E}rd\\H\nos-{H}erzog-{P}iranian lemniscate. (2009), 49--60.\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[Po59] Pommerenke, Ch., On some problems by Erd\\H{o}s, Herzog and Piranian. Michigan Math. J. (1959), 221-225.\n\n[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.\n\n[Ta25] T. Tao, The maximal length of the Erd\\H{o}s-Herzog-Piranian leminscate length in high degree. arXiv:2512.12455 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1938, "problem_number": "EP-117", "title": "Erdős Problem #117", "statement": "Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\neq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups.\nEstimate $h(n)$ as well as possible.", "background": "Pyber \\cite{Py87} has proved there exist constants $c_2>c_1>1$ such that $c_1^n0$ such that for infinitely many $n$ we have $M_n > n^c$?\nIs it true that there exists $c>0$ such that, for all large $n$, $ \\sum_{k\\leq n}M_k > n^{1+c}? $ ", "background": "This is Problem 4.1 in \\cite{Ha74} where it is attributed to Erd\\H{o}s.\nThe weaker conjecture that $\\limsup M_n=\\infty$ was proved by Wagner \\cite{Wa80}, who show that there is some $c>0$ with $M_n>(\\log n)^c$ infinitely often.\nThe second question was answered by Beck \\cite{Be91}, who proved that there exists some $c>0$ such that $ \\max_{n\\leq N} M_n > N^c. $ Erd\\H{o}s (e.g. see \\cite{Ha74}) gave a construction of a sequence with $M_n\\leq n+1$ for all $n$. Linden \\cite{Li77} improved this to give a sequence with $M_n\\ll n^{1-c}$ for some $c>0$.\nThe third question seems to remain open.\nReferences\n\n\n[Be91] Beck, J., The modulus of polynomials with zeros on the unit circle: A problem of Erd\\H{o}s. Annals of Math. (1991), 609-651.\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[Li77] Linden, C. N., The modulus of polynomials with zeros on the unit circle. Bull. London Math. Soc. (1977), 65--69.\n\n[Wa80] Wagner, Gerold, On a problem of {E}rd\\H{o}s in {D}iophantine approximation. Bull. London Math. Soc. (1980), 81--88.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1940, "problem_number": "EP-120", "title": "Erdős Problem #120", "statement": "Let $A\\subseteq\\mathbb{R}$ be an infinite set. Must there be a set $E\\subset \\mathbb{R}$ of positive measure which does not contain any set of the shape $aA+b$ for some $a,b\\in\\mathbb{R}$ and $a\neq 0$?", "background": "The Erd\\H{o}s similarity problem.\nThis is true if $A$ is unbounded or dense in some interval. It therefore suffices to prove this when $A=\\{a_1>a_2>\\cdots\\}$ is a countable strictly monotone sequence which converges to $0$.\nSteinhaus \\cite{St20} has proved this is false whenever $A$ is a finite set.\nThis conjecture is known in many special cases (but, for example, it is open when $A=\\{1,1/2,1/4,\\ldots\\}$, which is Problem 94 on Green's open problems list). For an overview of progress we recommend a nice survey by Svetic \\cite{Sv00} on this problem. A survey of more recent progress was written by Jung, Lai, and Mooroogen \\cite{JLM24}.\nReferences\n\n\n[JLM24] Y. Jung and C.-K. Lai and Y. Mooroogen, Some recent progress on the Erd\\H{o}s similarity conjecture. arXiv:2412.11062 (2024).\n\n[St20] Steinhaus, Hugo, Sur les distances des points dans les ensembles de measure positive. Fund. Math. (1920), 93-104.\n\n[Sv00] Svetic, R. E., The Erd\\H{o}s similarity problem: a survey. Real Anal. Exchange (2000/01), 525-539.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1941, "problem_number": "EP-122", "title": "Erdős Problem #122", "statement": "For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $f(n)/F(n)\\to 0$ for almost all $n$, there are infinitely many $x$ such that $ \\frac{\\#\\{ n\\in \\mathbb{N} : n+f(n)\\in (x,x+F(x))\\}}{F(x)}\\to \\infty? $ ", "background": "Asked by Erd\\H{o}s, Pomerance, and S\\'{a}rk\"{o}zy \\cite{EPS97} who prove that this is true when $f$ is the divisor function or the number of distinct prime divisors of $n$, but Erd\\H{o}s believed it is false when $f(n)=\\phi(n)$ or $\\sigma(n)$.\nReferences\n\n\n[EPS97] Erd\\H{o}s, Paul and Pomerance, Carl and S\\'{a}rk\"{o}zy, Andr\\'{a}s, On locally repeated values of certain arithmetic functions. IV. Ramanujan J. (1997), 227-241.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1942, "problem_number": "EP-123", "title": "Erdős Problem #123", "statement": "Let $a,b,c\\geq 1$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^m$ ($k,l,m\\geq 0$), none of which divide any other?", "background": "A sequence is said to be $d$-complete if every large integer is the sum of distinct integers from the sequence, none of which divide any other. This particular case of $d$-completeness was conjectured by Erd\\H{o}s and Lewin \\cite{ErLe96}, who (among other related results) prove this when $a=3$, $b=5$, and $c=7$.\nAs a partial record of progress so far, the sequence $\\{a^kb^lc^m\\}$ is known to be $d$-complete when:\n{UL}\n{LI}$a=3$, $b=5$, $c=7$ (Erd\\H{o}s and Lewin \\cite{ErLe96}).{/LI}\n{LI}$a=2$, $b=5$, $c\\in \\{7,11,13,17,19\\}$ (Erd\\H{o}s and Lewin \\cite{ErLe96}).{/LI}\n{LI}$a=2$, $b=5$, $c\\in \\{9,21,23,27,29,31\\}$ - more generally, $a=2$, $b=5$, and any $c>6$ with $(c,10)=1$ such that there exists $N$ where every integer in $(N,25cN)$ is the sum of distinct elements of $\\{2^k3^lc^m\\}$, none of which divide any other (Ma and Chen \\cite{MaCh16}).{/LI}\n{LI} $a=2$, $b=5$, $3\\leq c\\leq 87$ with $(c,10)=1$, or $a=2$, $b=7$, $3\\leq c\\leq 33$ with $(c,14)=1$, or $a=3$, $b=5$, $2\\leq c\\leq 14$ with $(c,15)=1$ (Chen and Yu \\cite{ChYu23b}).{/LI}\n{/UL}\nIn \\cite{Er92b} Erd\\H{o}s makes the stronger conjecture (for $a=2$, $b=3$, and $c=5$) that, for any $\\epsilon>0$, all large integers $n$ can be written as the sum of distinct integers $b_1<\\cdots 0$, of an infinite set of $d_i$ for which every sufficiently large integer can be written as a finite sum of the shape $\\sum_i c_ia_i$ where $c_i\\in \\{0,1\\}$ and $a_i\\in P(d_i,0)$ and yet $\\sum_{i}\\frac{1}{d_i-1}<\\epsilon$.\nSee also [125].\nReferences\n\n\n[BEGL96] Burr, S. A. and Erd\\H{o}s, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. (1996), 133-138.\n\n[Er97] Erd\\H{o}s, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160.\n\n[Er97e] Erd\\H{o}s, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.\n\n[Me04] Melfi, Giuseppe, On certain positive integer sequences. Riv. Mat. Univ. Parma (7) (2004), 253--260.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1944, "problem_number": "EP-125", "title": "Erdős Problem #125", "statement": "Let $A = \\{ \\sum\\epsilon_k3^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\\{ \\sum\\epsilon_k4^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $4$.\nDoes $A+B$ have positive density?", "background": "A problem of Burr, Erd\\H{o}s, Graham, and Li \\cite{BEGL96}. More generally, if $n_1<\\cdots1 $ and $A_i$ is the set of integers with only the digits $0,1$ in base $n_i$ then does $A_1+\\cdots+A_k$ have positive density? Melfi \\cite{Me01} noted this is false as written, with a counterexample given by $\\{3,9,81\\}$, but suggests it is true if we further insist that the $n_k$ are pairwise coprime.\nIf $C=A+B$ then Melfi \\cite{Me01} showed $\\lvert C\\cap[1,x]\\rvert \\gg x^{0.965}$ and Hasler and Melfi \\cite{HaMe24} improved this to $\\lvert C\\cap [1,x]\\rvert \\gg x^{0.9777}$. Hasler and Melfi also show that the lower density of $C$ is at most $ \\frac{1015}{1458}\\approx 0.69616. $ See also [124].\nReferences\n\n\n[BEGL96] Burr, S. A. and Erd\\H{o}s, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. (1996), 133-138.\n\n[HaMe24] M. Hasler and G. Melfi, On sums of distinct powers of $3$ and $4$. Combinatorics and Number Theory (2024).\n\n[Me01] Melfi, Giuseppe, An additive problem about powers of fixed integers. Rend. Circ. Mat. Palermo (2) (2001), 239--246.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1945, "problem_number": "EP-126", "title": "Erdős Problem #126", "statement": "Let $f(n)$ be maximal such that if $A\\subseteq\\mathbb{N}$ has $\\lvert A\\rvert=n$ then $\\prod_{a\neq b\\in A}(a+b)$ has at least $f(n)$ distinct prime factors. Is it true that $f(n)/\\log n\\to\\infty$?", "background": "Investigated by Erd\\H{o}s and Tur\\'{a}n \\cite{ErTu34} (prompted by a question of L\\'{a}z\\'{a}r and Gr\"{u}nwald) in their first joint paper, where they proved that $ \\log n \\ll f(n) \\ll n/\\log n $ (the upper bound is trivial, taking $A=\\{1,\\ldots,n\\}$). Erd\\H{o}s says that $f(n)=o(n/\\log n)$ has never been proved, but perhaps never seriously attacked.\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[ErTu34] Erd\\H{o}s, Paul and Turan, Paul, On a Problem in the Elementary Theory of Numbers. Amer. Math. Monthly (1934), 608-611.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1946, "problem_number": "EP-129", "title": "Erdős Problem #129", "statement": "Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy of $K_k$ in at least one of the $r$ colours. Prove that there is a constant $C=C(r)>1$ such that $ R(n;3,r) < C^{\\sqrt{n}}. $ ", "background": "Conjectured by Erd\\H{o}s and Gy\\'{a}rf\\'{a}s, who proved the existence of some $C>1$ such that $R(n;3,r)>C^{\\sqrt{n}}$. Note that when $r=k=2$ we recover the classic Ramsey numbers. Erd\\H{o}s thought it likely that for all $r,k\\geq 2$ there exists some $C_1,C_2>1$ (depending only on $r$) such that $ C_1^{n^{1/k-1}}< R(n;k,r) < C_2^{n^{1/k-1}}. $ Antonio Girao has pointed out that this problem as written is easily disproved, and indeed $R(n;3,2) \\geq C^{n}$:\nThe obvious probabilistic construction (randomly colour the edges red/blue independently uniformly at random) yields a 2-colouring of the edges of $K_N$ such every set on $n$ vertices contains a red triangle and a blue triangle (using that every set of $n$ vertices contains $\\gg n^2$ edge-disjoint triangles), provided $N \\leq C^n$ for some absolute constant $C>1$. This implies $R(n;3,2) \\geq C^{n}$, contradicting the conjecture.\nPerhaps Erd\\H{o}s had a different problem in mind, but it is not clear what that might be. It would presumably be one where the natural probabilistic argument would deliver a bound like $C^{\\sqrt{n}}$ as Erd\\H{o}s and Gy\\'{a}rf\\'{a}s claim to have achieved via the probabilistic method.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1947, "problem_number": "EP-130", "title": "Erdős Problem #130", "statement": "Let $A\\subset\\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertices the points in $A$, where two vertices are joined by an edge if and only if they are an integer distance apart.\nHow large can the chromatic number and clique number of this graph be? In particular, can the chromatic number be infinite?", "background": "Asked by Andr\\'{a}sfai and Erd\\H{o}s. Erd\\H{o}s \\cite{Er97b} also asked where such a graph could contain an infinite complete graph, but this is impossible by an earlier result of Anning and Erd\\H{o}s \\cite{AnEr45}.\nSee also [213].\nReferences\n\n\n[AnEr45] Anning, Norman H. and Erd\\H{o}s, Paul, Integral distances. Bull. Amer. Math. Soc. (1945), 598-600.\n\n[Er97b] Erd\\H{o}s, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1948, "problem_number": "EP-131", "title": "Erdős Problem #131", "statement": "Let $F(N)$ be the maximal size of $A\\subseteq\\{1,\\ldots,N\\}$ such that no $a\\in A$ divides the sum of any distinct elements of $A\\backslash\\{a\\}$. Estimate $F(N)$. In particular, is it true that $ F(N) > N^{1/2-o(1)}? $ ", "background": "This was studied by Erd\\H{o}s, Lev, Rauzy, S\\'{a}ndor, and S\\'{a}rk\"{o}zy \\cite{ELRSS99}, where they call such a property 'non-dividing', and prove the explicit bound $ F(N)<3N^{1/2}+1. $ In \\cite{Er97b} Erd\\H{o}s credits Csaba with a construction that proves $F(N) \\gg N^{1/5}$. Such a construction was also given in \\cite{ELRSS99}, where it is linked to the problem of non-averaging sets (see [186]).\nIndeed, every such set is non-averaging, and hence the result of Pham and Zakharov \\cite{PhZa24} implies $ F(N) \\leq N^{1/4+o(1)}. $ This shows the answer to the original question is no, but the general question of the correct growth of $F(N)$ remains open.\nIn \\cite{Er75b} Erd\\H{o}s writes that he originally thought $F(N) <(\\log N)^{O(1)}$, but that Straus proved that $ F(N) > \\exp((\\sqrt{\\tfrac{2}{\\log 2}}+o(1))\\sqrt{\\log N}). $ See also [13].\nThis is discussed in problem C16 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[ELRSS99] Erd\\H{o}s, P. and Lev, V. and Rauzy, G. and S\\'andor, C. and\nS\\'ark\"ozy, A., Greedy algorithm, arithmetic progressions, subset sums and\ndivisibility. Discrete Math. (1999), 119--135.\n\n[Er75b] Erd\\H{o}s, Paul, Problems and results in combinatorial number theory. Journ\\'{e}es Arithm\\'{e}tiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310.\n\n[Er97b] Erd\\H{o}s, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[PhZa24] Pham, H. T. and Zakharov, D., Sharp bound for the Erd\\H{o}s-Straus non-averaging set problem. arXiv:2410.14624 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1949, "problem_number": "EP-132", "title": "Erdős Problem #132", "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points. Must there be two distances which occur at least once but between at most $n$ pairs of points? Must the number of such distances $\\to \\infty$ as $n\\to \\infty$?", "background": "Asked by Erd\\H{o}s and Pach. Hopf and Pannowitz \\cite{HoPa34} proved that the largest distance between points of $A$ can occur at most $n$ times, but it is unknown whether a second such distance must occur.\nIt may be true that there are at least $n^{1-o(1)}$ many such distances. In \\cite{Er97e} Erd\\H{o}s offers \\$100 for 'any nontrivial result'.\nErd\\H{o}s \\cite{Er84c} believed that for $n\\geq 5$ there must always exist at least two such distances. This is false for $n=4$, as witnessed by two equilateral triangles of the same side-length glued together. Erd\\H{o}s and Fishburn \\cite{ErFi95} proved this is true for $n=5$ and $n=6$.\nClemen, Dumitrescu, and Liu \\cite{CDL25} have proved that there always at least two such distances if $A$ is in convex position (that is, no point lies inside the convex hull of the others). They also prove it is true if the set $A$ is 'not too convex', in a specific technical sense.\nSee also [223], [756], and [957].\nReferences\n\n\n[CDL25] F. Clemen, A. Dumitrescu, and D. Liu, On multiplicities of interpoint distances. arXiv:2505.04283 (2025).\n\n[Er84c] Erd\\H{o}s, Paul, Some old and new problems in combinatorial geometry. Convexity and graph theory (Jerusalem, 1981) (1984), 129-136.\n\n[Er97e] Erd\\H{o}s, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.\n\n[ErFi95] Erd\\H{o}s, Paul and Fishburn, Peter C., Multiplicities of interpoint distances in finite planar sets. Discrete Appl. Math. (1995), 141--147.\n\n[HoPa34] Hopf, H. and Pannwitz, E., Aufgabe 167. Jber. Deutsch. Math. Verein. (1934), 114.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1950, "problem_number": "EP-137", "title": "Erdős Problem #137", "statement": "We say that $N$ is powerful if whenever $p\\mid N$ we also have $p^2\\mid N$. Let $k\\geq 3$. Can the product of any $k$ consecutive positive integers ever be powerful?", "background": "Conjectured by Erd\\H{o}s and Selfridge. There are infinitely many $n$ such that $n(n+1)$ is powerful (see [364]). Erd\\H{o}s and Selfridge \\cite{ErSe75} proved that the product of $k\\geq 3$ consecutive positive integers can never be a perfect power. Erd\\H{o}s remarked that this 'seems hopeless at present'.\nIn \\cite{Er82c} he further conjectures that, if $k$ is fixed and $n$ is sufficiently large, then, for all $m$, there must be at least $k$ distinct primes $p$ such that $ p\\mid m(m+1)\\cdots (m+n) $ and yet $p^2$ does not divide the right-hand side.\nSee also [364].\nReferences\n\n\n[Er82c] Erd\\H{o}s, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45.\n\n[ErSe75] Erd\\H{o}s, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1951, "problem_number": "EP-138", "title": "Erdős Problem #138", "statement": "Let the van der Waerden number $W(k)$ be such that whenever $N\\geq W(k)$ and $\\{1,\\ldots,N\\}$ is $2$-coloured there must exist a monochromatic $k$-term arithmetic progression. Improve the bounds for $W(k)$ - for example, prove that $W(k)^{1/k}\\to \\infty$.", "background": "When $p$ is prime Berlekamp \\cite{Be68} has proved $W(p+1)\\geq p2^p$. Gowers \\cite{Go01} has proved $ W(k) \\leq 2^{2^{2^{2^{2^{k+9}}}}}. $ The best general lower bound is $W(k)\\gg 2^k$, due to Kozik and Shabanov \\cite{KoSh16}.\nIn \\cite{Er81} Erd\\H{o}s further asks whether $W(k+1)/W(k)\\to \\infty$, or $W(k+1)-W(k)\\to \\infty$.\nIn \\cite{Er80} Erd\\H{o}s asks whether $W(k)/2^k\\to \\infty$, and offers \\$500 for a proof or disproof of $W(k)^{1/k}\\to \\infty$.\nReferences\n\n\n[Be68] Berlekamp, E. R., A construction for partitions which avoid long arithmetic progressions. Canad. Math. Bull. (1968), 409-414.\n\n[Er80] Erd\\H{o}s, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Go01] Gowers, W. T., A new proof of Szemer\\'{e}di's theorem. Geom. Funct. Anal. (2001), 465-588.\n\n[KoSh16] Kozik, Jakub and Shabanov, Dmitry, Improved algorithms for colorings of simple hypergraphs and\napplications. J. Combin. Theory Ser. B (2016), 312--332.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1952, "problem_number": "EP-141", "title": "Erdős Problem #141", "statement": "Let $k\\geq 3$. Are there $k$ consecutive primes in arithmetic progression?", "background": "Green and Tao \\cite{GrTa08} have proved that there must always exist some $k$ primes in arithmetic progression, but these need not be consecutive. Erd\\H{o}s called this conjecture 'completely hopeless at present'.\nThe existence of such progressions for small $k$ has been verified for $k\\leq 10$, see the Wikipedia page. It is open, even for $k=3$, whether there are infinitely many such progressions.\nSee also [219].\nThis is discussed in problem A6 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[GrTa08] Green, Ben and Tao, Terence, The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) (2008), 481-547.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L3\"\n},{", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1953, "problem_number": "EP-142", "title": "Erdős Problem #142", "statement": "Let $r_k(N)$ be the largest possible size of a subset of $\\{1,\\ldots,N\\}$ that does not contain any non-trivial $k$-term arithmetic progression. Prove an asymptotic formula for $r_k(N)$.", "background": "Erd\\H{o}s remarked this is 'probably unattackable at present'. In \\cite{Er97c} Erd\\H{o}s offered \\$1000, but given that he elsewhere offered \\$5000 just for (essentially) showing that $r_k(N)=o_k(N/\\log N)$, that value seems odd. In \\cite{Er81} he offers \\$10000, stating it is 'probably enormously difficult'.\nThe best known upper bounds for $r_k(N)$ are due to Kelley and Meka \\cite{KeMe23} for $k=3$, Green and Tao \\cite{GrTa17} for $k=4$, and Leng, Sah, and Sawhney \\cite{LSS24} for $k\\geq 5$. An asymptotic formula is still far out of reach, even for $k=3$.\nSee also [3] and [139].\nReferences\n\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Er97c] Erd\\H{o}s, Paul, Some of my favorite problems and results. The mathematics of Paul Erd\\H{o}s, I (1997), 47-67.\n\n[GrTa17] Green, Ben and Tao, Terence, New bounds for Szemer\\'{e}di's theorem, III: a polylogarithmic bound for $r_4(N)$. Mathematika (2017), 944-1040.\n\n[KeMe23] Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).\n\n[LSS24] Leng, J., Sah, A. and Sawhney, M., Improved bounds for Szemer\\'{e}di's theorem. arXiv:2402.17995 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1954, "problem_number": "EP-143", "title": "Erdős Problem #143", "statement": "Let $A\\subset (1,\\infty)$ be a countably infinite set such that for all $x\neq y\\in A$ and integers $k\\geq 1$ we have $ \\lvert kx -y\\rvert \\geq 1. $ Does this imply that $A$ is sparse? In particular, does this imply that $ \\sum_{x\\in A}\\frac{1}{x\\log x}<\\infty $ or $ \\sum_{\\substack{x 0$ is some absolute constant and $c_0=1.26408\\cdots$ is the 'Vardi constant'. The lower bound is due to Konyagin \\cite{Ko14} and the upper bound to Elsholtz and Planitzer \\cite{ElPl21}.\nReferences\n\n\n[ElPl21] Elsholtz, Christian and Planitzer, Stefan, Sums of four and more unit fractions and approximate parametrizations. Bull. Lond. Math. Soc. (2021), 695-709.\n\n[Ko14] Konyagin, S. V., Double exponential lower bound for the number of representations of unity by Egyptian fractions. Math. Notes (2014), 277-281.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1958, "problem_number": "EP-149", "title": "Erdős Problem #149", "statement": "Let $G$ be a graph with maximum degree $\\Delta$. Is $G$ the union of at most $\\tfrac{5}{4}\\Delta^2$ sets of strongly independent edges (sets such that the induced subgraph is the union of vertex-disjoint edges)?", "background": "Asked by Erd\\H{o}s and Ne\\v{s}et\\v{r}il in 1985 (see \\cite{FGST89}). This is equivalent to asking whether the chromatic number of the square of the line graph $L(G)^2$ is at most $\\frac{5}{4}\\Delta^2$.\nThis bound would be the best possible, as witnessed by a blowup of $C_5$. The minimum number of such sets required is sometimes called the strong chromatic index of $G$.\nThe weaker conjecture that there exists some $c>0$ such that $(2-c)\\Delta^2$ sets suffice was proved by Molloy and Reed \\cite{MoRe97}, who proved that $1.998\\Delta^2$ sets suffice (for $\\Delta$ sufficiently large). This was improved to $1.93\\Delta^2$ by Bruhn and Joos \\cite{BrJo18} and to $1.835\\Delta^2$ by Bonamy, Perrett, and Postle \\cite{BPP22}. The best bound currently available is $ 1.772\\Delta^2, $ proved by Hurley, de Joannis de Verclos, and Kang \\cite{HJK22}. Mahdian has, in their Masters' thesis, proved an upper bound of $(2+o(1))\\frac{\\Delta^2}{\\log \\Delta}$ under the additional assumption that $G$ is $C_4$-free.\nErd\\H{o}s and Ne\\v{s}et\\v{r}il also asked the easier problem of whether $G$ containing at least $\\tfrac{5}{4}\\Delta^2$ many edges implies $G$ containing two strongly independent edges. This was proved by Chung, Gy\\'{a}rf\\'{a}s, Tuza, and Trotter \\cite{CGTT90}.\nIt is still open even whether the clique number of $L(G)^2$ at most $\\frac{5}{4}\\Delta^2$. Let $\\omega=\\omega(L(G)^2)$ be this clique number. \\'{S}leszy\\'{n}ska-Nowak \\cite{Sl15} proved $\\omega \\leq \\frac{3}{2}\\Delta^2$. Faron and Postle \\cite{FaPo19} proved $\\omega\\leq \\frac{4}{3}\\Delta^2$. Cames van Batenburg, Kang, and Pirot \\cite{CKP20} have proved $\\omega\\leq \\frac{5}{4}\\Delta^2$ under the additional assumption that $G$ is triangle-free (and $\\omega\\leq \\Delta^2$ if $G$ is $C_5$-free).\nReferences\n\n\n[BPP22] Bonamy, Marthe and Perrett, Thomas and Postle, Luke, Colouring graphs with sparse neighbourhoods: bounds and\napplications. J. Combin. Theory Ser. B (2022), 278-317.\n\n[BrJo18] Bruhn, Henning and Joos, Felix, A stronger bound for the strong chromatic index. Combin. Probab. Comput. (2018), 21-43.\n\n[CGTT90] Chung, F. R. K. and Gy\\'arf\\'as, A. and Tuza, Z. and Trotter,\nW. T., The maximum number of edges in {$2K_2$}-free graphs of bounded\ndegree. Discrete Math. (1990), 129--135.\n\n[CKP20] Cames van Batenburg, Wouter and Kang, Ross J. and Pirot,\nFran\\c cois, Strong cliques and forbidden cycles. Indag. Math. (N.S.) (2020), 64--82.\n\n[FGST89] Faudree, R. J. and Gy\\'{a}rf\\'{a}s, A. and Schelp, R. H. and Tuza,\nZs., Induced matchings in bipartite graphs. Discrete Math. (1989), 83-87.\n\n[FaPo19] Faron, Maxime and Postle, Luke, On the clique number of the square of a line graph and its\nrelation to maximum degree of the line graph. J. Graph Theory (2019), 261--274.\n\n[HJK22] Hurley, Eoin and de Joannis de Verclos, R\\'{e}mi and Kang, Ross\nJ., An improved procedure for colouring graphs of bounded local\ndensity. Adv. Comb. (2022), Paper No. 7, 33.\n\n[MoRe97] Molloy, Michael and Reed, Bruce, A bound on the strong chromatic index of a graph. J. Combin. Theory Ser. B (1997), 103-109.\n\n[Sl15] No reference found.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1959, "problem_number": "EP-151", "title": "Erdős Problem #151", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ on at least two vertices (sometimes called the clique transversal number).\nLet $H(n)$ be maximal such that every triangle-free graph on $n$ vertices contains an independent set on $H(n)$ vertices.\nIf $G$ is a graph on $n$ vertices then is $ \\tau(G)\\leq n-H(n)? $ ", "background": "It is easy to see that $\\tau(G) \\leq n-\\sqrt{n}$. Note also that if $G$ is triangle-free then trivially $\\tau(G)\\leq n-H(n)$.\nThis is listed in \\cite{Er88} as a problem of Erd\\H{o}s and Gallai, who were unable to make progress even assuming $G$ is $K_4$-free. There Erd\\H{o}s remarked that this conjecture is 'perhaps completely wrongheaded'.\nIt later appeared as Problem 1 in \\cite{EGT92}.\nThe general behaviour of $\\tau(G)$ is the subject of [610].\nReferences\n\n\n[EGT92] Erd\\H{o}s, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289.\n\n[Er88] Erd\\H{o}s, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1960, "problem_number": "EP-152", "title": "Erdős Problem #152", "statement": "For any $M\\geq 1$, if $A\\subset \\mathbb{N}$ is a sufficiently large finite Sidon set then there are at least $M$ many $a\\in A+A$ such that $a+1,a-1\not\\in A+A$.", "background": "There may even be $\\gg \\lvert A\\rvert^2$ many such $a$. A similar question can be asked for truncations of infinite Sidon sets.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1961, "problem_number": "EP-153", "title": "Erdős Problem #153", "statement": "Let $A$ be a finite Sidon set and $A+A=\\{s_1<\\cdots0$ such that\n$$R(C_4,K_n) \\ll n^{2-c}.$$", "background": "The current bounds are $ \\frac{n^{3/2}}{(\\log n)^{3/2}}\\ll R(C_4,K_n)\\ll \\frac{n^2}{(\\log n)^2}. $ The upper bound is due to Szemer\\'{e}di (mentioned in \\cite{EFRS78}), and the lower bound is due to Spencer \\cite{Sp77}.\nThis problem is #17 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[EFRS78] Erd\\H{o}s, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., On cycle-complete graph Ramsey numbers. J. Graph Theory (1978), 53-64.\n\n[Sp77] Spencer, J., Asymptotic lower bounds for Ramsey functions. Discrete Math. (1977), 69-76.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1966, "problem_number": "EP-160", "title": "Erdős Problem #160", "statement": "Let $h(N)$ be the smallest $k$ such that $\\{1,\\ldots,N\\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate $h(N)$.", "background": "Investigated by Erd\\H{o}s and Freud. This has been discussed on MathOverflow, where LeechLattice shows $ h(N) \\ll N^{2/3}. $ In the comments of this site Hunter improves this to $ h(N) \\ll N^{\\frac{\\log 3}{\\log 22}+o(1)} $ (note $\\frac{\\log 3}{\\log 22}\\approx 0.355$).\nThe observation of Zach Hunter in that question coupled with recent progress on the size of subsets without three-term arithmetic progression (see \\cite{BlSi23} which improves slightly on the bounds due to Kelley and Meka \\cite{KeMe23}) imply that $ h(N) \\gg \\exp(c(\\log N)^{1/9}) $ for some $c>0$.\nReferences\n\n\n[BlSi23] T. F. Bloom and O. Sisask, An improvement to the Kelley-Meka bounds on three-term arithmetic progressions. arXiv:2309.02353 (2023).\n\n[KeMe23] Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1967, "problem_number": "EP-161", "title": "Erdős Problem #161", "statement": "Let $\\alpha\\in[0,1/2)$ and $n,t\\geq 1$. Let $F^{(t)}(n,\\alpha)$ be the smallest $m$ such that we can $2$-colour the edges of the complete $t$-uniform hypergraph on $n$ vertices such that if $X\\subseteq [n]$ with $\\lvert X\\rvert \\geq m$ then there are at least $\\alpha \\binom{\\lvert X\\rvert}{t}$ many $t$-subsets of $X$ of each colour.\nFor fixed $n,t$ as we change $\\alpha$ from $0$ to $1/2$ does $F^{(t)}(n,\\alpha)$ increase continuously or are there jumps? Only one jump?", "background": "For $\\alpha=0$ this is the usual Ramsey function.\nA conjecture of Erd\\H{o}s, Hajnal, and Rado (see [562]) implies that $ F^{(t)}(n,0)\\asymp \\log_{t-1} n $ and results of Erd\\H{o}s and Spencer imply that $ F^{(t)}(n,\\alpha) \\gg_\\alpha (\\log n)^{\\frac{1}{t-1}} $ for all $\\alpha>0$, and a similar upper bound holds for $\\alpha$ close to $1/2$.\nErd\\H{o}s said in \\cite{Er90b}: 'If I can hazard a guess completely unsupported by evidence, I am afraid that the jump occurs all in one step at $0$. It would be much more interesting if my conjecture would be wrong and perhaps there is some hope for this for $t>3$. I know nothing and offer \\$500 to anybody who can clear up this mystery.'\nConlon, Fox, and Sudakov \\cite{CFS11} have proved that, for any fixed $\\alpha>0$, $ F^{(3)}(n,\\alpha) \\ll_\\alpha \\sqrt{\\log n}. $ Coupled with the lower bound above, this implies that there is only one jump for fixed $\\alpha$ when $t=3$, at $\\alpha=0$.\nFor all $\\alpha>0$ it is known that $ F^{(t)}(n,\\alpha)\\gg_t (\\log n)^{c_\\alpha}. $ See also [563] for more on the case $t=2$.\nThis problem is #40 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[CFS11] Conlon, David and Fox, Jacob and Sudakov, Benny, Large almost monochromatic subsets in hypergraphs. Israel J. Math. (2011), 423--432.\n\n[Er90b] Erd\\H{o}s, Paul, Problems and results on graphs and hypergraphs: similarities and differences. Mathematics of Ramsey theory (1990), 12-28.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1968, "problem_number": "EP-162", "title": "Erdős Problem #162", "statement": "Let $\\alpha>0$ and $n\\geq 1$. Let $F(n,\\alpha)$ be the largest $k$ such that there exists some 2-colouring of the edges of $K_n$ in which any induced subgraph $H$ on at least $k$ vertices contains more than $\\alpha\\binom{\\lvert H\\rvert}{2}$ many edges of each colour.\nProve that for every fixed $0\\leq \\alpha \\leq 1/2$, as $n\\to\\infty$, $ F(n,\\alpha)\\sim c_\\alpha \\log n $ for some constant $c_\\alpha$.", "background": "It is easy to show with the probabilistic method that there exist $c_1(\\alpha),c_2(\\alpha)$ such that $ c_1(\\alpha)\\log n < F(n,\\alpha) < c_2(\\alpha)\\log n. $ \",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1969, "problem_number": "EP-165", "title": "Erdős Problem #165", "statement": "Give an asymptotic formula for $R(3,k)$.", "background": "It is known that there exists some constant $c>0$ such that for large $k$ $ (c+o(1))\\frac{k^2}{\\log k}\\leq R(3,k) \\leq (1+o(1))\\frac{k^2}{\\log k}. $ The lower bound is due to Kim \\cite{Ki95}, the upper bound is due to Shearer \\cite{Sh83}, improving an earlier bound of Ajtai, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AKS80}.\nThe value of $c$ in the lower bound has seen a number of improvements. Kim's original proof gave $c\\geq 1/162$. The bound $c\\geq 1/4$ was proved independently by Bohman and Keevash \\cite{BoKe21} and Pontiveros, Griffiths and Morris \\cite{PGM20}. The latter collection of authors conjecture that this lower bound is the true order of magnitude.\nThis was, however, improved by Campos, Jenssen, Michelen, and Sahasrabudhe \\cite{CJMS25} to $c\\geq 1/3$, and further by Hefty, Horn, King, and Pfender \\cite{HHKP25} to $c\\geq 1/2$. Both of these papers conjecture that $c=1/2$ is the correct asymptotic.\nSee also [544], and [986] for the general case. See [1013] for a related function.\nReferences\n\n\n[AKS80] Ajtai, Mikl\\'{o}s and Koml\\'{o}s, J\\'{a}nos and Szemer\\'{e}di, Endre, A note on Ramsey numbers. J. Combin. Theory Ser. A (1980), 354-360.\n\n[BoKe21] Bohman, Tom and Keevash, Peter, Dynamic concentration of the triangle-free process. Random Structures Algorithms (2021), 221-293.\n\n[CJMS25] M. Campos, M. Jenssen, M. Michelen, and J. Sahasrabudhe, A new lower bound for the Ramsey numbers $R(3,k)$. arXiv:2505.13371 (2025).\n\n[HHKP25] Z. Hefty, P. Horn, D. King, and F. Pfender, Improving $R(3,k)$ in just two bites. arXiv:2510.19718 (2025).\n\n[Ki95] Kim, J. H., The Ramsey number $R(3,t)$ has order of magnitude $t^2/\\log t$. Random Structures and Algorithms (1995), 173-207.\n\n[PGM20] Fiz Pontiveros, Gonzalo and Griffiths, Simon and Morris, Robert, The triangle-free process and the Ramsey number $R(3,k)$. Mem. Amer. Math. Soc. (2020), v+125.\n\n[Sh83] Shearer J., A note on the independence number of triangle-free graphs. Discrete Math. (1983), 83-87.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1970, "problem_number": "EP-168", "title": "Erdős Problem #168", "statement": "Let $F(N)$ be the size of the largest subset of $\\{1,\\ldots,N\\}$ which does not contain any set of the form $\\{n,2n,3n\\}$. What is $ \\lim_{N\\to \\infty}\\frac{F(N)}{N}? $ Is this limit irrational?", "background": "This limit was proved to exist by Graham, Spencer, and Witsenhausen \\cite{GSW77}, who showed it is equal to $ \\frac{1}{3}\\sum_{k\\in K}\\frac{1}{d_k}, $ where $d_1f(k-1)$, where $f$ counts the largest subset of $\\{d_1,\\ldots,d_k\\}$ that avoids $\\{n,2n,3n\\}$.\nSimilar questions can be asked for the density or upper density of infinite sets without such configurations.\nThe limit can be estimated by elementary arguments (see the comments). Eberhard has used the formula of \\cite{GSW77} mentioned above to calculate the value of the limit as $ 0.800965\\cdots. $ This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[GSW77] Graham, R. and Spencer, J. and Witsenhausen, H., On Extremal Density Theorems for Linear Forms. Number Theory and Algebra (1977).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1971, "problem_number": "EP-169", "title": "Erdős Problem #169", "statement": "Let $k\\geq 3$ and $f(k)$ be the supremum of $\\sum_{n\\in A}\\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a $k$-term arithmetic progression. Estimate $f(k)$.\nIs $ \\lim_{k\\to \\infty}\\frac{f(k)}{\\log W(k)}=\\infty $ where $W(k)$ is the van der Waerden number?", "background": "Berlekamp \\cite{Be68} proved $f(k) \\geq \\frac{\\log 2}{2}k$. Gerver \\cite{Ge77} proved $ f(k) \\geq (1-o(1))k\\log k. $ It is trivial that $ \\frac{f(k)}{\\log W(k)}\\geq \\frac{1}{2}, $ but improving the right-hand side to any constant $>1/2$ is open.\nGerver also proved (see the comments for an alternative argument of Tao) that [3] is equivalent to $f(k)$ being finite for all $k$.\nThe current record for $f(3)$ is $f(3)\\geq 3.00849$, due to Wr\\'{o}blewski \\cite{Wr84}. Walker \\cite{Wa25} proved $f(4)\\geq 4.43975$.\nWalker \\cite{Wa25} has shown that it suffices to consider Kempner sets (that is, sets of integers defined as all those whose base $b$ digits are contained in some $S\\subset \\{0,\\ldots,b-1\\}$ for fixed $b$ and $S$), in the sense that for any $k\\geq 3$ and $\\epsilon>0$ there is a Kempner set $A$ lacking $k$-term arithmetic progressions such that $ \\sum_{n\\in A}\\frac{1}{n}\\geq f(k)-\\epsilon. $ \nReferences\n\n\n[Be68] Berlekamp, E. R., A construction for partitions which avoid long arithmetic progressions. Canad. Math. Bull. (1968), 409-414.\n\n[Ge77] Gerver, Joseph L., The sum of the reciprocals of a set of integers with no\narithmetic progression of {$k$} terms. Proc. Amer. Math. Soc. (1977), 211--214.\n\n[Wa25] A. Walker, Integer sets of large harmonic sum which avoid long arithmetic progressions. arXiv:2203.06045 (2025).\n\n[Wr84] No reference found.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1972, "problem_number": "EP-170", "title": "Erdős Problem #170", "statement": "Let $F(N)$ be the smallest possible size of $A\\subset \\{0,1,\\ldots,N\\}$ such that $\\{0,1,\\ldots,N\\}\\subset A-A$. Find the value of $ \\lim_{N\\to \\infty}\\frac{F(N)}{N^{1/2}}. $ ", "background": "The Sparse Ruler problem. R\\'{e}dei asked whether this limit exists, which was proved by Erd\\H{o}s and G\\'{a}l \\cite{ErGa48}. Bounds on the limit were improved by Leech \\cite{Le56}. The limit is known to be in the interval $[1.56,\\sqrt{3}]$. The lower bound is due to Leech \\cite{Le56}, the upper bound is due to Wichmann \\cite{Wi63}. Computational evidence by Pegg \\cite{Pe20} suggests that the upper bound is the truth. A similar question can be asked without the restriction $A\\subset \\{0,1,\\ldots,N\\}$.\nReferences\n\n\n[ErGa48] Erd\\H{o}s, P. and G\\'{a}l, I., On the representation of $1,2,\\ldots,N$ by differences. Nederl. Akad. Wetensch., Proc. (1948), 1155-1158.\n\n[Le56] Leech, J., On the representation of $1,2,\\ldots,n$ by differences. J. London Math. Soc. (1956), 160-169.\n\n[Pe20] Pegg, E., Hitting All the Marks: Exploring New Bounds for Sparse Rulers and a Wolfram Language Proof. https://blog.wolfram.com/2020/02/12/hitting-all-the-marks-exploring-new-bounds-for-sparse-rulers-and-a-wolfram-language-proof/ (2020).\n\n[Wi63] Wichmann, B., A note on restricted difference bases. J. London Math. Soc. (1963), 465-466.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1973, "problem_number": "EP-172", "title": "Erdős Problem #172", "statement": "Is it true that in any finite colouring of $\\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?", "background": "First asked by Hindman. Hindman \\cite{Hi80} has proved this is false (with 7 colours) if we ask for an infinite $A$. In \\cite{Er77c} Erd\\H{o}s asks about the case for an infinite $A$ with just $2$ colours.\nMoreira \\cite{Mo17} has proved that in any finite colouring of $\\mathbb{N}$ there exist $x,y$ such that $\\{x,x+y,xy\\}$ are all the same colour.\nAlweiss \\cite{Al23} has proved that, in any finite colouring of $\\mathbb{Q}\\backslash \\{0\\}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour. Bowen and Sabok \\cite{BoSa22} had proved this earlier for the first non-trivial case of $\\lvert A\\rvert=2$.\nReferences\n\n\n[Al23] R. Alweiss, Hindman's conjecture over the rationals. arXiv:2307.08901 (2023).\n\n[BoSa22] M. Bowen and M. Sabok, Monochromatic Sums and Products in the Rationals. arXiv:2210.12290 (2022).\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[Hi80] Hindman, Neil, Partitions and sums and products-two counterexamples. J. Combin. Theory Ser. A (1980), 113-120.\n\n[Mo17] Moreira, J., Monochromatic sums and products in $\\mathbbN$. Ann. Math. (2017), 1069-1090.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1974, "problem_number": "EP-173", "title": "Erdős Problem #173", "statement": "In any $2$-colouring of $\\mathbb{R}^2$, for all but at most one triangle $T$, there is a monochromatic congruent copy of $T$.", "background": "For some colourings a single equilateral triangle has to be excluded, considering the colouring by alternating strips. Shader \\cite{Sh76} has proved this is true if we just consider a single right-angled triangle.\nReferences\n\n\n[Sh76] Shader, L., All right triangles are Ramsey in $\\mathbbE^2$!. J. Comb. Th. A (1976), 385-389.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1975, "problem_number": "EP-174", "title": "Erdős Problem #174", "statement": "A finite set $A\\subset \\mathbb{R}^n$ is called Ramsey if, for any $k\\geq 1$, there exists some $d=d(A,k)$ such that in any $k$-colouring of $\\mathbb{R}^d$ there exists a monochromatic copy of $A$. Characterise the Ramsey sets in $\\mathbb{R}^n$.", "background": "Erd\\H{o}s, Graham, Montgomery, Rothschild, Spencer, and Straus \\cite{EGMRSS73} proved that every Ramsey set is 'spherical': it lies on the surface of some sphere. Graham has conjectured that every spherical set is Ramsey. Leader, Russell, and Walters \\cite{LRW12} have alternatively conjectured that a set is Ramsey if and only if it is 'subtransitive': it can be embedded in some higher-dimensional set on which rotations act transitively.\nSets known to be Ramsey include vertices of $k$-dimensional rectangles \\cite{EGMRSS73}, non-degenerate simplices \\cite{FrRo90}, trapezoids \\cite{Kr92}, and regular polygons/polyhedra \\cite{Kr91}.\nReferences\n\n\n[EGMRSS73] Erd\\H{o}s, P. and Graham, R. L. and Montgomery, P. and Rothschild, B. L. and Spencer, J. and Straus, E. G., Euclidean Ramsey Theorems I. J. Comb. Th. A (1973), 341-363.\n\n[FrRo90] Frankl, P. and R\"{o}dl, V., A partition property of simplices in Euclidean space. J. Amer. Math. Soc. (1990), 1-7.\n\n[Kr91] K\\v{r}\\'{\\i}\\v{z}, Igor, Permutation groups in Euclidean Ramsey theory. Proc. Amer. Math. Soc. (1991), 899-907.\n\n[Kr92] K\\v{r}\\'{\\i}\\v{z}, Igor, All trapezoids are Ramsey. Discrete Math. (1992), 59-62.\n\n[LRW12] Leader, Imre and Russell, Paul A. and Walters, Mark, Transitive sets in Euclidean Ramsey theory. J. Combin. Theory Ser. A (2012), 382-396.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1976, "problem_number": "EP-176", "title": "Erdős Problem #176", "statement": "Let $N(k,\\ell)$ be the minimal $N$ such that for any $f:\\{1,\\ldots,N\\}\\to\\{-1,1\\}$ there must exist a $k$-term arithmetic progression $P$ such that $ \\left\\lvert \\sum_{n\\in P}f(n)\\right\\rvert\\geq \\ell. $ Find good upper bounds for $N(k,\\ell)$. Is it true that for any $c>0$ there exists some $C>1$ such that $ N(k,ck)\\leq C^k? $ What about $ N(k,2)\\leq C^k $ or $ N(k,\\sqrt{k})\\leq C^k? $ ", "background": "When $\\ell=k$ this is the van der Waerden number $W(k)$ (see [138]). Spencer \\cite{Sp73} has proved that if $k=2^tm$ with $m$ odd then $ N(k,1)=2^t(k-1)+1. $ Erd\\H{o}s and Graham write that 'no decent bound' is known even for $N(k,2)$.\nErd\\H{o}s \\cite{Er63d} proved that, for every $c>0$, $ N(k,ck)> (1+\\alpha_c)^k $ where $\\alpha_c\\to 0$ as $c\\to 0$ and $\\alpha_c\\to \\sqrt{2}-1$ as $c\\to 1$.\nReferences\n\n\n[Er63d] Erd\\H{o}s, P\\'al, On combinatorial questions connected with a theorem of\n{R}amsey and van der {W}aerden. Mat. Lapok (1963), 29--37.\n\n[Sp73] J. Spencer, Problems 185. Bull. Canad. Math. Soc. (1973), 185.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1977, "problem_number": "EP-177", "title": "Erdős Problem #177", "statement": "Find the smallest $h(d)$ such that the following holds. There exists a function $f:\\mathbb{N}\\to\\{-1,1\\}$ such that, for every $d\\geq 1$, $ \\max_{P_d}\\left\\lvert \\sum_{n\\in P_d}f(n)\\right\\rvert\\leq h(d), $ where $P_d$ ranges over all finite arithmetic progressions with common difference $d$.", "background": "Cantor, Erd\\H{o}s, Schreiber, and Straus \\cite{Er66} proved that $h(d)\\ll d!$ is possible. Van der Waerden's theorem implies that $h(d)\\to \\infty$. Beck \\cite{Be17} has shown that $h(d) \\leq d^{8+\\epsilon}$ is possible for every $\\epsilon>0$. Roth's famous discrepancy lower bound \\cite{Ro64} implies that $h(d)\\gg d^{1/2}$.\nReferences\n\n\n[Be17] Beck, J\\'{o}zsef, A discrepancy problem: balancing infinite dimensional vectors. Number theory-Diophantine problems, uniform distribution\nand applications (2017), 61-82.\n\n[Er66] Erd\\H{o}s, P\\'al, Remarks on number theory. {V}. {E}xtremal problems in number\ntheory. {II}. Mat. Lapok (1966), 135--155.\n\n[Ro64] Roth, K. F., Remark concerning integer sequences. Acta Arith. (1964), 257-260.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1978, "problem_number": "EP-180", "title": "Erdős Problem #180", "statement": "If $\\mathcal{F}$ is a finite set of finite graphs then $\\mathrm{ex}(n;\\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\\mathcal{F}$. Note that it is trivial that $\\mathrm{ex}(n;\\mathcal{F})\\leq \\mathrm{ex}(n;G)$ for every $G\\in\\mathcal{F}$.\nIs it true that, for every $\\mathcal{F}$, there exists $G\\in\\mathcal{F}$ such that $ \\mathrm{ex}(n;G)\\ll_{\\mathcal{F}}\\mathrm{ex}(n;\\mathcal{F})? $ ", "background": "A problem of Erd\\H{o}s and Simonovits.\nThis is trivially true if $\\mathcal{F}$ does not contain any bipartite graphs, since by the Erd\\H{o}s-Stone theorem if $H\\in\\mathcal{F}$ has minimal chromatic number $r\\geq 2$ then $ \\mathrm{ex}(n;H)=\\mathrm{ex}(n;\\mathcal{F})=\\left(\\frac{r-2}{r-1}+o(1)\\right)\\binom{n}{2}. $ Erd\\H{o}s and Simonovits observe that this is false for infinite families $\\mathcal{F}$, e.g. the family of all cycles.\nHunter has provided the following 'folklore counterexample': if $\\mathcal{F}=\\{H_1,H_2\\}$ where $H_1$ is a star and $H_2$ is a matching, both with at least two edges, then $\\mathrm{ex}(n;\\mathcal{F})\\ll 1$, but $\\mathrm{ex}(n;H_i)\\asymp n$ for $1\\leq i\\leq 2$. This conjecture may still hold for all other $\\mathcal{F}$.\nSee also [575].\nThis problem is #47 in Extremal Graph Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1979, "problem_number": "EP-181", "title": "Erdős Problem #181", "statement": "Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Prove that $ R(Q_n) \\ll 2^n. $ ", "background": "Conjectured by Burr and Erd\\H{o}s, althouhg in \\cite{Er93} Erd\\H{o}s says the behaviour of $R(Q_n)$ was considered by himself and S\\'{o}s, who could not decide whether $R(Q_n)/2^n\\to \\infty$ or not.\nThe trivial bound is $ R(Q_n) \\leq R(K_{2^n})\\leq C^{2^n} $ for some constant $C>1$. This was improved a number of times; the current best bound due to Tikhomirov \\cite{Ti22} is $ R(Q_n)\\ll 2^{(2-c)n} $ for some small constant $c>0$. (In fact $c\\approx 0.03656$ is permissible.)\nThis problem is #20 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[Ti22] Tikhomirov, K., A remark on the Ramsey number of the hypercube. arXiv:2208.14568 (2022).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1980, "problem_number": "EP-183", "title": "Erdős Problem #183", "statement": "Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours then there must exist a monochromatic triangle. Determine $ \\lim_{k\\to \\infty}R(3;k)^{1/k}. $ ", "background": "Erd\\H{o}s offers \\$100 for showing that this limit is finite. An easy pigeonhole argument shows that $ R(3;k)\\leq 2+k(R(3;k-1)-1), $ from which $R(3;k)\\leq \\lceil e k!\\rceil$ immediately follows. The best-known upper bounds are all of the form $ck!+O(1)$, and arise from this type of inductive relationship and computational bounds for $R(3;k)$ for small $k$. The best-known lower bound (coming from lower bounds for Schur numbers) is $ R(3,k)\\geq (380)^{k/5}-O(1), $ due to Ageron, Casteras, Pellerin, Portella, Rimmel, and Tomasik \\cite{ACPPRT21} (improving previous bounds of Exoo \\cite{Ex94} and Fredricksen and Sweet \\cite{FrSw00}). Note that $380^{1/5}\\approx 3.2806$.\nSee also [483].\nThis problem is #21 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[ACPPRT21] R. Ageron, P. Casteras, T. Pellerin, Y. Portella, A. Rimmel, and J. Tomasik, New lower bounds for Schur and weak Schur numbers. arXiv:2112.03175 (2021).\n\n[Ex94] Exoo, G., A lower bound for Schur numbers and multicolor Ramsey numbers. Electronic J. of Combinatorics (1994).\n\n[FrSw00] Fredricksen, Harold and Sweet, Melvin M., Symmetric sum-free partitions and lower bounds for {S}chur\nnumbers. Electron. J. Combin. (2000), Research Paper 32, 9.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1981, "problem_number": "EP-184", "title": "Erdős Problem #184", "statement": "Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges.", "background": "Conjectured by Erd\\H{o}s and Gallai, who proved that $O(n\\log n)$ many cycles and edges suffices. The graph $K_{3,n-3}$ shows that at least $(1+c)n$ many cycles and edges are required, for some constant $c>0$. In \\cite{Er71} Erd\\H{o}s suggests that only $n-1$ many cycles and edges are required if we do not require them to be edge-disjoint.\nThe best bound available is due to Buci\\'{c} and Montgomery \\cite{BM22}, who prove that $O(n\\log^*n)$ many cycles and edges suffice, where $\\log^*$ is the iterated logarithm function.\nConlon, Fox, and Sudakov \\cite{CFS14} proved that $O_\\epsilon(n)$ cycles and edges suffice if $G$ has minimum degree at least $\\epsilon n$, for any $\\epsilon>0$.\nSee also [583] for an analogous problem decomposing into paths, and [1017] for decomposing into complete graphs.\nReferences\n\n\n[BM22] Buci\\'C, M. and Montgomery, R., Towards the Erd\\H{o}s-Gallai Cycle Decomposition Conjecture. arXiv:2211.07689 (2022).\n\n[CFS14] Conlon, David and Fox, Jacob and Sudakov, Benny, Cycle packing. Random Structures Algorithms (2014), 608-626.\n\n[Er71] Erd\\H{o}s, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc.\nConf., Oxford, 1969) (1971), 97-109.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1982, "problem_number": "EP-187", "title": "Erdős Problem #187", "statement": "Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common difference $d$ of length $f(d)$ for infinitely many $d$.", "background": "Originally asked by Cohen. Erd\\H{o}s observed that colouring according to whether $\\{ \\sqrt{2}n\\}<1/2$ or not implies $f(d) \\ll d$ (using the fact that $\\|\\sqrt{2}q\\| \\gg 1/q$ for all $q$, where $\\|x\\|$ is the distance to the nearest integer). Beck \\cite{Be80} has improved this using the probabilistic method, constructing a colouring that shows $f(d)\\leq (1+o(1))\\log_2 d$. Van der Waerden's theorem implies $f(d)\\to \\infty$ is necessary.\nReferences\n\n\n[Be80] Beck, J\\'{o}zsef, A remark concerning arithmetic progressions. J. Combin. Theory Ser. A (1980), 376-379.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1983, "problem_number": "EP-188", "title": "Erdős Problem #188", "statement": "What is the smallest $k$ such that $\\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic progression of blue points with distance $1$?", "background": "Erd\\H{o}s, Graham, Montgomery, Rothschild, Spencer, and Straus \\cite{EGMRSS75} proved $k\\geq 5$. Tsaturian \\cite{Ts17} improved this to $k\\geq 6$. Erd\\H{o}s and Graham claim that $k\\leq 10000000$ ('more or less'), but give no proof.\nErd\\H{o}s and Graham asked this with just any $k$-term arithmetic progression in blue (not necessarily with distance $1$), but Alon has pointed out that in fact no such $k$ exists: in any red/blue colouring of the integer points on a line either there are two red points distance $1$ apart, or else the set of blue points and the same set shifted by $1$ cover all integers, and hence by van der Waerden's theorem there are arbitrarily long blue arithmetic progressions.\nIt seems most likely, from context, that Erd\\H{o}s and Graham intended to restrict the blue arithmetic progression to have distance $1$ (although they do not write this restriction in their papers).\nReferences\n\n\n[EGMRSS75] Erd\\H{o}s, P. and Graham, R. L. and Montgomery, P. and\nRothschild, B. L. and Spencer, J. and Straus, E. G., Euclidean {R}amsey theorems. {II}. (1975), 529--557.\n\n[Ts17] Tsaturian, Sergei, A {E}uclidean {R}amsey result in the plane. Electron. J. Combin. (2017), Paper No. 4.35, 9.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1984, "problem_number": "EP-190", "title": "Erdős Problem #190", "statement": "Let $H(k)$ be the smallest $N$ such that in any finite colouring of $\\{1,\\ldots,N\\}$ (into any number of colours) there is always either a monochromatic $k$-term arithmetic progression or a rainbow arithmetic progression (i.e. all elements are different colours). Estimate $H(k)$. Is it true that $ H(k)^{1/k}/k \\to \\infty $ as $k\\to\\infty$?", "background": "This type of problem belongs to 'canonical' Ramsey theory. The existence of $H(k)$ follows from Szemer\\'{e}di's theorem, and it is easy to show that $H(k)^{1/k}\\to\\infty$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1985, "problem_number": "EP-193", "title": "Erdős Problem #193", "statement": "Let $S\\subseteq \\mathbb{Z}^3$ be a finite set and let $A=\\{a_1,a_2,\\ldots,\\}\\subset \\mathbb{Z}^3$ be an infinite $S$-walk, so that $a_{i+1}-a_i\\in S$ for all $i$. Must $A$ contain three collinear points?", "background": "Originally conjectured by Gerver and Ramsey \\cite{GeRa79}, who showed that the answer is yes for $\\mathbb{Z}^2$, and for $\\mathbb{Z}^3$ that the largest number of collinear points can be bounded.\nReferences\n\n\n[GeRa79] Gerver, Joseph L. and Ramsey, L. Thomas, On certain sequences of lattice points. Pacific J. Math. (1979), 357-363.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1986, "problem_number": "EP-195", "title": "Erdős Problem #195", "statement": "What is the largest $k$ such that in any permutation of $\\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1<\\cdotsj>k>l$ such that $x_i,x_j,x_k,x_l$ are an arithmetic progression?", "background": "Davis, Entringer, Graham, and Simmons \\cite{DEGS77} have shown that there must exist a monotone 3-term arithmetic progression and need not contain a 5-term arithmetic progression.\nSee also [194] and [195].\nReferences\n\n\n[DEGS77] Davis, J. A. and Entringer, R. C. and Graham, R. L. and\nSimmons, G. J., On permutations containing no long arithmetic progressions. Acta Arith. (1977/78), 81-90.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1988, "problem_number": "EP-197", "title": "Erdős Problem #197", "statement": "Can $\\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?", "background": "If three sets are allowed then this is possible.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1989, "problem_number": "EP-200", "title": "Erdős Problem #200", "statement": "Does the longest arithmetic progression of primes in $\\{1,\\ldots,N\\}$ have length $o(\\log N)$?", "background": "It follows from the prime number theorem that such a progression has length $\\leq(1+o(1))\\log N$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1990, "problem_number": "EP-201", "title": "Erdős Problem #201", "statement": "Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression. Determine the size of $G_k(N)$. How does it relate to $R_k(N)$, the size of the largest subset of $\\{1,\\ldots,N\\}$ without a $k$-term arithmetic progression? Is it true that $ \\lim_{N\\to \\infty}\\frac{R_3(N)}{G_3(N)}=1? $ ", "background": "First asked and investigated by Riddell \\cite{Ri69}. It is trivial that $G_k(N)\\leq R_k(N)$, and it is possible that $G_k(N) 0$, $ \\frac{N}{\\exp((\\log N)^{1/2+\\epsilon})} \\ll_\\epsilon f(N) < \\frac{N}{(\\log N)^c} $ for some $c>0$. Erd\\H{o}s believed the lower bound is closer to the truth.\nThese bounds were improved by Croot \\cite{Cr03b} who proved $ \\frac{N}{L(N)^{\\sqrt{2}+o(1)}}< f(N)<\\frac{N}{L(N)^{1/6-o(1)}}, $ where $L(N)=\\exp(\\sqrt{\\log N\\log\\log N})$. These bounds were further improved by Chen \\cite{Ch05} and then by de la Bret\\'{e}che, Ford, and Vandehey \\cite{BFV13} to $ \\frac{N}{L(N)^{1+o(1)}}0$ and large $n$, $ s_{n+1}-s_n \\ll_\\epsilon s_n^{\\epsilon}? $ Is it true that $ s_{n+1}-s_n \\leq (1+o(1))\\frac{\\pi^2}{6}\\frac{\\log s_n}{\\log\\log s_n}? $ ", "background": "Erd\\H{o}s \\cite{Er51} showed that there are infinitely many $n$ such that $ s_{n+1}-s_n > (1+o(1))\\frac{\\pi^2}{6}\\frac{\\log s_n}{\\log\\log s_n}, $ so this bound would be the best possible.\nIn \\cite{Er79} Erd\\H{o}s says perhaps $s_{n+1}-s_n \\ll \\log s_n$, but he is 'very doubtful'.\nFilaseta and Trifonov \\cite{FiTr92} proved an upper bound of $s_n^{1/5+o(1)}$. Pandey \\cite{Pa24} has improved this exponent to $1/5-c$ for some constant $c>0$.\nGranville \\cite{Gr98} showed that $s_{n+1}-s_n\\ll_\\epsilon s_n^\\epsilon$ for all $\\epsilon>0$ follows from the ABC conjecture.\nSee also [489] and [145]. A more general form of this problem is given in [1101].\nReferences\n\n\n[Er51] Erd\"{o}s, P., Some problems and results in elementary number theory. Publ. Math. Debrecen (1951), 103-109.\n\n[Er79] Erd\\H{o}s, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.\n\n[FiTr92] Filaseta, M. and Trifonov, O., On gaps between squarefree numbers II. J. London Math. Soc. (1992), 215-221.\n\n[Gr98] Granville, Andrew, {$ABC$} allows us to count squarefrees. Internat. Math. Res. Notices (1998), 991--1009.\n\n[Pa24] Pandey, M., Squarefree numbers in short intervals. arXiv:2401.13981 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1994, "problem_number": "EP-212", "title": "Erdős Problem #212", "statement": "Is there a dense subset of $\\mathbb{R}^2$ such that all pairwise distances are rational?", "background": "Conjectured by Ulam. Erd\\H{o}s believed there cannot be such a set. This problem is discussed in a blogpost by Terence Tao, in which he shows that there cannot be such a set, assuming the Bombieri-Lang conjecture. The same conclusion was independently obtained by Shaffaf \\cite{Sh18}.\nIndeed, Shaffaf and Tao actually proved that such a rational distance set must be contained in a finite union of real algebraic curves. Solymosi and de Zeeuw \\cite{SdZ10} then proved (unconditionally) that a rational distance set contained in a real algebraic curve must be finite, unless the curve contains a line or a circle.\nAscher, Braune, and Turchet \\cite{ABT20} observed that, combined, these facts imply that a rational distance set in general position must be finite (conditional on the Bombieri-Lang conjecture).\nIn \\cite{Er87b} Erd\\H{o}s mentions that Besicovitch conjectured that the limit points of a rational distance set cannot contain arbitrarily large convex sets.\nReferences\n\n\n[ABT20] Ascher, K. and Braune, L. and Turchet, A., The Erd\\H{o}s-Ulam problem, Lang's conjecture, and uniformity. arXiv:1901.02616 (2020).\n\n[Er87b] Erd\\H{o}s, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Si\\'{o}fok, 1985) (1987), 167-177.\n\n[SdZ10] Solymosi, Jozsef and de Zeeuw, Frank, On a question of Erd\\H{o}s and Ulam. Discrete Comput. Geom. (2010), 393-401.\n\n[Sh18] Shaffaf, Jafar, A solution of the Erd\\H{o}s-Ulam problem on rational\ndistance sets assuming the Bombieri-Lang conjecture. Discrete Comput. Geom. (2018), 283-293.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1995, "problem_number": "EP-213", "title": "Erdős Problem #213", "statement": "Let $n\\geq 4$. Are there $n$ points in $\\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers?", "background": "Anning and Erd\\H{o}s \\cite{AnEr45} proved there cannot exist an infinite such set. Harborth constructed such a set when $n=5$. The best construction to date, due to Kreisel and Kurz \\cite{KK08}, has $n=7$.\nAscher, Braune, and Turchet \\cite{ABT20} have shown that there is a uniform upper bound on the size of such a set, conditional on the Bombieri-Lang conjecture. Greenfeld, Iliopoulou, and Peluse \\cite{GIP24} have shown (unconditionally) that any such set must be very sparse, in that if $S\\subseteq [-N,N]^2$ has no three on a line and no four on a circle, and all pairwise distances integers, then $ \\lvert S\\rvert \\ll (\\log N)^{O(1)}. $ See also [130].\nReferences\n\n\n[ABT20] Ascher, K. and Braune, L. and Turchet, A., The Erd\\H{o}s-Ulam problem, Lang's conjecture, and uniformity. arXiv:1901.02616 (2020).\n\n[AnEr45] Anning, Norman H. and Erd\\H{o}s, Paul, Integral distances. Bull. Amer. Math. Soc. (1945), 598-600.\n\n[GIP24] Greenfeld, R. and Iliopoulou, M. and Peluse, S., On integer distance sets. arXiv:2401.10821 (2024).\n\n[KK08] Kreisel, Tobias and Kurz, Sascha, There are integral heptagons, no three points on a line, on four on a circle. Discrete Comput. Geom. (2008), 786-790.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1996, "problem_number": "EP-217", "title": "Erdős Problem #217", "statement": "For which $n$ are there $n$ points in $\\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and so that (in some ordering of the distances) the $i$th distance occurs $i$ times?", "background": "An example with $n=4$ is an isosceles triangle with the point in the centre. Erd\\H{o}s originally believed this was impossible for $n\\geq 5$, but Pomerance constructed a set with $n=5$ (see \\cite{Er83c} for a description), and Pal\\'{a}sti has proved such sets exist for all $n\\leq 8$.\nErd\\H{o}s believed this is impossible for all sufficiently large $n$. This would follow from $h(n)\\geq n$ for sufficiently large $n$, where $h(n)$ is as in [98].\nReferences\n\n\n[Er83c] Erd\\H{o}s, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1997, "problem_number": "EP-218", "title": "Erdős Problem #218", "statement": "Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$.", "background": "In \\cite{Er85c} Erd\\H{o}s also conjectures that $d_n=d_{n+1}=\\cdots=d_{n+k}$ is solvable for every $k$ (which is equivalent to $k$ consecutive primes in arithmetic progression, see [141]).\nReferences\n\n\n[Er85c] Erd\\H{o}s, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 1998, "problem_number": "EP-222", "title": "Erdős Problem #222", "statement": "Let $n_10$.\nThe sequence of values of $f(n)$ is A109925 on the OEIS.\nSee also [237].\nReferences\n\n\n[Er50] Erd\"{o}s, P., On integers of the form $2^k+p$ and some related problems. Summa Brasil. Math. (1950), 113-123.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[MiWe69] Mientka, Walter E. and Weitzenkamp, Roger C., On {$f$}-plentiful numbers. J. Combinatorial Theory (1969), 374--377.\n\n[Va73] Vaughan, R. C., Some applications of {M}ontgomery's sieve. J. Number Theory (1973), 64--79.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2002, "problem_number": "EP-238", "title": "Erdős Problem #238", "statement": "Let $c_1,c_2>0$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\\log x$ many consecutive primes $\\leq x$ such that the difference between any two is $>c_2$?", "background": "Erd\\H{o}s \\cite{Er49c} proved this is true for any $c_2>0$ if $c_1>0$ is sufficiently small (depending on $c_1$).\nReferences\n\n\n[Er49c] Erd\\H{o}s, P., On some applications of {B}run's method. Acta Univ. Szeged. Sect. Sci. Math. (1949), 57--63.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2003, "problem_number": "EP-241", "title": "Erdős Problem #241", "statement": "Let $f(N)$ be the maximum size of $A\\subseteq \\{1,\\ldots,N\\}$ such that the sums $a+b+c$ with $a,b,c\\in A$ are all distinct (aside from the trivial coincidences). Is it true that $ f(N)\\sim N^{1/3}? $ ", "background": "Originally asked to Erd\\H{o}s by Bose. Bose and Chowla \\cite{BoCh62} provided a construction proving one half of this, namely $ (1+o(1))N^{1/3}\\leq f(N). $ The best upper bound known to date is due to Green \\cite{Gr01}, $ f(N) \\leq ((7/2)^{1/3}+o(1))N^{1/3} $ (note that $(7/2)^{1/3}\\approx 1.519$).\nMore generally, Bose and Chowla conjectured that the maximum size of $A\\subseteq \\{1,\\ldots,N\\}$ with all $r$-fold sums distinct (aside from the trivial coincidences) then $ \\lvert A\\rvert \\sim N^{1/r}. $ This is known only for $r=2$ (see [30]).\nThis is discussed in problem C11 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[BoCh62] Bose, R. C. and Chowla, S., Theorems in the additive theory of numbers. Comment. Math. Helv. (1962/63), 141-147.\n\n[Gr01] Green, Ben, The number of squares and {$B_h[g]$} sets. Acta Arith. (2001), 365-390.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2004, "problem_number": "EP-243", "title": "Erdős Problem #243", "statement": "Let $1\\leq a_10. $ A sequence satisfying the reucrrence $a_n = a_{n-1}^2-a_{n-1}+1$ is known as Sylvester's sequence.\nDuverney \\cite{Du01} proved a weaker version of this problem: if $ \\sum_{n\\geq 0}\\left(\\frac{a_{n+1}}{a_n^2}-1\\right) $ converges then $\\sum \\frac{1}{a_n}$ is rational if and only if $ a_{n}=a_{n-1}^2-a_{n-1}+1 $ for all large $n$.\nReferences\n\n\n[Du01] Duverney, Daniel, Irrationality of fast converging series of rational numbers. J. Math. Sci. Univ. Tokyo (2001), 275--316.\n\n[ErSt64] Erd\\H{o}s, P. and Straus, E. G., On the irrationality of certain {A}hmes series. J. Indian Math. Soc. (N.S.) (1964), 129--133.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2005, "problem_number": "EP-244", "title": "Erdős Problem #244", "statement": "Let $C>1$. Does the set of integers of the form $p+\\lfloor C^k\\rfloor$, for some prime $p$ and $k\\geq 0$, have density $>0$?", "background": "Originally asked to Erd\\H{o}s by Kalm\\'{a}r. Erd\\H{o}s believed the answer is yes. Romanoff \\cite{Ro34} proved that the answer is yes if $C$ is an integer.\nDing \\cite{Di25} has proved that this is true for almost all $C>1$.\nReferences\n\n\n[Di25] Y. Ding, On a Romanoff type problem of Erd\\H{o}s and Kalm\\'{a}r. arXiv:2503.22700 (2025).\n\n[Ro34] Romanoff, N. P., \"{U}ber einige S\"Atze der additiven Zahlentheorie. Math. Ann. (1934), 668-678.\n\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2006, "problem_number": "EP-247", "title": "Erdős Problem #247", "statement": "Let $1\\leq a_1cn^2$ then $\\sum_{n=1}^\\infty \\frac{1}{2^{a_n}}$ is not the root of any quadratic polynomial'.\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[Er75c] Erd\\H{o}s, P., Some problems and results on the irrationality of the sum of infinite series. J. Math. Sci. (1975), 1-7 (1976).\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\n\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2007, "problem_number": "EP-249", "title": "Erdős Problem #249", "statement": "Is $ \\sum_n \\frac{\\phi(n)}{2^n} $ irrational? Here $\\phi$ is the Euler totient function.", "background": "The decimal expansion of this sum is A256936 on the OEIS.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2008, "problem_number": "EP-251", "title": "Erdős Problem #251", "statement": "Is $ \\sum \\frac{p_n}{2^n} $ irrational? (Here $p_n$ is the $n$th prime.)", "background": "Erd\\H{o}s \\cite{Er58b} proved that $\\sum \\frac{p_n^k}{n!}$ is irrational for every $k\\geq 1$.\nIn \\cite{Er88c} he further conjectures that $\\sum \\frac{p_n^k}{2^n}$ is irrational for every $k$, and that if $g_n\\geq 2$ and $g_n=o(p_n)$ then $ \\sum_{n=1}^\\infty \\frac{p_n}{g_1\\cdots g_n} $ is irrational. (The example $g_n=p_n+1$ shows that some condition on the growth of the $g_n$ is necessary here.)\nThe decimal expansion of this sum is A098990 on the OEIS.\nReferences\n\n\n[Er58b] Erd\\H{o}s, Paul, Sur certaines s\\'{e}ries \\`a{} valeur irrationnelle. Enseign. Math. (2) (1958), 93--100.\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2009, "problem_number": "EP-252", "title": "Erdős Problem #252", "statement": "Let $k\\geq 1$ and $\\sigma_k(n)=\\sum_{d\\mid n}d^k$. Is $ \\sum \\frac{\\sigma_k(n)}{n!} $ irrational?", "background": "This is known now for $1\\leq k\\leq 4$. The cases $k=1,2$ are reasonably straightforward, as observed by Erd\\H{o}s \\cite{Er52}. The case $k=3$ was proved independently by Schlage-Puchta \\cite{ScPu06} and Friedlander, Luca, and Stoiciu \\cite{FLC07}. The case $k=4$ was proved by Pratt \\cite{Pr22}.\nIt is known that this sum is irrational for all $k\\geq 1$ conditional on either Schinzel's conjecture (Schlage-Puchta \\cite{ScPu06}) or the prime tuples conjecture (Friedlander, Luca, and Stoiciu \\cite{FLC07}).\nThis is discussed in problem B14 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er52] Erd\\H{o}s, P., Problem 4493. Amer. Math. Monthly (1952), 557-558.\n\n[FLC07] Friedlander, J. B. and Luca, F. and Stoiciu, M., On the irrationality of a divisor function series. Integers (2007).\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Pr22] Pratt, K., The irrationality of a divisor function series of Erd\\H{o}s and Kac. arXiv:2209.11124 (2022).\n\n[ScPu06] Schlage-Puchta, J. C., The irrationality of a number theoretical series. Ramanujan J. (2006), 455-460.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2010, "problem_number": "EP-254", "title": "Erdős Problem #254", "statement": "Let $A\\subseteq \\mathbb{N}$ be such that $ \\lvert A\\cap [1,2x]\\rvert -\\lvert A\\cap [1,x]\\rvert \\to \\infty\\textrm{ as }x\\to \\infty $ and $ \\sum_{n\\in A} \\{ \\theta n\\}=\\infty $ for every $\\theta\\in (0,1)$, where $\\{x\\}$ is the distance of $x$ from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of $A$.", "background": "Cassels \\cite{Ca60} proved this under the alternative hypotheses $ \\lim \\frac{\\lvert A\\cap [1,2x]\\rvert -\\lvert A\\cap [1,x]\\rvert}{\\log\\log x}=\\infty $ and $ \\sum_{n\\in A} \\{ \\theta n\\}^2=\\infty $ for every $\\theta\\in (0,1)$.\nReferences\n\n\n[Ca60] Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2011, "problem_number": "EP-256", "title": "Erdős Problem #256", "statement": "Let $n\\geq 1$ and $f(n)$ be maximal such that for any integers $1\\leq a_1\\leq \\cdots \\leq a_n$ we have $ \\max_{\\lvert z\\rvert=1}\\left\\lvert \\prod_{i}(1-z^{a_i})\\right\\rvert\\geq f(n). $ Estimate $f(n)$ - in particular, is it true that there exists some constant $c>0$ such that $ \\log f(n) \\gg n^c? $ ", "background": "Erd\\H{o}s and Szekeres \\cite{ErSz59} proved that $\\lim f(n)^{1/n}=1$ and $f(n)>\\sqrt{2n}$. Erd\\H{o}s proved an upper bound of $\\log f(n) \\ll n^{1-c}$ for some constant $c>0$ with probabilistic methods. Atkinson \\cite{At61} showed that $\\log f(n) \\ll n^{1/2}\\log n$.\nThis was improved to $ \\log f(n) \\ll n^{1/3}(\\log n)^{4/3} $ by Odlyzko \\cite{Od82}.\nIf we denote by $f^*(n)$ the analogous quantity with the assumption that $a_1<\\cdots0 $ for some $\\epsilon>0$ then the above folklore result implies that $a_n$ is such an irrationality sequence.\nReferences\n\n\n[KoTa24] Kova\\vC, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2017, "problem_number": "EP-264", "title": "Erdős Problem #264", "statement": "Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\neq 0$ and $b_n\neq 0$ for all $n$) the sum $ \\sum \\frac{1}{a_n+b_n} $ is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence?", "background": "A possible definition of an 'irrationality sequence' (see also [262] and [263]). One example is $a_n=2^{2^n}$. In \\cite{ErGr80} they also ask whether such a sequence can have polynomial growth, but Erd\\H{o}s later retracted this in \\cite{Er88c}, claiming 'It is not hard to show that it cannot increase slower than exponentially'.\nKova\\v{c} and Tao \\cite{KoTa24} have proved that $2^n$ is not such an irrationality sequence. More generally, they prove that any strictly increasing sequence of positive integers such that $\\sum\\frac{1}{a_n}$ converges and $ \\liminf \\left(a_n^2\\sum_{k>n}\\frac{1}{a_k^2}\\right) >0 $ is not such an irrationality sequence. In particular, any strictly increasing sequence with $\\limsup a_{n+1}/a_n <\\infty$ is not such an irrationality sequence.\nOn the other hand, Kova\\v{c} and Tao do prove that for any function $F$ with $\\lim F(n+1)/F(n)=\\infty$ there exists such an irrationality sequence with $a_n\\sim F(n)$.\nReferences\n\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[KoTa24] Kova\\vC, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2018, "problem_number": "EP-265", "title": "Erdős Problem #265", "statement": "Let $1\\leq a_11$.\nIt remains open whether one can achieve $ \\limsup a_n^{1/2^n}>1. $ A folklore result states that $\\sum \\frac{1}{a_n}$ is irrational whenever $\\lim a_n^{1/2^n}=\\infty$, and hence such a sequence cannot grow faster than doubly exponentially - the remaining question is the precise exponent possible.\nReferences\n\n\n[KoTa24] Kova\\vC, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2019, "problem_number": "EP-267", "title": "Erdős Problem #267", "statement": "Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_11$. Must $ \\sum_k\\frac{1}{F_{n_k}} $ be irrational?", "background": "It may be sufficient to have $n_k/k\\to \\infty$. Good \\cite{Go74} and Bicknell and Hoggatt \\cite{BiHo76} have shown that $\\sum \\frac{1}{F_{2^n}}$ is irrational - in fact, $ \\sum \\frac{1}{F_{2^n}}=\\frac{7-\\sqrt{5}}{2}. $ Badea \\cite{Ba87} proved that $\\sum \\frac{1}{F_{2^n+1}}$ is irrational.\nThe sum $\\sum \\frac{1}{F_n}$ itself was proved to be irrational by Andr\\'{e}-Jeannin \\cite{An89}.\nThe main problem has been proved for $c\\geq 2$ by Badea \\cite{Ba93}. It remains open for $10$, $a_k\\leq (\\frac{1}{2}+\\epsilon)k^2$ for all sufficiently large $k$. van Doorn and Sothanaphan have noted in the comment section that Moy's proof can be upgraded to give a fully explicit result of $ a_k\\leq \\frac{(k-1)(k+2)}{2}+n $ for all $k\\geq 0$.\nIn general, sequences which begin with some initial segment and thereafter are continued in a greedy fashion to avoid three-term arithmetic progressions are known as Stanley sequences.\nReferences\n\n\n[Li90] S. Lindhurst, An investigation of several interesting sets of numbers generated by the greedy\nalgorithm. Senior thesis at Princeton University (1990).\n\n[Mo11] Moy, Richard A., On the growth of the counting function of Stanley sequences. Discrete Math. (2011), 560-562.\n\n[OdSt78] A. Odlyzko and R. Stanley, Some curious sequences constructed with the greedy algorithm. Bell Laboratories internal memorandum (1978).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2022, "problem_number": "EP-272", "title": "Erdős Problem #272", "statement": "Let $N\\geq 1$. What is the largest $t$ such that there are $A_1,\\ldots,A_t\\subseteq \\{1,\\ldots,N\\}$ with $A_i\\cap A_j$ a non-empty arithmetic progression for all $i\neq j$?", "background": "Simonovits and S\\'{o}s \\cite{SiSo81} have shown that $t\\ll N^2$.\nErd\\H{o}s and Graham asked whether the maximal $t$ is achieved when we take the $A_i$ to be all arithmetic progressions in $\\{1,\\ldots,N\\}$ containing some fixed element, 'presumably the integer $\\lfloor N/2\\rfloor$'. This was disproved by Simonovits and S\\'{o}s \\cite{SiSo81}, who observed that taking all sets containing at most $3$ elements, containing some fixed element, produces $\\binom{N}{2}+1$ many such sets, which is asymptotically greater than the number of arithmetic progressions containing a fixed element, which is $\\sim \\frac{\\pi^2}{24}N^2$.\nIf we drop the non-empty requirement then Graham, Simonovits, and S\\'{o}s \\cite{GSS80} have shown that $ t\\leq \\binom{N}{3}+\\binom{N}{2}+\\binom{N}{1}+1 $ and this is best possible.\nSzabo \\cite{Sz99} proved that the maximal such $t$ is equal to $ \\frac{N^2}{2}+O(N^{5/3}(\\log N)^3), $ resolving the asymptotic question. On the other hand, Szabo showed that the conjecture of Simonovits and S\\'{o}s that $\\binom{n}{2}+1$ is best possible is false, giving a construction which yields $ t \\geq \\binom{N}{2}+\\left\\lfloor\\frac{N-1}{4}\\right\\rfloor+1. $ Szabo conjectures that the asymptotic $t=\\binom{N}{2}+O(N)$ holds, and that in any extremal example there is an integer contained in all sets.\nReferences\n\n\n[GSS80] Graham, R. L. and Simonovits, M. and S\\'{o}s, V. T., A note on the intersection properties of subsets of integers. J. Combin. Theory Ser. A (1980), 106-110.\n\n[SiSo81] Simonovits, Mikl\\'{o}s and S\\'{o}s, Vera T., Intersection properties of subsets of integers. European J. Combin. (1981), 363-372.\n\n[Sz99] Szab\\'o, Tibor, Intersection properties of subsets of integers. European J. Combin. (1999), 429--444.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2023, "problem_number": "EP-273", "title": "Erdős Problem #273", "statement": "Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\\geq 5$?", "background": "Selfridge has found an example using divisors of $360$ if $p=3$ is allowed.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2024, "problem_number": "EP-274", "title": "Erdős Problem #274", "statement": "If $G$ is a group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)", "background": "A question of Herzog and Sch\"{onheim}, who conjectured more generally that if $G$ is any (not necessarily finite) group and $a_1G_1,\\ldots,a_kG_k$ are finitely many cosets of subgroups of $G$ with distinct indices $[G:G_i]$ then the $a_iG_i$ cannot form a partition of $G$.\nThis conjecture was proved in the case when all the $G_i$ are subnormal in $G$ by Sun \\cite{Su04}. In particular if $G$ is abelian (which was the special case asked about in \\cite{Er77c} and \\cite{ErGr80}) the answer to the original question is no.\nMargolis and Schnabel \\cite{MaSc19} proved this conjecture for all groups $G$ of size $<1440$.\nReferences\n\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[MaSc19] Margolis, Leo and Schnabel, Ofir, The {H}erzog-{S}ch\"onheim conjecture for small groups and\nharmonic subgroups. Beitr. Algebra Geom. (2019), 399--418.\n\n[Su04] Sun, Zhi-Wei, On the {H}erzog-{S}ch\"onheim conjecture for uniform covers of\ngroups. J. Algebra (2004), 153--175.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2025, "problem_number": "EP-276", "title": "Erdős Problem #276", "statement": "Is there an infinite Lucas sequence $a_0,a_1,\\ldots$ where $a_{n+2}=a_{n+1}+a_n$ for $n\\geq 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every term of the sequence?", "background": "Whether such a composite Lucas sequence even exists was open for a while, but using covering systems Graham \\cite{Gr64} showed that $ a_0 = 1786772701928802632268715130455793 $ and $ a_1 = 1059683225053915111058165141686995 $ generate such a sequence. This problem asks whether one can have a composite Lucas sequence without 'an underlying system of covering congruences responsible'.\nThis problem has been 'conjecturally solved' by Ismailescu and Son \\cite{IsSo14}, in that they provide an explicit infinite Lucas sequence in which all the terms are composite, and believe that no covering system is responsible for this. See the comment by van Doorn below for more details.\nSee also [1113] for another problem in which the question is whether covering systems are always responsible.\nReferences\n\n\n[Gr64] Graham, R. L., A Fibonacci-Like Sequence of Composite Numbers. Math. Mag. (1964), 322-324.\n\n[IsSo14] Ismailescu, Dan and Son, Jaesung, A new kind of {F}ibonacci-like sequence of composite numbers. J. Integer Seq. (2014), Article 14.8.2, 9.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2026, "problem_number": "EP-278", "title": "Erdős Problem #278", "statement": "Let $A=\\{n_1<\\cdots0$ there exists some $k$ such that, for every choice of congruence classes $a_i$, the density of integers not satisfying any of the congruences $a_i\\pmod{n_i}$ for $1\\leq i\\leq k$ is less than $\\epsilon$?", "background": "The latter condition is clearly sufficient, the problem is if it's also necessary. The assumption implies $\\sum \\frac{1}{n_i}=\\infty$. If the $n_i$ are pairwise relatively prime then it is sufficient that $\\sum \\frac{1}{n_i}=\\infty$.\nThis is true - a proof is given in the comments by Somani (using ChatGPT).\nAn alternative elementary proof was noted by KoishiChan in the comments: let $\\delta$ be the lower density of the set of integers divisible by some $n\\in A$, and $\\delta_k$ be the density of the set of integers divisible by at least one of $n_1,\\ldots,n_k$. A theorem of Davenport and Erd\\H{o}s \\cite{DaEr36} states that $ \\delta = \\lim_{k\\to \\infty}\\delta_k. $ In the present case $\\delta=1$ and hence for every $\\epsilon>0$ there exists $k$ such that $\\delta_k>1-\\epsilon$. In other words, the density of those integers not satisfying $a_i\\equiv 0\\pmod{n_i}$ for $1\\leq i\\leq k$ is $<\\epsilon$. By a theorem of Rogers (which first appeared in print in Chapter V.3 of the book of Halberstam and Roth \\cite{HaRo66}) the density of those integers not satisfying any of the congruences $a_i\\pmod{n_i}$ for $1\\leq i\\leq k$ is maximised when $a_i\\equiv 0$, which concludes the proof.\nGiven that both Rogers' result and the Davenport-Erd\\H{o}s theorem mentioned above must have been very familiar to Erd\\H{o}s in 1980, it is strange that this natural argument was overlooked.\nReferences\n\n\n[DaEr36] Davenport, H. and Erd\\H{o}s, P., On sequences of positive integers. Acta Arithmetica (1936), 147-151.\n\n[HaRo66] Halberstam, H. and Roth, K. F., Sequences. {V}ol. {I}. (1966), xx+291.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2029, "problem_number": "EP-282", "title": "Erdős Problem #282", "statement": "Let $A\\subseteq \\mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\\in (0,1)$: choose the minimal $n\\in A$ such that $n\\geq 1/x$ and repeat with $x$ replaced by $x-\\frac{1}{n}$. If this terminates after finitely many steps then this produces a representation of $x$ as the sum of distinct unit fractions with denominators from $A$.\nDoes this process always terminate if $x$ has odd denominator and $A$ is the set of odd numbers? More generally, for which pairs $x$ and $A$ does this process terminate?", "background": "In 1202 Fibonacci observed that this process terminates for any $x$ when $A=\\mathbb{N}$. The problem when $A$ is the set of odd numbers is due to Stein.\nGraham \\cite{Gr64b} has shown that $\\frac{m}{n}$ is the sum of distinct unit fractions with denominators $\\equiv a\\pmod{d}$ if and only if $ \\left(\\frac{n}{(n,(a,d))},\\frac{d}{(a,d)}\\right)=1. $ Does the greedy algorithm always terminate in such cases?\nGraham \\cite{Gr64c} has also shown that $x$ is the sum of distinct unit fractions with square denominators if and only if $x\\in [0,\\pi^2/6-1)\\cup [1,\\pi^2/6)$. Does the greedy algorithm for this always terminate? Erd\\H{o}s and Graham believe not - indeed, perhaps it fails to terminate almost always.\nSee also [206].\nReferences\n\n\n[Gr64b] Graham, R. L., On finite sums of unit fractions. Proc. London Math. Soc. (3) (1964), 193-207.\n\n[Gr64c] Graham, R. L., On finite sums of reciprocals of distinct nth powers. Pacific J. Math. (1964), 85-92.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2030, "problem_number": "EP-283", "title": "Erdős Problem #283", "statement": "Let $p:\\mathbb{Z}\\to \\mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\\geq 2$ with $d\\mid p(n)$ for all $n\\geq 1$. Is it true that, for all sufficiently large $m$, there exist integers $1\\leq n_1<\\cdots 0$ (provided $m$ is taken sufficiently large depending on $\\alpha$).\nCassels \\cite{Ca60} has proved that these conditions on the polynomial imply every sufficiently large integer is the sum of $p(n_i)$ with distinct $n_i$. Burr has proved this if $p(x)=x^k$ with $k\\geq 1$ and if we allow $n_i=n_j$.\nAlekseyev \\cite{Al19} has proved this when $p(x)=x^2$, for all $m>8542$. For example, $ 1=\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{6}+\\frac{1}{12} $ and $ 200 = 2^2+4^2+6^2+12^2. $ van Doorn \\cite{vD25} has investigated the question of what 'sufficiently large' means for $p(x)=x$. van Doorn has also proved the original conjecture for many linear and quadratic polynomials, for example $p(x)=x+5$ or $p(x)=x^2+100$ - see the comments section.\nReferences\n\n\n[Al19] Alekseyev, Max A., On partitions into squares of distinct integers whose\nreciprocals sum to 1. (2019), 213--221.\n\n[Ca60] Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.\n\n[Gr63] Graham, R. L., A theorem on partitions. J. Austral. Math. Soc. (1963), 435-441.\n\n[vD25] W. van Doorn, Partitions with prescribed sum of rationals: asymptotic bounds. arXiv:2502.02200 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2031, "problem_number": "EP-288", "title": "Erdős Problem #288", "statement": "Is it true that there are only finitely many pairs of intervals $I_1,I_2$ such that $ \\sum_{n_1\\in I_1}\\frac{1}{n_1}+\\sum_{n_2\\in I_2}\\frac{1}{n_2}\\in \\mathbb{N}? $ ", "background": "For example, $ \\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}+\\frac{1}{6}+\\frac{1}{20}=1. $ This is still open even if $\\lvert I_2\\rvert=1$. It is perhaps true with two intervals replaced by any $k$ intervals.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2032, "problem_number": "EP-289", "title": "Erdős Problem #289", "statement": "Is it true that, for all sufficiently large $k$, there exist finite intervals $I_1,\\ldots,I_k\\subset \\mathbb{N}$, distinct, not overlapping or adjacent, with $\\lvert I_i\\rvert \\geq 2$ for $1\\leq i\\leq k$ such that $ 1=\\sum_{i=1}^k \\sum_{n\\in I_i}\\frac{1}{n}? $ ", "background": "Erd\\H{o}s and Graham posed this in \\cite{ErGr80} without the stipulation the intervals be distinct, non-overlapping, or adjacent, but Kovac in the comments has provided a simple argument showing that it is easily possible without this restriction, and likely \\cite{ErGr80} just forgot to mention this natural restriction.\nAs an example representing $2$ rather than $1$, Hickerson and Montgomery, in the solution to AMS Monthly problem E2689 proposed by Hahn, found $ 2=\\sum_{i=1}^5 \\sum_{n\\in I_i}\\frac{1}{n} $ where $I_1=[2,7]$, $I_2=[9,10]$, $I_3=[17,18]$, $I_4=[34,35]$, and $I_5=[84,85]$.\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2033, "problem_number": "EP-291", "title": "Erdős Problem #291", "statement": "Let $n\\geq 1$ and define $L_n$ to be the least common multiple of $\\{1,\\ldots,n\\}$ and $a_n$ by $ \\sum_{1\\leq k\\leq n}\\frac{1}{k}=\\frac{a_n}{L_n}. $ Is it true that $(a_n,L_n)=1$ and $(a_n,L_n)>1$ both occur for infinitely many $n$?", "background": "Steinerberger has observed that the answer to the second question is trivially yes: for example, any $n$ which begins with a $2$ in base $3$ has $3\\mid (a_n,L_n)$.\nMore generally, if the leading digit of $n$ in base $p$ is $p-1$ then $p\\mid (a_n,L_n)$. There is in fact a necessary and sufficient condition: a prime $p\\leq n$ divides $(a_n,L_n)$ if and only if $p$ divides the numerator of $1+\\cdots+\\frac{1}{k}$, where $k$ is the leading digit of $n$ in base $p$. This can be seen by writing $ a_n = \\frac{L_n}{1}+\\cdots+\\frac{L_n}{n} $ and observing that the right-hand side is congruent to $1+\\cdots+1/k$ modulo $p$. (The previous claim about $p-1$ follows immediately from Wolstenholme's theorem.)\nThis leads to a heuristic prediction (see for example a preprint of Shiu \\cite{Sh16}) of $\\asymp\\frac{x}{\\log x}$ for the number of $n\\in [1,x]$ such that $(a_n,L_n)=1$. In particular, there should be infinitely many $n$, but the set of such $n$ should have density zero. Unfortunately this heuristic is difficult to turn into a proof.\nWu and Yan \\cite{WuYa22} have proved, conditional on $\\frac{1}{\\log p}$ being linearly independent over $\\mathbb{Q}$ for any finite collection of primes $p$ (itself a consequence of Schanuel's conjecture), that the set of $n$ for which $(a_n,L_n)>1$ has upper density $1$.\nReferences\n\n\n[Sh16] P. Shiu, The denominators of harmonic numbers. arXiv:1607.02863 (2016).\n\n[WuYa22] Wu, Bing-Ling and Yan, Xiao-Hui, On the denominators of harmonic numbers. {IV}. C. R. Math. Acad. Sci. Paris (2022), 53--57.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2034, "problem_number": "EP-293", "title": "Erdős Problem #293", "statement": "Let $k\\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to $ 1=\\frac{1}{n_1}+\\cdots+\\frac{1}{n_k} $ with $1\\leq n_1<\\cdots 0$, and noted a close connection to [304]. In particular, if $N(b)\\ll \\log\\log b$ as in [304] then it is likely the methods of \\cite{vDTa25b} prove $v(k) \\geq e^{e^{ck}}$ for some $c>0$.\nReferences\n\n\n[BlEr75] Bleicher, M. N. and Erd\\H{o}s, P., The number of distinct subsums of $\\sum \\sb{1}\\spN\\,1/i$. Math. Comp. (1975), 29-42.\n\n[vDTa25b] W. van Doorn and Q. Tang, The smallest denominator not contained in a unit fraction decomposition of 1 with fixed length. arXiv:2512.22083 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2035, "problem_number": "EP-295", "title": "Erdős Problem #295", "statement": "Let $N\\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\\leq n_1<\\cdots 0$ such that $ -c < k(N)-(e-1)N \\ll \\frac{N}{\\log N}. $ \nReferences\n\n\n[ErSt71b] Erd\\H{o}s, P. and Straus, E. G., Solution to Problem. Amer. Math. Monthly (1971), 302-303.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2036, "problem_number": "EP-301", "title": "Erdős Problem #301", "statement": "Let $f(N)$ be the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there are no solutions to $ \\frac{1}{a}= \\frac{1}{b_1}+\\cdots+\\frac{1}{b_k} $ with distinct $a,b_1,\\ldots,b_k\\in A$?\nEstimate $f(N)$. In particular, is it true that $f(N)=(\\tfrac{1}{2}+o(1))N$?", "background": "The example $A=(N/2,N]\\cap \\mathbb{N}$ shows that $f(N)\\geq N/2$.\nWouter van Doorn has given an elementary argument that proves $ f(N)\\leq (25/28+o(1))N. $ Indeed, consider the sets $S_a=\\{2a,3a,4a,6a,12a\\}\\cap [1,N]$ as $a$ ranges over all integers of the form $8^b9^cd$ with $(d,6)=1$. All such $S_a$ are disjoint and, if $A$ has no solutions to the given equation, then $A$ must omit at least two elements of $S_a$ when $a\\leq N/12$ and at least one element of $S_a$ when $N/120}$ with $b$ squarefree. Are there integers $10$.\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2041, "problem_number": "EP-312", "title": "Erdős Problem #312", "statement": "Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\\sum_{n\\in A}\\frac{1}{n}>K$ there exists some $S\\subseteq A$ such that $ 1-e^{-cK} < \\sum_{n\\in S}\\frac{1}{n}\\leq 1? $ ", "background": "Erd\\H{o}s and Graham knew this with $e^{-cK}$ replaced by $c/K^2$.\n\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2042, "problem_number": "EP-313", "title": "Erdős Problem #313", "statement": "Are there infinitely many solutions to $ \\frac{1}{p_1}+\\cdots+\\frac{1}{p_k}=1-\\frac{1}{m}, $ where $m\\geq 2$ is an integer and $p_1<\\cdots0$ such that for every $n\\geq 1$ there exists some $\\delta_k\\in \\{-1,0,1\\}$ for $1\\leq k\\leq n$ with $ 0< \\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert < \\frac{c}{2^n}? $ Is it true that for sufficiently large $n$, for any $\\delta_k\\in \\{-1,0,1\\}$, $ \\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert > \\frac{1}{[1,\\ldots,n]} $ whenever the left-hand side is not zero?", "background": "Inequality is obvious for the second claim, the problem is strict inequality. This fails for small $n$, for example $ \\frac{1}{2}-\\frac{1}{3}-\\frac{1}{4}=-\\frac{1}{12}. $ Arguments of Kovac and van Doorn in the comment section prove a weak version of the first question, with an upper bound of $ 2^{-n\\frac{(\\log\\log\\log n)^{1+o(1)}}{\\log n}}, $ and van Doorn gives a heuristic that suggests this may be the true order of magnitude.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2044, "problem_number": "EP-318", "title": "Erdős Problem #318", "statement": "Let $A\\subseteq \\mathbb{N}$ be an infinite arithmetic progression and $f:A\\to \\{-1,1\\}$ be a non-constant function. Must there exist a finite non-empty $S\\subset A$ such that $ \\sum_{n\\in S}\\frac{f(n)}{n}=0? $ What about if $A$ is an arbitrary set of positive density? What if $A$ is the set of squares excluding $1$?", "background": "Erd\\H{o}s and Straus \\cite{ErSt75} proved this when $A=\\mathbb{N}$. Sattler \\cite{Sa75} proved this when $A$ is the set of odd numbers. For the squares $1$ must be excluded or the result is trivially false, since $ \\sum_{k\\geq 2}\\frac{1}{k^2}<1. $ This is false for some sets $A$ of positive density - indeed, it fails for any set $A$ containing exactly one even number. (Sattler \\cite{Sa82} credits this observation to Erd\\H{o}s, who presumably found this after \\cite{ErGr80}.)\nSattler \\cite{Sa82b} proved the answer to the original question is yes, in that any arithmetic progression has this property.\nThe final question of the set of squares excluding $1$ appears to be open - Sattler announced a proof in \\cite{Sa82} and \\cite{Sa82b}, but this never appeared.\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[ErSt75] Erd\\H{o}s, P. and Straus, E. G., Solution to Problem 387. Nieuw Arch. Wisk. (1975), 183.\n\n[Sa75] Sattler, R., Solution to Problem 387. Nieuw Arch. Wisk. (1975), 184-189.\n\n[Sa82] Sattler, R., On {E}rd\\H{o}s property {${\\rm P}\\sb{1}$}\\ for the sequence of\nsquarefree numbers. Nederl. Akad. Wetensch. Indag. Math. (1982), 341--346.\n\n[Sa82b] Sattler, R., On {E}rd\\H{o}s property {${\\rm P}\\sb{1}$}\\ for the arithmetical\nsequence. Nederl. Akad. Wetensch. Indag. Math. (1982), 347--352.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2045, "problem_number": "EP-319", "title": "Erdős Problem #319", "statement": "What is the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there is a function $\\delta:A\\to \\{-1,1\\}$ such that $ \\sum_{n\\in A}\\frac{\\delta_n}{n}=0 $ and $ \\sum_{n\\in A'}\\frac{\\delta_n}{n}\neq 0 $ for all non-empty $A'\\subsetneq A$?", "background": "Adenwalla has observed that a lower bound of $ \\lvert A\\rvert\\geq (1-\\tfrac{1}{e}+o(1))N $ follows from the main result of Croot \\cite{Cr01}, which states that there exists some set of integers $B\\subset [(\\frac{1}{e}-o(1))N,N]$ such that $\\sum_{b\\in B}\\frac{1}{b}=1$. Since the sum of $\\frac{1}{m}$ for $m\\in [c_1N,c_2N]$ is asymptotic to $\\log(c_2/c_1)$ we must have $\\lvert B\\rvert \\geq (1-\\tfrac{1}{e}+o(1))N$.\nWe may then let $A=B\\cup\\{1\\}$ and choose $\\delta(n)=-1$ for all $n\\in B$ and $\\delta(1)=1$.\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[Cr01] Croot, III, Ernest S., On unit fractions with denominators in short intervals. Acta Arith. (2001), 99-114.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2046, "problem_number": "EP-320", "title": "Erdős Problem #320", "statement": "Let $S(N)$ count the number of distinct sums of the form $\\sum_{n\\in A}\\frac{1}{n}$ for $A\\subseteq \\{1,\\ldots,N\\}$. Estimate $S(N)$.", "background": "Bleicher and Erd\\H{o}s \\cite{BlEr75} proved the lower bound $ \\log S(N)\\geq \\frac{N}{\\log N}\\left(\\log 2\\prod_{i=3}^k\\log_iN\\right), $ valid for $k\\geq 4$ and $\\log_kN\\geq k$, and also \\cite{BlEr76b} proved the upper bound $ \\log S(N)\\leq \\frac{N}{\\log N}\\left(\\log_r N \\prod_{i=3}^r \\log_iN\\right), $ valid for $r\\geq 1$ and $\\log_{2r}N\\geq 1$. (In these bounds $\\log_in$ denotes the $i$-fold iterated logarithm.)\nBettin, Greni\\'{e}, Molteni, and Sanna \\cite{BGMS25} improved the lower bound to $ \\log S(N) \\geq \\frac{N}{\\log N}\\left(2\\log 2\\left(1-\\frac{3/2}{\\log_kN}\\right)\\prod_{i=3}^k\\log_iN\\right), $ valid for $k\\geq 4$ and $\\log_kN\\geq 3/2$. (In particular this goes to infinity faster than the lower bound of Bleicher and Erd\\H{o}s.)\nSee also [321].\nReferences\n\n\n[BGMS25] S. Bettin, L. Greni\\'{e}, G. Molteni, and C. Sanna, A lower bound for the number of Egyptian fractions. arXiv:2509.10030 (2025).\n\n[BlEr75] Bleicher, M. N. and Erd\\H{o}s, P., The number of distinct subsums of $\\sum \\sb{1}\\spN\\,1/i$. Math. Comp. (1975), 29-42.\n\n[BlEr76b] Bleicher, Michael N. and Erd\\H{o}s, Paul, Denominators of Egyptian fractions. II. Illinois J. Math. (1976), 598-613.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2047, "problem_number": "EP-321", "title": "Erdős Problem #321", "statement": "What is the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that all sums $\\sum_{n\\in S}\\frac{1}{n}$ are distinct for $S\\subseteq A$?", "background": "Let $R(N)$ be the maximal such size. Results of Bleicher and Erd\\H{o}s from \\cite{BlEr75} and \\cite{BlEr76b} imply that $ \\frac{N}{\\log N}\\prod_{i=3}^k\\log_iN\\leq R(N)\\leq \\frac{1}{\\log 2}\\log_r N\\left(\\frac{N}{\\log N} \\prod_{i=3}^r \\log_iN\\right), $ valid for any $k\\geq 4$ with $\\log_kN\\geq k$ and any $r\\geq 1$ with $\\log_{2r}N\\geq 1$. (In these bounds $\\log_in$ denotes the $i$-fold iterated logarithm.)\nSee also [320].\nReferences\n\n\n[BlEr75] Bleicher, M. N. and Erd\\H{o}s, P., The number of distinct subsums of $\\sum \\sb{1}\\spN\\,1/i$. Math. Comp. (1975), 29-42.\n\n[BlEr76b] Bleicher, Michael N. and Erd\\H{o}s, Paul, Denominators of Egyptian fractions. II. Illinois J. Math. (1976), 598-613.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2048, "problem_number": "EP-322", "title": "Erdős Problem #322", "statement": "Let $k\\geq 3$ and $A\\subset \\mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations of $n$ as the sum of $k$ many $k$th powers? Does there exist some $c>0$ and infinitely many $n$ such that $ 1_A^{(k)}(n) >n^c? $ ", "background": "Connected to Waring's problem. The famous Hypothesis $K$ of Hardy and Littlewood was that $1_A^{(k)}(n)\\leq n^{o(1)}$, but this was disproved by Mahler \\cite{Ma36} for $k=3$, who constructed infinitely many $n$ such that $ 1_A^{(3)}(n)\\gg n^{1/12} $ (where $A$ is the set of cubes). Erd\\H{o}s believed Hypothesis $K$ fails for all $k\\geq 4$, but this is unknown. Hardy and Littlewood made the weaker Hypothesis $K^*$ that for all $N$ and $\\epsilon>0$ $ \\sum_{n\\leq N}1_A^{(k)}(n)^2 \\ll_\\epsilon N^{1+\\epsilon}. $ Erd\\H{o}s and Graham remark: 'This is probably true but no doubt very deep. However, it would suffice for most applications.'\nIndependently Erd\\H{o}s \\cite{Er36} and Chowla proved that for all $k\\geq 3$ and infinitely many $n$ $ 1_A^{(k)}(n) \\gg n^{c/\\log\\log n} $ for some constant $c>0$ (depending on $k$). In \\cite{Er65b} Erd\\H{o}s claims an unpublished proof that, if $B$ is the set of $k$th powers of any set of positive density, then $ \\limsup 1_B^{(k)}(n)=\\infty. $ This is discussed in problem D4 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er36] Erd\"{o}s, Paul, On the Representation of an Integer as the Sum of k k-th\nPowers. J. London Math. Soc. (1936), 133-136.\n\n[Er65b] Erd\\H{o}s, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ma36] Mahler, Kurt, Note on Hypothesis K of Hardy and Littlewood. J. London Math. Soc. (1936), 136-138.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2049, "problem_number": "EP-323", "title": "Erdős Problem #323", "statement": "Let $1\\leq m\\leq k$ and $f_{k,m}(x)$ denote the number of integers $\\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that $ f_{k,k}(x) \\gg_\\epsilon x^{1-\\epsilon} $ for all $\\epsilon>0$? Is it true that if $m0$.\nFor $k>2$ it is not known if $f_{k,k}(x)=o(x)$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2050, "problem_number": "EP-324", "title": "Erdős Problem #324", "statement": "Does there exist a polynomial $f(x)\\in\\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $af(n)$.)", "background": "Erd\\H{o}s originally asked if even $f(n)\\leq n^{1/3}$ is true. This is known, and the best bound is due to Balog \\cite{Ba89} who proved that $ f(n) \\ll_\\epsilon n^{\\frac{4}{9\\sqrt{e}}+\\epsilon} $ for all $\\epsilon>0$. (Note $\\frac{4}{9\\sqrt{e}}=0.2695\\ldots$.)\nIt is likely that $f(n)\\leq n^{o(1)}$, or even $f(n)\\leq e^{O(\\sqrt{\\log n})}$.\nSee also Problem 59 on Green's open problems list.\nReferences\n\n\n[Ba89] Balog, A., On additive representation of integers. Acta Math. Hungar. (1989), 297-301.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2058, "problem_number": "EP-335", "title": "Erdős Problem #335", "statement": "Let $d(A)$ denote the density of $A\\subseteq \\mathbb{N}$. Characterise those $A,B\\subseteq \\mathbb{N}$ with positive density such that $ d(A+B)=d(A)+d(B). $ ", "background": "One way this can happen is if there exists $\\theta>0$ such that $ A=\\{ n>0 : \\{ n\\theta\\} \\in X_A\\}\\textrm{ and }B=\\{ n>0 : \\{n\\theta\\} \\in X_B\\} $ where $\\{x\\}$ denotes the fractional part of $x$ and $X_A,X_B\\subseteq \\mathbb{R}/\\mathbb{Z}$ are such that $\\mu(X_A+X_B)=\\mu(X_A)+\\mu(X_B)$. Are all possible $A$ and $B$ generated in a similar way (using other groups)?\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2059, "problem_number": "EP-336", "title": "Erdős Problem #336", "statement": "For $r\\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\\subseteq \\mathbb{N}$ of order $r$ (so every large integer is the sum of at most $r$ integers from $A$) and exact order $k$ (so every large integer is the sum of exactly $k$ integers from $A$).\nFind the value of $ \\lim_r \\frac{h(r)}{r^2}. $ ", "background": "A simple example of the order of a basis differing from the exact order is given by $A=\\cup_{k\\geq 0}(2^{2k},2^{2k+1}]$, which has order $2$ but exact order $3$.\nErd\\H{o}s and Graham \\cite{ErGr80b} have shown that a basis $A$ has an exact order if and only if $a_2-a_1,a_3-a_2,a_4-a_3,\\ldots$ are coprime. They also proved that $ \\frac{1}{4}\\leq \\lim_r \\frac{h(r)}{r^2}\\leq \\frac{5}{4}. $ The best bounds known for the limit are $ \\frac{1}{3}\\leq \\lim_r \\frac{h(r)}{r^2}\\leq \\frac{1}{2}, $ the lower bound originally due to Grekos \\cite{Gr88} and the upper bound to Nash \\cite{Na93}. Improved bounds in the lower order terms were given by Plagne \\cite{Pl04}.\nErd\\H{o}s and Graham \\cite{ErGr80b} showed $h(2)=4$. Nash \\cite{Na93} showed $h(3)=7$. The value of $h(4)$ is unknown, but it is known \\cite{Pl04} that $10\\leq h(4)\\leq 11$.\nReferences\n\n\n[ErGr80b] Erd\\H{o}s, P. and Graham, R. L., On bases with an exact order. Acta Arith. (1980), 201-207.\n\n[Gr88] Grekos, Georges, Sur l'ordre d'une base additive. ([1988?]), Exp. No. 31, 13.\n\n[Na93] Nash, John C. M., Some applications of a theorem of {M}. {K}neser. J. Number Theory (1993), 1--8.\n\n[Pl04] Plagne, Alain, \\`A{} propos de la fonction {$X$} d'{E}rd\\H{o}s et {G}raham. Ann. Inst. Fourier (Grenoble) (2004), 1717--1767.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2060, "problem_number": "EP-338", "title": "Erdős Problem #338", "statement": "The restricted order of a basis is the least integer $t$ (if it exists) such that every large integer is the sum of at most $t$ distinct summands from $A$. What are necessary and sufficient conditions that this exists? Can it be bounded (when it exists) in terms of the order of the basis? What are necessary and sufficient conditions that this is equal to the order of the basis?", "background": "Bateman has observed that for $h\\geq 3$ there is a basis of order $h$ with no restricted order, taking $ A=\\{1\\}\\cup \\{x>0 : h\\mid x\\}. $ Kelly \\cite{Ke57} has shown that any basis of order $2$ has restricted order at most $4$ and conjectured it always has restricted order at most $3$ (which he proved under the additional assumption that the basis has positive lower density). Kelly's conjecture was disproved by Hennecart \\cite{He05}, who constructed a basis of order $2$ with restricted order $4$.\nThe set of squares has order $4$ and restricted order $5$ (see \\cite{Pa33}) and the set of triangular numbers has order $3$ and restricted order $3$ (see \\cite{Sc54}).\nIs it true that if $A\\backslash F$ is a basis for all finite sets $F$ then $A$ must have a restricted order? What if they are all bases of the same order?\nHegyv\\'{a}ri, Hennecart, and Plagne \\cite{HHP07} have shown that for all $k\\geq2$ there exists a basis of order $k$ which has restricted order at least $ 2^{k-2}+k-1. $ \nReferences\n\n\n[HHP07] Hegyv\\'ari, Norbert and Hennecart, Fran\\c cois and Plagne,\nAlain, Answer to a question by {B}urr and {E}rd\\H{o}s on restricted\naddition, and related results. Combin. Probab. Comput. (2007), 747--756.\n\n[He05] Hennecart, Fran\\c cois, On the restricted order of asymptotic bases of order two. Ramanujan J. (2005), 123--130.\n\n[Ke57] Kelly, John B., Restricted bases. Amer. J. Math. (1957), 258-264.\n\n[Pa33] Pall, Gordon, On Sums of Squares. Amer. Math. Monthly (1933), 10-18.\n\n[Sc54] Schinzel, A., Sur la d\\'{e}composition des nombres naturels en sommes de nombres triangulaires distincts. Bull. Acad. Polon. Sci. Cl. III. (1954), 409-410.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2061, "problem_number": "EP-340", "title": "Erdős Problem #340", "statement": "Let $A=\\{1,2,4,8,13,21,31,45,66,81,97,\\ldots\\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a+b=c+d$). What is the order of growth of $A$? Is it true that $ \\lvert A\\cap \\{1,\\ldots,N\\}\\rvert \\gg N^{1/2-\\epsilon} $ for all $\\epsilon>0$ and large $N$?", "background": "This sequence is sometimes called the Mian-Chowla sequence. It is trivial that this sequence grows at least like $\\gg N^{1/3}$.\nErd\\H{o}s and Graham \\cite{ErGr80} also asked about the difference set $A-A$, whether this has positive density, and whether this contains $22$. It does contain $22$, since $a_{15}-a_{14}=204-182=22$. The smallest integer which is unknown to be in $A-A$ is $33$ (see A080200). It may be true that all or almost all integers are in $A-A$.\nThis sequence is at OEIS A005282.\nSee also [156].\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2062, "problem_number": "EP-341", "title": "Erdős Problem #341", "statement": "Let $A=\\{a_1<\\cdotsa_n$ which can be expressed uniquely as $a_i+a_j$ for $iT(n^{k+1})$?", "background": "Erd\\H{o}s and Graham \\cite{ErGr80} remark that very little is known about $T(A)$ in general. It is known that $ T(n)=1, T(n^2)=128, T(n^3)=12758, $ $ T(n^4)=5134240,\\textrm{ and }T(n^5)=67898771. $ Erd\\H{o}s and Graham remark that a good candidate for the $n$ in the question are $k=2^t$ for large $t$, perhaps even $t=3$, because of the highly restricted values of $n^{2^t}$ modulo $2^{t+1}$.\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2065, "problem_number": "EP-346", "title": "Erdős Problem #346", "statement": "Let $A=\\{1\\leq a_1< a_2<\\cdots\\}$ be a set of integers such that\n{UL}\n{LI} $A\\backslash B$ is complete for any finite subset $B$ and {/LI}\n{LI} $A\\backslash B$ is not complete for any infinite subset $B$.{/LI}\n{/UL}\n(Here 'complete' means all sufficiently large integers can be written as a sum of distinct members of the sequence.)\nIs it true that if $a_{n+1}/a_n \\geq 1+\\epsilon$ for some $\\epsilon>0$ and all $n$ then $ \\lim_n \\frac{a_{n+1}}{a_n}=\\frac{1+\\sqrt{5}}{2}? $ ", "background": "Graham \\cite{Gr64d} has shown that the sequence $a_n=F_n-(-1)^{n}$, where $F_n$ is the $n$th Fibonacci number, has these properties. Erd\\H{o}s and Graham \\cite{ErGr80} remark that it is easy to see that if $a_{n+1}/a_n>\\frac{1+\\sqrt{5}}{2}$ then the second property is automatically satisfied, and that it is not hard to construct very irregular sequences satisfying both properties.\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[Gr64d] Graham, R. L., A property of Fibonacci numbers. Fibonacci Quart. (1964), 1-10.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2066, "problem_number": "EP-348", "title": "Erdős Problem #348", "statement": "For what values of $0\\leq m0$ and all $1<\\alpha < \\frac{1+\\sqrt{5}}{2}$. Proving this seems very difficult, since we do not even know whether $\\lfloor (3/2)^n\\rfloor$ is odd or even infinitely often.\nReferences\n\n\n[Gr64e] Graham, R. L., On a conjecture of Erd\\H{o}s in additive number theory. Acta Arith. (1964/65), 63-70.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2068, "problem_number": "EP-351", "title": "Erdős Problem #351", "statement": "Let $p(x)\\in \\mathbb{Q}[x]$. Is it true that $ A=\\{ p(n)+1/n : n\\in \\mathbb{N}\\} $ is strongly complete, in the sense that, for any finite set $B$, $ \\left\\{\\sum_{n\\in X}n : X\\subseteq A\\backslash B\\textrm{ finite }\\right\\} $ contains all sufficiently large integers?", "background": "Graham \\cite{Gr63} proved this is true when $p(n)=n$. Erd\\H{o}s and Graham also ask which rational functions $r(x)\\in\\mathbb{Z}(x)$ force $\\{ r(n) : n\\in\\mathbb{N}\\}$ to be complete?\nGraham \\cite{Gr64f} gave a complete characterisation of which polynomials $r\\in \\mathbb{R}[x]$ are such that $\\{ r(n) : n\\in \\mathbb{N}\\}$ is complete.\nIn the comments van Doorn has noted that a positive solution for $p(n)=n^2$ follows from \\cite{Gr63} together with result of Alekseyev \\cite{Al19} mentioned in [283].\nReferences\n\n\n[Al19] Alekseyev, Max A., On partitions into squares of distinct integers whose\nreciprocals sum to 1. (2019), 213--221.\n\n[Gr63] Graham, R. L., A theorem on partitions. J. Austral. Math. Soc. (1963), 435-441.\n\n[Gr64f] Graham, R. L., Complete sequences of polynomial values. Duke Math. J. (1964), 275-285.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2069, "problem_number": "EP-352", "title": "Erdős Problem #352", "statement": "Is there some $c>0$ such that every measurable $A\\subseteq \\mathbb{R}^2$ of measure $\\geq c$ contains the vertices of a triangle of area 1?", "background": "Erd\\H{o}s (unpublished) proved that this is true if $A$ has infinite measure, or if $A$ is an unbounded set of positive measure (stating in \\cite{Er78d} and \\cite{Er83d} it 'follows easily from the Lebesgue density theorem').\nIn \\cite{Er78d} and \\cite{Er83d} he speculated that perhaps $C=4\\pi/\\sqrt{27}\\approx 2.418$ works, which would be the best possible, as witnessed by a circle of radius $<2\\cdot 3^{-3/4}$.\nFurther evidence for this is given by a result of Freiling and Mauldin \\cite{Ma02}, who proved that if $A$ has outer measure $>4\\pi/\\sqrt{27}$ then $A$ contains the vertices of a triangle with area $>1$. This also proves the same threshold for the original problem under the assumption that $A$ is a compact convex set.\nMauldin also discusses this problem in \\cite{Ma13}, in which he notes that it suffices to prove this under the assumption that $A$ is the union of the interiors of $n<\\infty$ many compact convex sets. Freiling and Mauldin (see \\cite{Ma13}) have proved this conjecture if $1\\leq n\\leq 3$.\nReferences\n\n\n[Er78d] Erd\\H{o}s, P., Set-theoretic, measure-theoretic, combinatorial, and\nnumber-theoretic problems concerning point sets in Euclidean\nspace. Real Anal. Exchange (1978/79), 113-138.\n\n[Er83d] Erd\\H{o}s, Paul, Some combinatorial, geometric and set theoretic problems in measure theory. Measure Theory, Oberwolfach 1983: Proceedings of the Conference held at Oberwolfach, June 26-July 2, 1983 (1984), 321-327.\n\n[Ma02] Mauldin, R. D., Some problems in set theory, analysis and geometry. (2002), 493--506.\n\n[Ma13] Mauldin, R. Daniel, Some problems and ideas of {E}rd\\H{o}s in analysis and\ngeometry. (2013), 365--376.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2070, "problem_number": "EP-354", "title": "Erdős Problem #354", "statement": "Let $\\alpha,\\beta\\in \\mathbb{R}_{>0}$ such that $\\alpha/\\beta$ is irrational. Is the multiset $ \\{ \\lfloor \\alpha\\rfloor,\\lfloor 2\\alpha\\rfloor,\\lfloor 4\\alpha\\rfloor,\\ldots\\}\\cup \\{ \\lfloor \\beta\\rfloor,\\lfloor 2\\beta\\rfloor,\\lfloor 4\\beta\\rfloor,\\ldots\\} $ complete? That is, can all sufficiently large natural numbers $n$ be written as $ n=\\sum_{s\\in S}\\lfloor 2^s\\alpha\\rfloor+\\sum_{t\\in T}\\lfloor 2^t\\beta\\rfloor $ for some finite $S,T\\subset \\mathbb{N}$?\nWhat if $2$ is replaced by some $\\gamma\\in(1,2)$?", "background": "This question was first mentioned by Graham \\cite{Gr71}.\nHegyv\\'{a}ri \\cite{He89} proved that this holds if $\\alpha=m/2^n$ is a dyadic rational and $\\beta$ is not. He later \\cite{He91} proved that, for any fixed $\\alpha>0$, the set of $\\beta$ for which this holds either has measure $0$ or infinite measure. In \\cite{He94} he proved that the set of $(\\alpha,\\beta)$ for which the corresponding set of sums does not contain an infinite arithmetic progression has cardinality continuum.\nHegyv\\'{a}ri \\cite{He89} proved that the sequence is not complete if $\\alpha\\geq 2$ and $\\beta =2^k\\alpha$ for some $k\\geq 0$. Jiang and Ma \\cite{JiMa24} and Fang and He \\cite{FaHe25} prove that the sequence is not complete if $1<\\alpha<2$ and $\\beta=2^k\\alpha$ for some sufficiently large $k$.\nIt is likely (and Hegyv\\'{a}ri conjectures) that the assumption $\\alpha/\\beta$ irrational can be weakened to $\\alpha/\\beta \neq 2^k$ and either $\\alpha$ or $\\beta$ not a dyadic rational.\nIn the comments van Doorn proves the sequence is complete if $\\alpha < 2<\\beta<3$, and also proves that if either $\\alpha$ or $\\beta$ is not a dyadic rational then the corresponding sequence with ceiling functions replacing the floor functions is complete.\nReferences\n\n\n[FaHe25] Fang, J.-H. and He, J.-Y., On a problem of {E}rd\\H{o}s and {G}raham. Acta Math. Hungar. (2025), 532--542.\n\n[Gr71] Graham, R. L., On sums of integers taken from a fixed sequence. (1971), 22--40.\n\n[He89] Hegyv\\'ari, N., Some remarks on a problem of {E}rd\\H{o}s and {G}raham. Acta Math. Hungar. (1989), 149--154.\n\n[He91] Hegyv\\'ari, N., On complete sequences. Ann. Univ. Sci. Budapest. E\"otv\"os Sect. Math. (1991), 7--10.\n\n[He94] Hegyv\\'ari, Norbert, On sumset of certain sets. Publ. Math. Debrecen (1994), 115--122.\n\n[JiMa24] Jiang, Xing-Wang and Ma, Wu-Xia, A conjecture of {H}egyv\\'ari. Int. J. Number Theory (2024), 915--933.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2071, "problem_number": "EP-357", "title": "Erdős Problem #357", "statement": "Let $1\\leq a_1<\\cdots 0$?", "background": "A problem of MacMahon, studied by Andrews \\cite{An75}. When $n=1$ this sequence begins $ 1,2,4,5,8,10,14,15,\\ldots. $ This sequence is A002048 in the OEIS. Andrews conjectures $ a_k\\sim \\frac{k\\log k}{\\log\\log k}. $ Porubsky \\cite{Po77} proved that, for any $\\epsilon>0$, there are infinitely many $k$ such that $ a_k < (\\log k)^\\epsilon \\frac{k\\log k}{\\log\\log k}, $ and also that if $A(x)$ counts the number of $a_i\\leq x$ then $ \\limsup \\frac{A(x)}{\\pi(x)}\\geq \\frac{1}{\\log 2} $ where $\\pi(x)$ counts the number of primes $\\leq x$.\nSee also [839].\nReferences\n\n\n[An75] Andrews, George E., Research Problems: Mac Mahon's Prime Numbers of Measurement. Amer. Math. Monthly (1975), 922-923.\n\n[Po77] Porubsk\\'y, \\v S., On {M}ac{M}ahon's segmented numbers and related sequences. Nieuw Arch. Wisk. (3) (1977), 403--408.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2074, "problem_number": "EP-361", "title": "Erdős Problem #361", "statement": "Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\\subseteq \\{1,\\ldots,\\lfloor cn\\rfloor\\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2075, "problem_number": "EP-365", "title": "Erdős Problem #365", "statement": "Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations? In other words, must either $n$ or $n+1$ be a square?\nIs the number of such $n\\leq x$ bounded by $(\\log x)^{O(1)}$?", "background": "Erd\\H{o}s originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation $x^2=2^3y^2+1$.\nThe list of $n$ such that $n$ and $n+1$ are both powerful is A060355 in the OEIS.\nThe answer to the first question is no: Golomb \\cite{Go70} observed that both $12167=23^3$ and $12168=2^33^213^2$ are powerful. Walker \\cite{Wa76} proved that the equation $ 7^3x^2=3^3y^2+1 $ has infinitely many solutions, giving infinitely many counterexamples.\nSee also [364].\nThis is discussed in problem B16 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Go70] Golomb, S. W., Powerful numbers. Amer. Math. Monthly (1970), 848-855.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Wa76] Walker, David T., Consecutive integer pairs of powerful numbers and related\nDiophantine equations. Fibonacci Quart. (1976), 111-116.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2076, "problem_number": "EP-367", "title": "Erdős Problem #367", "statement": "Let $B_2(n)$ be the 2-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true that, for every fixed $k\\geq 1$, $ \\prod_{n\\leq m0$, $ \\limsup \\frac{\\prod_{n\\leq m0$, there are infinitely many $n$ such that $F(n) <(\\log n)^{2+\\epsilon}$.\nPasten \\cite{Pa24b} has proved that $ F(n) \\gg \\frac{(\\log\\log n)^2}{\\log\\log\\log n}. $ The largest prime factors of $n(n+1)$ are listed as A074399 in the OEIS.\nReferences\n\n\n[Er76d] Erd\\H{o}s, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\n\n[Ma35] Mahler, Kurt, \"{U}ber den gr\"{o}ssten Primteiler spezieller Polynome zweiten Grades. Archiv f\"{u}r math. og naturvid (1935).\n\n[Pa24b] Pasten, Hector, The largest prime factor of {$n^2+1$} and improvements on\nsubexponential {$ABC$}. Invent. Math. (2024), 373--385.\n\n[Po18] P\\'{o}lya, Georg, Zur arithmetischen {U}ntersuchung der {P}olynome. Math. Z. (1918), 143--148.\n\n[Sc67b] Schinzel, A., On two theorems of Gelfond and some of their applications. Acta Arith. (1967/68), 177-236.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2078, "problem_number": "EP-369", "title": "Erdős Problem #369", "statement": "Let $\\epsilon>0$ and $k\\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\\{1,\\ldots,n\\}$ all of which are $n^\\epsilon$-smooth?", "background": "Erd\\H{o}s and Graham state that this is open even for $k=2$ and 'the answer should be affirmative but the problem seems very hard'.\nUnfortunately the problem is trivially true as written (simply taking $\\{1,\\ldots,k\\}$ and $n>k^{1/\\epsilon}$). There are (at least) two possible variants which are non-trivial, and it is not clear which Erd\\H{o}s and Graham meant. Let $P$ be the sequence of $k$ consecutive integers sought for. The potential strengthenings which make this non-trivial are:\n{UL}\n{LI}Each $m\\in P$ must be $m^\\epsilon$-smooth. If this is the problem then the answer is yes, which follows from a result of Balog and Wooley \\cite{BaWo98}: for any $\\epsilon>0$ and $k\\geq 2$ there exist infinitely many $m$ such that $m+1,\\ldots,m+k$ are all $m^\\epsilon$-smooth.{/LI}\n{LI}Each $m\\in P$ must be in $[n/2,n]$ (say). In this case a positive answer also follows from the result of Balog and Wooley \\cite{BaWo98} for infinitely many $n$, but the case of all sufficiently large $n$ is open.{/LI}\n{/UL}\nSee also [370] and [928].\nReferences\n\n\n[BaWo98] Balog, Antal and Wooley, Trevor D., On strings of consecutive integers with no large prime factors. J. Austral. Math. Soc. Ser. A (1998), 266-276.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2079, "problem_number": "EP-371", "title": "Erdős Problem #371", "statement": "Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n) (0.2017-o(1))x, $ and the same lower bound for the complement.\nIn \\cite{Er79e} Erd\\H{o}s also asks whether, for every $\\alpha$, the density of the set of $n$ where $ P(n+1)>P(n)n^\\alpha $ exists.\nTer\"{a}v\"{a}inen \\cite{Te18} has proved that the logarithmic density of the set of $n$ for which $P(n)P(n)n^\\alpha$ exists and is equal to $ \\int_{[0,1]^2}1_{y\\geq x+\\alpha}u(x)u(y)\\mathrm{d}x\\mathrm{d}y $ where $u(x)=x^{-1}\\rho(x^{-1}-1)$ and $\\rho$ is the Dickman function. Wang \\cite{Wa21} has proved the same value holds for the asymptotic density (and in particular provided an affirmative answer to the original question) conditional on the Elliott-Halberstam conjecture for friable integers.\nThe sequence of such $n$ is A070089 in the OEIS.\nSee also [372] and [928].\nReferences\n\n\n[Er79e] Erd\\H{o}s, Paul, Some unconventional problems in number theory. Ast\\'{e}risque (1979), 73-82.\n\n[ErPo78] Erd\\H{o}s, Paul and Pomerance, Carl, On the largest prime factors of {$n$} and {$n+1$}. Aequationes Math. (1978), 311-321.\n\n[LuWa25] L\"u, Xiaodong and Wang, Zhiwei, On the largest prime factors of consecutive integers. Monatsh. Math. (2025), 403--418.\n\n[TaTe19] Tao, Terence and Ter\"{a}v\"{a}inen, Joni, The structure of correlations of multiplicative functions at\nalmost all scales, with applications to the {C}howla and\n{E}lliott conjectures. Algebra Number Theory (2019), 2103--2150.\n\n[Te18] Ter\"{a}v\"{a}inen, Joni, On binary correlations of multiplicative functions. Forum Math. Sigma (2018), Paper No. e10, 41.\n\n[Wa21] Wang, Zhiwei, Three conjectures on {$P^+(n)$} and {$P^+(n+1)$} hold under\nthe {E}lliott-{H}alberstam conjecture for friable integers. J. Number Theory (2021), 1--11.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2080, "problem_number": "EP-373", "title": "Erdős Problem #373", "statement": "Show that the equation $ n! = a_1!a_2!\\cdots a_k!, $ with $n-1>a_1\\geq a_2\\geq \\cdots \\geq a_k\\geq 2$, has only finitely many solutions.", "background": "This would follow if $P(n(n+1))/\\log n\\to \\infty$, where $P(m)$ denotes the largest prime factor of $m$ (see Problem [368]). Erd\\H{o}s \\cite{Er76d} proved that this problem would also follow from showing that $P(n(n-1))>4\\log n$.\nThe condition $a_1a_1\\geq a_2$ then $ a_1\\geq n-5\\log\\log n, $ and says it 'would be nice' to prove $a_1\\geq n-o(\\log\\log n)$. Bhat and Ramachandra \\cite{BhRa10} replace the $5$ with $(1+o(1))\\frac{1}{\\log 2}$, and also prove that the same bound holds for arbitrary $k\\geq 2$.\nNumerical investigations on solutions to $n!=a_1!a_2!$ have been carried out by Caldwell \\cite{Ca94} and Habsieger \\cite{Ha}, and it is known that there are no solutions aside from $10!=6!7!$ for $n\\leq 10^{3000}$.\nReferences\n\n\n[BhRa10] Bhat, K. Dzh. and Ramachandra, K., A remark on factorials that are products of factorials. Mat. Zametki (2010), 350--354.\n\n[Ca94] C. Caldwell, The Diophantine equation $A!B!=C!$. J. Recreat. Math. (1994), 128-133.\n\n[Er76d] Erd\\H{o}s, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ha] Haight, J. A., Metric Diophantine approximation and related topics. PhD thesis ().\n\n[Lu07b] Luca, Florian, On factorials which are products of factorials. Math. Proc. Cambridge Philos. Soc. (2007), 533--542.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2081, "problem_number": "EP-374", "title": "Erdős Problem #374", "statement": "For any $m\\in \\mathbb{N}$, let $F(m)$ be the minimal $k\\geq 2$ (if it exists) such that there are $a_1<\\cdots 1\\}$,{/LI}\n{LI} $\\lvert D_3\\cap \\{1,\\ldots,n\\}\\rvert = o(\\lvert D_4\\cap \\{1,\\ldots,n\\}\\rvert)$,{/LI}\n{LI} the least element of $D_6$ is $527$, and{/LI}\n{LI} $D_k=\\emptyset$ for $k>6$.{/LI}\n{/UL}\nReferences\n\n\n[ErGr76] Erd\\H{o}s, P. and Graham, R. L., On products of factorials. Bull. Inst. Math. Acad. Sinica (1976), 337-355.\n\n[LSS14] Luca, F. and Saradha, N. and Shorey, T. N., Squares and factorials in products of factorials. Monatsh. Math. (2014), 385-400.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2082, "problem_number": "EP-376", "title": "Erdős Problem #376", "statement": "Are there infinitely many $n$ such that $\\binom{2n}{n}$ is coprime to $105$?", "background": "Erd\\H{o}s, Graham, Ruzsa, and Straus \\cite{EGRS75} have shown that, for any two odd primes $p$ and $q$, there are infinitely many $n$ such that $\\binom{2n}{n}$ is coprime to $pq$.\nThis is equivalent (via Kummer's theorem) to whether there are infinitely many $n$ which have only digits $0,1$ in base $3$, digits $0,1,2$ in base $5$, and digits $0,1,2,3$ in base $7$.\nThe sequence of such $n$ is A030979 in the OEIS.\nThe best result in this direction is due to Bloom and Croot \\cite{BlCr25}, who proved that, if $p_1,p_2,p_3$ are sufficiently large primes, then there are infinitely many $n$ such that almost all of the base $p_i$ digits are $0$, there are infinitely many $n$ such that $\\binom{2n}{n}$ is coprime to $p_1p_2p_3$, except for a factor of size $\\leq n^\\epsilon$.\nThis is mentioned in problem B33 of Guy's collection \\cite{Gu04}. It is also discussed in an article of Pomerance \\cite{Po15c}.\nGraham offered \\$1000 for a solution to this problem (as mentioned in \\cite{Gu04} and \\cite{BeHa98}).\nReferences\n\n\n[BeHa98] Berend, Daniel and Harmse, J\\o rgen E., On some arithmetical properties of middle binomial\ncoefficients. Acta Arith. (1998), 31--41.\n\n[BlCr25] T. F. Bloom and E. Croot, Integers with small digits in multiple bases. arXiv:2509.02835 (2025).\n\n[EGRS75] Erd\\H{o}s, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Po15c] Pomerance, Carl, Divisors of the middle binomial coefficient. Amer. Math. Monthly (2015), 636--644.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2083, "problem_number": "EP-377", "title": "Erdős Problem #377", "statement": "Is there some absolute constant $C>0$ such that $ \\sum_{p\\leq n}1_{p\nmid \\binom{2n}{n}}\\frac{1}{p}\\leq C $ for all $n$ (where the summation is restricted to primes $p\\leq n$)?", "background": "A question of Erd\\H{o}s, Graham, Ruzsa, and Straus \\cite{EGRS75}, who proved that if $f(n)$ is the sum in question then $ \\lim_{x\\to \\infty}\\frac{1}{x}\\sum_{n\\leq x}f(n) = \\sum_{k=2}^\\infty \\frac{\\log k}{2^k}=\\gamma_0 $ and $ \\lim_{x\\to \\infty}\\frac{1}{x}\\sum_{n\\leq x}f(n)^2 = \\gamma_0^2, $ so that for almost all integers $f(m)=\\gamma_0+o(1)$. They also prove that, for all large $n$, $ f(n) \\leq c\\log\\log n $ for some constant $c<1$. (It is trivial from Mertens estimates that $f(n)\\leq (1+o(1))\\log\\log n$.)\nA positive answer would imply that $ \\sum_{p\\leq n}1_{p\\mid \\binom{2n}{n}}\\frac{1}{p}=(1-o(1))\\log\\log n, $ and Erd\\H{o}s, Graham, Ruzsa, and Straus say there is 'no doubt' this latter claim is true.\nThis is mentioned in problem B33 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[EGRS75] Erd\\H{o}s, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2084, "problem_number": "EP-380", "title": "Erdős Problem #380", "statement": "We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\\prod_{u\\leq m\\leq v}m$ occurs with an exponent greater than $1$. Let $B(x)$ count the number of $n\\leq x$ which are contained in at least one bad interval. Is it true that $ B(x)\\sim \\#\\{ n\\leq x: P(n)^2\\mid n\\}, $ where $P(n)$ is the largest prime factor of $n$?", "background": "Erd\\H{o}s and Graham only knew that $B(x) > x^{1-o(1)}$. Similarly, we call an interval $[u,v]$ 'very bad' if $\\prod_{u\\leq m\\leq v}m$ is powerful. The number of integers $n\\leq x$ contained in at least one very bad interval should be $\\ll x^{1/2}$. In fact, it should be asymptotic to the number of powerful numbers $\\leq x$.\nWe have $ \\#\\{ n\\leq x: P(n)^2\\mid n\\}=\\frac{x}{\\exp((c+o(1))\\sqrt{\\log x\\log\\log x})} $ for some constant $c>0$.\nTao notes in the comments that if $[u,v]$ is bad then it cannot contain any primes, and hence certainly $v<2u$, and in general $v-u$ must be small (for example, assuming Cramer's conjecture, $v-u\\ll (\\log u)^2$).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2085, "problem_number": "EP-382", "title": "Erdős Problem #382", "statement": "Let $u\\leq v$ be such that the largest prime dividing $\\prod_{u\\leq m\\leq v}m$ appears with exponent at least $2$. Is it true that $v-u=v^{o(1)}$? Can $v-u$ be arbitrarily large?", "background": "Erd\\H{o}s and Graham report it follows from results of Ramachandra that $v-u\\leq v^{1/2+o(1)}$.\nCambie has observed that the first question boils down to some old conjectures on prime gaps.\nBy Cram\\'{er's conjecture}, for every $\\epsilon>0,$ for every $u$ sufficiently large there is a prime between $u$ and $u+u^\\epsilon$.\nThus for $u+u^\\epsilon0$. For any fixed $k$, there is therefore a positive 'probability' that there are $k$ consecutive integers around $q^2$ (for a prime $q$) all of whose prime divisors are bounded above by $q$, when $v-u\\geq k$. See [383] for a conjecture along these lines. A similar argument applies if we replace multiplicity $2$ with multiplicity $r$, for any fixed $r\\geq 2$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2086, "problem_number": "EP-383", "title": "Erdős Problem #383", "statement": "Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of $ \\prod_{0\\leq i\\leq k}(p^2+i) $ is $p$?", "background": "A positive answer to this would give an answer to the second part of [382]. Heuristically, the 'probability' that $n$ has no prime divisors $\\geq n^{1/2}$ is $1-\\log 2>0$, so standard heuristics predict the answer to this is yes.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2087, "problem_number": "EP-385", "title": "Erdős Problem #385", "statement": "Let $ F(n) = \\max_{\\substack{mn$ for all sufficiently large $n$? Does $F(n)-n\\to \\infty$ as $n\\to\\infty$?", "background": "A question of Erd\\H{o}s, Eggleton, and Selfridge, who write that 'plausible conjectures on primes' imply that $F(n)\\leq n$ for only finitely many $n$, and in fact it is possible that this quantity is always at least $n+(1-o(1))\\sqrt{n}$ (note that it is trivially $\\leq n+\\sqrt{n}$).\nTao has discussed this problem in a blog post.\nSarosh Adenwalla has observed that the first question is equivalent to [430]. Indeed, if $n$ is large and $a_i$ is the sequence defined in the latter problem, then [430] implies that there is a composite $a_j$ such that $a_j-p(a_j)>n$ and hence $F(n)>n$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2088, "problem_number": "EP-386", "title": "Erdős Problem #386", "statement": "Let $2\\leq k\\leq n-2$. Can $\\binom{n}{k}$ be the product of consecutive primes infinitely often? For example $ \\binom{21}{2}=2\\cdot 3\\cdot 5\\cdot 7. $ ", "background": "Erd\\H{o}s and Graham write that 'a proof that this cannot happen infinitely often for $\\binom{n}{2}$ seems hopeless; probably this can never happen for $\\binom{n}{k}$ if $3\\leq k\\leq n-3$.'\nWeisenberg has provided four easy examples that show Erd\\H{o}s and Graham were too optimistic here: $ \\binom{7}{3}=5\\cdot 7, $ $ \\binom{10}{4}= 2\\cdot 3\\cdot 5\\cdot 7, $ $ \\binom{14}{4} = 7\\cdot 11\\cdot 13, $ and $ \\binom{15}{6}=5\\cdot 7\\cdot 11\\cdot 13. $ The known values of $n$ for which $\\binom{n}{2}$ is the product of consecutive primes are $4,6,15,21,715$ (see A280992).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2089, "problem_number": "EP-387", "title": "Erdős Problem #387", "statement": "Is there an absolute constant $c>0$ such that, for all $1\\leq k< n$, the binomial coefficient $\\binom{n}{k}$ has a divisor in $(cn,n]$?", "background": "Erd\\H{o}s once conjectured that $\\binom{n}{k}$ must always have a divisor in $(n-k,n]$, but this was disproved by Schinzel and Erd\\H{o}s \\cite{Sc58}. A counterexample is given by $n=99215$ and $k=15$. Schinzel conjectured (see problem B34 of \\cite{Gu04}) that, for all sufficiently large $k$ which are not prime powers, there exists an $n$ such that $\\binom{n}{k}$ is not divisible by any integer in $(n-k,n]$.\nIt is easy to see that $\\binom{n}{k}$ always has a divisor in $[n/k,n]$.\nFaulkner \\cite{Fa66} proved that, if $p$ is the least prime $>2k$ and $n\\geq p$, then $\\binom{n}{k}$ has a prime divisor $\\geq p$ (except $\\binom{9}{2}$ and $\\binom{10}{3}$).\nThis is discussed in problems B33 and B34 of Guy's collection \\cite{Gu04}, who says that Erd\\H{o}s conjectured this is true for any $c<1$ (if $n$ is sufficiently large).\nReferences\n\n\n[Fa66] Faulkner, M., On a theorem of {S}ylvester and {S}chur. J. London Math. Soc. (1966), 107--110.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Sc58] Schinzel, A., Sur un probl\\`eme de {P}. {E}rd\\H{o}s. Colloq. Math. (1958), 198--204.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2090, "problem_number": "EP-388", "title": "Erdős Problem #388", "statement": "Can one classify all solutions of $ \\prod_{1\\leq i\\leq k_1}(m_1+i)=\\prod_{1\\leq j\\leq k_2}(m_2+j) $ where $k_1,k_2>3$ and $m_1+k_1\\leq m_2$? Are there only finitely many solutions?", "background": "More generally, if $k_1>2$ then for fixed $a$ and $b$ $ a\\prod_{1\\leq i\\leq k_1}(m_1+i)=b\\prod_{1\\leq j\\leq k_2}(m_2+j) $ should have only a finite number of solutions.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2091, "problem_number": "EP-389", "title": "Erdős Problem #389", "statement": "Is it true that for every $n\\geq 1$ there is a $k$ such that $ n(n+1)\\cdots(n+k-1)\\mid (n+k)\\cdots (n+2k-1)? $ ", "background": "Asked by Erd\\H{o}s and Straus.\nFor example when $n=2$ we have $k=5$: $ 2\\times 3 \\times 4 \\times 5\\times 6 \\mid 7 \\times 8 \\times 9\\times 10\\times 11. $ and when $n=3$ we have $k=4$: $ 3\\times 4\\times 5\\times 6 \\mid 7\\times 8\\times 9\\times 10. $ Bhavik Mehta has computed the minimal such $k$ for $1\\leq n\\leq 18$ (now available as A375071 on the OEIS).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2092, "problem_number": "EP-390", "title": "Erdős Problem #390", "statement": "Let $f(n)$ be the minimal $m$ such that $ n! = a_1\\cdots a_k $ with $n< a_1<\\cdots 0$?\nIs it true that, for $k\\geq 2$, $ \\sum_{n\\leq x}t_{k+1}(n) =o\\left(\\sum_{n\\leq x}t_k(n)\\right)? $ ", "background": "In \\cite{ErGr80} they mention a conjecture of Erd\\H{o}s that the sum is $o(x^2)$. This was proved by Erd\\H{o}s and Hall \\cite{ErHa78}, who proved that in fact $ \\sum_{n\\leq x}t_2(n)\\ll \\frac{\\log\\log\\log x}{\\log\\log x}x^2. $ Erd\\H{o}s and Hall conjecture that the sum is $o(x^2/(\\log x)^c)$ for any $c<\\log 2$.\nSince $t_2(p)=p-1$ for prime $p$ it is trivial that $ \\sum_{n\\leq x}t_2(n)\\gg \\frac{x^2}{\\log x}. $ Erd\\H{o}s and Hall \\cite{ErHa78} also note that $t_{n-1}(n!)=2$ and $t_{n-2}(n!)\\ll n$, which $n=2^r$ shows is the best possible. They ask about the behaviour of $t_{n-3}(n!)$ and also ask ask whether, for infinitely many $n$, $ t_k(n!)< t_{k-1}(n!)-1 $ for all $1\\leq k0$ such that there are infinitely many $n$ where $m+\\epsilon \\omega(m)\\leq n$ for all $m\\phi(m+2)>\\cdots \\phi(m+k)? $ Is it true that 'natural' ordering which mimics what happens to $\\phi(1),\\ldots,\\phi(k)$ is the most likely to appear?", "background": "Erd\\H{o}s \\cite{Er36b} proved that $ F(n)\\asymp \\log\\log\\log n, $ and similarly if we replace $\\phi$ with $\\sigma$ or $\\tau$ or $\nu$ or any 'decent' additive or multiplicative function.\nWeisenberg has observed that the same questions could be asked for ordering patterns which allow equality (indeed, the final problem only makes sense if we allow equality).\nReferences\n\n\n[Er36b] Erd\\H{o}s, P., On a problem of Chowla and some related problems. Proc. Cambridge Philos. Soc. (1936), 530-540.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2107, "problem_number": "EP-416", "title": "Erdős Problem #416", "statement": "Let $V(x)$ count the number of $n\\leq x$ such that $\\phi(m)=n$ is solvable. Does $V(2x)/V(x)\\to 2$? Is there an asymptotic formula for $V(x)$?", "background": "Pillai \\cite{Pi29} proved $V(x)=o(x)$. Erd\\H{o}s \\cite{Er35b} proved $V(x)=x(\\log x)^{-1+o(1)}$.\nThe behaviour of $V(x)$ is now almost completely understood. Maier and Pomerance \\cite{MaPo88} proved $ V(x)=\\frac{x}{\\log x}e^{(C+o(1))(\\log\\log\\log x)^2}, $ for some explicit constant $C>0$. Ford \\cite{Fo98} improved this to $ V(x)\\asymp\\frac{x}{\\log x}e^{C_1(\\log\\log\\log x-\\log\\log\\log\\log x)^2+C_2\\log\\log\\log x-C_3\\log\\log\\log\\log x} $ for some explicit constants $C_1,C_2,C_3>0$. Unfortunately this falls just short of an asymptotic formula for $V(x)$ and determining whether $V(2x)/V(x)\\to 2$.\nIn \\cite{Er79e} Erd\\H{o}s asks further to estimate the number of $n\\leq x$ such that the smallest solution to $\\phi(m)=n$ satisfies $kx1$?", "background": "It is trivial that $V'(x) \\leq V(x)$. In \\cite{Er98} Erd\\H{o}s suggests the limit may be infinite. See also [416].\nReferences\n\n\n[Er98] Erd\\H{o}s, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2109, "problem_number": "EP-420", "title": "Erdős Problem #420", "statement": "If $\\tau(n)$ counts the number of divisors of $n$ then let $ F(f,n)=\\frac{\\tau((n+\\lfloor f(n)\\rfloor)!)}{\\tau(n!)}. $ Is it true that $ \\lim_{n\\to \\infty}F((\\log n)^C,n)=\\infty $ for large $C$?\nIs it true that $F(\\log n,n)$ is everywhere dense in $(1,\\infty)$?\nMore generally, if $f(n)\\leq \\log n$ is a monotonic function such that $f(n)\\to \\infty$ as $n\\to \\infty$, then is $F(f,n)$ everywhere dense?", "background": "Erd\\H{o}s and Graham write that it is easy to show that $\\lim F(n^{1/2},n)=\\infty$, and in fact the $n^{1/2}$ can be replaced by $n^{1/2-c}$ for some small constant $c>0$.\nErd\\H{o}s, Graham, Ivi\\'{c}, and Pomerance \\cite{EGIP96} have proved that $ \\liminf F(c\\log n, n) = 1 $ for any $c>0$, and $ \\lim F(n^{4/9},n)=\\infty. $ (The exponent $4/9$ can be improved slightly.) They also prove that, if $f(n)=o((\\log n)^2)$, then for almost all $n$ $ F(f,n)\\sim 1. $ van Doorn notes in the comments that the existence of infinitely many bounded prime gaps implies $ \\limsup_{n\\to \\infty}F(g(n),n)=\\infty $ for any $g(n)\\to \\infty$, and that Cram\\'{e}r's conjecture implies $ \\lim F(g(n)(\\log n)^2, n)=\\infty $ for any $g(n)\\to \\infty$>\nReferences\n\n\n[EGIP96] Erd\\H{o}s, Paul and Graham, S. W. and Ivi\\'c, Aleksandar and\nPomerance, Carl, On the number of divisors of {$n!$}. (1996), 337--355.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2110, "problem_number": "EP-421", "title": "Erdős Problem #421", "statement": "Is there a sequence $1\\leq d_11/e-\\epsilon$ for any $\\epsilon>0$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2111, "problem_number": "EP-422", "title": "Erdős Problem #422", "statement": "Let $f(1)=f(2)=1$ and for $n>2$ $ f(n) = f(n-f(n-1))+f(n-f(n-2)). $ Does $f(n)$ miss infinitely many integers? What is its behaviour?", "background": "Asked by Hofstadter. The sequence begins $1,1,2,3,3,4,\\ldots$ and is A005185 in the OEIS. It is not even known whether $f(n)$ is well-defined for all $n$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2112, "problem_number": "EP-423", "title": "Erdős Problem #423", "statement": "Let $a_1=1$ and $a_2=2$ and for $k\\geq 3$ choose $a_k$ to be the least integer $>a_{k-1}$ which is the sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence?", "background": "Asked by Hofstadter (in \\cite{Er77c} Erd\\H{o}s says Hofstadter was inspired by a similar question of Ulam). The sequence begins $ 1,2,3,5,6,8,10,11,\\ldots $ and is A005243 in the OEIS.\nBolan and Tang have independently proved that there are infinitely many integers which do not appear in this sequence. In fact, the sequence $a_n-n$ is nondecreasing and unbounded.\nReferences\n\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2113, "problem_number": "EP-424", "title": "Erdős Problem #424", "statement": "Let $a_1=2$ and $a_2=3$ and continue the sequence by appending to $a_1,\\ldots,a_n$ all possible values of $a_ia_j-1$ with $i\neq j$. Is it true that the set of integers which eventually appear has positive density?", "background": "Asked by Hofstadter. The sequence begins $2,3,5,9,14,17,26,\\ldots$ and is A005244 in the OEIS. This problem is also discussed in section E31 of Guy's book Unsolved Problems in Number Theory.\nIn \\cite{ErGr80} (and in Guy's book) this problem as written is asking for whether almost all integers appear in this sequence, but the answer to this is trivially no (as pointed out to me by Steinerberger): no integer $\\equiv 1\\pmod{3}$ is ever in the sequence, so the set of integers which appear has density at most $2/3$. This is easily seen by induction, and the fact that if $a,b\\in \\{0,2\\}\\pmod{3}$ then $ab-1\\in \\{0,2\\}\\pmod{3}$.\nPresumably it is the weaker question of whether a positive density of integers appear (as correctly asked in \\cite{Er77c}) that was also intended in \\cite{ErGr80}.\nSee also Problem 63 of Green's open problems list.\nReferences\n\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2114, "problem_number": "EP-425", "title": "Erdős Problem #425", "statement": "Let $F(n)$ be the maximum possible size of a subset $A\\subseteq\\{1,\\ldots,N\\}$ such that the products $ab$ are distinct for all $a0? $ ", "background": "Erd\\H{o}s and Graham could show this is true (assuming the prime $k$-tuple conjecture) if we replace $\\liminf$ by $\\limsup$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2116, "problem_number": "EP-430", "title": "Erdős Problem #430", "statement": "Fix some integer $n$ and define a decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\\geq 2$, letting $a_k$ be the greatest integer in $[1,a_{k-1})$ such that all of the prime factors of $a_k$ are $>n-a_k$.\nIs it true that, for sufficiently large $n$, not all of this sequence can be prime?", "background": "Erd\\H{o}s and Graham write 'preliminary calculations made by Selfridge indicate that this is the case but no proof is in sight'. For example if $n=8$ we have $a_1=7$ and $a_2=5$ and then must stop.\nSarosh Adenwalla has observed that this problem is equivalent to (the first part of) [385]. Indeed, assuming a positive answer to that, for all large $n$, there exists a composite $mn-m$. It follows that such an $m$ is equal to some $a_i$ in the sequence defined for $[1,n)$, and $m$ is composite by assumption.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2117, "problem_number": "EP-431", "title": "Erdős Problem #431", "statement": "Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the set of prime numbers up to finitely many exceptions?", "background": "A problem of Ostmann, sometimes known as the 'inverse Goldbach problem'. The answer is surely no. The best result in this direction is due to Elsholtz and Harper \\cite{ElHa15}, who showed that if $A,B$ are such sets then for all large $x$ we must have $ \\frac{x^{1/2}}{\\log x\\log\\log x} \\ll \\lvert A \\cap [1,x]\\rvert \\ll x^{1/2}\\log\\log x $ and similarly for $B$.\nElsholtz \\cite{El01} has proved there are no sets $A,B,C$ (all of size at least $2$) such that $A+B+C$ agrees with the set of prime numbers up to finitely many exceptions.\nGranville \\cite{Gr90} proved, conditional on the prime $k$-tuples conjecture, that there are infinite sets $B$ and $C$ such that $ \\{ \\tfrac{b+c}{2}: b\\in B, c\\in C\\} $ is a subset of the primes. Tao and Ziegler \\cite{TaZi23} gave an unconditional proof that there are infinite sets $B=\\{b_1<\\cdots\\}$ and $C=\\{c_1<\\cdots\\}$ such that $ \\{ b_i+c_j : b_i\\in B, c_j\\in C, i1/2$, if $p$ is a sufficiently large prime then, for any $n\\geq 0$, there exist $a,b\\in(n,n+p^c)$ such that $ab\\equiv 1\\pmod{p}$?", "background": "Heilbronn (unpublished) proved this for $c$ sufficiently close to $1$. Heath-Brown \\cite{He00} used Kloosterman sums to prove this for all $c>3/4$.\nThis is discussed in this MathOverflow question.\nReferences\n\n\n[He00] Heath-Brown, D. R., Arithmetic applications of {K}loosterman sums. Nieuw Arch. Wiskd. (5) (2000), 380--384.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2121, "problem_number": "EP-450", "title": "Erdős Problem #450", "statement": "How large must $y=y(\\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\\epsilon y$?", "background": "It is not clear what the intended quantifier on $x$ is. Cambie has observed that if this is intended to hold for all $x$ then, provided $ \\epsilon(\\log n)^\\delta (\\log\\log n)^{3/2}\\to \\infty $ as $n\\to \\infty$, where $\\delta=0.086\\cdots$, there is no such $y$, which follows from an averaging argument and the work of Ford \\cite{Fo08}.\nOn the other hand, Cambie has observed that if $\\epsilon\\ll 1/n$ then $y(\\epsilon,n)\\sim 2n$: indeed, if $y<2n$ then this is impossible taking $x+n$ to be a multiple of the lowest common multiple of $\\{n+1,\\ldots,2n-1\\}$. On the other hand, for every fixed $\\delta\\in (0,1)$ and $n$ large every $2(1+\\delta)n$ consecutive elements contains many elements which are a multiple of an element in $(n,2n)$.\nReferences\n\n\n[Fo08] Ford, Kevin, The distribution of integers with a divisor in a given\ninterval. Ann. of Math. (2) (2008), 367-433.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2122, "problem_number": "EP-451", "title": "Erdős Problem #451", "statement": "Estimate $n_k$, the smallest integer $>2k$ such that $\\prod_{1\\leq i\\leq k}(n_k-i)$ has no prime factor in $(k,2k)$.", "background": "Erd\\H{o}s and Graham write 'we can prove $n_k>k^{1+c}$ but no doubt much more is true'.\nIn \\cite{Er79d} Erd\\H{o}s writes that probably $n_kk^d$ for all constant $d$.\nAdenwalla observes that an easy upper bound is $n_k\\leq \\prod_{k\\log\\log n$ for all $n\\in I$?", "background": "Erd\\H{o}s \\cite{Er37} proved that the density of integers $n$ with $\\omega(n)>\\log\\log n$ is $1/2$. The Chinese remainder theorem implies that there is such an interval with $ \\lvert I\\rvert \\geq (1+o(1))\\frac{\\log x}{(\\log\\log x)^2}. $ It could be true that there is such an interval of length $(\\log x)^{k}$ for arbitrarily large $k$.\nReferences\n\n\n[Er37] Erd\"{o}s, Paul, Note on the number of prime divisors of integers. J. London Math. Soc. (1937), 308-314.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2124, "problem_number": "EP-454", "title": "Erdős Problem #454", "statement": "Let $ f(n) = \\min_{i0.352\\cdots. $ \nReferences\n\n\n[Ri76] Richter, Bernd, \"{U}ber die Monotonie von Differenzenfolgen. Acta Arith. (1976), 225-227.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2126, "problem_number": "EP-456", "title": "Erdős Problem #456", "statement": "Let $p_n$ be the smallest prime $\\equiv 1\\pmod{n}$ and let $m_n$ be the smallest integer such that $n\\mid \\phi(m_n)$.\nIs it true that $m_n0$ such that there are infinitely many $n$ where all primes $p\\leq (2+\\epsilon)\\log n$ divide $ \\prod_{1\\leq i\\leq \\log n}(n+i)? $ ", "background": "A problem of Erd\\H{o}s and Pomerance.\nMore generally, let $q(n,k)$ denote the least prime which does not divide $\\prod_{1\\leq i\\leq k}(n+i)$. This problem asks whether $q(n,\\log n)\\geq (2+\\epsilon)\\log n$ infinitely often. Taking $n$ to be the product of primes between $\\log n$ and $(2+o(1))\\log n$ gives an example where $ q(n,\\log n)\\geq (2+o(1))\\log n. $ Can one prove that $q(n,\\log n)<(1-\\epsilon)(\\log n)^2$ for all large $n$ and some $\\epsilon>0$?\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2128, "problem_number": "EP-460", "title": "Erdős Problem #460", "statement": "Let $a_0=0$ and $a_1=1$, and in general define $a_k$ to be the least integer $>a_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $0\\leq in$ (this observation was made by Svyable using ChatGPT).\nUnfortunately, although both sources mention a forthcoming paper of Eggleton, Erd\\H{o}s, and Selfridge, I cannot find a candidate paper of theirs with this problem in, and hence the motivation behind this problem, and what the precise problem intended was, is unclear.\nChojecki has noted in the comments that a positive solution to the main problem would follow if $ f(n) = \\sum_{aa}\\frac{1}{a}\\to \\infty, $ where $P^-(\\cdot)$ is the least prime factor. Standard estimates on rough numbers show that $\\frac{1}{N}\\sum_{n\\leq N}f(n)\\gg \\log\\log N$, so $f(n)$ does diverge on average, but it is unclear whether $f(n)\\to \\infty$ for all $n$.\nReferences\n\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2129, "problem_number": "EP-461", "title": "Erdős Problem #461", "statement": "Let $s_t(n)$ be the $t$-smooth component of $n$ - that is, the product of all primes $p$ (with multiplicity) dividing $n$ such that $p0$ such that $ \\sum_{\\substack{n0$ such that $ \\sum_{x\\leq n\\leq x+Cx^{1/2}(\\log x)^2}\\frac{p(n)}{n} \\gg 1 $ for all large $x$?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2131, "problem_number": "EP-463", "title": "Erdős Problem #463", "statement": "Is there a function $f$ with $f(n)\\to \\infty$ as $n\\to \\infty$ such that, for all large $n$, there is a composite number $m$ such that $ n+f(n)n}(m-p(m)), $ and whether $n-F(n)\\sim cn^{1/2}$ for some $c>0$.\nSee also [385].\nReferences\n\n\n[Er92e] Erd\\H{o}s, P\\'{a}l, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2132, "problem_number": "EP-467", "title": "Erdős Problem #467", "statement": "Prove the following for all large $x$: there is a choice of congruence classes $a_p$ for all primes $p\\leq x$ and a decomposition $\\{p\\leq x\\}=A\\sqcup B$ into two non-empty sets such that, for all $n1$, so the restriction $k\neq 1$ is necessary. Erd\\H{o}s and Graham report that Graham, Lehmer, and Lehmer have proved this for $k=2^i$ for $i\\geq 1$, or if $k=-1$, but I cannot find such a paper. Tang has written a short note giving a proof for this case.\nAs an indication of the difficulty, when $k=3$ the smallest $n$ such that $2^n\\equiv 3\\pmod{n}$ is $n=4700063497$.\nThe minimal such $n$ for each $k$ is A036236 in the OEIS.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2140, "problem_number": "EP-483", "title": "Erdős Problem #483", "statement": "Let $f(k)$ be the minimal $N$ such that if $\\{1,\\ldots,N\\}$ is $k$-coloured then there is a monochromatic solution to $a+b=c$. Estimate $f(k)$. In particular, is it true that $f(k) < c^k$ for some constant $c>0$?", "background": "The values of $f(k)$ are known as Schur numbers. The best-known bounds for large $k$ are $ (380)^{k/5}-O(1)\\leq f(k) \\leq \\lfloor(e-\\tfrac{1}{24}) k!\\rfloor-1. $ The lower bound is due to Ageron, Casteras, Pellerin, Portella, Rimmel, and Tomasik \\cite{ACPPRT21} (improving previous bounds of Exoo \\cite{Ex94} and Fredricksen and Sweet \\cite{FrSw00}) and the upper bound is due to Whitehead \\cite{Wh73}. Note that $380^{1/5}\\approx 3.2806$.\nThe known values of $f$ are $f(1)=2$, $f(2)=5$, $f(3)=14$, $f(4)=45$, and $f(5)=161$ (see A030126). (The equality $f(5)=161$ was established by Heule \\cite{He17}).\nSee also [183] (in particular a folklore observation gives $f(k)\\leq R(3;k)-1$).\nReferences\n\n\n[ACPPRT21] R. Ageron, P. Casteras, T. Pellerin, Y. Portella, A. Rimmel, and J. Tomasik, New lower bounds for Schur and weak Schur numbers. arXiv:2112.03175 (2021).\n\n[Ex94] Exoo, G., A lower bound for Schur numbers and multicolor Ramsey numbers. Electronic J. of Combinatorics (1994).\n\n[FrSw00] Fredricksen, Harold and Sweet, Melvin M., Symmetric sum-free partitions and lower bounds for {S}chur\nnumbers. Electron. J. Combin. (2000), Research Paper 32, 9.\n\n[He17] M. Heuele, Schur Number Five. arXiv:1711.08076 (2017).\n\n[Wh73] Whitehead, Jr., Earl Glen, The {R}amsey number {$N(3,\\,3,\\,3,\\,3;\\,2)$}. Discrete Math. (1973), 389--396.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2141, "problem_number": "EP-486", "title": "Erdős Problem #486", "statement": "Let $A\\subseteq \\mathbb{N}$, and for each $n\\in A$ choose some $X_n\\subseteq \\mathbb{Z}/n\\mathbb{Z}$. Let $ B = \\{ m\\in \\mathbb{N} : m\not\\in X_n\\pmod{n}\\textrm{ for all }n\\in A\\textrm{ with }m>n\\}. $ Must $B$ have a logarithmic density, i.e. is it true that $ \\lim_{x\\to \\infty} \\frac{1}{\\log x}\\sum_{\\substack{m\\in B\\\\ mn\\geq \\max(A)$, $ \\frac{\\lvert B\\cap [1,m]\\rvert }{m}< 2\\frac{\\lvert B\\cap [1,n]\\rvert}{n}? $ ", "background": "The constant $2$ would be the best possible here, as witnessed by taking $A=\\{a\\}$, $n=2a-1$, and $m=2a$.\nThis problem is also discussed in problem E5 of Guy's collection \\cite{Gu04}.\nIn \\cite{Er61} this problem is as stated above, but with $a\\mid n$ in the definition of $B$ replaced by $a\nmid n$. This is most likely a typo (especially since the problem is also given as stated above in \\cite{Er66}). There have been several counterexamples given for this alternate problem. Cambie has observed that, if $A$ is the set of primes bounded above by $n$, and $m=2n$, then $ \\frac{\\lvert B\\cap [1,m]\\rvert }{m}=\\frac{\\pi(2n)-\\pi(n)+1}{2n}\\sim \\frac{1}{2\\log n} $ while $ \\frac{\\lvert B\\cap [1,n]\\rvert}{n}=\\frac{1}{n}. $ Further concrete counterexamples, found by Alexeev and Aristotle, are given in the comments section.\nReferences\n\n\n[Er61] Erd\\H{o}s, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutat\\'{o} Int. K\"{o}zl. (1961), 221-254.\n\n[Er66] Erd\\H{o}s, P\\'al, Remarks on number theory. {V}. {E}xtremal problems in number\ntheory. {II}. Mat. Lapok (1966), 135--155.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2143, "problem_number": "EP-489", "title": "Erdős Problem #489", "statement": "Let $A\\subseteq \\mathbb{N}$ be a set such that $\\lvert A\\cap [1,x]\\rvert=o(x^{1/2})$. Let $ B=\\{ n\\geq 1 : a\nmid n\\textrm{ for all }a\\in A\\}. $ If $B=\\{b_10$ and $\\theta$ such that $ \\sum_{n\\in A}\\cos(n\\theta) < -cN^{1/2}? $ ", "background": "Chowla's cosine problem. Ruzsa \\cite{Ru04} (improving on an earlier result of Bourgain \\cite{Bo86}), proved an upper bound of $ -\\exp(O(\\sqrt{\\log N})). $ Polynomial bounds were proved independently by Bedert \\cite{Be25c} and Jin, Milojevi\\'{c}, Tomon, and Zhang \\cite{JMTZ25}. The best bound follows from the method of Bedert \\cite{Be25c}, which proved the existence of some $c>0$ such that, for all $A$ of size $N$, $ \\sum_{n\\in A}\\cos(n\\theta) < -cN^{1/7}. $ The example $A=B-B$, where $B$ is a Sidon set, shows that $N^{1/2}$ would be the best possible here.\nThis problem is Problem 81 on Green's open problems list.\nThis is related to [256].\nReferences\n\n\n[Be25c] B. Bedert, Polynomial bounds for the Chowla Cosine Problem. arXiv:2509.05260 (2025).\n\n[Bo86] Bourgain, J., Sur le minimum d'une somme de cosinus. Acta Arith. (1986), 381-389.\n\n[JMTZ25] Z. Jin, A. Milojevi\\'{c}, I. Tomon, and S. Zhang, From small eigenvalues to large cuts, and Chowla's cosine problem. arXiv:2509.03490 (2025).\n\n[Ru04] Ruzsa, Imre Z., Negative values of cosine sums. Acta Arith. (2004), 179-186.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2152, "problem_number": "EP-513", "title": "Erdős Problem #513", "statement": "Let $f=\\sum_{n=0}^\\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of $ \\liminf_{r\\to \\infty} \\frac{\\max_n\\lvert a_nr^n\\rvert}{\\max_{\\lvert z\\rvert=r}\\lvert f(z)\\rvert}? $ ", "background": "It is trivial that this value is in $[1/2,1)$. K\"{o}v\\'{a}ri (unpublished) observed that it must be $>1/2$. Clunie and Hayman \\cite{ClHa64} showed that it is $\\leq 2/\\pi-c$ for some absolute constant $c>0$. Some other results on this quantity were established by Gray and Shah \\cite{GrSh63}.\nSee also [227].\nReferences\n\n\n[ClHa64] Clunie, J. and Hayman, W. K., The maximum term of a power series. J. Analyse Math. (1964), 143-186.\n\n[GrSh63] Gray, Alfred and Shah, S. M., A note on entire functions and a conjecture of Erd\\H{o}s. Bull. Amer. Math. Soc. (1963), 573-577.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2153, "problem_number": "EP-514", "title": "Erdős Problem #514", "statement": "Let $f(z)$ be an entire transcendental function. Does there exist a path $L$ so that, for every $n$, $ \\lvert f(z)/z^n\\rvert \\to \\infty $ as $z\\to \\infty$ along $L$?\nCan the length of this path be estimated in terms of $M(r)=\\max_{\\lvert z\\rvert=r}\\lvert f(z)\\rvert$? Does there exist a path along which $\\lvert f(z)\\rvert$ tends to $\\infty$ faster than a fixed function of $M(r)$ (such that $M(r)^\\epsilon$)?", "background": "Boas (unpublished) has proved the first part, that such a path must exist.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2154, "problem_number": "EP-517", "title": "Erdős Problem #517", "statement": "Let $f(z)=\\sum_{k=1}^\\infty a_kz^{n_k}$ be an entire function (with $a_k\neq 0$ for all $k\\geq 1$). Is it true that if $n_k/k\\to \\infty$ then $f(z)$ assumes every value infinitely often?", "background": "A conjecture of Fej\\'{e}r and P\\'{o}lya.\nFej\\'{e}r \\cite{Fe08} proved that if $\\sum\\frac{1}{n_k}<\\infty$ then $f(z)$ assumes every value at least once, and Biernacki \\cite{Bi28} proved that if $\\sum\\frac{1}{n_k}<\\infty$ then $f(z)$ assumes every value infinitely often.\nP\\'{o}lya \\cite{Po29} proved that if $f$ has finite order then $f(z)$ assumes every value infinitely often under the assumption that $\\limsup (n_{k+1}-n_k)=\\infty$.\nReferences\n\n\n[Bi28] Biernacki, Mi\\'{e}cislas, Sur les \\'{e}quations alg\\'{e}briques contenant des param\\'{e}tres arbitraires. (1928), 145.\n\n[Fe08] Fej\\'{e}r, Leopold, \"{U}ber die Wurzel vom kleinsten absoluten Betrage einer algebraischen Gleichung. Math. Ann. (1908), 413-423.\n\n[Po29] P\\'olya, G., Untersuchungen \"uber {L}\"ucken und {S}ingularit\"{a}ten von\n{P}otenzreihen. Math. Z. (1929), 549--640.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2155, "problem_number": "EP-520", "title": "Erdős Problem #520", "statement": "Let $f$ be a Rademacher multiplicative function: a random $\\{-1,0,1\\}$-valued multiplicative function, where for each prime $p$ we independently choose $f(p)\\in \\{-1,1\\}$ uniformly at random, and for square-free integers $n$ we extend $f(p_1\\cdots p_r)=f(p_1)\\cdots f(p_r)$ (and $f(n)=0$ if $n$ is not squarefree). Does there exist some constant $c>0$ such that, almost surely, $ \\limsup_{N\\to \\infty}\\frac{\\sum_{m\\leq N}f(m)}{\\sqrt{N\\log\\log N}}=c? $ ", "background": "Note that if we drop the multiplicative assumption, and simply assign $f(m)=\\pm 1$ at random, then this statement is true (with $c=\\sqrt{2}$), the law of the iterated logarithm.\nWintner \\cite{Wi44} proved that, almost surely, $ \\sum_{m\\leq N}f(m)\\ll N^{1/2+o(1)}, $ and Erd\\H{o}s improved the right-hand side to $N^{1/2}(\\log N)^{O(1)}$. Lau, Tenenbaum, and Wu \\cite{LTW13} have shown that, almost surely, $ \\sum_{m\\leq N}f(m)\\ll N^{1/2}(\\log\\log N)^{2+o(1)}. $ Caich \\cite{Ca24b} has improved this to $ \\sum_{m\\leq N}f(m)\\ll N^{1/2}(\\log\\log N)^{3/4+o(1)}. $ Harper \\cite{Ha13} has shown that the sum is almost surely not $O(N^{1/2}/(\\log\\log N)^{5/2+o(1)})$, and conjectured that in fact Erd\\H{o}s' conjecture is false, and almost surely $ \\sum_{m\\leq N}f(m) \\ll N^{1/2}(\\log\\log N)^{1/4+o(1)}. $ \nReferences\n\n\n[Ca24b] R. Caich, Almost sure upper bound for random multiplicative functions. arXiv:2304.00943 (2024).\n\n[Ha13] Harper, Adam J., Bounds on the suprema of Gaussian processes, and omega\nresults for the sum of a random multiplicative function. Ann. Appl. Probab. (2013), 584-616.\n\n[LTW13] Lau, Yuk-Kam and Tenenbaum, G\\'{e}rald and Wu, Jie, On mean values of random multiplicative functions. Proc. Amer. Math. Soc. (2013), 409-420.\n\n[Wi44] Wintner, Aurel, Random factorizations and Riemann's hypothesis. Duke Math. J. (1944), 267-275.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2156, "problem_number": "EP-521", "title": "Erdős Problem #521", "statement": "Let $(\\epsilon_k)_{k\\geq 0}$ be independently uniformly chosen at random from $\\{-1,1\\}$. If $R_n$ counts the number of real roots of $f_n(z)=\\sum_{0\\leq k\\leq n}\\epsilon_k z^k$ then is it true that, almost surely, $ \\lim_{n\\to \\infty}\\frac{R_n}{\\log n}=\\frac{2}{\\pi}? $ ", "background": "Erd\\H{o}s and Offord \\cite{EO56} showed that the number of real roots of a random degree $n$ polynomial with $\\pm 1$ coefficients is $(\\frac{2}{\\pi}+o(1))\\log n$.\nIt is ambiguous in \\cite{Er61} whether Erd\\H{o}s intended the coefficients to be uniformly chosen from $\\{-1,1\\}$ or $\\{0,1\\}$. In the latter case, the constant $\\frac{2}{\\pi}$ should be $\\frac{1}{\\pi}$ (see the discussion in the comments).\nIn the case of $\\{-1,1\\}$ Do \\cite{Do24} proved that, if $R_n[-1,1]$ counts the number of roots in $[-1,1]$, then, almost surely, $ \\lim_{n\\to \\infty}\\frac{R_n[-1,1]}{\\log n}=\\frac{1}{\\pi}. $ See also [522].\nReferences\n\n\n[Do24] Y. Do, A strong law of large numbers for real roots of random polynomials. arXiv:2403.06353 (2024).\n\n[EO56] Erd\"{o}s, Paul and Offord, A. C., On the number of real roots of a random algebraic equation. Proc. London Math. Soc. (3) (1956), 139-160.\n\n[Er61] Erd\\H{o}s, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutat\\'{o} Int. K\"{o}zl. (1961), 221-254.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2157, "problem_number": "EP-522", "title": "Erdős Problem #522", "statement": "Let $f(z)=\\sum_{0\\leq k\\leq n} \\epsilon_k z^k$ be a random polynomial, where $\\epsilon_k\\in \\{-1,1\\}$ independently uniformly at random for $0\\leq k\\leq n$.\nIs it true that, if $R_n$ is the number of roots of $f(z)$ in $\\{ z\\in \\mathbb{C} : \\lvert z\\rvert \\leq 1\\}$, then $ \\frac{R_n}{n/2}\\to 1 $ almost surely?", "background": "Random polynomials with independently identically distributed coefficients are sometimes called Kac polynomials - this problem considers the case of Rademacher coefficients, i.e. independent uniform $\\pm 1$ values. Erd\\H{o}s and Offord \\cite{EO56} showed that the number of real roots of a random degree $n$ polynomial with $\\pm 1$ coefficients is $(\\frac{2}{\\pi}+o(1))\\log n$.\nThere is some ambiguity whether Erd\\H{o}s intended the coefficients to be in $\\{-1,1\\}$ or $\\{0,1\\}$ - see the comments section.\nA weaker version of this was solved by Yakir \\cite{Ya21}, who proved that $ \\frac{R_n}{n/2}\\to 1 $ in probability. (This weaker claim was also asked by Erd\\H{o}s, and also appears in a book of Hayman \\cite{Ha67}.) More precisely, $ \\lim_{n\\to \\infty} \\mathbb{P}(\\lvert R_n-n/2\\rvert \\geq n^{9/10}) =0. $ See also [521].\nReferences\n\n\n[EO56] Erd\"{o}s, Paul and Offord, A. C., On the number of real roots of a random algebraic equation. Proc. London Math. Soc. (3) (1956), 139-160.\n\n[Ha67] Hayman, W. K., Research problems in function theory. (1967), vii+56.\n\n[Ya21] Yakir, Oren, Approximately half of the roots of a random {L}ittlewood\npolynomial are inside the disk. Studia Math. (2021), 227--240.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2158, "problem_number": "EP-524", "title": "Erdős Problem #524", "statement": "For any $t\\in (0,1)$ let $t=\\sum_{k=1}^\\infty \\epsilon_k(t)2^{-k}$ (where $\\epsilon_k(t)\\in \\{0,1\\}$). What is the correct order of magnitude (for almost all $t\\in(0,1)$) for $ M_n(t)=\\max_{x\\in [-1,1]}\\left\\lvert \\sum_{k\\leq n}(-1)^{\\epsilon_k(t)}x^k\\right\\rvert? $ ", "background": "A problem of Salem and Zygmund \\cite{SaZy54}. Chung showed that, for almost all $t$, there exist infinitely many $n$ such that $ M_n(t) \\ll \\left(\\frac{n}{\\log\\log n}\\right)^{1/2}. $ Erd\\H{o}s (unpublished) showed that for almost all $t$ and every $\\epsilon>0$ we have $\\lim_{n\\to \\infty}M_n(t)/n^{1/2-\\epsilon}=\\infty$.\nReferences\n\n\n[SaZy54] Salem, R. and Zygmund, A., Some properties of trigonometric series whose terms have\nrandom signs. Acta Math. (1954), 245-301.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2159, "problem_number": "EP-528", "title": "Erdős Problem #528", "statement": "Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\\mathbb{Z}^k$ (i.e. those walks which do not intersect themselves). Determine $ C_k=\\lim_{n\\to\\infty}f(n,k)^{1/n}. $ ", "background": "The constant $C_k$ is sometimes known as the connective constant. Hammersley and Morton \\cite{HM54} showed that this limit exists, and it is trivial that $k\\leq C_k\\leq 2k-1$.\nKesten \\cite{Ke63} proved that $C_k=2k-1-1/2k+O(1/k^2)$, and more precise asymptotics are given by Clisby, Liang, and Slade \\cite{CLS07}.\nConway and Guttmann \\cite{CG93} showed that $C_2\\geq 2.62$ and Alm \\cite{Al93} showed that $C_2\\leq 2.696$. Jacobsen, Scullard, and Guttmann \\cite{JSG16} have computed the first few decimal places of $C_2$, showing that $ C_2 = 2.6381585303279\\cdots. $ See also [529].\nReferences\n\n\n[Al93] Alm, Sven Erick, Upper bounds for the connective constant of self-avoiding\nwalks. Combin. Probab. Comput. (1993), 115-136.\n\n[CG93] Conway, A. R. and Guttmann, A. J., Lower bound on the connective constant for square lattice\nself-avoiding walks. J. Phys. A (1993), 3719-3724.\n\n[CLS07] Clisby, Nathan and Liang, Richard and Slade, Gordon, Self-avoiding walk enumeration via the lace expansion. J. Phys. A (2007), 10973-11017.\n\n[HM54] Hammersley, J. M. and Morton, K. W., Poor man's Monte Carlo. J. Roy. Statist. Soc. Ser. B (1954), 23-38; discussion 61-75.\n\n[JSG16] Jacobsen, Jesper Lykke and Scullard, Christian R. and\nGuttmann, Anthony J., On the growth constant for square-lattice self-avoiding walks. J. Phys. A (2016), 494004, 18.\n\n[Ke63] Kesten, Harry, On the number of self-avoiding walks. J. Mathematical Phys. (1963), 960-969.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2160, "problem_number": "EP-529", "title": "Erdős Problem #529", "statement": "Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\\mathbb{Z}^k$ (conditional on no self intersections) - that is, a self-avoiding walk. Is it true that $ \\lim_{n\\to \\infty}\\frac{d_2(n)}{n^{1/2}}= \\infty? $ Is it true that $ d_k(n)\\ll n^{1/2} $ for $k\\geq 3$?", "background": "Slade \\cite{Sl87} proved that, for $k$ sufficiently large, $d_k(n)\\sim Dn^{1/2}$ for some constant $D>0$ (independent of $k$). Hara and Slade (\\cite{HaSl91} and \\cite{HaSl92}) proved this for all $k\\geq 5$.\nFor $k=2$ Duminil-Copin and Hammond \\cite{DuHa13} have proved that $d_2(n)=o(n)$.\nIt is now conjectured that $d_k(n)\\ll n^{1/2}$ is false for $k=3$ and $k=4$, and more precisely (see for example Section 1.4 of \\cite{MaSl93}) that $d_2(n)\\sim Dn^{3/4}$, $d_3(n)\\sim n^{\nu}$ where $\nu\\approx 0.59$, and $d_4(n)\\sim D(\\log n)^{1/8}n^{1/2}$.\nMadras and Slade \\cite{MaSl93} have a monograph on the topic of self-avoiding walks.\nSee also [528].\nReferences\n\n\n[DuHa13] Duminil-Copin, Hugo and Hammond, Alan, Self-avoiding walk is sub-ballistic. Comm. Math. Phys. (2013), 401--423.\n\n[HaSl91] Hara, Takashi and Slade, Gordon, Critical behaviour of self-avoiding walk in five or more\ndimensions. Bull. Amer. Math. Soc. (N.S.) (1991), 417--423.\n\n[HaSl92] Hara, Takashi and Slade, Gordon, Self-avoiding walk in five or more dimensions. {I}. {T}he\ncritical behaviour. Comm. Math. Phys. (1992), 101--136.\n\n[MaSl93] Madras, Neal and Slade, Gordon, The self-avoiding walk. (1993), xiv+425.\n\n[Sl87] Slade, Gordon, The diffusion of self-avoiding random walk in high dimensions. Comm. Math. Phys. (1987), 661--683.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2161, "problem_number": "EP-530", "title": "Erdős Problem #530", "statement": "Let $\\ell(N)$ be maximal such that in any finite set $A\\subset \\mathbb{R}$ of size $N$ there exists a Sidon subset $S$ of size $\\ell(N)$ (i.e. the only solutions to $a+b=c+d$ in $S$ are the trivial ones). Determine the order of $\\ell(N)$.", "background": "In particular, is it true that $\\ell(N)\\sim N^{1/2}$?\nOriginally asked by Riddell \\cite{Ri69}. Erd\\H{o}s noted the bounds $ N^{1/3} \\ll \\ell(N) \\leq (1+o(1))N^{1/2} $ (the upper bound following from the case $A=\\{1,\\ldots,N\\}$). The lower bound was improved to $N^{1/2}\\ll \\ell(N)$ by Koml\\'{o}s, Sulyok, and Szemer\\'{e}di \\cite{KSS75}. The correct constant is unknown, but it is likely that the upper bound is true, so that $\\ell(N)\\sim N^{1/2}$.\nIn \\cite{AlEr85} Alon and Erd\\H{o}s make the stronger conjecture that perhaps $A$ can always be written as the union of at most $(1+o(1))N^{1/2}$ many Sidon sets. (This is easily verified for $A=\\{1,\\ldots,N\\}$ using standard constructions of Sidon sets.)\nThis is discussed in problem C9 of Guy's collection \\cite{Gu04}.\nSee also [1088] for a higher-dimensional generalisation.\nReferences\n\n\n[AlEr85] Alon, Noga and Erd\\H{o}s, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[KSS75] Koml\\'{o}s, J. and Sulyok, M. and Szemeredi, E., Linear problems in combinatorial number theory. Acta Math. Acad. Sci. Hungar. (1975), 113-121.\n\n[Ri69] Riddell, J., On sets of numbers containing no $l$ terms in arithmetic progression. Nieuw Arch. Wisk. (3) (1969), 204-209.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2162, "problem_number": "EP-531", "title": "Erdős Problem #531", "statement": "Let $F(k)$ be the minimal $N$ such that if we two-colour $\\{1,\\ldots,N\\}$ there is a set $A$ of size $k$ such that all subset sums $\\sum_{a\\in S}a$ (for $\\emptyset\neq S\\subseteq A$) are monochromatic. Estimate $F(k)$.", "background": "The existence of $F(k)$ was established by Sanders and Folkman, and it also follows from Rado's theorem. It is commonly known as Folkman's theorem.\nErd\\H{o}s and Spencer \\cite{ErSp89} proved that $ F(k) \\geq 2^{ck^2/\\log k} $ for some constant $c>0$. Balogh, Eberhrad, Narayanan, Treglown, and Wagner \\cite{BENTW17} have improved this to $ F(k) \\geq 2^{2^{k-1}/k}. $ \nReferences\n\n\n[BENTW17] Balogh, J\\'{o}zsef and Eberhard, Sean and Narayanan, Bhargav and Treglown, Andrew and Wagner, Adam Zsolt, An improved lower bound for Folkman's theorem. Bull. Lond. Math. Soc. (2017), 745-747.\n\n[ErSp89] Erd\\H{o}s, Paul and Spencer, Joel, Monochromatic sumsets. J. Combin. Theory Ser. A (1989), 162-163.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2163, "problem_number": "EP-533", "title": "Erdős Problem #533", "statement": "Let $\\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_5$ and at least $\\delta n^2$ edges then $G$ contains a set of $\\gg_\\delta n$ vertices containing no triangle.", "background": "A problem of Erd\\H{o}s, Hajnal, Simonovits, S\\'{o}s, and Szemer\\'{e}di, who could prove this is true for $\\delta>1/16$, and could further prove it for $\\delta>0$ if we replace $K_5$ with $K_4$.\nThey further observed that it fails for $\\delta =1/4$ if we replace $K_5$ with $K_7$: by a construction of Erd\\H{o}s and Rogers \\cite{ErRo62} (see [620]) there exists some constant $c>0$ such that, for all large $n$, there is a graph on $n$ vertices which contains no $K_4$ and every set of at least $n^{1-c}$ vertices contains a triangle. If we take two vertex disjoint copies of this graph and add all edges between the two copies then this yields a graph on $2n$ vertices with $\\geq n^2$ edges, which contains no $K_7$, yet every set of at least $2n^{1-c}$ vertices contains a triangle.\nSee also [579] and the entry in the graphs problem collection.\nReferences\n\n\n[ErRo62] Erd\\H{o}s, P. and Rogers, C. A., The construction of certain graphs. Canadian J. Math. (1962), 702-707.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2164, "problem_number": "EP-535", "title": "Erdős Problem #535", "statement": "Let $r\\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\\{1,\\ldots,N\\}$ such that no subset of size $r$ has the same pairwise greatest common divisor between all elements. Estimate $f_r(N)$.", "background": "Erd\\H{o}s \\cite{Er64} proved that $ f_r(N) \\leq N^{\\frac{3}{4}+o(1)}, $ and Abbott and Hanson \\cite{AbHa70} improved this exponent to $1/2$. Erd\\H{o}s \\cite{Er64} proved the lower bound $ f_3(N) > N^{\\frac{c}{\\log\\log N}} $ for some constant $c>0$, and conjectured this should also be an upper bound.\nErd\\H{o}s writes this is 'intimately connected' with the sunflower problem [20]. Indeed, the conjectured upper bound would follow from the following stronger version of the sunflower problem: estimate the size of the largest set of integers $A$ such that $\\omega(n)=k$ for all $n\\in A$ and there does not exist $a_1,\\ldots,a_r\\in A$ and an integer $d$ such that $(a_i,a_j)=d$ for all $i\neq j$ and $(a_i/d,d)=1$ for all $i$. The conjectured upper bound for $f_r(N)$ would follow if the size of such an $A$ must be at most $c_r^k$. The original sunflower proof of Erd\\H{o}s and Rado gives the upper bound $c_r^kk!$.\nSee also [536].\nReferences\n\n\n[AbHa70] Abbott, H. L. and Hanson, D., An extremal problem in number theory. Bull. London Math. Soc. (1970), 324-326.\n\n[Er64] Erd\\H{o}s, P., On a problem in elementary number theory and a combinatorial problem. Math. Comp. (1964), 644-646.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2165, "problem_number": "EP-536", "title": "Erdős Problem #536", "statement": "Let $\\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $\\epsilon N$ then there must be distinct $a,b,c\\in A$ such that $ [a,b]=[b,c]=[a,c], $ where $[a,b]$ denotes the least common multiple?", "background": "This is false if we ask for four elements with the same pairwise least common multiple, as shown by Erd\\H{o}s \\cite{Er62} (with a proof given in \\cite{Er70}).\nThis was also asked by Erd\\H{o}s at the 1991 problem session of West Coast Number Theory.\nIn the comments Weisenberg sketches a construction of a set $A\\subseteq [1,N]$ without this property such that $ \\lvert A\\rvert \\gg (\\log\\log N)^{f(N)}\\frac{N}{\\log N} $ for some $f(N)\\to \\infty$. Weisenberg also sketches a proof of the main problem when $\\epsilon>\\frac{221}{225}$.\nSee also [535], [537], and [856]. A related combinatorial problem is asked at [857].\nReferences\n\n\n[Er62] Erd\\H{o}s, P\\'{a}l, Remarks on number theory. IV. Extremal problems in number theory. I. Mat. Lapok (1962), 228-255.\n\n[Er70] Erd\\H{o}s, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2166, "problem_number": "EP-538", "title": "Erdős Problem #538", "statement": "Let $r\\geq 2$ and suppose that $A\\subseteq\\{1,\\ldots,N\\}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and $a\\in A$. Give the best possible upper bound for $ \\sum_{n\\in A}\\frac{1}{n}. $ ", "background": "Erd\\H{o}s observed that $ \\sum_{n\\in A}\\frac{1}{n}\\sum_{p\\leq N}\\frac{1}{p}\\leq r\\sum_{m\\leq N^2}\\frac{1}{m}\\ll r\\log N, $ and hence $ \\sum_{n\\in A}\\frac{1}{n} \\ll r\\frac{\\log N}{\\log\\log N}. $ See also [536] and [537].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2167, "problem_number": "EP-539", "title": "Erdős Problem #539", "statement": "Let $h(n)$ be such that, for any set $A\\subseteq \\mathbb{N}$ of size $n$, the set $ \\left\\{ \\frac{a}{(a,b)}: a,b\\in A\\right\\} $ has size at least $h(n)$. Estimate $h(n)$.", "background": "Erd\\H{o}s and Szemer\\'{e}di proved that $ n^{1/2} \\ll h(n) \\ll n^{1-c} $ for some constant $c>0$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2168, "problem_number": "EP-543", "title": "Erdős Problem #543", "statement": "Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size $N$ and $A\\subseteq G$ is a random set of size $k$ then, with probability $\\geq 1/2$, all elements of $G$ can be written as $\\sum_{x\\in S}x$ for some $S\\subseteq A$. Is $ f(N) \\leq \\log_2 N+o(\\log\\log N)? $ ", "background": "Erd\\H{o}s and R\\'{e}nyi \\cite{ErRe65} proved that $ f(N) \\leq \\log_2N+O(\\log\\log N). $ Erd\\H{o}s believed improving this to $o(\\log\\log N)$ is impossible.\nReferences\n\n\n[ErRe65] Erd\\H{o}s, P. and R\\'{e}nyi, A., Probabilistic methods in group theory. J. Analyse Math. (1965), 127-138.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2169, "problem_number": "EP-544", "title": "Erdős Problem #544", "statement": "Show that $ R(3,k+1)-R(3,k)\\to\\infty $ as $k\\to \\infty$. Similarly, prove or disprove that $ R(3,k+1)-R(3,k)=o(k). $ ", "background": "A problem of Erd\\H{o}s and S\\'{o}s.\nThis problem is #8 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2170, "problem_number": "EP-545", "title": "Erdős Problem #545", "statement": "Let $G$ be a graph with $m$ edges and no isolated vertices. Is the Ramsey number $R(G)$ maximised when $G$ is 'as complete as possible'? That is, if $m=\\binom{n}{2}+t$ edges with $0\\leq t0$, there are infinitely many $n$ such that $ R(C_4,S_n)\\leq n+\\sqrt{n}-c? $ ", "background": "A problem of Burr, Erd\\H{o}s, Faudree, Rousseau, and Schelp \\cite{BEFRS89}. Erd\\H{o}s often asked about $R(C_4,S_n)$ in the equivalent formulation of asking for a bound on the minimum degree of a graph which would guarantee the existence of a $C_4$ (see [85]).\nIt is known that $ n+\\sqrt{n}-6n^{11/40} \\leq R(C_4,S_n)\\leq n+\\lceil\\sqrt{n}\\rceil+1. $ The lower bound is due to \\cite{BEFRS89}, the upper bound is due to Parsons \\cite{Pa75}. The lower bound of \\cite{BEFRS89} is related to gaps between primes, and assuming e.g. Cramer's conjecture on gaps between primes their lower bound would be $n+\\sqrt{n}-n^{o(1)}$.\nErd\\H{o}s offered \\$100 for a proof or disproof of the second question in \\cite{BEFRS89}. In \\cite{Er96} Erd\\H{o}s asks (an equivalent formulation of) whether $R(C_4,S_n)\\geq n+\\sqrt{n}-O(1)$, but says this is probably 'too optimistic'.\nThey also ask, if $f(n)=R(C_4,S_n)$, whether $f(n+1)=f(n)$ infinitely often, and is the density of such $n$ $0$? Also, is it true that $f(n+1)\\leq f(n)+2$ for all $n$? A similar question about an equivalent function is the subject of [85].\nParsons \\cite{Pa75} proved that $ R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil $ whenever $n=q^2+1$ for a prime power $q$ and $ R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil+1 $ whenever $n=q^2$ for a prime power $q$ (in particular both equalities occur infinitely often).\nThis has been extended in various works, all in the cases $n=q^2\\pm t$ for some $0\\leq t\\leq q$ and prime power $q$. We refer to the work of Parsons \\cite{Pa76}, Wu, Sun, Zhang, and Radziszowski \\cite{WSZR15}, and Zhang, Chen, and Cheng (\\cite{ZCC17} and \\cite{ZCC17b}) for a precise description. In every known case $ R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil+\\{0,1\\}, $ and Zhang, Chen, and Cheng \\cite{ZCC17} speculate whether this is in fact true for all $n\\geq 2$ (whence the answer to the question above would be no).\nThis problem is #19 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[BEFRS89] Burr, S. and Erd\"{o}s, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Some complete bipartite graph-tree Ramsey numbers. Graph theory in memory of G. A. Dirac (Sandbjerg,\n1985) (1989), 79-89.\n\n[Er96] Erd\\H{o}s, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9.\n\n[Pa75] Parsons, T. D., Ramsey graphs and block designs. {I}. Trans. Amer. Math. Soc. (1975), 33--44.\n\n[Pa76] No reference found.\n\n\n[WSZR15] Wu, Yali and Sun, Yongqi and Zhang, Rui and Radziszowski,\nStanis\\l aw P., Ramsey numbers of {$C_4$} versus wheels and stars. Graphs Combin. (2015), 2437--2446.\n\n[ZCC17] Zhang, Xuemei and Chen, Yaojun and Cheng, T. C. Edwin, Some values of {R}amsey numbers for {$C_4$} versus stars. Finite Fields Appl. (2017), 73--85.\n\n[ZCC17b] Zhang, Xuemei and Chen, Yaojun and Cheng, T. C. Edwin, Polarity graphs and {R}amsey numbers for {$C_4$} versus stars. Discrete Math. (2017), 655--660.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2173, "problem_number": "EP-554", "title": "Erdős Problem #554", "statement": "Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Show that $ \\lim_{k\\to \\infty}\\frac{R(C_{2n+1};k)}{R(K_3;k)}=0 $ for any $n\\geq 2$.", "background": "A problem of Erd\\H{o}s and Graham. The problem is open even for $n=2$.\nThis problem is #23 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2174, "problem_number": "EP-555", "title": "Erdős Problem #555", "statement": "Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine the value of $ R(C_{2n};k). $ ", "background": "A problem of Erd\\H{o}s and Graham. Erd\\H{o}s \\cite{Er81c} gives the bounds $ k^{1+\\frac{1}{2n}}\\ll R(C_{2n};k)\\ll k^{1+\\frac{1}{n-1}}. $ Chung and Graham \\cite{ChGr75} showed that $ R(C_4;k)>k^2-k+1 $ when $k-1$ is a prime power and $ R(C_4;k)\\leq k^2+k+1 $ for all $k$.\nThis problem is #24 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[ChGr75] Chung, Fan R. K. and Graham, R. L., On multicolor Ramsey numbers for complete bipartite graphs. J. Combinatorial Theory Ser. B (1975), 164-169.\n\n[Er81c] Erd\\H{o}s, Paul, Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (1981), 9-17.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2175, "problem_number": "EP-557", "title": "Erdős Problem #557", "statement": "Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Is it true that $ R(T;k)\\leq kn+O(1) $ for any tree $T$ on $n$ vertices?", "background": "A problem of Erd\\H{o}s and Graham. Implied by [548].\nThis would be best possible since, for example, $R(S_n,k)\\geq kn-O(k)$ if $S_n=K_{1,n-1}$ is a star on $n$ vertices.\nThis problem is #26 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2176, "problem_number": "EP-558", "title": "Erdős Problem #558", "statement": "Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine $ R(K_{s,t};k) $ where $K_{s,t}$ is the complete bipartite graph with $s$ vertices in one component and $t$ in the other.", "background": "Chung and Graham \\cite{ChGr75} prove the general bounds $ (2\\pi\\sqrt{st})^{\\frac{1}{s+t}}\\left(\\frac{s+t}{e^2}\\right)k^{\\frac{st-1}{s+t}}\\leq R(K_{s,t};k)\\leq (t-1)(k+k^{1/s})^s $ and determined $ R(K_{2,2},k)=(1+o(1))k^2. $ Alon, R\\'{o}nyai, and Szab\\'{o} \\cite{ARS99} have proved that $ R(K_{3,3},k)=(1+o(1))k^3 $ and that if $s\\geq (t-1)!+1$ then $ R(K_{s,t},k)\\asymp k^t. $ This problem is #27 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[ARS99] Alon, Noga and R\\'{o}nyai, Lajos and Szab\\'{o}, Tibor, Norm-graphs: variations and applications. J. Combin. Theory Ser. B (1999), 280-290.\n\n[ChGr75] Chung, Fan R. K. and Graham, R. L., On multicolor Ramsey numbers for complete bipartite graphs. J. Combinatorial Theory Ser. B (1975), 164-169.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2177, "problem_number": "EP-560", "title": "Erdős Problem #560", "statement": "Let $\\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$.\nDetermine $ \\hat{R}(K_{n,n}), $ where $K_{n,n}$ is the complete bipartite graph with $n$ vertices in each component.", "background": "We know that $ \\frac{1}{60}n^22^n<\\hat{R}(K_{n,n})< \\frac{3}{2}n^32^n. $ The lower bound (which holds for $n\\geq 6$) was proved by Erd\\H{o}s and Rousseau \\cite{ErRo93}. The upper bound was proved by Erd\\H{o}s, Faudree, Rousseau, and Schelp \\cite{EFRS78b} and Ne\\v{s}et\\v{r}il and R\"{o}dl \\cite{NeRo78}.\nConlon, Fox, and Wigderson \\cite{CFW23} have proved that, for any $s\\leq t$, $ \\hat{R}(K_{s,t})\\gg s^{2-\\frac{s}{t}}t2^s, $ and prove that when $t\\gg s\\log s$ we have $\\hat{R}(K_{s,t})\\asymp s^2t2^s$. They conjecture that this should hold for all $s\\leq t$, and so in particular we should have $\\hat{R}(K_{n,n})\\asymp n^32^n$.\nThis problem is #29 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[CFW23] Conlon, David and Fox, Jacob and Wigderson, Yuval, Three early problems on size Ramsey numbers. Combinatorica (2023), 743-768.\n\n[EFRS78b] Erd\\H{o}s, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., The size Ramsey number. Period. Math. Hungar. (1978), 145-161.\n\n[ErRo93] Erd\\H{o}s, P. and Rousseau, C. C., The size Ramsey number of a complete bipartite graph. Discrete Math. (1993), 259-262.\n\n[NeRo78] Ne\\vSet\\v{r}il, J. and R\"{o}dl, V., The structure of critical Ramsey graphs. Acta Math. Acad. Sci. Hungar. (1978), 295-300.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2178, "problem_number": "EP-561", "title": "Erdős Problem #561", "statement": "Let $\\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$.\nLet $F_1$ and $F_2$ be the union of stars. More precisely, let $F_1=\\cup_{i\\leq s} K_{1,n_i}$ and $F_2=\\cup_{j\\leq t} K_{1,m_j}$. Prove that $ \\hat{R}(F_1,F_2) = \\sum_{2\\leq k\\leq s+2}\\max\\{n_i+m_j-1 : i+j=k\\}. $ ", "background": "Burr, Erd\\H{o}s, Faudree, Rousseau, and Schelp \\cite{BEFRS78} proved this when all the $n_i$ are identical and all the $m_i$ are identical.\nThis problem is #30 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[BEFRS78] Burr, S. A. and Erd\\H{o}s, P. and Faudree, R. J. and Rousseau,\nC. C. and Schelp, R. H., Ramsey-minimal graphs for multiple copies. Nederl. Akad. Wetensch. Indag. Math. (1978), 187-195.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2179, "problem_number": "EP-562", "title": "Erdős Problem #562", "statement": "Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some monochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices.\nProve that, for $r\\geq 3$, $ \\log_{r-1} R_r(n) \\asymp_r n, $ where $\\log_{r-1}$ denotes the $(r-1)$-fold iterated logarithm. That is, does $R_r(n)$ grow like $ 2^{2^{\\cdots n}} $ where the tower of exponentials has height $r-1$?", "background": "A problem of Erd\\H{o}s, Hajnal, and Rado \\cite{EHR65}. A generalisation of [564].\nThis problem is #38 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[EHR65] Erd\\H{o}s, P. and Hajnal, A. and Rado, R., Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. (1965), 93-196.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2180, "problem_number": "EP-563", "title": "Erdős Problem #563", "statement": "Let $F(n,\\alpha)$ denote the smallest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\\subseteq [n]$ with $\\lvert X\\rvert\\geq m$ contains more than $\\alpha \\binom{\\lvert X\\rvert}{2}$ many edges of each colour.\nProve that, for every $0\\leq \\alpha< 1/2$, $ F(n,\\alpha)\\sim c_\\alpha\\log n $ for some constant $c_\\alpha$ depending only on $\\alpha$.", "background": "It is easy to show via the probabilistic method that, for every $0\\leq \\alpha<1/2$, $ F(n,\\alpha)\\asymp_\\alpha \\log n. $ Note that when $\\alpha=0$ this is just asking for a $2$-colouring of the edges of $K_n$ which contains no monochromatic clique of size $m$, and hence we recover the classical Ramsey numbers.\nSee also [161] for a generalisation to hypergraphs.\nThis problem is #39 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2181, "problem_number": "EP-564", "title": "Erdős Problem #564", "statement": "Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform hypergraph on $n$ vertices.\nIs there some constant $c>0$ such that $ R_3(n) \\geq 2^{2^{cn}}? $ ", "background": "A special case of [562]. A problem of Erd\\H{o}s, Hajnal, and Rado \\cite{EHR65}, who prove the bounds $ 2^{cn^2}< R_3(n)< 2^{2^{n}} $ for some constant $c>0$.\nErd\\H{o}s, Hajnal, M\\'{a}t\\'{e}, and Rado \\cite{EHMR84} have proved a doubly exponential lower bound for the corresponding problem with $4$ colours.\nThis problem is #37 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[EHMR84] Erd\\H{o}s, Paul and Hajnal, Andr\\'{a}s and M\\'{a}t\\'{e}, Attila and Rado, Richard, Combinatorial set theory: partition relations for cardinals. (1984), 347.\n\n[EHR65] Erd\\H{o}s, P. and Hajnal, A. and Rado, R., Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. (1965), 93-196.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2182, "problem_number": "EP-566", "title": "Erdős Problem #566", "statement": "Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then $ R(G,H)\\ll m? $ ", "background": "In other words, is $G$ Ramsey size linear? This fails for a graph $G$ with $n$ vertices and $2n-2$ edges (for example with $H=K_n$). Erd\\H{o}s, Faudree, Rousseau, and Schelp \\cite{EFRS93} have shown that any graph $G$ with $n$ vertices and at most $n+1$ edges is Ramsey size linear.\nImplies [567].\nThis problem is #31 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[EFRS93] Erd\\H{o}s, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2183, "problem_number": "EP-567", "title": "Erdős Problem #567", "statement": "Let $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords to $C_5$). Is it true that, if $H$ has $m$ edges and no isolated vertices, then $ R(G,H)\\ll m? $ ", "background": "In other words, is $G$ Ramsey size linear? A special case of [566]. In \\cite{Er95} Erd\\H{o}s specifically asks about the case $G=K_{3,3}$.\nThe graph $H_5$ can also be described as $K_4^*$, obtained from $K_4$ by subdividing one edge. ($K_4$ itself is not Ramsey size linear, since $R(4,n)\\gg n^{3-o(1)}$, see [166].) Brada\\'{c}, Gishboliner, and Sudakov \\cite{BGS23} have shown that every subdivision of $K_4$ on at least $6$ vertices is Ramsey size linear, and also that $R(H_5,H) \\ll m$ whenever $H$ is a bipartite graph with $m$ edges and no isolated vertices.\nThis problem is #32 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[BGS23] Brada\\'C, D. and Gishboliner, L. and Sudakov, B., On Ramsey size-linear graphs and related questions. arXiv:2202.10388 (2023).\n\n[Er95] Erd\\H{o}s, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2184, "problem_number": "EP-568", "title": "Erdős Problem #568", "statement": "Let $G$ be a graph such that $R(G,T_n)\\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\\ll n^2$. Is it true that, for any $H$ with $m$ edges and no isolated vertices, $ R(G,H)\\ll m? $ ", "background": "In other words, is $G$ Ramsey size linear?\nThis problem is #33 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2185, "problem_number": "EP-569", "title": "Erdős Problem #569", "statement": "Let $k\\geq 1$. What is the best possible $c_k$ such that $ R(C_{2k+1},H)\\leq c_k m $ for any graph $H$ on $m$ edges without isolated vertices?", "background": "This problem is #34 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2186, "problem_number": "EP-571", "title": "Erdős Problem #571", "statement": "Show that for any rational $\\alpha \\in [1,2)$ there exists a bipartite graph $G$ such that $ \\mathrm{ex}(n;G)\\asymp n^{\\alpha}. $ ", "background": "A problem of Erd\\H{o}s and Simonovits.\nBukh and Conlon \\cite{BuCo18} proved that this holds if we weaken asking for the extremal number of a single graph to asking for the extremal number of a finite family of graphs.\nA rational $\\alpha\\in [1,2)$ for which this holds is known as a Tur\\'{a}n exponent. Known Tur\\'{a}n exponents are:\n{UL}\n{LI} $\\frac{3}{2}-\\frac{1}{2s}$ for $s\\geq 2$ (Conlon, Janzer, and Lee \\cite{CJL21}).{/LI}\n{LI} $\\frac{4}{3}-\\frac{1}{3s}$ and $\\frac{5}{4}-\\frac{1}{4s}$ for $s\\geq 2$ (Jiang and Qiu \\cite{JiQi20}).{/LI}\n{LI} $2-\\frac{a}{b}$ for $\\lfloor b/a\\rfloor^3 \\leq a\\leq \\frac{b}{\\lfloor b/a\\rfloor+1}+1$ (Jiang, Jiang, and Ma \\cite{JJM20}).{/LI}\n{LI} $2-\\frac{a}{b}$ with $b>a\\geq 1$ and $b\\equiv \\pm 1\\pmod{a}$ (Kang, Kim, and Liu \\cite{KKL21}).{/LI}\n{LI} $1+a/b$ with $b>a^2$ (Jiang and Qiu \\cite{JiQi23}),{/LI}\n{LI} $2-\\frac{2}{2b+1}$ for $b\\geq 2$ or $7/5$ (Jiang, Ma, and Yepremyan \\cite{JMY22}).{/LI}\n{LI} $2-a/b$ with $b\\geq (a-1)^2$ (Conlon and Janzer \\cite{CoJa22}).{/LI}\n{/UL}\nSee also [713].\nThis problem is #45 in Extremal Graph Theory in the graphs problem collection.\nReferences\n\n\n[BuCo18] Bukh, Boris and Conlon, David, Rational exponents in extremal graph theory. J. Eur. Math. Soc. (JEMS) (2018), 1747-1757.\n\n[CJL21] Conlon, David and Janzer, Oliver and Lee, Joonkyung, More on the extremal number of subdivisions. Combinatorica (2021), 465-494.\n\n[CoJa22] Conlon, David and Janzer, Oliver, Rational exponents near two. Adv. Comb. (2022), Paper No. 9, 10.\n\n[JJM20] Jiang, Tao and Jiang, Zilin and Ma, Jie, Negligible obstructions and Tur\\'{a}n exponents. arXiv:2007.02975 (2020).\n\n[JMY22] Jiang, Tao and Ma, Jie and Yepremyan, Liana, On Tur\\'{a}n exponents of bipartite graphs. Combin. Probab. Comput. (2022), 333-344.\n\n[JiQi20] Jiang, Tao and Qiu, Yu, Tur\\'{a}n numbers of bipartite subdivisions. SIAM J. Discrete Math. (2020), 556-570.\n\n[JiQi23] Jiang, Tao and Qiu, Yu, Many Tur\\'{a}n exponents via subdivisions. Combin. Probab. Comput. (2023), 134-150.\n\n[KKL21] Kang, Dong Yeap and Kim, Jaehoon and Liu, Hong, On the rational Tur\\'{a}n exponents conjecture. J. Combin. Theory Ser. B (2021), 149-172.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2187, "problem_number": "EP-572", "title": "Erdős Problem #572", "statement": "Show that for $k\\geq 3$ $ \\mathrm{ex}(n;C_{2k})\\gg n^{1+\\frac{1}{k}}. $ ", "background": "It is easy to see that $\\mathrm{ex}(n;C_{2k+1})=\\lfloor n^2/4\\rfloor$ for any $k\\geq 1$ (and $n>2k+1$) (since no bipartite graph contains an odd cycle). Erd\\H{o}s and Klein \\cite{Er38} proved $\\mathrm{ex}(n;C_4)\\asymp n^{3/2}$.\nErd\\H{o}s \\cite{Er64c} and Bondy and Simonovits \\cite{BoSi74} showed that $ \\mathrm{ex}(n;C_{2k})\\ll kn^{1+\\frac{1}{k}}. $ Benson \\cite{Be66} has proved this conjecture for $k=3$ and $k=5$. Lazebnik, Ustimenko, and Woldar \\cite{LUW95} have shown that, for arbitrary $k\\geq 3$, $ \\mathrm{ex}(n;C_{2k})\\gg n^{1+\\frac{2}{3k-3+\nu}}, $ where $\nu=0$ if $k$ is odd and $\nu=1$ if $k$ is even. See \\cite{LUW99} for further history and references.\nSee also [765].\nThis problem is #46 in Extremal Graph Theory in the graphs problem collection.\nReferences\n\n\n[Be66] Benson, Clark T., Minimal regular graphs of girths eight and twelve. Canadian J. Math. (1966), 1091-1094.\n\n[BoSi74] Bondy, J. A. and Simonovits, M., Cycles of even length in graphs. J. Combinatorial Theory Ser. B (1974), 97-105.\n\n[Er38] P. Erd\\H{o}s, On sequences of integers no one of which divides the product of two others and on related problems. Tomsk. Gos. Univ. Ucen Zap. (1938), 74-82.\n\n[Er64c] Erd\\H{o}s, P., Extremal problems in graph theory. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963) (1964), 29-36.\n\n[LUW95] Lazebnik, F. and Ustimenko, V. A. and Woldar, A. J., A new series of dense graphs of high girth. Bull. Amer. Math. Soc. (N.S.) (1995), 73-79.\n\n[LUW99] Lazebnik, Felix and Ustimenko, Vasiliy A. and Woldar, Andrew\nJ., Polarities and $2k$-cycle-free graphs. Discrete Math. (1999), 503-513.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2188, "problem_number": "EP-573", "title": "Erdős Problem #573", "statement": "Is it true that $ \\mathrm{ex}(n;\\{C_3,C_4\\})\\sim (n/2)^{3/2}? $ ", "background": "A problem of Erd\\H{o}s and Simonovits, who proved that $ \\mathrm{ex}(n;\\{C_4,C_5\\})=(n/2)^{3/2}+O(n). $ K\"{o}v\\'{a}ri, S\\'{o}s, and Tur\\'{a}n \\cite{KST54} proved that the extremal number of edges for containing either $C_4$ or an odd cycle of any length is $\\sim (n/2)^{3/2}$. This problem is therefore asking whether the threshold is the same if we just forbid odd cycles of length $3$.\nSee also [574] for the general case, and [765] for $\\mathrm{ex}(n;C_4)$.\nThis problem is #48 in Extremal Graph Theory in the graphs problem collection.\nReferences\n\n\n[KST54] K\"{o}vari, T. and S\\'{o}s, V. T. and Tur\\'{a}n, P., On a problem of K. Zarankiewicz. Colloq. Math. (1954), 50-57.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2189, "problem_number": "EP-574", "title": "Erdős Problem #574", "statement": "Is it true that, for $k\\geq 2$, $ \\mathrm{ex}(n;\\{C_{2k-1},C_{2k}\\})=(1+o(1))(n/2)^{1+\\frac{1}{k}}. $ ", "background": "A problem of Erd\\H{o}s and Simonovits.\nSee also [573] for the specific case of $k=2$.\nThis problem is #49 in Extremal Graph Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2190, "problem_number": "EP-575", "title": "Erdős Problem #575", "statement": "If $\\mathcal{F}$ is a finite set of finite graphs then $\\mathrm{ex}(n;\\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\\mathcal{F}$. Note that it is trivial that $\\mathrm{ex}(n;\\mathcal{F})\\leq \\mathrm{ex}(n;G)$ for every $G\\in\\mathcal{F}$.\nIs it true that, for every $\\mathcal{F}$, if there is a bipartite graph in $\\mathcal{F}$ then there exists some bipartite $G\\in\\mathcal{F}$ such that $ \\mathrm{ex}(n;G)\\ll_{\\mathcal{F}}\\mathrm{ex}(n;\\mathcal{F})? $ ", "background": "A problem of Erd\\H{o}s and Simonovits.\nSee also [180].\nThis problem is #51 in Extremal Graph Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2191, "problem_number": "EP-576", "title": "Erdős Problem #576", "statement": "Let $Q_k$ be the $k$-dimensional hypercube graph (so that $Q_k$ has $2^k$ vertices and $k2^{k-1}$ edges). Determine the behaviour of $ \\mathrm{ex}(n;Q_k). $ ", "background": "Erd\\H{o}s and Simonovits \\cite{ErSi70} proved that $ (\\tfrac{1}{2}+o(1))n^{3/2}\\leq \\mathrm{ex}(n;Q_3) \\ll n^{8/5}. $ (In \\cite{ErSi70} they mention that Erd\\H{o}s had originally conjectured that $ \\mathrm{ex}(n;Q_3)\\gg n^{5/3}$.) Erd\\H{o}s and Simonovits also proved that, if $G$ is the graph $Q_3$ with a missing edge, then $\\mathrm{ex}(n;G)\\asymp n^{3/2}$.\nIn \\cite{Er74c}, \\cite{Er81}, and \\cite{Er93} Erd\\H{o}s asked whether it is $\\mathrm{ex}(n;Q_3)\\asymp n^{8/5}$.\nA theorem of Sudakov and Tomon \\cite{SuTo22} implies $ \\mathrm{ex}(n;Q_k)=o(n^{2-\\frac{1}{k}}). $ Janzer and Sudakov \\cite{JaSu22} have improved this to $ \\mathrm{ex}(n;Q_k)\\ll_k n^{2-\\frac{1}{k-1}+\\frac{1}{(k-1)2^{k-1}}}. $ See also [1035].\nThis problem is #52 in Extremal Graph Theory in the graphs problem collection.\nReferences\n\n\n[Er74c] Erd\\H{o}s, Paul, Extremal problems on graphs and hypergraphs. (1974), 75-84.\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[ErSi70] Erd\\H{o}s, P. and Simonovits, M., Some extremal problems in graph theory. Combinatorial theory and its applications, I-III (Proc. Colloq., Balatonf\"{u}red, 1969) (1970), 377-390.\n\n[JaSu22] Janzer, O. and Sudakov, B., On the Tur\\'{a}n number of the hypercube. arXiv:2211.02015 (2024).\n\n[SuTo22] Sudakov, Benny and Tomon, Istv\\'{a}n, The extremal number of tight cycles. Int. Math. Res. Not. IMRN (2022), 9663-9684.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2192, "problem_number": "EP-579", "title": "Erdős Problem #579", "statement": "Let $\\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_{2,2,2}$ and at least $\\delta n^2$ edges then $G$ contains an independent set of size $\\gg_\\delta n$.", "background": "A problem of Erd\\H{o}s, Hajnal, S\\'{o}s, and Szemer\\'{e}di, who could prove this is true for $\\delta>1/8$.\nSee also [533] and the entry in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2193, "problem_number": "EP-584", "title": "Erdős Problem #584", "statement": "Let $G$ be a graph with $n$ vertices and $\\delta n^{2}$ edges. Are there subgraphs $H_1,H_2\\subseteq G$ such that\n{UL}\n{LI}$H_1$ has $\\gg \\delta^3n^2$ edges and every two edges in $H_1$ are contained in a cycle of length at most $6$, and furthermore if two edges share a vertex they are on a cycle of length $4$, and\n{LI}$H_2$ has $\\gg \\delta^2n^2$ edges and every two edges in $H_2$ are contained in a cycle of length at most $8$.\n{/UL}", "background": "A problem of Erd\\H{o}s, Duke, and R\"{o}dl. Duke and Erd\\H{o}s \\cite{DuEr83}, who proved the first if $n$ is sufficiently large depending on $\\delta$. The real challenge is to prove this when $\\delta=n^{-c}$ for some $c>0$. Duke, Erd\\H{o}s, and R\"{o}dl \\cite{DER84} proved the first statement with a $\\delta^5$ in place of a $\\delta^3$.\nFox and Sudakov \\cite{FoSu08b} have proved the second statement when $\\delta >n^{-1/5}$.\nSee also the entry in the graphs problem collection.\nReferences\n\n\n[DER84] Duke, Richard and Erd\\H{o}s, Paul and R\"{o}dl, Vojt\\vEch, More results on subgraphs with many short cycles. Proceedings of the fifteenth Southeastern conference on\ncombinatorics, graph theory and computing (Baton Rouge,\nLa., 1984) (1984), 295-300.\n\n[DuEr83] No reference found.\n\n\n[FoSu08b] Fox, Jacob and Sudakov, Benny, On a problem of Duke-Erd\\H{o}s-R\"{o}dl on cycle-connected subgraphs. J. Combin. Theory Ser. B (2008), 1056-1062.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2194, "problem_number": "EP-585", "title": "Erdős Problem #585", "statement": "What is the maximum number of edges that a graph on $n$ vertices can have if it does not contain two edge-disjoint cycles with the same vertex set?", "background": "Pyber, R\"{o}dl, and Szemer\\'{e}di \\cite{PRS95} constructed such a graph with $\\gg n\\log\\log n$ edges.\nChakraborti, Janzer, Methuku, and Montgomery \\cite{CJMM24} have shown that such a graph can have at most $n(\\log n)^{O(1)}$ many edges. Indeed, they prove that there exists a constant $C>0$ such that for any $k\\geq 2$ there is a $c_k$ such that if a graph has $n$ vertices and at least $c_kn(\\log n)^{C}$ many edges then it contains $k$ pairwise edge-disjoint cycles with the same vertex set.\nReferences\n\n\n[CJMM24] Chakraborti, D. and Janzer, O. and Methuku, A. and Montgomery, R., Edge-disjoint cycles with the same vertex set. arXiv:2404.07190 (2024).\n\n[PRS95] Pyber, L. and R\"{o}dl, V. and Szemer\\'{e}di, E., Dense subgraphs without 3-regular subgraphs. Journal of Combinatorial Theory, Series B (1995), 41-54.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2195, "problem_number": "EP-588", "title": "Erdős Problem #588", "statement": "Let $f_k(n)$ be minimal such that if $n$ points in $\\mathbb{R}^2$ have no $k+1$ points on a line then there must be at most $f_k(n)$ many lines containing at least $k$ points. Is it true that $ f_k(n)=o(n^2) $ for $k\\geq 4$?", "background": "A generalisation of [101] (which asks about $k=4$).\nThe restriction to $k\\geq 4$ is necessary since Sylvester has shown that $f_3(n)= n^2/6+O(n)$. (See also Burr, Gr\"{u}nbaum, and Sloane \\cite{BGS74} and F\"{u}redi and Pal\\'{a}sti \\cite{FuPa84} for constructions which show that $f_3(n)\\geq(1/6+o(1))n^2$.)\nFor $k\\geq 4$, K\\'{a}rteszi \\cite{Ka63} proved $ f_k(n)\\gg_k n\\log n $ (resolving a conjecture of Erd\\H{o}s that $f_k(n)/n\\to \\infty$). Gr\"{u}nbaum \\cite{Gr76} proved $ f_k(n) \\gg_k n^{1+\\frac{1}{k-2}}. $ Erd\\H{o}s speculated this may be the correct order of magnitude, but Solymosi and Stojakovi\\'{c} \\cite{SoSt13} give a construction which shows $ f_k(n)\\gg_k n^{2-O_k(1/\\sqrt{\\log n})} $ \nReferences\n\n\n[BGS74] Burr, Stefan A. and Gr\"{u}nbaum, Branko and Sloane, N. J. A., The orchard problem. Geometriae Dedicata (1974), 397-424.\n\n[FuPa84] F\"{u}redi, Z. and Pal\\'{a}sti, I., Arrangements of lines with a large number of triangles. Proc. Amer. Math. Soc. (1984), 561-566.\n\n[Gr76] Gr\"{u}nbaum, Branko, New views on some old questions of combinatorial geometry. Colloquio Internazionale sulle Teorie Combinatorie\n(Roma, 1973), Tomo I (1976), 451-468.\n\n[Ka63] F. K\\'{a}rteszi, Sylvester egy t\\'{e}tel\\'{e}r\\H{o}l \\'{e}s Erd\\H{o}s egy sejt\\'{e}s\\'{e}r\\H{o}l. Matematikai Lapok (1963), 3-10.\n\n[SoSt13] Solymosi, J\\'{o}zsef and Stojakovi\\'C, Milo\\vS, Many collinear {$k$}-tuples with no {$k+1$} collinear points. Discrete Comput. Geom. (2013), 811-820.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2196, "problem_number": "EP-589", "title": "Erdős Problem #589", "statement": "Let $g(n)$ be maximal such that in any set of $n$ points in $\\mathbb{R}^2$ with no four points on a line there exists a subset on $g(n)$ points with no three points on a line. Estimate $g(n)$.", "background": "The trivial greedy algorithm gives $g(n)\\gg n^{1/2}$. A similar question can be asked for a set with no $k$ points on a line, searching for a subset with no $l$ points on a line, for any $3\\leq l\\aleph_0$.", "background": "Similar problems were investigated by Erd\\H{o}s, Galvin, and Hajnal \\cite{EGH75}. Erd\\H{o}s claims that for graphs the problem is completely solved: a graph of chromatic number $\\geq \\aleph_1$ must contain all finite bipartite graphs but need not contain any fixed odd cycle.\nReferences\n\n\n[EGH75] Erd\\H{o}s, P. and Galvin, F. and Hajnal, A., On set-systems having large chromatic number and not containing prescribed subsystems. (1975), 425--513.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2200, "problem_number": "EP-595", "title": "Erdős Problem #595", "statement": "Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs?", "background": "A problem of Erd\\H{o}s and Hajnal. Folkman \\cite{Fo70} and Ne\\v{s}et\\v{r}il and R\"{o}dl \\cite{NeRo75} have proved that for every $n\\geq 1$ there is a graph $G$ which contains no $K_4$ and is not the union of $n$ triangle-free graphs.\nSee also [582] and [596].\nReferences\n\n\n[Fo70] Folkman, Jon, Graphs with monochromatic complete subgraphs in every edge\ncoloring. SIAM J. Appl. Math. (1970), 19-24.\n\n[NeRo75] Ne\\u set\\u ril, Jaroslav and R\"odl, Vojt\\v ech, Type theory of partition properties of graphs. (1975), 405-412.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2201, "problem_number": "EP-596", "title": "Erdős Problem #596", "statement": "For which graphs $G_1,G_2$ is it true that\n{UL}\n{LI} for every $n\\geq 1$ there is a graph $H$ without a $G_1$ but if the edges of $H$ are $n$-coloured then there is a monochromatic copy of $G_2$, and yet{/LI}\n{LI} for every graph $H$ without a $G_1$ there is an $\\aleph_0$-colouring of the edges of $H$ without a monochromatic $G_2$.\n{/UL}", "background": "Erd\\H{o}s and Hajnal originally conjectured that there are no such $G_1,G_2$, but in fact $G_1=C_4$ and $G_2=C_6$ is an example. Indeed, for this pair Ne\\v{s}et\\v{r}il and R\"{o}dl established the first property and Erd\\H{o}s and Hajnal the second (in fact every $C_4$-free graph is a countable union of trees).\nWhether this is true for $G_1=K_4$ and $G_2=K_3$ is the content of [595].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2202, "problem_number": "EP-597", "title": "Erdős Problem #597", "statement": "Let $G$ be a graph on at most $\\aleph_1$ vertices which contains no $K_4$ and no $K_{\\aleph_0,\\aleph_0}$ (the complete bipartite graph with $\\aleph_0$ vertices in each class). Is it true that $ \\omega_1^2 \\to (\\omega_1\\omega, G)^2? $ What about finite $G$?", "background": "Erd\\H{o}s and Hajnal proved that $\\omega_1^2 \\to (\\omega_1\\omega,3)^2$. Erd\\H{o}s originally asked this with just the assumption that $G$ is $K_4$-free, but Baumgartner proved that $\\omega_1^2 \not\\to (\\omega_1\\omega, K_{\\aleph_0,\\aleph_0})^2$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2203, "problem_number": "EP-598", "title": "Erdős Problem #598", "statement": "Let $m$ be an infinite cardinal and $\\kappa$ be the successor cardinal of $2^{\\aleph_0}$. Can one colour the countable subsets of $m$ using $\\kappa$ many colours so that every $X\\subseteq m$ with $\\lvert X\\rvert=\\kappa$ contains subsets of all possible colours?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2204, "problem_number": "EP-600", "title": "Erdős Problem #600", "statement": "Let $e(n,r)$ be minimal such that every graph on $n$ vertices with at least $e(n,r)$ edges, each edge contained in at least one triangle, must have an edge contained in at least $r$ triangles. Let $r\\geq 2$. Is it true that $ e(n,r+1)-e(n,r)\\to \\infty $ as $n\\to \\infty$? Is it true that $ \\frac{e(n,r+1)}{e(n,r)}\\to 1 $ as $n\\to \\infty$?", "background": "Ruzsa and Szemer\\'{e}di \\cite{RuSz78} proved that $e(n,r)=o(n^2)$ for any fixed $r$.\nSee also [80].\nReferences\n\n\n[RuSz78] Ruzsa, I. Z. and Szemer\\'{e}di, E., Triple systems with no six points carrying three triangles. Combinatorics (Proc. Fifth Hungarian Colloq.,\nKeszthely, 1976), Vol. II (1978), 939-945.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2205, "problem_number": "EP-601", "title": "Erdős Problem #601", "statement": "For which limit ordinals $\\alpha$ is it true that if $G$ is a graph with vertex set $\\alpha$ then $G$ must have either an infinite path or independent set on a set of vertices with order type $\\alpha$?", "background": "A problem of Erd\\H{o}s, Hajnal, and Milner \\cite{EHM70}, who proved this is true for $\\alpha < \\omega_1^{\\omega+2}$.\nIn \\cite{Er82e} Erd\\H{o}s offers \\$250 for showing what happens when $\\alpha=\\omega_1^{\\omega+2}$ and \\$500 for settling the general case.\nLarson \\cite{La90} proved this is true for all $\\alpha<2^{\\aleph_0}$ assuming Martin's axiom.\nReferences\n\n\n[EHM70] Erd\\H{o}s, P. and Hajnal, A. and Milner, E. C., Set mappings and polarized partition relations. Combinatorial theory and its applications, I-III (Proc.\nColloq., Balatonf\"{u}red, 1969) (1970), 327-363.\n\n[Er82e] Erd\\H{o}s, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.\n\n[La90] Larson, Jean A., Martin's axiom and ordinal graphs: large independent sets or infinite paths. Ann. Pure Appl. Logic (1990), 31-39.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2206, "problem_number": "EP-602", "title": "Erdős Problem #602", "statement": "Let $(A_i)$ be a family of sets with $\\lvert A_i\\rvert=\\aleph_0$ for all $i$, such that for any $i\neq j$ we have $\\lvert A_i\\cap A_j\\rvert$ finite and $\neq 1$. Is there a $2$-colouring of $\\cup A_i$ such that no $A_i$ is monochromatic?", "background": "A problem of Komj\\'{a}th. The existence of such a $2$-colouring is sometimes known as Property B.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2207, "problem_number": "EP-603", "title": "Erdős Problem #603", "statement": "Let $(A_i)$ be a family of countably infinite sets such that $\\lvert A_i\\cap A_j\\rvert \neq 2$ for all $i\neq j$. Find the smallest cardinal $C$ such that $\\cup A_i$ can always be coloured with at most $C$ colours so that no $A_i$ is monochromatic.", "background": "A problem of Komj\\'{a}th. If instead we have $\\lvert A_i\\cap A_j\\rvert \neq 1$ then Komj\\'{a}th showed that this is possible with at most $\\aleph_0$ colours.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2208, "problem_number": "EP-604", "title": "Erdős Problem #604", "statement": "Given $n$ distinct points $A\\subset\\mathbb{R}^2$ must there be a point $x\\in A$ such that $ \\#\\{ d(x,y) : y \\in A\\} \\gg n^{1-o(1)}? $ Or even $\\gg n/\\sqrt{\\log n}$?", "background": "The pinned distance problem, a stronger form of [89]. The example of an integer grid show that $n/\\sqrt{\\log n}$ would be best possible.\nIt may be true that there are $\\gg n$ many such points, or that this is true on average - for example, if $d(x)$ counts the number of distinct distances from $x$ then in \\cite{Er75f} Erd\\H{o}s conjectured $ \\sum_{x\\in A}d(x) \\gg \\frac{n^2}{\\sqrt{\\log n}}, $ where $A\\subset \\mathbb{R}^2$ is any set of $n$ points.\nIn \\cite{Er97e} Erd\\H{o}s offers \\$500 for a solution to this problem, but it is unclear whether he intended this for proving the existence of a single such point or for $\\gg n$ many such points.\nIn \\cite{Er97e} Erd\\H{o}s wrote that he initially 'overconjectured' and thought that the answer to this problem is the same as for the number of distinct distances between all pairs (see [89]), but this was disproved by Harborth. It could be true that the answers are the same up to an additive factor of $n^{o(1)}$.\nThe best known bound is $ \\gg n^{c-o(1)}, $ due to Katz and Tardos \\cite{KaTa04}, where $ c=\\frac{48-14e}{55-16e}=0.864137\\cdots. $ \nReferences\n\n\n[Er75f] Erd\\H{o}s, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\n\n[Er97e] Erd\\H{o}s, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.\n\n[KaTa04] Katz, Nets Hawk and Tardos, G\\'{a}bor, A new entropy inequality for the Erd\\H{o}s distance problem. Towards a theory of geometric graphs (2004), 119-126.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2209, "problem_number": "EP-609", "title": "Erdős Problem #609", "statement": "Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$. Estimate $f(n)$.", "background": "A problem of Erd\\H{o}s and Graham. The edges of $K_{2^n}$ can be $n$-coloured to avoid odd cycles of any length. It can be shown that $C_5$ and $C_7$ can be avoided for large $n$.\nChung \\cite{Ch97} asked whether $f(n)\\to \\infty$ as $n\\to \\infty$. Day and Johnson \\cite{DaJo17} proved this is true, and that $ f(n)\\geq 2^{c\\sqrt{\\log n}} $ for some constant $c>0$. The trivial upper bound is $2^n$.\nGir\\~{a}o and Hunter \\cite{GiHu24} have proved that $ f(n) \\ll \\frac{2^n}{n^{1-o(1)}}. $ Janzer and Yip \\cite{JaYi25} have improved this to $ f(n) \\ll n^{3/2}2^{n/2}. $ See also the entry in the graphs problem collection.\nReferences\n\n\n[Ch97] Chung, F. R. K., Open problems of {P}aul Erd\\H{o}s in graph theory. J. Graph Theory (1997), 3--36.\n\n[DaJo17] Day, A. Nicholas and Johnson, J. Robert, Multicolour Ramsey numbers of odd cycles. J. Combin. Theory Ser. B (2017), 56-63.\n\n[GiHu24] A. Gir\\~Ao and Z. Hunter, Monochromatic odd cycles in edge-coloured complete graphs. arXiv:2412.07708 (2024).\n\n[JaYi25] O. Janzer and F. Yip, Short monochromatic odd cycles. arXiv:2506.14910 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2210, "problem_number": "EP-610", "title": "Erdős Problem #610", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number).\nEstimate $\\tau(G)$. In particular, is it true that if $G$ has $n$ vertices then $ \\tau(G) \\leq n-\\omega(n)\\sqrt{n} $ for some $\\omega(n)\\to \\infty$, or even $ \\tau(G) \\leq n-c\\sqrt{n\\log n} $ for some absolute constant $c>0$?", "background": "A problem of Erd\\H{o}s, Gallai, and Tuza \\cite{EGT92}, who proved that $ \\tau(G) \\leq n-\\sqrt{2n}+O(1). $ This would be best possible, since there exist triangle-free graphs with all independent sets of size $O(\\sqrt{n\\log n})$, which follows from the lower bound for $R(3,k)$ by Kim \\cite{Ki95} (see [165]).\nIndeed, Erd\\H{o}s, Gallai, and Tuza speculate that if $f(n)$ is the largest $k$ such that every triangle-free graph on $n$ vertices contains an independent set on $f(n)$ vertices, then $\\tau(G)\\leq n-f(n)$.\nA positive answer to this problem would follow from a positive answer to [151] (since Ajtai, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AKS80} have proved that the $H(n)$ defined there satisfies $H(n)\\gg \\sqrt{n\\log n}$).\nSee also [151], [611], this entry and and this entry in the graphs problem collection.\nReferences\n\n\n[AKS80] Ajtai, Mikl\\'{o}s and Koml\\'{o}s, J\\'{a}nos and Szemer\\'{e}di, Endre, A note on Ramsey numbers. J. Combin. Theory Ser. A (1980), 354-360.\n\n[EGT92] Erd\\H{o}s, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289.\n\n[Ki95] Kim, J. H., The Ramsey number $R(3,t)$ has order of magnitude $t^2/\\log t$. Random Structures and Algorithms (1995), 173-207.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2211, "problem_number": "EP-611", "title": "Erdős Problem #611", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number).\nIs it true that if all maximal cliques in $G$ have at least $cn$ vertices then $\\tau(G)=o_c(n)$?\nSimilarly, estimate for $c>0$ the minimal $k_c(n)$ such that if every maximal clique in $G$ has at least $k_c(n)$ vertices then $\\tau(G)<(1-c)n$.", "background": "A problem of Erd\\H{o}s, Gallai, and Tuza \\cite{EGT92}, who proved for the latter question that $k_c(n) \\geq n^{c'/\\log\\log n}$ for some $c'>0$, and that if every clique has size least $k$ then $\\tau(G) \\leq n-(kn)^{1/2}$. Bollob\\'{a}s and Erd\\H{o}s proved that if every maximal clique has at least $n+3-2\\sqrt{n}$ vertices then $\\tau(G)=1$ (and this threshold is best possible).\nSee also [610] and the entry in the graphs problem collection.\nReferences\n\n\n[EGT92] Erd\\H{o}s, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2212, "problem_number": "EP-612", "title": "Erdős Problem #612", "statement": "Let $G$ be a connected graph with $n$ vertices, minimum degree $d$, and diameter $D$. Show if that $G$ contains no $K_{2r}$ and $(r-1)(3r+2)\\mid d$ then $ D\\leq \\frac{2(r-1)(3r+2)}{2r^2-1}\\frac{n}{d}+O(1), $ and if $G$ contains no $K_{2r+1}$ and $3r-1 \\mid d$ then $ D\\leq \\frac{3r-1}{r}\\frac{n}{d}+O(1). $ ", "background": "A problem of Erd\\H{o}s, Pach, Pollack, and Tuza \\cite{EPPT89}, who gave constructions showing that the above bounds would be sharp, and proved the case $2r+1=3$. It is known (see \\cite{EPPT89} for example) that any connected graph on $n$ vertices with minimum degree $d$ has diameter $ D\\leq 3\\frac{n}{d+1}+O(1). $ This was disproven for the case of $K_{2r}$-free graphs with $r\\geq 2$ by Czabarka, Singgih, and Sz\\'{e}kely \\cite{CSS21}, who constructed arbitrarily large connected graphs on $n$ vertices which contain no $K_{2r}$ and have minimum degree $d$, and diameter $ \\frac{6r-5}{(2r-1)d+2r-3}n+O(1), $ which contradicts the above conjecture for each fixed $r$ as $d\\to \\infty$.\nThey suggest the amended conjecture, which no longer divides into two cases, that if $G$ is a connected graph on $n$ vertices with minimum degree $d$ which contains no $K_{k+1}$ then the diameter of $G$ is at most $ (3-\\tfrac{2}{k})\\frac{n}{d}+O(1). $ This bound is known under the weaker assumption that $G$ is $k$-colourable when $k=3$ and $k=4$, shown by Czabarka, Dankelmann, and Sz\\'{e}kely \\cite{CDS09} and Czabarka, Smith, and Sz\\'{e}kely \\cite{CSS23}.\nCambie and Jooken \\cite{CaJo25} have given an example that shows the diameter for $K_4$-free graphs with minimum degree $16$ is at least $\\frac{31}{216}n+O(1)$, giving another counterexample to the original conjecture.\nSee also the entry in the graphs problem collection.\nReferences\n\n\n[CDS09] Czabarka, \\'{e}. and Dankelmann, P. and Sz\\'{e}kely, L. A., Diameter of 4-colourable graphs. European J. Combin. (2009), 1082--1089.\n\n[CSS21] Czabarka, \\'{e}va and Singgih, Inne and Sz\\'{e}kely, L\\'aszl\\'{o}{}\nA., Counterexamples to a conjecture of {E}rd\\H{o}s, {P}ach,\n{P}ollack and {T}uza. J. Combin. Theory Ser. B (2021), 38--45.\n\n[CSS23] Czabarka, \\'{e}va and Smith, Stephen J. and Sz\\'{e}kely,\nL\\'aszl\\'{o}, Maximum diameter of 3- and 4-colorable graphs. J. Graph Theory (2023), 262--270.\n\n[CaJo25] S. Cambie and J. Jooken, Sharp results for the Erd\\H{o}s, Pach, Pollack and Tuza problem. arXiv:2502.08626 (2025).\n\n[EPPT89] No reference found.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2213, "problem_number": "EP-614", "title": "Erdős Problem #614", "statement": "Let $f(n,k)$ be minimal such that there is a graph with $n$ vertices and $f(n,k)$ edges where every set of $k+2$ vertices induces a subgraph with maximum degree at least $k$. Determine $f(n,k)$.\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2214, "problem_number": "EP-616", "title": "Erdős Problem #616", "statement": "Let $r\\geq 3$. For an $r$-uniform hypergraph $G$ let $\\tau(G)$ denote the covering number (or transversal number), the minimum size of a set of vertices which includes at least one from each edge in $G$.\nDetermine the best possible $t$ such that, if $G$ is an $r$-uniform hypergraph $G$ where every subgraph $G'$ on at most $3r-3$ vertices has $\\tau(G')\\leq 1$, we have $\\tau(G)\\leq t$.", "background": "Erd\\H{o}s, Hajnal, and Tuza \\cite{EHT91} proved that this $t$ satisfies $ \\frac{3}{16}r+\\frac{7}{8}\\leq t \\leq \\frac{1}{5}r. $ \nReferences\n\n\n[EHT91] Erd\\H{o}s, Paul and Hajnal, Andr\\'{a}s and Tuza, Zsolt, Local constraints ensuring small representing sets. J. Combin. Theory Ser. A (1991), 78-84.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2215, "problem_number": "EP-619", "title": "Erdős Problem #619", "statement": "For a triangle-free graph $G$ let $h_r(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $r$ (while preserving the property of being triangle-free).\nIs it true that there exists a constant $c>0$ such that if $G$ is a connected graph on $n$ vertices then $h_4(G)<(1-c)n$?", "background": "A problem of Erd\\H{o}s, Gy\\'{a}rf\\'{a}s, and Ruszink\\'{o} \\cite{EGR98} who proved that $h_3(G)\\leq n$ and $h_5(G) \\leq \\frac{n-1}{2}$ and there exist connected graphs $G$ on $n$ vertices with $h_3(G)\\geq n-c$ for some constant $c>0$.\nIf we omit the condition that the graph must remain triangle-free then Alon, Gy\\'{a}rf\\'{a}s, and Ruszink\\'{o} \\cite{AGR00} have proved that adding $n/2$ edges always suffices to obtain diameter at most $4$.\nSee also [134] and [618].\nReferences\n\n\n[AGR00] Alon, Noga and Gy\\'{a}rf\\'{a}s, Andr\\'{a}s and Ruszink\\'{o}, Mikl\\'{o}s, Decreasing the diameter of bounded degree graphs. J. Graph Theory (2000), 161--172.\n\n[EGR98] Erd\\H{o}s, Paul and Gy\\'{a}rf\\'{a}s, Andr\\'{a}s and\nRuszink\\'{o}, Mikl\\'{o}s, How to decrease the diameter of triangle-free graphs. Combinatorica (1998), 493-501.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2216, "problem_number": "EP-620", "title": "Erdős Problem #620", "statement": "If $G$ is a graph on $n$ vertices without a $K_4$ then how large a triangle-free induced subgraph must $G$ contain?", "background": "This was first asked by Erd\\H{o}s and Rogers \\cite{ErRo62}, and is generally known as the Erd\\H{o}s-Rogers problem. Let $f(n)$ be such that every such graph contains a triangle-free subgraph with at least $f(n)$ vertices.\nIt is now known that $f(n)=n^{1/2+o(1)}$. Bollob\\'{a}s and Hind \\cite{BoHi91} proved $ n^{1/2} \\ll f(n) \\ll n^{7/10+o(1)}. $ Krivelevich \\cite{Kr94} improved this to $ n^{1/2}(\\log\\log n)^{1/2} \\ll f(n) \\ll n^{2/3}(\\log n)^{1/3}. $ Wolfovitz \\cite{Wo13} proved $ f(n) \\ll n^{1/2}(\\log n)^{120}. $ The best bounds currently known are $ n^{1/2}\\frac{(\\log n)^{1/2}}{\\log\\log n}\\ll f(n) \\ll n^{1/2}\\log n. $ The lower bound follows from results of Shearer \\cite{Sh95}, and the upper bound was proved by Mubayi and Verstraete \\cite{MuVe24}.\nReferences\n\n\n[BoHi91] Bollob\\'{a}s, B. and Hind, H. R., Graphs without large triangle free subgraphs. Discrete Math. (1991), 119-131.\n\n[ErRo62] Erd\\H{o}s, P. and Rogers, C. A., The construction of certain graphs. Canadian J. Math. (1962), 702-707.\n\n[Kr94] Krivelevich, Michael, {$K^s$}-free graphs without large {$K^r$}-free subgraphs. Combin. Probab. Comput. (1994), 349-354.\n\n[MuVe24] D. Mubayi and J. Verstraete, On the order of Erd\\H{o}s-Rogers functions. arXiv:2401.02548 (2024).\n\n[Sh95] Shearer, James B., On the independence number of sparse graphs. Random Structures Algorithms (1995), 269--271.\n\n[Wo13] Wolfovitz, Guy, {$K_4$}-free graphs without large induced triangle-free\nsubgraphs. Combinatorica (2013), 623-631.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2217, "problem_number": "EP-623", "title": "Erdős Problem #623", "statement": "Let $X$ be a set of cardinality $\\aleph_\\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\not\\in A$ for all $A$. Must there exist an infinite $Y\\subseteq X$ that is independent - that is, for all finite $B\\subset Y$ we have $f(B)\not\\in Y$?", "background": "A problem of Erd\\H{o}s and Hajnal \\cite{ErHa58}, who proved that if $\\lvert X\\rvert <\\aleph_\\omega$ then the answer is no. Erd\\H{o}s suggests in \\cite{Er99} that this problem is 'perhaps undecidable'.\nReferences\n\n\n[Er99] Erd\\H{o}s, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6.\n\n[ErHa58] Erd\\H{o}s, P. and Hajnal, A., On the structure of set mappings. Acta Math. Acad. Sci. Hungar. (1958), 111-133.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2218, "problem_number": "EP-624", "title": "Erdős Problem #624", "statement": "Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:\\{A : A\\subseteq X\\}\\to X$ so that for every $Y\\subseteq X$ with $\\lvert Y\\rvert \\geq H(n)$ we have $ \\{ f(A) : A\\subseteq Y\\}=X. $ Prove that $ H(n)-\\log_2 n \\to \\infty. $ ", "background": "A problem of Erd\\H{o}s and Hajnal \\cite{ErHa68} who proved that $ \\log_2 n \\leq H(n) < \\log_2n +(3+o(1))\\log_2\\log_2n. $ Erd\\H{o}s said that even the weaker statement that for $n=2^k$ we have $H(n)\\geq k+1$ is open, but Alon has provided the following simple proof: by the pigeonhole principle there are $\\frac{n-1}{2}$ subsets $A_i$ of size $2$ such that $f(A_i)$ is the same. Any set $Y$ of size $k$ containing at least $k/2$ of them can have at most $ 2^k-\\lfloor k/2\\rfloor+1< 2^k=n $ distinct elements in the union of the images of $f(A)$ for $A\\subseteq Y$.\nFor this weaker statement, Erd\\H{o}s and Gy\\'{a}rf\\'{a}s conjectured the stronger form that if $\\lvert X\\rvert=2^k$ then, for any $f:\\{A : A\\subseteq X\\}\\to X$, there must exist some $Y\\subset X$ of size $k$ such that $ \\#\\{ f(A) : A\\subseteq Y\\}< 2^k-k^C $ for every $C$ (with $k$ sufficiently large depending on $C$). This was proved by Alon (personal communication), who proved the stronger version that there exists some absolute constant $c>0$ such that, if $k$ is large enough, there must exist some $Y\\subset X$ of size $k$ such that $ \\#\\{ f(A) : A\\subseteq Y\\}<(1-c)2^k. $ Alon also proved that, provided $k$ is large enough, if $\\lvert X\\rvert=2^k$ there exists some $f:\\{A: A\\subseteq X\\}\\to X$ such that, if $Y\\subset X$ with $\\lvert Y\\rvert=k$, then $ \\#\\{ f(A) : A\\subseteq Y\\}>\\tfrac{1}{4}2^k. $ \nReferences\n\n\n[ErHa68] Erd\\H{o}s, P. and Hajnal, A., On a combinatorial problem. Mat. Lapok (1968), 345-348.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2219, "problem_number": "EP-625", "title": "Erdős Problem #625", "statement": "The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. Let $\\chi(G)$ denote the chromatic number.\nIf $G$ is a random graph with $n$ vertices and each edge included independently with probability $1/2$ then is it true that almost surely $ \\chi(G) - \\zeta(G) \\to \\infty $ as $n\\to \\infty$?", "background": "A problem of Erd\\H{o}s and Gimbel (see also \\cite{Gi16}). At a conference on random graphs in Poznan, Poland (most likely in 1989) Erd\\H{o}s offered \\$100 for a proof that this is true, and \\$1000 for a proof that this is false (although later told Gimbel that \\$1000 was perhaps too much).\nIt is known that almost surely $ \\frac{n}{2\\log_2n}\\leq \\zeta(G)\\leq \\chi(G)\\leq (1+o(1))\\frac{n}{2\\log_2n}. $ (The final upper bound is due to Bollob\\'{a}s \\cite{Bo88}. The first inequality follows from the fact that almost surely $G$ has clique number and independence number $< 2\\log_2n$.)\nHeckel \\cite{He24} and, independently, Steiner \\cite{St24b} have shown that it is not the case that $\\chi(G)-\\zeta(G)$ is bounded with high probability, and in fact if $\\chi(G)-\\zeta(G) \\leq f(n)$ with high probability then $f(n)\\geq n^{1/2-o(1)}$ along an infinite sequence of $n$. Heckel conjectures that, with high probability, $ \\chi(G)-\\zeta(G) \\asymp \\frac{n}{(\\log n)^3}. $ Heckel \\cite{He24c} further proved that, for any $\\epsilon>0$, we have $ \\chi(G) -\\zeta(G) \\geq n^{1-\\epsilon} $ for roughly $95\\%$ of all $n$.\nReferences\n\n\n[Bo88] Bollob\\'{a}s, B., The chromatic number of random graphs. Combinatorica (1988), 49-55.\n\n[Gi16] J. Gimbel, Some of my favorite coloring problems for graphs and digraphs. Graph Theory: Favorite conjectures and open problems (2016), 95-108.\n\n[He24] A. Heckel, On a question of Erd\\H{o}s and Gimbel on the cochromatic number. arXiv:2408.13839 (2024).\n\n[He24c] A. Heckel, The difference between the chromatic and the cochromatic number of a random graph. arXiv:2409.17614 (2024).\n\n[St24b] R. Steiner, On the difference between the chromatic and cochromatic number. arXiv:2408.02400 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2220, "problem_number": "EP-626", "title": "Erdős Problem #626", "statement": "Let $k\\geq 4$ and $g_k(n)$ denote the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ and girth $>m$ (i.e. contains no cycle of length $\\leq m$). Does $ \\lim_{n\\to \\infty}\\frac{g_k(n)}{\\log n} $ exist?\nConversely, if $h^{(m)}(n)$ is the maximal chromatic number of a graph on $n$ vertices with girth $>m$ then does $ \\lim_{n\\to \\infty}\\frac{\\log h^{(m)}(n)}{\\log n} $ exist, and what is its value?", "background": "It is known that $ \\frac{1}{4\\log k}\\log n\\leq g_k(n) \\leq \\frac{2}{\\log(k-2)}\\log n+1, $ the lower bound due to Kostochka \\cite{Ko88} and the upper bound to Erd\\H{o}s \\cite{Er59b}.\nErd\\H{o}s \\cite{Er59b} proved that $ \\lim_{n\\to \\infty}\\frac{\\log h^{(m)}(n)}{\\log n}\\gg \\frac{1}{m} $ and, for odd $m$, $ \\lim_{n\\to \\infty}\\frac{\\log h^{(m)}(n)}{\\log n}\\leq \\frac{2}{m+1}, $ and conjectured this is sharp. He had no good guess for the value of the limit for even $m$, other that it should lie in $[\\frac{2}{m+2},\\frac{2}{m}]$, but could not prove this even for $m=4$.\nSee also the entry in the graphs problem collection.\nReferences\n\n\n[Er59b] Erd\\H{o}s, P., Graph theory and probability. Canadian J. Math. (1959), 34-38.\n\n[Ko88] Kostochka, A. V., Upper bounds on the chromatic number of graphs. Trudy Inst. Mat. (Novosibirsk) (1988), 204-226, 265.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2221, "problem_number": "EP-627", "title": "Erdős Problem #627", "statement": "Let $\\omega(G)$ denote the clique number of $G$ and $\\chi(G)$ the chromatic number. If $f(n)$ is the maximum value of $\\chi(G)/\\omega(G)$, as $G$ ranges over all graphs on $n$ vertices, then does $ \\lim_{n\\to\\infty}\\frac{f(n)}{n/(\\log n)^2} $ exist?", "background": "Tutte and Zykov \\cite{Zy52} independently proved that for every $k$ there is a graph with $\\omega(G)=2$ and $\\chi(G)=k$. Erd\\H{o}s \\cite{Er61d} proved that for every $n$ there is a graph on $n$ vertices with $\\omega(G)=2$ and $\\chi(G)\\gg n^{1/2}/\\log n$, whence $f(n) \\gg n^{1/2}/\\log n$.\nErd\\H{o}s \\cite{Er67c} proved that $ f(n) \\asymp \\frac{n}{(\\log n)^2} $ and that the limit in question, if it exists, must be in $ (\\log 2)^2\\cdot [1/4,1]. $ See also the entry in the graphs problem collection.\nReferences\n\n\n[Er61d] Erd\\H{o}s, P., Graph theory and probability. II. Canadian J. Math. (1961), 346-352.\n\n[Er67c] Erd\\H{o}s, P., Some remarks on chromatic graphs. Colloq. Math. (1967), 253-256.\n\n[Zy52] Zykov, A. A., On some properties of linear complexes. Amer. Math. Soc. Translation (1952), 33.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2222, "problem_number": "EP-629", "title": "Erdős Problem #629", "statement": "The list chromatic number $\\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours.\nDetermine the minimal number of vertices $n(k)$ of a bipartite graph $G$ such that $\\chi_L(G)>k$.", "background": "A problem of Erd\\H{o}s, Rubin, and Taylor \\cite{ERT80}, who proved that $ 2^{k-1}0$: take $A$ to be the union of all odd numbers together with numbers of the shape $2^k$ with $k$ odd.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2226, "problem_number": "EP-638", "title": "Erdős Problem #638", "statement": "Let $S$ be a family of finite graphs such that for every $n$ there is some $G_n\\in S$ such that if the edges of $G_n$ are coloured with $n$ colours then there is a monochromatic triangle.\nIs it true that for every infinite cardinal $\\aleph$ there is a graph $G$ of which every finite subgraph is in $S$ and if the edges of $G$ are coloured with $\\aleph$ many colours then there is a monochromatic triangle.", "background": "Erd\\H{o}s writes 'if the answer is affirmative many extensions and generalisations will be possible'.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2227, "problem_number": "EP-640", "title": "Erdős Problem #640", "statement": "Is there some function $f$ such that for all $k\\geq 3$ if a finite graph $G$ has chromatic number $\\geq f(k)$ then $G$ must contain some odd cycle whose vertices span a graph of chromatic number $\\geq k$?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2228, "problem_number": "EP-642", "title": "Erdős Problem #642", "statement": "Let $f(n)$ be the maximal number of edges in a graph on $n$ vertices such that all cycles have more vertices than diagonals. Is it true that $f(n)\\ll n$?", "background": "A problem of Hamburger and Szegedy.\nChen, Erd\\H{o}s, and Staton \\cite{CES96} proved $f(n) \\ll n^{3/2}$. Dragani\\'{c}, Methuku, Munh\\'{a} Correia, and Sudakov \\cite{DMMS24} have improved this to $ f(n) \\ll n(\\log n)^8. $ \nReferences\n\n\n[CES96] Chen, Guantao and Erd\\H{o}s, Paul and Staton, William, Proof of a conjecture of {B}ollob\\'as on nested cycles. J. Combin. Theory Ser. B (1996), 38--43.\n\n[DMMS24] Dragani\\'c, Nemanja and Methuku, Abhishek and Munh\\'a{}\nCorreia, David and Sudakov, Benny, Cycles with many chords. Random Structures Algorithms (2024), 3--16.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2229, "problem_number": "EP-643", "title": "Erdős Problem #643", "statement": "Let $f(n;t)$ be minimal such that if a $t$-uniform hypergraph on $n$ vertices contains at least $f(n;t)$ edges then there must be four edges $A,B,C,D$ such that $ A\\cup B= C\\cup D $ and $ A\\cap B=C\\cap D=\\emptyset. $ Estimate $f(n;t)$ - in particular, is it true that for $t\\geq 3$ $ f(n;t)=(1+o(1))\\binom{n}{t-1}? $ ", "background": "For $t=2$ this is asking for the maximal number of edges on a graph which contains no $C_4$, and so $f(n;2)=(1/2+o(1))n^{3/2}$.\nF\"{u}redi \\cite{Fu84} proved that $f(n;3) \\ll n^2$ and $f(n;3) > \\binom{n}{2}$ for infinitely many $n$. Pikhurko and Verstra\"{e}te \\cite{PiVe09} have proved $f(n;3)\\leq \\frac{13}{9}\\binom{n}{2}$ for all $n$.\nMore generally, F\"{u}redi \\cite{Fu84} proved that $ \\binom{n-1}{t-1}+\\left\\lfloor\\frac{n-1}{t}\\right\\rfloor\\leq f(n;t) < \\frac{7}{2}\\binom{n}{t-1}, $ and conjectured the lower bound is sharp for $t\\geq 4$. Pikhurko and Verstra\"{e}te \\cite{PiVe09} have proved that $ 1 \\leq \\limsup_{n\\to \\infty} \\frac{f(n;t)}{\\binom{n}{t-1}}\\leq \\min\\left(\\frac{7}{4},1+\\frac{2}{\\sqrt{t}}\\right) $ for all $t\\geq 3$.\nF\"{u}redi \\cite{Fu84} proved that $f(n;3)/\\binom{n}{2}$ converges as $n\\to \\infty$, but the existence of the limit for $t\\geq 4$ is unknown.\nReferences\n\n\n[Fu84] F\"uredi, Z., Hypergraphs in which all disjoint pairs have distinct unions. Combinatorica (1984), 161--168.\n\n[PiVe09] Pikhurko, Oleg and Verstra\"{e}te, Jacques, The maximum size of hypergraphs without generalized 4-cycles. J. Combin. Theory Ser. A (2009), 637--649.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2230, "problem_number": "EP-644", "title": "Erdős Problem #644", "statement": "Let $f(k,r)$ be minimal such that if $A_1,A_2,\\ldots$ is a family of sets, all of size $k$, such that for every collection of $r$ of the $A_is$ there is some pair $\\{x,y\\}$ which intersects all of the $A_j$, then there is some set of size $f(k,r)$ which intersects all of the sets $A_i$. Is it true that $ f(k,7)=(1+o(1))\\frac{3}{4}k? $ Is it true that for any $r\\geq 3$ there exists some constant $c_r$ such that $ f(k,r)=(1+o(1))c_rk? $ ", "background": "A problem of Erd\\H{o}s, Fon-Der-Flaass, Kostochka, and Tuza \\cite{EFKT92}, who proved that $f(k,3)=2k$ and $f(k,4)=\\lfloor 3k/2\\rfloor$ and $f(k,5)=\\lfloor 5k/4\\rfloor$, and further that $f(k,6)=k$.\nReferences\n\n\n[EFKT92] Erd\"{o}s, P. and Fon-Der-Flaass, D. and Kostochka, A. V. and\nTuza, Zs., Small transversals in uniform hypergraphs. Siberian Adv. Math. (1992), 82-88.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2231, "problem_number": "EP-650", "title": "Erdős Problem #650", "statement": "Let $f(m)$ be such that if $A\\subseteq \\{1,\\ldots,N\\}$ has $\\lvert A\\rvert=m$ then every interval in $[1,\\infty)$ of length $2N$ contains $\\geq f(m)$ many distinct integers $b_1,\\ldots,b_r$ where each $b_i$ is divisible by some $a_i\\in A$, where $a_1,\\ldots,a_r$ are distinct.\nEstimate $f(m)$. In particular is it true that $f(m)\\ll m^{1/2}$?", "background": "Erd\\H{o}s and Sar\\'{a}nyi \\cite{ErSa59} proved that $f(m)\\gg m^{1/2}$.\nReferences\n\n\n[ErSa59] Erd\\H{o}s, P. and Sar\\'{a}nyi, Megjegyz\\'{e}sek egy versenyfeladathoz. Matematikai Lapok (1959).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2232, "problem_number": "EP-652", "title": "Erdős Problem #652", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ and let $R(x_i)=\\#\\{ \\lvert x_j-x_i\\rvert : j\neq i\\}$, where the points are ordered such that $ R(x_1)\\leq \\cdots \\leq R(x_n). $ Let $\\alpha_k$ be minimal such that, for all large enough $n$, there exists a set of $n$ points with $R(x_k)<\\alpha_kn^{1/2}$. Is it true that $\\alpha_k\\to \\infty$ as $k\\to \\infty$?", "background": "It is trivial that $R(x_1)=1$ is possible, and that $R(x_2) \\ll n^{1/2}$ is also possible, but we always have $ R(x_1)R(x_2)\\gg n. $ Erd\\H{o}s originally conjectured that $R(x_3)/n^{1/2}\\to \\infty$ as $n\\to \\infty$, but Elekes proved that for every $k$ and $n$ sufficiently large there exists some set of $n$ points with $R(x_k)\\ll_k n^{1/2}$.\nMathialagan \\cite{Ma21} proved that given a set $P$ of $k$ points and a set $Q$ of $n$ points, with $2\\leq k\\leq n^{1/3}$, there exists a point in $P$ which determines $\\gg (kn)^{1/2}$ distances to points in $Q$. This immediately implies $R(x_k)\\gg (kn)^{1/2}$ for $2\\leq k\\leq n^{1/3}$.\nReferences\n\n\n[Ma21] Mathialagan, Surya, On bipartite distinct distances in the plane. Electron. J. Combin. (2021), Paper No. 4.33, 25.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2233, "problem_number": "EP-653", "title": "Erdős Problem #653", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ and let $R(x_i)=\\#\\{ \\lvert x_j-x_i\\rvert : j\neq i\\}$, where the points are ordered such that $ R(x_1)\\leq \\cdots \\leq R(x_n). $ Let $g(n)$ be the maximum number of distinct values the $R(x_i)$ can take. Is it true that $g(n) \\geq (1-o(1))n$?", "background": "Erd\\H{o}s and Fishburn proved $g(n)>\\frac{3}{8}n$ and Csizmadia proved $g(n)>\\frac{7}{10}n$. Both groups proved $g(n) < n-cn^{2/3}$ for some constant $c>0$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2234, "problem_number": "EP-654", "title": "Erdős Problem #654", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ with no four points on a circle. Must there exist some $x_i$ with at least $(1-o(1))n$ distinct distances to other $x_i$?", "background": "It is clear that every point has at least $\\frac{n-1}{3}$ distinct distances to other points in the set.\nIn \\cite{Er87b} and \\cite{ErPa90} Erd\\H{o}s and Pach ask this under the additional assumption that there are no three points on a line (so that the points are in general position), although they only ask the weaker question whether there is a lower bound of the shape $(\\tfrac{1}{3}+c)n$ for some constant $c>0$.\nThey suggest the lower bound $(1-o(1))n$ is true under the assumption that any circle around a point $x_i$ contains at most $2$ other $x_j$.\nReferences\n\n\n[Er87b] Erd\\H{o}s, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Si\\'{o}fok, 1985) (1987), 167-177.\n\n[ErPa90] Erd\\H{o}s, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2235, "problem_number": "EP-655", "title": "Erdős Problem #655", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points. Are there at least $ (1+c)\\frac{n}{2} $ distinct distances determined between the $x_i$, for some constant $c>0$ and all $n$ sufficiently large?", "background": "A problem of Erd\\H{o}s and Pach. It is easy to see that this assumption implies that there are at least $\\frac{n-1}{2}$ distinct distances determined by every point.\nZach Hunter has observed that taking $n$ points equally spaced on a circle disproves this conjecture. In the spirit of related conjectures of Erd\\H{o}s and others, presumably some kind of assumption that the points are in general position (e.g. no three on a line and no four on a circle) was intended.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2236, "problem_number": "EP-657", "title": "Erdős Problem #657", "statement": "Is it true that if $A\\subset \\mathbb{R}^2$ is a set of $n$ points such that every subset of $3$ points determines $3$ distinct distances (i.e. $A$ has no isosceles triangles) then $A$ must determine at least $f(n)n$ distinct distances, for some $f(n)\\to \\infty$?", "background": "In \\cite{Er73} Erd\\H{o}s attributes this problem (more generally in $\\mathbb{R}^k$) to himself and Davies. In \\cite{Er97e} he does not mention Davis, but says this problem was investigated by himself, F\"{u}redi, Ruzsa, and Pach.\nIn \\cite{Er73} Erd\\H{o}s says it is not even known in $\\mathbb{R}$ whether $f(n)\\to \\infty$. Sarosh Adenwalla has observed that this is equivalent to minimising the number of distinct differences in a set $A\\subset \\mathbb{R}$ of size $n$ without three-term arithmetic progressions. Dumitrescu \\cite{Du08} proved that, in these terms, $ (\\log n)^c \\leq f(n) \\leq 2^{O(\\sqrt{\\log n})} $ for some constant $c>0$.\nHunter observed in the comments that a result of Ruzsa coupled with standard tools of additive combinatorics (with details given by Alfaiz and Tang) allow recent progress on the size of subsets without three-term arithmetic progression (see \\cite{BlSi23} which improves slightly on the bounds due to Kelley and Meka \\cite{KeMe23}) yield $ 2^{c(\\log n)^{1/9}}\\leq f(n) $ for some constant $c>0$.\nStraus has observed that if $2^k\\geq n$ then there exist $n$ points in $\\mathbb{R}^k$ which contain no isosceles triangle and determine at most $n-1$ distances.\nSee also [135].\nReferences\n\n\n[BlSi23] T. F. Bloom and O. Sisask, An improvement to the Kelley-Meka bounds on three-term arithmetic progressions. arXiv:2309.02353 (2023).\n\n[Du08] Dumitrescu, Adrian, On distinct distances and {$\\lambda$}-free point sets. Discrete Math. (2008), 6533--6538.\n\n[Er73] Erd\\H{o}s, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n\n[Er97e] Erd\\H{o}s, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.\n\n[KeMe23] Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2237, "problem_number": "EP-660", "title": "Erdős Problem #660", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^3$ be the vertices of a convex polyhedron. Are there at least $ (1-o(1))\\frac{n}{2} $ many distinct distances between the $x_i$?", "background": "For the similar problem in $\\mathbb{R}^2$ there are always at least $n/2$ distances, as proved by Altman \\cite{Al63} (see [93]). In \\cite{Er75f} Erd\\H{o}s claims that Altman proved that the vertices determine $\\gg n$ many distinct distances, but gives no reference.\nReferences\n\n\n[Al63] Altman, E., On a problem of P. Erd\\H{o}s. Amer. Math. Monthly (1963), 148-157.\n\n[Er75f] Erd\\H{o}s, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2238, "problem_number": "EP-661", "title": "Erdős Problem #661", "statement": "Are there, for all large $n$, some points $x_1,\\ldots,x_n,y_1,\\ldots,y_n\\in \\mathbb{R}^2$ such that the number of distinct distances $d(x_i,y_j)$ is $ o\\left(\\frac{n}{\\sqrt{\\log n}}\\right)? $ ", "background": "One can also ask this for points in $\\mathbb{R}^3$. In $\\mathbb{R}^4$ Lenz observed that there are $x_1,\\ldots,x_n,y_1,\\ldots,y_n\\in \\mathbb{R}^4$ such that $d(x_i,y_j)=1$ for all $i,j$, taking the points on two orthogonal circles.\nMore generally, if $F(2n)$ is the minimal number of such distances, and $f(2n)$ is minimal number of distinct distances between any $2n$ points in $\\mathbb{R}^2$, then is $F =o(f)$?\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2239, "problem_number": "EP-662", "title": "Erdős Problem #662", "statement": "Consider the triangular lattice with minimal distance between two points $1$. Denote by $f(t)$ the number of distances from any points $\\leq t$. For example $f(1)=6$, $f(\\sqrt{3})=12$, and $f(3)=18$.\nLet $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that $d(x_i,x_j)\\geq 1$ for all $i\neq j$. Is it true that, provided $n$ is sufficiently large depending on $t$, the number of distances $d(x_i,x_j)\\leq t$ is less than or equal to $f(t)$ with equality perhaps only for the triangular lattice?\nIn particular, is it true that the number of distances $\\leq \\sqrt{3}-\\epsilon$ is less than $1$?", "background": "A problem of Erd\\H{o}s, Lov\\'{a}sz, and Vesztergombi.\nThis is essentially verbatim the problem description in \\cite{Er97e}, but this does not make sense as written; there must be at least one typo. Suggestions about what this problem intends are welcome.\nErd\\H{o}s also goes on to write 'Perhaps the following stronger conjecture holds: Let $t_10$ and, for all large $n$, a pairwise balanced design such that $ \\lvert A_i\\rvert > n^{1/2}-C $ for all $1\\leq i\\leq m$?", "background": "A problem of Erd\\H{o}s and Larson \\cite{ErLa82}. In general, as Erd\\H{o}s asks in \\cite{Er97f}, find the slowest growing function $h$ such that, for all large $n$, there exists a pairwise balanced design with $ \\lvert A_i\\rvert > n^{1/2}-h(n) $ for all $1\\leq i\\leq m$.\nThe problem above asks whether $h(n)\\ll 1$. Erd\\H{o}s and Larson prove that $h(n) \\ll n^{1/2-c}$ for some constant $c>0$, and note this can be improved to $h(n)\\ll (\\log n)^2$ assuming Cramer-type bounds on the difference between consecutive primes.\nShrikhande and Singhi \\cite{ShSi85} have proved that the answer is no conditional on the conjecture that the order of every projective plane is a prime power (see [723]), by proving that every pairwise balanced design on $n$ points in which each block is of size $\\geq n^{1/2}-c$ can be embedded in a projective plane of order $n+i$ for some $i\\leq c+2$, if $n$ is sufficiently large.\nIn general, if $H(n)$ is the largest prime gap $\\leq n$, then the above reuslts show that, assuming the prime power conjecture, $H(n)\\asymp h(n)$.\nReferences\n\n\n[Er97f] Erd\\H{o}s, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.\n\n[ErLa82] Erd\\H{o}s, P. and Larson, J., On pairwise balanced block designs with the sizes of blocks as uniform as possible. Annals of Discrete Mathematics (1982), 129-134.\n\n[ShSi85] S. S. Shrikhande and N. M. Singhi, On a problem of Erd\\H{o}s and Larson. Combinatorica (1985), 351-358.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2242, "problem_number": "EP-667", "title": "Erdős Problem #667", "statement": "Let $p,q\\geq 1$ be fixed integers. We define $H(n)=H(N;p,q)$ to be the largest $m$ such that any graph on $n$ vertices where every set of $p$ vertices spans at least $q$ edges must contain a complete graph on $m$ vertices.\nIs $ c(p,q)=\\liminf \\frac{\\log H(n)}{\\log n} $ a strictly increasing function of $q$ for $1\\leq q\\leq \\binom{p-1}{2}+1$?", "background": "A problem of Erd\\H{o}s, Faudree, Rousseau, and Schelp.\nWhen $q=1$ this corresponds exactly to the classical Ramsey problem, and hence for example $ \\frac{1}{p-1}\\leq c(p,1) \\leq \\frac{2}{p+1}. $ It is easy to see that if $q=\\binom{p-1}{2}+1$ then $c(p,q)=1$. Erd\\H{o}s, Faudree, Rousseau, and Schelp have shown that $c(p,\\binom{p-1}{2})\\leq 1/2$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2243, "problem_number": "EP-668", "title": "Erdős Problem #668", "statement": "Is it true that the number of incongruent sets of $n$ points in $\\mathbb{R}^2$ which maximise the number of unit distances tends to infinity as $n\\to\\infty$? Is it always $>1$ for $n>3$?", "background": "In fact this is $=1$ also for $n=4$, the unique example given by two equilateral triangles joined by an edge.\nComputational evidence of Engel, Hammond-Lee, Su, Varga, and Zs\\'{a}mboki \\cite{EHSVZ25} and Alexeev, Mixon, and Parshall \\cite{AMP25} suggests that this count is $=1$ for various other $5\\leq n\\leq 21$ (although these calculations were checking only up to graph isomorphism, rather than congruency).\nThe actual maximal number of unit distances is the subject of [90].\nReferences\n\n\n[AMP25] B. Alexeev, D. Mixon, and H. Parshall, The Erd\\H{o}s unit distance problem for small point sets. arXiv:2412.11914 (2025).\n\n[EHSVZ25] P. Engel, O. Hammond-Lee, Y. Su, D. Varga, and P. Zs\\'{a}mboki, Diverse beam search to find densest-known planar unit distance graphs. arXiv:2406.15317 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2244, "problem_number": "EP-669", "title": "Erdős Problem #669", "statement": "Let $F_k(n)$ be minimal such that for any $n$ points in $\\mathbb{R}^2$ there exist at most $F_k(n)$ many distinct lines passing through at least $k$ of the points, and $f_k(n)$ similarly but with lines passing through exactly $k$ points.\nEstimate $f_k(n)$ and $F_k(n)$ - in particular, determine $\\lim F_k(n)/n^2$ and $\\lim f_k(n)/n^2$.", "background": "Trivially $f_k(n)\\leq F_k(n)$ and $f_2(n)=F_2(n)=\\binom{n}{2}$. The problem with $k=3$ is the classical 'Orchard problem' of Sylvester. Burr, Gr\"{u}nbaum, and Sloane \\cite{BGS74} have proved that $ f_3(n)=\\frac{n^2}{6}-O(n) $ and $ F_3(n)=\\frac{n^2}{6}-O(n). $ There is a trivial upper bound of $F_k(n) \\leq \\binom{n}{2}/\\binom{k}{2}$, and hence $ \\lim F_k(n)/n^2 \\leq \\frac{1}{k(k-1)}. $ See also [101].\nReferences\n\n\n[BGS74] Burr, Stefan A. and Gr\"{u}nbaum, Branko and Sloane, N. J. A., The orchard problem. Geometriae Dedicata (1974), 397-424.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2245, "problem_number": "EP-670", "title": "Erdős Problem #670", "statement": "Let $A\\subseteq \\mathbb{R}^d$ be a set of $n$ points such that all pairwise distances differ by at least $1$. Is the diameter of $A$ at least $(1+o(1))n^2$?", "background": "The lower bound of $\\binom{n}{2}$ for the diameter is trivial. Erd\\H{o}s \\cite{Er97f} proved the claim when $d=1$.\nReferences\n\n\n[Er97f] Erd\\H{o}s, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2246, "problem_number": "EP-671", "title": "Erdős Problem #671", "statement": "Given $a_{i}^n\\in [-1,1]$ for all $1\\leq i\\leq n<\\infty$ we define $p_{i}^n$ as the unique polynomial of degree $n-1$ such that $p_{i}^n(a_{i}^n)=1$ and $p_{i}^n(a_{i'}^n)=0$ if $1\\leq i'\\leq n$ with $i\neq i'$. We similarly define $ \\mathcal{L}^nf(x) = \\sum_{1\\leq i\\leq n}f(a_i^n)p_i^n(x), $ the unique polynomial of degree $n-1$ which agrees with $f$ on $a_i^n$ for $1\\leq i\\leq n$ (that is, the sequence of Lagrange interpolation polynomials).", "background": "Is there such a sequence of $a_i^n$ such that for every continuous $f:[-1,1]\\to \\mathbb{R}$ there exists some $x\\in [-1,1]$ where $ \\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty $ and yet $ \\mathcal{L}^nf(x) \\to f(x)? $ Is there such a sequence such that $ \\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty $ for every $x\\in [-1,1]$ and yet for every continuous $f:[-1,1]\\to \\mathbb{R}$ there exists $x\\in [-1,1]$ with $ \\mathcal{L}^nf(x) \\to f(x)? $ \nBernstein \\cite{Be31} proved that for any choice of $a_i^n$ there exists $x_0\\in [-1,1]$ such that $ \\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty. $ Erd\\H{o}s and V\\'{e}rtesi \\cite{ErVe80} proved that for any choice of $a_i^n$ there exists a continuous $f:[-1,1]\\to \\mathbb{R}$ such that $ \\limsup_{n\\to \\infty} \\lvert \\mathcal{L}^nf(x)\\rvert=\\infty $ for almost all $x\\in [-1,1]$.\nReferences\n\n\n[Be31] S. Bernstein, Sur la limitation des valeurs d'un polynome $P_n(x)$ de degr\\'{e} n sur tout un segment par ses valeurs en $(n+1)$ points du segment. Izv. Akad. Nauk. SSSR (1931), 1025-1050.\n\n[ErVe80] Erd\\H{o}s, P. and V\\'{e}rtesi, P., On the almost everywhere divergence of Lagrange\ninterpolatory polynomials for arbitrary system of nodes. Acta Math. Acad. Sci. Hungar. (1980), 71-89.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2247, "problem_number": "EP-675", "title": "Erdős Problem #675", "statement": "We say that $A\\subset \\mathbb{N}$ has the translation property if, for every $n$, there exists some integer $t_n\\geq 1$ such that, for all $1\\leq a\\leq n$, $ a\\in A\\quad\\textrm{ if and only if }\\quad a+t_n\\in A. $ {UL}", "background": "{LI}Does the set of the sums of two squares have the translation property?{/LI}\n{LI}If we partition all primes into $P\\sqcup Q$, such that each set contains $\\gg x/\\log x$ many primes $\\leq x$ for all large $x$, then can the set of integers only divisible by primes from $P$ have the translation property?{/LI}\n{LI}If $A$ is the set of squarefree numbers then how fast does the minimal such $t_n$ grow? Is it true that $t_n>\\exp(n^c)$ for some constant $c>0$?{/LI}\n{/UL}\nElementary sieve theory implies that the set of squarefree numbers has the translation property.\nMore generally, Brun's sieve can be used to prove that if $B\\subseteq \\mathbb{N}$ is a set of pairwise coprime integers with $\\sum_{b0$. Erd\\H{o}s \\cite{Er79} believed it is 'rather unlikely' that all large integers are of this form.\nWhat if the condition that $p$ is prime is omitted? Selfridge and Wagstaff made a 'preliminary computer search' and suggested that there are infinitely many $n$ not of this form even without the condition that $p$ is prime. It should be true that the number of exceptions in $[1,x]$ is $1$? Erd\\H{o}s expects very few (and none when $l\\geq k$).\nThe only solutions Erd\\H{o}s knew were $M(4,3)=M(13,2)$ and $M(3,4)=M(19,2)$.\nIn \\cite{Er79d} Erd\\H{o}s conjectures the stronger fact that (aside from a finite number of exceptions) if $k>2$ and $m\\geq n+k$ then $\\prod_{i\\leq k}(n+i)$ and $\\prod_{i\\leq k}(m+i)$ cannot have the same set of prime factors.\nSee also [678], [686], and [850].\nThis is discussed in problem B35 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er79d] Erd\\H{o}s, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2250, "problem_number": "EP-679", "title": "Erdős Problem #679", "statement": "Let $\\epsilon>0$ and $\\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that $ \\omega(n-k) < (1+\\epsilon)\\frac{\\log k}{\\log\\log k} $ for all $k0$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2251, "problem_number": "EP-680", "title": "Erdős Problem #680", "statement": "Is it true that, for all sufficiently large $n$, there exists some $k$ such that $ p(n+k)>k^2+1, $ where $p(m)$ denotes the least prime factor of $m$?\nCan one prove this is false if we replace $k^2+1$ by $e^{(1+\\epsilon)\\sqrt{k}}+C_\\epsilon$, for all $\\epsilon>0$, where $C_\\epsilon>0$ is some constant?", "background": "This follows from 'plausible assumptions on the distribution of primes' (as does the question with $k^2$ replaced by $k^d$ for any $d$); the challenge is to prove this unconditionally.\nErd\\H{o}s observed that Cramer's conjecture $ \\limsup_{k\\to \\infty} \\frac{p_{k+1}-p_k}{(\\log k)^2}=1 $ implies that for all $\\epsilon>0$ and all sufficiently large $n$ there exists some $k$ such that $ p(n+k)>e^{(1-\\epsilon)\\sqrt{k}}. $ There is now evidence, however, that Cramer's conjecture is false; a more refined heuristic by Granville \\cite{Gr95} suggests this $\\limsup$ is $2e^{-\\gamma}\\approx 1.119\\cdots$, and so perhaps the $1+\\epsilon$ in the second question should be replaced by $2e^{-\\gamma}+\\epsilon$.\nSee also [681] and [682].\nReferences\n\n\n[Gr95] Granville, Andrew, Harald {C}ram\\'{e}r and the distribution of prime numbers. Scand. Actuar. J. (1995), 12--28.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2252, "problem_number": "EP-681", "title": "Erdős Problem #681", "statement": "Is it true that for all large $n$ there exists $k$ such that $n+k$ is composite and $ p(n+k)>k^2, $ where $p(m)$ is the least prime factor of $m$?", "background": "Related to questions of Erd\\H{o}s, Eggleton, and Selfridge. This may be true with $k^2$ replaced by $k^d$ for any $d$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2253, "problem_number": "EP-683", "title": "Erdős Problem #683", "statement": "Is it true that for every $1\\leq k\\leq n$ the largest prime divisor of $\\binom{n}{k}$, say $P(\\binom{n}{k})$, satisfies $ P\\left(\\binom{n}{k}\\right)\\geq \\min(n-k+1, k^{1+c}) $ for some constant $c>0$?", "background": "A theorem of Sylvester and Schur (see \\cite{Er34}) states that $P(\\binom{n}{k})>k$ if $k\\leq n/2$. Erd\\H{o}s \\cite{Er55d} proved that there exists some $c>0$ such that, whenever $k\\leq n/2$, $ P\\left(\\binom{n}{k}\\right)\\gg k\\log k. $ Erd\\H{o}s \\cite{Er79d} writes it 'seems certain' that this holds for every $c>0$, with only a finite number of exceptions (depending on $c$). Standard heuristics on prime gaps suggest that the largest prime divisor of $\\binom{n}{k}$ is, for $k\\leq n/2$, in fact $ >e^{c\\sqrt{k}} $ for some constant $c>0$.\nThis is essentially equivalent to [961].\nReferences\n\n\n[Er34] Erd\\H{o}s, Paul, A {T}heorem of {S}ylvester and {S}chur. J. London Math. Soc. (1934), 282--288.\n\n[Er55d] Erd\\H{o}s, P., On consecutive integers. Nieuw Arch. Wisk. (3) (1955), 124--128.\n\n[Er79d] Erd\\H{o}s, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2254, "problem_number": "EP-684", "title": "Erdős Problem #684", "statement": "For $0\\leq k\\leq n$ write $ \\binom{n}{k} = uv $ where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$.\nLet $f(n)$ be the smallest $k$ such that $u>n^2$. Give bounds for $f(n)$.", "background": "A classical theorem of Mahler states that for any $\\epsilon>0$ and integers $k$ and $l$ then, writing $ (n+1)\\cdots (n+k) = ab $ where the only primes dividing $a$ are $\\leq l$ and the only primes dividing $b$ are $>l$, we have $a < n^{1+\\epsilon}$ for all sufficiently large (depending on $\\epsilon,k,l$) $n$.\nMahler's theorem implies $f(n)\\to \\infty$ as $n\\to \\infty$, but is ineffective, and so gives no bounds on the growth of $f(n)$.\nOne can similarly ask for estimates on the smallest integer $f(n,k)$ such that if $m$ is the factor of $\\binom{n}{k}$ containing all primes $\\leq f(n,k)$ then $m > n^2$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2255, "problem_number": "EP-685", "title": "Erdős Problem #685", "statement": "Let $\\epsilon>0$ and $n$ be large depending on $\\epsilon$. Is it true that for all $n^\\epsilon\\frac{\\log \\binom{n}{k}}{\\log n}, $ and this inequality becomes (asymptotic) equality if $k>n^{1-o(1)}$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2256, "problem_number": "EP-686", "title": "Erdős Problem #686", "statement": "Can every integer $N\\geq 2$ be written as $ N=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)} $ for some $k\\geq 2$ and $m\\geq n+k$?", "background": "If $n$ and $k$ are fixed then can one say anything about the set of integers so represented?\nSee also [677].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2257, "problem_number": "EP-687", "title": "Erdős Problem #687", "statement": "Let $Y(x)$ be the maximal $y$ such that there exists a choice of congruence classes $a_p$ for all primes $p\\leq x$ such that every integer in $[1,y]$ is congruent to at least one of the $a_p\\pmod{p}$.\nGive good estimates for $Y(x)$. In particular, can one prove that $Y(x)=o(x^2)$ or even $Y(x)\\ll x^{1+o(1)}$?", "background": "This function (associated with Jacobsthal) is closely related to the problem of gaps between primes (see [4]). The best known upper bound is due to Iwaniec \\cite{Iw78}, $ Y(x) \\ll x^2. $ The best lower bound is due to Ford, Green, Konyagin, Maynard, and Tao \\cite{FGKMT18}, $ Y(x) \\gg x\\frac{\\log x\\log\\log\\log x}{\\log\\log x}, $ improving on a previous bound of Rankin \\cite{Ra38}.\nMaier and Pomerance have conjectured that $Y(x)\\ll x(\\log x)^{2+o(1)}$.\nIn \\cite{Er80} he writes 'It is not clear who first formulated this problem - probably many of us did it independently. I offer the maximum of \\$1000 dollars and $1/2$ my total savings for clearing up of this problem.'\nIn \\cite{Er80} Erd\\H{o}s also asks about a weaker variant in which all except $o(y/\\log y)$ of the integers in $[1,y]$ are congruent to at least one of the $a_p\\pmod{p}$, and in particular asks if the answer is very different.\nSee also [688] and [689]. A more general Jacobsthal function is the focus of [970].\nReferences\n\n\n[Er80] Erd\\H{o}s, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.\n\n[FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence, Long gaps between primes. J. Amer. Math. Soc. (2018), 65-105.\n\n[Iw78] Iwaniec, Henryk, On the problem of {J}acobsthal. Demonstratio Math. (1978), 225--231.\n\n[Ra38] Rankin, R. A., The Difference between Consecutive Prime Numbers. J. London Math. Soc. (1938), 242-247.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2258, "problem_number": "EP-688", "title": "Erdős Problem #688", "statement": "Define $\\epsilon_n$ to be maximal such that there exists some choice of congruence class $a_p$ for all primes $n^{\\epsilon_n}\\alpha$.\nTenenbaum notes in \\cite{Te96} that this is certainly not true as written since if the $n_j$ grow sufficiently quickly then this sequence is never Behrend, for any choice of $\\eta_k$. He then writes 'we understand from subsequent discussions with Erd\\H{o}s that he had actually in mind a two-sided condition on' $n_{j+1}/n_j$.\nTenenbaum \\cite{Te96} proves this conjecture: if there are constants $1\\log 2$.\nReferences\n\n\n[Te96] Tenenbaum, G., On block {B}ehrend sequences. Math. Proc. Cambridge Philos. Soc. (1996), 355--367.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2262, "problem_number": "EP-693", "title": "Erdős Problem #693", "statement": "Let $k\\geq 2$ and $n$ be sufficiently large depending on $k$. Let $A=\\{a_1n^{1/2}$?{/LI}\n{LI} Is it true that, for every composite $n$, $ f(n) \\ll_A \\frac{n}{(\\log n)^A} $ for every $A>0$?{/LI}\n{/UL}", "background": "A problem of Erd\\H{o}s and Szekeres. It is easy to see that $f(n)\\leq n/P(n)$ for composite $n$, since if $j=p^k$ where $p^k\\mid n$ and $p^{k+1}\nmid n$ then $\\textrm{gcd}\\left(n,\\binom{n}{j}\\right)=n/p^k$. This implies $ f(n) \\leq (1+o(1))\\frac{n}{\\log n}. $ It is known that $f(n)=n/P(n)$ when $n$ is the product of two primes. Another example is $n=30$.\nFor the second problem, it is easy to see that for any $n$ we have $f(n)\\geq p(n)$, where $p(n)$ is the smallest prime dividing $n$, and hence there are infinitely many $n$ (those $=p^2)$ such that $f(n)\\geq n^{1/2}$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2267, "problem_number": "EP-701", "title": "Erdős Problem #701", "statement": "Let $\\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if $B\\subseteq A\\in\\mathcal{F}$ then $B\\in \\mathcal{F}$). There exists some element $x$ such that whenever $\\mathcal{F}'\\subseteq \\mathcal{F}$ is an intersecting subfamily we have $ \\lvert \\mathcal{F}'\\rvert \\leq \\lvert \\{ A\\in \\mathcal{F} : x\\in A\\}\\rvert. $ ", "background": "A problem of Chv\\'{a}tal \\cite{Ch74}, who proved it replacing the closed under subsets condition with the (stronger) condition that, assuming all sets in $\\mathcal{F}$ are subsets of $\\{1,\\ldots,n\\}$, whenever $A\\in \\mathcal{F}$ and there is an injection $f:B\\to A$ such that $x\\leq f(x)$ for all $x\\in B$, then $B\\in \\mathcal{F}$.\nSterboul \\cite{St74} proved this when, letting $\\mathcal{G}$ be the maximal sets (under inclusion) in $\\mathcal{F}$, all sets in $\\mathcal{G}$ have the same size, $\\lvert A\\cap B\\rvert\\leq 1$ for all $A\neq B\\in \\mathcal{G}$, and at least two sets in $\\mathcal{G}$ have non-empty intersection.\nFrankl and Kupavskii \\cite{FrKu23} have proved this when $\\mathcal{F}$ has covering number $2$.\nBorg \\cite{Bo11} has proposed a weighted generalisation of this conjecture, which he proves under certain additional assumptions.\nReferences\n\n\n[Bo11] Borg, Peter, On Chv\\'{a}tal's conjecture and a conjecture on families of\nsigned sets. European J. Combin. (2011), 140-145.\n\n[Ch74] Chv\\'{a}tal, V., Intersecting families of edges in hypergraphs having the\nhereditary property. (1974), 61-66.\n\n[FrKu23] Frankl, Peter and Kupavskii, Andrey, Perfect matchings in down-sets. Discrete Math. (2023), Paper No. 113323, 7.\n\n[St74] Sterboul, F., Sur une conjecture de V. Chv\\'{a}tal. (1974), 152-164.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2268, "problem_number": "EP-704", "title": "Erdős Problem #704", "statement": "Let $G_n$ be the unit distance graph in $\\mathbb{R}^n$, with two vertices joined by an edge if and only if the distance between them is $1$.\nEstimate the chromatic number $\\chi(G_n)$. Does it grow exponentially in $n$? Does $ \\lim_{n\\to \\infty}\\chi(G_n)^{1/n} $ exist?", "background": "A generalisation of the Hadwiger-Nelson problem (which addresses $n=2$). Frankl and Wilson \\cite{FrWi81} proved exponential growth: $ \\chi(G_n) \\geq (1+o(1))1.2^n. $ The trivial colouring (by tiling with cubes) gives $ \\chi(G_n) \\leq (2+\\sqrt{n})^n. $ Larman and Rogers \\cite{LaRo72} improved this to $ \\chi(G_n) \\leq (3+o(1))^n, $ and conjecture the truth may be $(2^{3/2}+o(1))^n$. Prosanov \\cite{Pr20} has given an alternative proof of this upper bound.\nSee also [508], [705], and [706].\nReferences\n\n\n[FrWi81] Frankl, P. and Wilson, R. M., Intersection theorems with geometric consequences. Combinatorica (1981), 357-368.\n\n[LaRo72] Larman, D. G. and Rogers, C. A., The realization of distances within sets in Euclidean space. Mathematika (1972), 1-24.\n\n[Pr20] Prosanov, Roman, A new proof of the Larman-Rogers upper bound for the\nchromatic number of the Euclidean space. Discrete Appl. Math. (2020), 115-120.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2269, "problem_number": "EP-705", "title": "Erdős Problem #705", "statement": "Let $G$ be a finite unit distance graph in $\\mathbb{R}^2$ (i.e. the vertices are a finite collection of points in $\\mathbb{R}^2$ and there is an edge between two points if and only if the distance between them is $1$).\nIs there some $k$ such that if $G$ has girth $\\geq k$ (i.e. $G$ contains no cycles of length $p_\\ell^2$.\nGallai was the first to consider problems of this type, and observed that $g(2)=2$ and $g(3)\\geq 4$.\nIn \\cite{Er92c} Erd\\H{o}s offers '100 dollars or 1000 rupees', whichever is more, for a proof or disproof. (In 1992 1000 rupees was worth approximately \\$38.60.)\nErd\\H{o}s and Sur\\'{a}nyi similarly asked what is the smallest $c_n\\geq 1$ such that in any interval $I\\subset [0,\\infty)$ of length $c_n\\max(A)$ there exists some $B\\subseteq I\\cap \\mathbb{N}$ with $\\lvert B\\rvert=n$ such that $ \\prod_{a\\in A} a \\mid \\prod_{b\\in B}b. $ They prove $c_2=1$ and $c_3=\\sqrt{2}$, but have no good upper or lower bounds in general.\nSee also [709].\nReferences\n\n\n[Er92c] Erd\"{o}s, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50.\n\n[ErSu59] Erd\\H{o}s, P\\'{a}l and Sur\\'{a}nyi, J\\'{a}nos, Bemerkungen zu einer Aufgabe eines mathematischen\n{W}ettbewerbs. Mat. Lapok (1959), 39-48.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2272, "problem_number": "EP-709", "title": "Erdős Problem #709", "statement": "Let $f(n)$ be minimal such that, for any $A=\\{a_1,\\ldots,a_n\\}\\subseteq [2,\\infty)\\cap\\mathbb{N}$ of size $n$, in any interval $I$ of $f(n)\\max(A)$ consecutive integers there exist distinct $x_1,\\ldots,x_n\\in I$ such that $a_i\\mid x_i$.\nObtain good bounds for $f(n)$, or even an asymptotic formula.", "background": "A problem of Erd\\H{o}s and Sur\\'{a}nyi \\cite{ErSu59}, who proved $ (\\log n)^c \\ll f(n) \\ll n^{1/2} $ for some constant $c>0$.\nSee also [708].\nReferences\n\n\n[ErSu59] Erd\\H{o}s, P\\'{a}l and Sur\\'{a}nyi, J\\'{a}nos, Bemerkungen zu einer Aufgabe eines mathematischen\n{W}ettbewerbs. Mat. Lapok (1959), 39-48.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2273, "problem_number": "EP-710", "title": "Erdős Problem #710", "statement": "Let $f(n)$ be minimal such that in $(n,n+f(n))$ there exist distinct integers $a_1,\\ldots,a_n$ such that $k\\mid a_k$ for all $1\\leq k\\leq n$. Obtain an asymptotic formula for $f(n)$.", "background": "A problem of Erd\\H{o}s and Pomerance \\cite{ErPo80}, who proved $ (2/\\sqrt{e}+o(1))n\\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/2}\\leq f(n)\\leq (1.7398\\cdots+o(1))n(\\log n)^{1/2}. $ In \\cite{Er92c} Erd\\H{o}s offered 2000 rupees for an asymptotic formula; for uniform comparison across prizes I have converted this using the 1992 exchange rates.\nSee also [711].\nReferences\n\n\n[Er92c] Erd\"{o}s, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50.\n\n[ErPo80] P. Erd\\H{o}s and C. Pomerance, Matching the natural numbers up to $n$ with distinct multiples of another interval. Indigationes Math. (1980), 147-151.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2274, "problem_number": "EP-711", "title": "Erdős Problem #711", "statement": "Let $f(n,m)$ be minimal such that in $(m,m+f(n,m))$ there exist distinct integers $a_1,\\ldots,a_n$ such that $k\\mid a_k$ for all $1\\leq k\\leq n$. Prove that $ \\max_m f(n,m) \\leq n^{1+o(1)} $ and that $ \\max_m (f(n,m)-f(n,n))\\to \\infty. $ ", "background": "A problem of Erd\\H{o}s and Pomerance \\cite{ErPo80}, who proved that $ \\max_m f(n,m) \\ll n^{3/2} $ and $ n\\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/2} \\ll f(n,n)\\ll n(\\log n)^{1/2}. $ In \\cite{Er92c} Erd\\H{o}s offered 1000 rupees for a proof of either; for uniform comparison across prizes I have converted this using the 1992 exchange rates.\nvan Doorn \\cite{vD26} has provided an affirmative answer to the second question, proving that, for all large $n$, there exists $m=m(n)$ such that $ f(n,m)-f(n,n) \\gg \\frac{\\log n}{\\log\\log n}n. $ See also [710].\nReferences\n\n\n[Er92c] Erd\"{o}s, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50.\n\n[ErPo80] P. Erd\\H{o}s and C. Pomerance, Matching the natural numbers up to $n$ with distinct multiples of another interval. Indigationes Math. (1980), 147-151.\n\n[vD26] W. van Doorn, On the length of an interval that contains distinct multiples of the first $n$ positive integers. Integers (2026), #A7.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2275, "problem_number": "EP-712", "title": "Erdős Problem #712", "statement": "Determine, for any $k>r>2$, the value of $ \\frac{\\mathrm{ex}_r(n,K_k^r)}{\\binom{n}{r}}, $ where $\\mathrm{ex}_r(n,K_k^r)$ is the largest number of $r$-edges which can placed on $n$ vertices so that there exists no set of $k$ vertices which is covered by all $\\binom{k}{r}$ possible $r$-edges.", "background": "Tur\\'{an proved} that, when $r=2$, this limit is $ \\frac{1}{2}\\left(1-\\frac{1}{k-1}\\right). $ Erd\\H{o}s \\cite{Er81} offered \\$500 for the determination of this value for any fixed $k>r>2$, and \\$1000 for 'clearing up the whole set of problems'.\nSee also [500] for the case $r=3$ and $k=4$.\nReferences\n\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2276, "problem_number": "EP-713", "title": "Erdős Problem #713", "statement": "Is it true that, for every bipartite graph $G$, there exists some $\\alpha\\in [1,2)$ and $c>0$ such that $ \\mathrm{ex}(n;G)\\sim cn^\\alpha? $ Must $\\alpha$ be rational?", "background": "A problem of Erd\\H{o}s and Simonovits. Erd\\H{o}s sometimes asked this in the weaker version with just $ \\mathrm{ex}(n;G)\\asymp n^{\\alpha}. $ Erd\\H{o}s \\cite{Er67d} had initially conjectured that, for any bipartite graph $G$, $\\mathrm{ex}(n;G)\\sim cn^{\\alpha}$ for some constant $c>0$ and $\\alpha$ of the shape $1+\\frac{1}{k}$ or $2-\\frac{1}{k}$ for some integer $k\\geq 2$. This was disproved by Erd\\H{o}s and Simonovits \\cite{ErSi70}.\nThe analogous statement is not true for hypergraphs, as shown by Frankl and F\"{u}redi \\cite{FrFu87}, who proved that if $G$ is the $5$-uniform hypergraph on $8$ vertices with edges $\\{12346,12457,12358\\}$ then $\\mathrm{ex}(n;G)=o(n^5)$ but $\\mathrm{ex}(n;G)\neq O(n^c)$ for any $c<5$.\nA simplified proof was given by F\"{u}redi and Gerbner \\cite{FuGe21}, who extended it to a counterexample for all $k\\geq 5$. It remains open whether it is true for $k=3$ and $k=4$ (though F\"{u}redi and Gerbner conjecture it is not).\nSee also [571].\nReferences\n\n\n[Er67d] Erd\\H{o}s, P., Some recent results on extremal problems in graph theory.\n{R}esults. (1967), 117--123 (English); pp. 124--130 (French).\n\n[ErSi70] Erd\\H{o}s, P. and Simonovits, M., Some extremal problems in graph theory. Combinatorial theory and its applications, I-III (Proc. Colloq., Balatonf\"{u}red, 1969) (1970), 377-390.\n\n[FrFu87] Frankl, P. and F\"uredi, Z., Exact solution of some {T}ur\\'an-type problems. J. Combin. Theory Ser. A (1987), 226--262.\n\n[FuGe21] F\"uredi, Zolt\\'an and Gerbner, D\\'aniel, Hypergraphs without exponents. J. Combin. Theory Ser. A (2021), Paper No. 105517, 9.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2277, "problem_number": "EP-714", "title": "Erdős Problem #714", "statement": "Is it true that $ \\mathrm{ex}(n; K_{r,r}) \\gg n^{2-1/r}? $ ", "background": "K\"{o}v\\'{a}ri, S\\'{o}s, and Tur\\'{a}n \\cite{KST54} proved $ \\mathrm{ex}(n; K_{r,r}) \\ll n^{2-1/r} $ for all $r\\geq 2$. Brown \\cite{Br66} and, independently, Erd\\H{o}s, R\\'{e}nyi, and S\\'{o}s \\cite{ERS66}, proved the conjectured lower bound when $r=3$.\nWhen $r=2$ it is known that $ \\mathrm{ex}(n;K_{2,2})=\\left(\\frac{1}{2}+o(1)\\right)n^{3/2} $ (see [768], since $K_{2,2}=C_4$).\nSee also [147].\nReferences\n\n\n[Br66] Brown, W. G., On graphs that do not contain a Thomsen graph. Canad. Math. Bull. (1966), 281-285.\n\n[ERS66] Erd\\H{o}s, P. and R\\'{e}nyi, A. and S\\'os, V. T., On a problem of graph theory. Studia Sci. Math. Hungar. (1966), 215--235.\n\n[KST54] K\"{o}vari, T. and S\\'{o}s, V. T. and Tur\\'{a}n, P., On a problem of K. Zarankiewicz. Colloq. Math. (1954), 50-57.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2278, "problem_number": "EP-719", "title": "Erdős Problem #719", "statement": "Let $\\mathrm{ex}_r(n;K_{r+1}^r)$ be the maximum number of $r$-edges that can be placed on $n$ vertices without forming a $K_{r+1}^r$ (the $r$-uniform complete graph on $r+1$ vertices).\nIs every $r$-hypergraph $G$ on $n$ vertices the union of at most $\\mathrm{ex}_{r}(n;K_{r+1}^r)$ many copies of $K_r^r$ and $K_{r+1}^r$, no two of which share a $K_r^r$?\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2279, "problem_number": "EP-724", "title": "Erdős Problem #724", "statement": "Let $f(n)$ be the maximum number of mutually orthogonal Latin squares of order $n$. Is it true that $ f(n) \\gg n^{1/2}? $ ", "background": "Euler conjectured that $f(n)=1$ when $n\\equiv 2\\pmod{4}$, but this was disproved by Bose, Parker, and Shrikhande \\cite{BPS60} who proved $f(n)\\geq 2$ for $n\\geq 7$.\nChowla, Erd\\H{o}s, and Straus \\cite{CES60} proved $f(n) \\gg n^{1/91}$. Wilson \\cite{Wi74} proved $f(n) \\gg n^{1/17}$. Beth \\cite{Be83c} proved $f(n) \\gg n^{1/14.8}$.\nThe sequence of $f(n)$ is A001438 in the OEIS.\nReferences\n\n\n[BPS60] Bose, R. C. and Shrikhande, S. S. and Parker, E. T., Further results on the construction of mutually orthogonal\nLatin squares and the falsity of Euler's conjecture. Canadian J. Math. (1960), 189-203.\n\n[Be83c] Beth, Thomas, Eine Bemerkung zur Absch\"{a}tzung der Anzahl orthogonaler\nlateinischer Quadrate mittels Siebverfahren. Abh. Math. Sem. Univ. Hamburg (1983), 284-288.\n\n[CES60] Chowla, S. and Erd\\H{o}s, P. and Straus, E. G., On the maximal number of pairwise orthogonal Latin squares\nof a given order. Canadian J. Math. (1960), 204-208.\n\n[Wi74] Wilson, Richard M., Concerning the number of mutually orthogonal Latin squares. Discrete Math. (1974), 181-198.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2280, "problem_number": "EP-725", "title": "Erdős Problem #725", "statement": "Give an asymptotic formula for the number of $k\\times n$ Latin rectangles.", "background": "Erd\\H{o}s and Kaplansky \\cite{ErKa46} proved the count is $ \\sim e^{-\\binom{k}{2}}(n!)^k $ when $k=o((\\log n)^{3/2-\\epsilon})$. Yamamoto \\cite{Ya51} extended this to $k\\leq n^{1/3-o(1)}$.\nThe count of such Latin rectangles is A001009 in the OEIS.\nReferences\n\n\n[ErKa46] Erd\"{o}s, Paul and Kaplansky, Irving, The asymptotic number of Latin rectangles. Amer. J. Math. (1946), 230-236.\n\n[Ya51] Yamamoto, Koichi, On the asymptotic number of Latin rectangles. Jpn. J. Math. (1951), 113-119.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2281, "problem_number": "EP-726", "title": "Erdős Problem #726", "statement": "As $n\\to \\infty$ ranges over integers $ \\sum_{p\\leq n}1_{n\\in (p/2,p)\\pmod{p}}\\frac{1}{p}\\sim \\frac{\\log\\log n}{2}. $ ", "background": "A conjecture of Erd\\H{o}s, Graham, Ruzsa, and Straus \\cite{EGRS75}. For comparison the classical estimate of Mertens states that $ \\sum_{p\\leq n}\\frac{1}{p}\\sim \\log\\log n. $ By $n\\in (p/2,p)\\pmod{p}$ we mean $n\\equiv r\\pmod{p}$ for some integer $r$ with $p/20$).\nErd\\H{o}s \\cite{Er68c} proved that if $a!b!\\mid n!$ then $a+b\\leq n+O(\\log n)$.\nReferences\n\n\n[Ba29] H. Balakran, On the values of $n$ which make $(2n)!/(n+1)!(n+1)!$ an integer. J. Indian Math. Soc. (1929), 97-100.\n\n[EGRS75] Erd\\H{o}s, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\n\n[Er68c] P. Erd\\H{o}s, Aufgabe 557. Elemente Math. (1968), 111-113.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2283, "problem_number": "EP-730", "title": "Erdős Problem #730", "statement": "Are there infinitely many pairs of integers $n\neq m$ such that $\\binom{2n}{n}$ and $\\binom{2m}{m}$ have the same set of prime divisors?", "background": "A problem of Erd\\H{o}s, Graham, Ruzsa, and Straus \\cite{EGRS75}, who believed there is 'no doubt' that the answer is yes.\nFor example $(87,88)$ and $(607,608)$. Those $n$ such that there exists some suitable $m>n$ are listed as A129515 in the OEIS.\nA triple of such $n$ for which $\\binom{2n}{n}$ all share the same set of prime divisors is $(10003,10004,10005)$. It is not known whether there are such pairs of the shape $(n,n+k)$ for every $k\\geq 1$.\nReferences\n\n\n[EGRS75] Erd\\H{o}s, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2284, "problem_number": "EP-731", "title": "Erdős Problem #731", "statement": "Find some reasonable function $f(n)$ such that, for almost all integers $n$, the least integer $m$ such that $m\nmid \\binom{2n}{n}$ satisfies $ m\\sim f(n). $ ", "background": "A problem of Erd\\H{o}s, Graham, Ruzsa, and Straus \\cite{EGRS75}, who say it is 'not hard to show that', for almost all $n$, the minimal such $m$ satisfies $ m=\\exp((\\log n)^{1/2+o(1)}). $ \nReferences\n\n\n[EGRS75] Erd\\H{o}s, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2285, "problem_number": "EP-734", "title": "Erdős Problem #734", "statement": "Find, for all large $n$, a non-trivial pairwise balanced block design $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ such that, for all $t$, there are $O(n^{1/2})$ many $i$ such that $\\lvert A_i\\rvert=t$.", "background": "$A_1,\\ldots,A_m$ is a pairwise balanced block design if every pair in $\\{1,\\ldots,n\\}$ is contained in exactly one of the $A_i$.\nErd\\H{o}s \\cite{Er81} writes 'this will be probably not be very difficult to prove but so far I was not successful'.\nErd\\H{o}s and de Bruijn \\cite{dBEr48} proved that if $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ is a pairwise balanced block design then $m\\geq n$, and this implies there must be some $t$ such that there are $\\gg n^{1/2}$ many $t$ with $\\lvert A_i\\rvert=t$.\nReferences\n\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[dBEr48] de Bruijn, N. G. and Erd\\H{o}s, P., On a combinatorial problem. Nederl. Akad. Wetensch., Proc. (1948), 1277--1279 = Indagationes Math. 10, 421--423.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2286, "problem_number": "EP-738", "title": "Erdős Problem #738", "statement": "If $G$ has infinite chromatic number and is triangle-free (contains no $K_3$) then must $G$ contain every tree as an induced subgraph?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2287, "problem_number": "EP-740", "title": "Erdős Problem #740", "statement": "Let $\\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\\mathfrak{m}$. Let $r\\geq 1$. Must $G$ contain a subgraph of chromatic number $\\mathfrak{m}$ which does not contain any odd cycle of length $\\leq r$?", "background": "A question of Erd\\H{o}s and Hajnal. R\"{o}dl proved this is true if $\\mathfrak{m}=\\aleph_0$ and $r=3$ (see [108] for the finitary version).\nMore generally, Erd\\H{o}s and Hajnal asked must there exist (for every cardinal $\\mathfrak{m}$ and integer $r$) some $f_r(\\mathfrak{m})$ such that every graph with chromatic number $\\geq f_r(\\mathfrak{m})$ contains a subgraph with chromatic number $\\mathfrak{m}$ with no odd cycle of length $\\leq r$?\nErd\\H{o}s \\cite{Er95d} claimed that even the $r=3$ case of this is open: must every graph with sufficiently large chromatic number contain a triangle free graph with chromatic number $\\mathfrak{m}$?\nIn \\cite{Er81} Erd\\H{o}s also asks the same question but with girth (i.e. the subgraph does not contain any cycle at all of length $\\leq C$).\nReferences\n\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Er95d] Erd\\H{o}s, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) (1995), 61-65.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2288, "problem_number": "EP-741", "title": "Erdős Problem #741", "statement": "Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive density. Can one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both have positive density?\nIs there a basis $A$ of order $2$ such that if $A=A_1\\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$ cannot both have bounded gaps?", "background": "A problem of Burr and Erd\\H{o}s. Erd\\H{o}s \\cite{Er94b} thought he could construct a basis as in the second question, but 'could never quite finish the proof'.\nReferences\n\n\n[Er94b] Erd\\H{o}s, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2289, "problem_number": "EP-749", "title": "Erdős Problem #749", "statement": "Let $\\epsilon>0$. Does there exist $A\\subseteq \\mathbb{N}$ such that the lower density of $A+A$ is at least $1-\\epsilon$ and yet $1_A\\ast 1_A(n) \\ll_\\epsilon 1$ for all $n$?", "background": "A similar question can be asked for upper density.\nSee also [28].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2290, "problem_number": "EP-750", "title": "Erdős Problem #750", "statement": "Let $f(m)$ be some function such that $f(m)\\to \\infty$ as $m\\to \\infty$. Does there exist a graph $G$ of infinite chromatic number such that every subgraph on $m$ vertices contains an independent set of size at least $\\frac{m}{2}-f(m)$?", "background": "In \\cite{Er69b} Erd\\H{o}s conjectures this for $f(m)=\\epsilon m$ for any fixed $\\epsilon>0$. This follows from a result of Erd\\H{o}s, Hajnal, and Szemer\\'{e}di \\cite{EHS82}, as described by msellke in the comments.\nIn \\cite{ErHa67b} Erd\\H{o}s and Hajnal prove this for $f(m)\\geq cm$ for all $c>1/4$.\nSee also [75].\nReferences\n\n\n[EHS82] Erd\\H{o}s, P. and Hajnal, A. and Szemer\\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123.\n\n[Er69b] Erd\\H{o}s, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\n\n[ErHa67b] Erd\\H{o}s, P. and Hajnal, Andr\\'as, On chromatic graphs. Mat. Lapok (1967), 1--4.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2291, "problem_number": "EP-757", "title": "Erdős Problem #757", "statement": "Let $A\\subset \\mathbb{R}$ be a set of size $n$ such that every subset $B\\subseteq A$ with $\\lvert B\\rvert =4$ has $\\lvert B-B\\rvert\\geq 11$. Find the best constant $c>0$ such that $A$ must always contain a Sidon set of size $\\geq cn$.", "background": "For comparison, note that if $B$ were a Sidon set then $\\lvert B-B\\rvert=13$, so this condition is saying that at most one difference is 'missing' from $B-B$. Equivalently, one can view $A$ as a set such that every four points determine at least five distinct distances, and ask for a subset with all distances distinct.\nWithout loss of generality, one can assume $A\\subset \\mathbb{N}$.\nErd\\H{o}s and S\\'{o}s proved that $c\\geq 1/2$. Gy\\'{a}rf\\'{a}s and Lehel \\cite{GyLe95} proved $ \\frac{1}{2}1$ such that $d\\equiv 1\\pmod{p}$. Is it true that there exists some constant $c>0$ such that for all large $N$ $ \\frac{\\lvert A\\cap [1,N]\\rvert}{N}=\\exp(-(c+o(1))\\sqrt{\\log N}\\log\\log N). $ ", "background": "Erd\\H{o}s could prove that there exists some constant $c>0$ such that for all large $N$ $ \\exp(-c\\sqrt{\\log N}\\log\\log N)\\leq \\frac{\\lvert A\\cap [1,N]\\rvert}{N} $ and $ \\frac{\\lvert A\\cap [1,N]\\rvert}{N}\\leq \\exp(-(1+o(1))\\sqrt{\\log N\\log\\log N}). $ Erd\\H{o}s asked about this because $\\lvert A\\cap [1,N]\\rvert$ provides an upper bound for the number of integers $n\\leq N$ for which there is a non-cyclic simple group of order $n$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2295, "problem_number": "EP-769", "title": "Erdős Problem #769", "statement": "Let $c(n)$ be minimal such that if $k\\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes. Give good bounds for $c(n)$ - in particular, is it true that $c(n) \\gg n^n$?", "background": "A problem first investigated by Hadwiger, who proved the lower bound $ c(n) \\geq 2^n+2^{n-1}. $ It is easy to see that $c(2)=6$. Meier conjectured $c(3)=48$. Burgess and Erd\\H{o}s \\cite{Er74b} proved $ c(n) \\ll n^{n+1}. $ Erd\\H{o}s wrote 'I am certain that if $n+1$ is a prime then $c(n)>n^n$.'\nHudelson \\cite{Hu98} proved that if $(2^n-1,3^n-1)=1$ then $c(n) < 6^n$, and in general $c(n) \\ll (2n)^{n-1}$. Connor and Marmorino \\cite{CoMa18} proved that $ c(n) \\geq 2^{n+1}-1 $ for all $n\\geq 3$, $ c(n) \\leq 1.8n^{n+1} $ if $n+1$ is prime, and $ c(n) \\leq e^2n^n $ otherwise.\nReferences\n\n\n[CoMa18] Connor, Peter and Marmorino, Phillip, Decomposing cubes into smaller cubes. J. Geom. (2018), Paper No. 19, 11.\n\n[Er74b] Erd\\H{o}s, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.\n\n[Hu98] Hudelson, Matthew, Dissecting {$d$}-cubes into smaller {$d$}-cubes. J. Combin. Theory Ser. A (1998), 190--200.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2296, "problem_number": "EP-770", "title": "Erdős Problem #770", "statement": "Let $h(n)$ be minimal such that $2^n-1,3^n-1,\\ldots,h(n)^n-1$ are mutually coprime.\nDoes, for every prime $p$, the density $\\delta_p$ of integers with $h(n)=p$ exist? Does $\\liminf h(n)=\\infty$? Is it true that if $p$ is the greatest prime such that $p-1\\mid n$ and $p>n^\\epsilon$ then $h(n)=p$?", "background": "It is easy to see that $h(n)=n+1$ if and only if $n+1$ is prime, and that $h(n)$ is unbounded for odd $n$.\nIt is probably true that $h(n)=3$ for infinitely many $n$.\nSee also [820].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2297, "problem_number": "EP-773", "title": "Erdős Problem #773", "statement": "What is the size of the largest Sidon subset $A\\subseteq\\{1,2^2,\\ldots,N^2\\}$? Is it $N^{1-o(1)}$?", "background": "A question of Alon and Erd\\H{o}s \\cite{AlEr85}, who proved $\\lvert A\\rvert \\geq N^{2/3-o(1)}$ is possible (via a random subset), and observed that $ \\lvert A\\rvert \\ll \\frac{N}{(\\log N)^{1/4}}, $ since (as shown by Landau) the density of the sums of two squares decays like $(\\log N)^{-1/2}$. The lower bound was improved to $ \\lvert A\\rvert \\gg N^{2/3} $ by Lefmann and Thiele \\cite{LeTh95}.\nReferences\n\n\n[AlEr85] Alon, Noga and Erd\\H{o}s, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203.\n\n[LeTh95] Lefmann, Hanno and Thiele, Torsten, Point sets with distinct distances. Combinatorica (1995), 379--408.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2298, "problem_number": "EP-774", "title": "Erdős Problem #774", "statement": "We call $A\\subset \\mathbb{N}$ dissociated if $\\sum_{n\\in X}n\neq \\sum_{m\\in Y}m$ for all finite $X,Y\\subset A$ with $X\neq Y$.\nLet $A\\subset \\mathbb{N}$ be an infinite set. We call $A$ proportionately dissociated if every finite $B\\subset A$ contains a dissociated set of size $\\gg \\lvert B\\rvert$.\nIs every proportionately dissociated set the union of a finite number of dissociated sets?", "background": "This question appears in a paper of Alon and Erd\\H{o}s \\cite{AlEr85}, although the general topic was first considered by Pisier \\cite{Pi83}, who observed that the converse holds, and proved that being proportionately dissociated is equivalent to being a 'Sidon set' in the harmonic analysis sense; that is, whenever $f:A\\to \\mathbb{C}$ there exists some $\\theta\\in [0,1]$ such that $ \\| f\\|_1 \\ll \\left\\lvert\\sum_{n\\in A} f(n)e(n\\theta)\\right\\rvert, $ where $e(x)=e^{2\\pi ix}$.\nAlon and Erd\\H{o}s write that it 'seems unlikely that [this] is also sufficient'. They also point out the same question can be asked replacing dissociated with Sidon (in the additive combinatorial sense) (see [328]). This latter question was resolved in the negative by Ne\\v{s}et\\v{r}il, R\"{o}dl, and Sales \\cite{NRS24}.\nReferences\n\n\n[AlEr85] Alon, Noga and Erd\\H{o}s, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203.\n\n[NRS24] Ne\\v set\\v ril, Jaroslav and R\"odl, Vojt\\v ech and Sales,\nMarcelo, On {P}isier type theorems. Combinatorica (2024), 1211--1232.\n\n[Pi83] Pisier, Gilles, Arithmetic characterizations of Sidon sets. Bull. Amer. Math. Soc. (N.S.) (1983), 87-89.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2299, "problem_number": "EP-776", "title": "Erdős Problem #776", "statement": "Let $r\\geq 2$ and $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ be such that $A_i\not\\subseteq A_j$ for all $i\neq j$ and for any $t$ if there exists some $i$ with $\\lvert A_i\\rvert=t$ then there must exist at least $r$ sets of that size.\nHow large must $n$ be (as a function of $r$) to ensure that there is such a family which achieves $n-3$ distinct sizes of sets?", "background": "A problem of Erd\\H{o}s and Trotter. For $r=1$ and $n>3$ the maximum possible is $n-2$. For $r>1$ and $n$ sufficiently large $n-3$ is achievable, but $n-2$ is never achievable.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2300, "problem_number": "EP-778", "title": "Erdős Problem #778", "statement": "Alice and Bob play a game on the edges of $K_n$, alternating colouring edges by red (Alice) and blue (Bob). Alice goes first, and wins if at the end the largest red clique is larger than any of the blue cliques.\nDoes Bob have a winning strategy for $n\\geq 3$? (Erd\\H{o}s believed the answer is yes.)", "background": "If we change the game so that Bob colours two edges after each edge that Alice colours, but now require Bob's largest clique to be strictly larger than Alice's, then does Bob have a winning strategy for $n>3$?\nFinally, consider the game when Alice wins if the maximum degree of the red subgraph is larger than the maximum degree of the blue subgraph. Who wins?\nMalekshahian and Spiro \\cite{MaSp24} have proved that, for the first game, the set of $n$ for which Bob wins has density at least $3/4$ - in fact they prove that if Alice wins at $n$ then Bob wins at $n+1,n+2,n+3$.\nSimilarly, for the third game they prove that the set of $n$ for which Bob wins has density at least $2/3$, and prove the stronger statement that if Alice wins at $n$ then Bob wins at $n+1,n+2$.\nReferences\n\n\n[MaSp24] Malekshahian, A. and Spiro, S., On a clique-building game of Erd\\H{o}s. arXiv:2410.18304 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2301, "problem_number": "EP-782", "title": "Erdős Problem #782", "statement": "Do the squares contain arbitrarily long quasi-progressions? That is, does there exist some constant $C>0$ such that, for any $k$, the squares contain a sequence $x_1,\\ldots,x_k$ where, for some $d$ and all $1\\leq i0$ and let $n$ be large. Let $A\\subseteq \\{2,\\ldots,n\\}$ be such that $(a,b)=1$ for all $a\neq b\\in A$ and $\\sum_{n\\in A}\\frac{1}{n}\\leq C$.\nWhat choice of such an $A$ minimises the number of integers $m\\leq n$ not divisible by any $a\\in A$? Is this minimised by letting $n\\geq q_1>q_2>\\cdots$ be the consecutive primes in decreasing order and choosing $A=\\{q_1,\\ldots,q_k\\}$ where $k$ is maximal such that $ \\sum_{i=1}^k\\frac{1}{q_i}\\leq C? $ \",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2303, "problem_number": "EP-786", "title": "Erdős Problem #786", "statement": "Let $\\epsilon>0$. Is there some set $A\\subset \\mathbb{N}$ of density $>1-\\epsilon$ such that $a_1\\cdots a_r=b_1\\cdots b_s$ with $a_i,b_j\\in A$ can only hold when $r=s$?\nSimilarly, can one always find a set $A\\subset\\{1,\\ldots,N\\}$ with this property of size $\\geq (1-o(1))N$?", "background": "An example of such a set with density $1/4$ is given by the integers $\\equiv 2\\pmod{4}$.\nSelfridge constructed such a set with density $1/e-\\epsilon$ for any $\\epsilon>0$: let $p_1<\\cdotsN^{1/2}$ give an example of a set with size $\\geq (\\log 2)N$. Erd\\H{o}s could improve this constant slightly.\nIn \\cite{Er65} Erd\\H{o}s reports that Ruzsa proved the maximal size of such an $A$ is $\\leq (1-c)N$ for some constant $c>0$ for large $N$, but the proof 'is not yet published'. As far as I know, no such proof was ever published.\nSee also [421] and [795].\nReferences\n\n\n[Er65] Erd\\H{o}s, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2304, "problem_number": "EP-787", "title": "Erdős Problem #787", "statement": "Let $g(n)$ be maximal such that given any set $A\\subset \\mathbb{R}$ with $\\lvert A\\rvert=n$ there exists some $B\\subseteq A$ of size $\\lvert B\\rvert\\geq g(n)$ such that $b_1+b_2\not\\in A$ for all $b_1\neq b_2\\in B$.\nEstimate $g(n)$.", "background": "This function was considered by Erd\\H{o}s and Moser. Choi observed that, without loss of generality, one can assume that $A\\subset \\mathbb{Z}$.\nKlarner proved $g(n) \\gg \\log n$ (indeed, a greedy construction suffices). Choi \\cite{Ch71} proved $g(n) \\ll n^{2/5+o(1)}$. The current best bounds known are $ (\\log n)^{1+c} \\ll g(n) \\ll \\exp(\\sqrt{\\log n}) $ for some constant $c>0$, the lower bound due to Sanders \\cite{Sa21} and the upper bound due to Ruzsa \\cite{Ru05}. Beker \\cite{Be25} has proved $ (\\log n)^{1+\\tfrac{1}{68}+o(1)} \\ll g(n). $ \nReferences\n\n\n[Be25] A. Beker, The Erd\\H{o}s-Moser sum-free set problem via improved bounds for $k$-configurations. arXiv:2501.10203 (2025).\n\n[Ch71] Choi, S. L. G., On a combinatorial problem in number theory. Proc. London Math. Soc. (3) (1971), 629-642.\n\n[Ru05] Ruzsa, Imre Z., Sum-avoiding subsets. Ramanujan J. (2005), 77-82.\n\n[Sa21] Sanders, Tom, The Erd\\H{o}s-Moser sum-free set problem. Canad. J. Math. (2021), 63-107.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2305, "problem_number": "EP-788", "title": "Erdős Problem #788", "statement": "Let $f(n)$ be maximal such that if $B\\subset (2n,4n)\\cap \\mathbb{N}$ there exists some $C\\subset (n,2n)\\cap \\mathbb{N}$ such that $c_1+c_2\not\\in B$ for all $c_1\neq c_2\\in C$ and $\\lvert C\\rvert+\\lvert B\\rvert \\geq f(n)$.\nEstimate $f(n)$. In particular is it true that $f(n)\\leq n^{1/2+o(1)}$?", "background": "A conjecture of Choi \\cite{Ch71}, who proved $f(n) \\ll n^{3/4}$. Adenwalla in the comments has provided a simple construction that proves $f(n) \\gg n^{1/2}$.\nHunter in the comments has sketched an argument that gives $f(n) \\ll n^{2/3+o(1)}$. The bound $ f(n) \\ll (n\\log n)^{2/3} $ was proved by Baltz, Schoen, and Srivastav \\cite{BSS00}.\nReferences\n\n\n[BSS00] Baltz, Andreas and Schoen, Tomasz and Srivastav, Anand, Probabilistic construction of small strongly sum-free sets via\nlarge {S}idon sets. Colloq. Math. (2000), 171--176.\n\n[Ch71] Choi, S. L. G., On a combinatorial problem in number theory. Proc. London Math. Soc. (3) (1971), 629-642.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2306, "problem_number": "EP-789", "title": "Erdős Problem #789", "statement": "Let $h(n)$ be maximal such that if $A\\subseteq \\mathbb{Z}$ with $\\lvert A\\rvert=n$ then there is $B\\subseteq A$ with $\\lvert B\\rvert \\geq h(n)$ such that if $a_1+\\cdots+a_r=b_1+\\cdots+b_s$ with $a_i,b_i\\in B$ then $r=s$.\nEstimate $h(n)$.", "background": "Erd\\H{o}s \\cite{Er62c} proved $h(n) \\ll n^{5/6}$. Straus \\cite{St66} proved $h(n) \\ll n^{1/2}$. Erd\\H{o}s noted the bound $h(n)\\gg n^{1/3}$, taking $ B=\\{ a: \\{ \\alpha a\\} \\in n^{-1/3}+\\tfrac{1}{2} (-n^{-2/3},n^{-2/3})\\} $ for a random $\\alpha\\in [0,1]$. \\cite{Er62c} and Choi \\cite{Ch74b} improved this to $h(n) \\gg (n\\log n)^{1/3}$.\nSee also [186] and [874].\nReferences\n\n\n[Ch74b] Choi, S. L. G., On an extremal problem in number theory. J. Number Theory (1974), 105--111.\n\n[Er62c] Erd\\H{o}s, P\\'{a}l, Some remarks on number theory. {III}. Mat. Lapok (1962), 28--38.\n\n[St66] Straus, E. G., On a problem in combinatorial number theory. J. Math. Sci. (1966), 77--80.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2307, "problem_number": "EP-790", "title": "Erdős Problem #790", "statement": "Let $l(n)$ be maximal such that if $A\\subset\\mathbb{Z}$ with $\\lvert A\\rvert=n$ then there exists a sum-free $B\\subseteq A$ with $\\lvert B\\rvert \\geq l(n)$ - that is, $B$ is such that there are no solutions to $ a_1=a_2+\\cdots+a_r $ with $a_i\\in B$ all distinct.\nEstimate $l(n)$. In particular, is it true that $l(n)n^{-1/2}\\to \\infty$? Is it true that $l(n)< n^{1-c}$ for some $c>0$?", "background": "Erd\\H{o}s observed that $l(n)\\geq (n/2)^{1/2}$, which Choi improved to $l(n)>(1+c)n^{1/2}$ for some $c>0$. Erd\\H{o}s \\cite{Er73} thought he could prove $l(n)=o(n)$ but had 'difficulties in reconstructing [his] proof'. (In \\cite{Er65} he wrote 'by complicated arguments we can show $l(n)=o(n)$'.)\nChoi, Koml\\'{o}s, and Szemer\\'{e}di \\cite{CKS75} proved $ \\left(\\frac{\\log n}{\\log\\log n}n\\right)^{1/2}\\ll l(n) \\ll \\frac{n}{\\log n}. $ They further conjecture that $l(n)\\geq n^{1-o(1)}$.\nSee also [876].\nReferences\n\n\n[CKS75] Choi, S. L. G. and Koml\\'os, J. and Szemer\\'{e}di, E., On sum-free subsequences. Trans. Amer. Math. Soc. (1975), 307--313.\n\n[Er65] Erd\\H{o}s, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.\n\n[Er73] Erd\\H{o}s, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2308, "problem_number": "EP-791", "title": "Erdős Problem #791", "statement": "Let $g(n)$ be minimal such that there exists $A\\subseteq \\{0,\\ldots,n\\}$ of size $g(n)$ with $\\{0,\\ldots,n\\}\\subseteq A+A$. Estimate $g(n)$. In particular is it true that $g(n)\\sim 2n^{1/2}$?", "background": "Such a set is often called a finite additive $2$-basis. A problem of Rohrbach, who proved in \\cite{Ro37} $ (2+c)n \\leq g(n)^2 \\leq 4n $ for some small constant $c>0$. The current best-known bounds are $ (2.181\\cdots+o(1))n\\leq g(n)^2 \\leq (3.458\\cdots+o(1))n. $ The lower bound is due to Yu \\cite{Yu15}, and the upper bound is due to Kohonen \\cite{Ko17}. (The disproof of $g(n)\\sim 2n^{1/2}$ was accomplished by Mrose \\cite{Mr79}, who gave a construction implying $g(n)^2 \\leq \\frac{7}{2}n$.)\nReferences\n\n\n[Ko17] Kohonen, Jukka, An improved lower bound for finite additive 2-bases. J. Number Theory (2017), 518--524.\n\n[Mr79] Mrose, Arnulf, Untere {S}chranken f\"ur die {R}eichweiten von {E}xtremalbasen\nfester {O}rdnung. Abh. Math. Sem. Univ. Hamburg (1979), 118--124.\n\n[Ro37] Rohrbach, Hans, Ein {B}eitrag zur additiven {Z}ahlentheorie. Math. Z. (1937), 1--30.\n\n[Yu15] Yu, Gang, A new upper bound for finite additive {$h$}-bases. J. Number Theory (2015), 95--104.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2309, "problem_number": "EP-792", "title": "Erdős Problem #792", "statement": "Let $f(n)$ be maximal such that in any $A\\subset \\mathbb{Z}$ with $\\lvert A\\rvert=n$ there exists some sum-free subset $B\\subseteq A$ with $\\lvert B\\rvert \\geq f(n)$, so that there are no solutions to $ a+b=c $ with $a,b,c\\in B$. Estimate $f(n)$.", "background": "Erd\\H{o}s \\cite{Er65} gave a simple proof that shows $f(n) \\geq n/3$. Alon and Kleitman \\cite{AlKl90} improved this to $f(n)\\geq \\frac{n+1}{3}$, and Bourgain \\cite{Bo97} further improved this to $\\frac{n+2}{3}$. The best lower bound known is $ f(n)\\geq \\frac{n}{3}+c\\log\\log n $ for some constant $c>0$, due to Bedert \\cite{Be25b}. The best upper bound known is $ f(n) \\leq \\frac{n}{3}+o(n), $ due to Eberhard, Green, and Manners \\cite{EGM14}.\nThis problem is Problem 1 on Green's open problems list.\nReferences\n\n\n[AlKl90] Alon, N. and Kleitman, D. J., Sum-free subsets. (1990), 13--26.\n\n[Be25b] B. Bedert, Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric sums. arXiv:2502.08624 (2025).\n\n[Bo97] Bourgain, Jean, Estimates related to sumfree subsets of sets of integers. Israel J. Math. (1997), 71-92.\n\n[EGM14] Eberhard, Sean and Green, Ben and Manners, Freddie, Sets of integers with no large sum-free subset. Ann. of Math. (2) (2014), 621-652.\n\n[Er65] Erd\\H{o}s, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2310, "problem_number": "EP-793", "title": "Erdős Problem #793", "statement": "Let $F(n)$ be the maximum possible size of a subset $A\\subseteq\\{1,\\ldots,n\\}$ such that $a\nmid bc$ whenever $a,b,c\\in A$ with $a\neq b$ and $a\neq c$. Is there a constant $C$ such that $ F(n)=\\pi(n)+(C+o(1))n^{2/3}(\\log n)^{-2}? $ ", "background": "Erd\\H{o}s \\cite{Er38} proved there exist constants $0g(n)\\geq (\\log n)^2$ is there a graph on $n$ vertices in which every induced subgraph on $g(n)$ vertices contains a clique of size $\\geq \\log n$ and an independent set of size $\\geq \\log n$?\nIn particular, is there such a graph for $g(n)=(\\log n)^3$?", "background": "A problem of Erd\\H{o}s and Hajnal, who thought that there is no such graph for $g(n)=(\\log n)^3$. Alon and Sudakov \\cite{AlSu07} proved that there is no such graph with $ g(n)=\\frac{c}{\\log\\log n}(\\log n)^3 $ for some constant $c>0$.\nAlon, Buci\\'{c}, and Sudakov \\cite{ABS21} construct such a graph with $ g(n)\\leq 2^{2^{(\\log\\log n)^{1/2+o(1)}}}. $ See also [804].\nReferences\n\n\n[ABS21] Alon, Noga and Buci\\'c, Matija and Sudakov, Benny, Large cliques and independent sets all over the place. Proc. Amer. Math. Soc. (2021), 3145-3157.\n\n[AlSu07] Alon, Noga and Sudakov, Benny, On graphs with subgraphs having large independence numbers. J. Graph Theory (2007), 149-157.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2314, "problem_number": "EP-809", "title": "Erdős Problem #809", "statement": "Let $k\\geq 3$ and define $F_k(n)$ to be the minimal $r$ such that there is a graph $G$ on $n$ vertices with $\\lfloor n^2/4\\rfloor+1$ many edges such that the edges can be $r$-coloured so that every subgraph isomorphic to $C_{2k+1}$ has no colour repeating on the edges.\nIs it true that $ F_k(n)\\sim n^2/8? $ ", "background": "A problem of Burr, Erd\\H{o}s, Graham, and S\\'{o}s, who proved that $ F_k(n)\\gg n^2. $ See also [810].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2315, "problem_number": "EP-810", "title": "Erdős Problem #810", "statement": "Does there exist some $\\epsilon>0$ such that, for all sufficiently large $n$, there exists a graph $G$ on $n$ vertices with at least $\\epsilon n^2$ many edges such that the edges can be coloured with $n$ colours so that every $C_4$ receives $4$ distinct colours?", "background": "A problem of Burr, Erd\\H{o}s, Graham, and S\\'{o}s.\nSee also [809].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2316, "problem_number": "EP-811", "title": "Erdős Problem #811", "statement": "Suppose $n\\equiv 1\\pmod{m}$. We say that an edge-colouring of $K_n$ using $m$ colours is balanced if every vertex sees exactly $\\lfloor n/m\\rfloor$ many edges of each colours.\nFor which graphs $G$ is it true that, if $m=e(G)$, for all large $n\\equiv 1\\pmod{m}$, every balanced edge-colouring of $K_n$ with $m$ colours contains a rainbow copy of $G$? (That is, a subgraph isomorphic to $G$ where each edge receives a different colour.)", "background": "In \\cite{Er91} Erd\\H{o}s credits this problem to himself, Pyber, and Tuza. This problem was explored in a paper of Erd\\H{o}s and Tuza \\cite{ErTu93}. In \\cite{Er96} Erd\\H{o}s seems to suggest that this might be true for every graph $G$, and specifically asks specific challenge posed in \\cite{Er91} and \\cite{Er96} is whether, in any balanced edge-colouring of $K_{6n+1}$ by $6$ colours there must exist a rainbow $C_6$ and $K_4$.\nIn general, one can ask for a quantitative version, defining $d_G(n)$ to be minimal (if it exists) such that if $n$ is sufficiently large and the edges of $K_n$ are coloured with $e(G)$ many colours such that the minimum degree of each colour class is $\\geq d_G(n)$ then there is a rainbow copy of $G$. Erd\\H{o}s and Tuza \\cite{ErTu93} proved that $ \\lfloor n/6\\rfloor \\leq d_{C_4}(n) \\leq \\left(\\frac{1}{4}-c\\right)n $ for some constant $c>0$.\nAxenovich and Clemen \\cite{AxCl24} have proved that there exist infinitely many graphs without this property. In particular, they show that for any odd $\\ell \\geq 3$ and $m=\\lfloor \\sqrt{\\ell}+3.5\\rfloor$ there exist arbitrarily large $n$ such that $K_n$ has a balanced edge-colouring using $\\ell$ colours which contains no rainbow $K_m$. They conjecture that $K_m$ lacks this property for all $m\\geq 4$.\nClemen and Wagner \\cite{ClWa23} proved that $K_4$ does lack this property.\nReferences\n\n\n[AxCl24] Axenovich, Maria and Clemen, Felix C., Rainbow subgraphs in edge-colored complete graphs: answering\ntwo questions by {E}rd\\H{o}s and {T}uza. J. Graph Theory (2024), 57--66.\n\n[ClWa23] Clemen, Felix Christian and Wagner, Adam Zsolt, Balanced edge-colorings avoiding rainbow cliques of size four. Electron. J. Combin. (2023), Paper No. 3.17, 3.\n\n[Er91] Erd\"{o}s, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.\n\n[Er96] Erd\\H{o}s, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9.\n\n[ErTu93] Erd\\H{o}s, Paul and Tuza, Zsolt, Rainbow subgraphs in edge-colorings of complete graphs. (1993), 81--88.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2317, "problem_number": "EP-812", "title": "Erdős Problem #812", "statement": "Is it true that $ \\frac{R(n+1)}{R(n)}\\geq 1+c $ for some constant $c>0$, for all large $n$? Is it true that $ R(n+1)-R(n) \\gg n^2? $ ", "background": "Burr, Erd\\H{o}s, Faudree, and Schelp \\cite{BEFS89} proved that $ R(n+1)-R(n) \\geq 4n-8 $ for all $n\\geq 2$. The lower bound of [165] implies that $ R(n+2)-R(n) \\gg n^{2-o(1)}. $ \nReferences\n\n\n[BEFS89] Burr, S. A. and Erd\\H{o}s, P. and Faudree, R. J. and Schelp, R.\nH., On the difference between consecutive {R}amsey numbers. Utilitas Math. (1989), 115--118.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2318, "problem_number": "EP-813", "title": "Erdős Problem #813", "statement": "Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle (a copy of $K_3$) must contain a clique on at least $h(n)$ vertices. Estimate $h(n)$ - in particular, do there exist constants $c_1,c_2>0$ such that $ n^{1/3+c_1}\\ll h(n) \\ll n^{1/2-c_2}? $ ", "background": "A problem of Erd\\H{o}s and Hajnal, who could prove that $ n^{1/3}\\ll h(n) \\ll n^{1/2}. $ Buci\\'{c} and Sudakov \\cite{BuSu23} have proved $ h(n) \\gg n^{5/12-o(1)}. $ \nReferences\n\n\n[BuSu23] M. Buci\\'C and B. Sudakov, Large independent sets from local considerations. arXiv:2007.03667 (2023).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2319, "problem_number": "EP-817", "title": "Erdős Problem #817", "statement": "Let $k\\geq 3$ and define $g_k(n)$ to be the minimal $N$ such that $\\{1,\\ldots,N\\}$ contains some $A$ of size $\\lvert A\\rvert=n$ such that $ \\langle A\\rangle = \\left\\{\\sum_{a\\in A}\\epsilon_aa: \\epsilon_a\\in \\{0,1\\}\\right\\} $ contains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In particular, is it true that $ g_3(n) \\gg 3^n? $ ", "background": "A problem of Erd\\H{o}s and S\\'{a}rk\"{o}zy who proved $ g_3(n) \\gg \\frac{3^n}{n^{O(1)}}. $ \",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2320, "problem_number": "EP-819", "title": "Erdős Problem #819", "statement": "Let $f(N)$ be maximal such that there exists $A\\subseteq \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert=\\lfloor N^{1/2}\\rfloor$ such that $\\lvert (A+A)\\cap [1,N]\\rvert=f(N)$. Estimate $f(N)$.", "background": "Erd\\H{o}s and Freud \\cite{ErFr91} proved $ \\left(\\frac{3}{8}-o(1)\\right)N \\leq f(N) \\leq \\left(\\frac{1}{2}+o(1)\\right)N, $ and note that it is closely connected to the size of the largest quasi-Sidon set (see [840]).\nReferences\n\n\n[ErFr91] Erd\\H{o}s, P. and Freud, R., On sums of a {S}idon-sequence. J. Number Theory (1991), 196--205.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2321, "problem_number": "EP-820", "title": "Erdős Problem #820", "statement": "Let $H(n)$ be the smallest integer $l$ such that there exist $k0$ such that, for all $\\epsilon>0$, $ H(n) > \\exp(n^{(c-\\epsilon)/\\log\\log n}) $ for infinitely many $n$ and $ H(n) < \\exp(n^{(c+\\epsilon)/\\log\\log n}) $ for all large enough $n$?\nDoes a similar upper bound hold for the smallest $k$ such that $(k^n-1,2^n-1)=1$?", "background": "Erd\\H{o}s \\cite{Er74b} proved that there exists a constant $c>0$ such that $ H(n) > \\exp(n^{c/(\\log\\log n)^2}) $ for infinitely many $n$.\nvan Doorn in the comments sketches a proof of the lower bound: that there exists some constant $c>0$ and infinitely many $n$ such that $ H(n) > \\exp(n^{c/\\log\\log n}). $ The sequence $H(n)$ for $1\\leq n\\leq 10$ is $ 3,3,3,6,3,18,3,6,3,12. $ The sequence of $n$ for which $(2^n-1,3^n-1)=1$ is A263647 in the OEIS.\nSee also [770].\nReferences\n\n\n[Er74b] Erd\\H{o}s, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2322, "problem_number": "EP-821", "title": "Erdős Problem #821", "statement": "Let $g(n)$ count the number of $m$ such that $\\phi(m)=n$. Is it true that, for every $\\epsilon>0$, there exist infinitely many $n$ such that $ g(n) > n^{1-\\epsilon}? $ ", "background": "Pillai proved that $\\limsup g(n)=\\infty$ and Erd\\H{o}s \\cite{Er35b} proved that there exists some constant $c>0$ such that $g(n) >n^c$ for infinitely many $n$.\nThis conjecture would follow if we knew that, for every $\\epsilon>0$, there are $\\gg_\\epsilon \\frac{x}{\\log x}$ many primes $p n^{0.71568\\cdots}, $ obtained by Lichtman \\cite{Li22} as a consequence of proving that there are $\\geq \\frac{x}{(\\log x)^{O(1)}}$ many primes $p\\leq x$ such that all prime factors of $p-1$ are $\\leq x^{0.2843\\cdots}$ (which improves a number of previous exponents, most recently Baker and Harman \\cite{BaHa98}).\nThe average size of $g(n)$ was investigated by Luca and Pollack \\cite{LuPo11}.\nSee also [416].\nReferences\n\n\n[BaHa98] Baker, R. C. and Harman, G., Shifted primes without large prime factors. Acta Arith. (1998), 331--361.\n\n[Er35b] Erd\\H{o}s, P., On the normal number of prime factors of $p-1$ and some related problems concerning Euler's $\\varphi$-function. Quart. J. Math. (1935), 205-213.\n\n[Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022).\n\n[LuPo11] Luca, Florian and Pollack, Paul, An arithmetic function arising from {C}armichael's conjecture. J. Th\\'{e}or. Nombres Bordeaux (2011), 697--714.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2323, "problem_number": "EP-824", "title": "Erdős Problem #824", "statement": "Let $h(x)$ count the number of integers $1\\leq ax^{2-o(1)}$?", "background": "Erd\\H{o}s \\cite{Er74b} proved that $\\limsup h(x)/x= \\infty$, and claimed a similar proof for this problem. A complete proof that $h(x)/x\\to \\infty$ was provided by Pollack and Pomerance \\cite{PoPo16}.\nA similar question can be asked if we replace the condition $(a,b)=1$ with the condition that $a$ and $b$ are squarefree. Weisenberg suggests another variant, with the condition that there are no proper factors $u\\mid a$ and $v\\mid b$ such that $\\sigma(u)=\\sigma(v)$ and $(u,a/u)=(v,b/v)=1$, which is the weakest restriction one can impose that is still strong enough to eliminate trivial duplicates.\nReferences\n\n\n[Er74b] Erd\\H{o}s, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.\n\n[PoPo16] Pollack, Paul and Pomerance, Carl, Some problems of Erd\\H{o}s on the sum-of-divisors function. Trans. Amer. Math. Soc. Ser. B (2016), 1-26.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2324, "problem_number": "EP-825", "title": "Erdős Problem #825", "statement": "Is there an absolute constant $C>0$ such that every integer $n$ with $\\sigma(n)>Cn$ is the distinct sum of proper divisors of $n$?", "background": "A problem of Benkoski and Erd\\H{o}s. In other words, this problem asks for an upper bound for the abundancy index of weird numbers. This could be true with $C=3$. We must have $C>2$ since $\\sigma(70)=144$ but $70$ is not the distinct sum of integers from $\\{1,2,5,7,10,14,35\\}$.\nErd\\H{o}s suggested that as $C\\to \\infty$ only divisors at most $\\epsilon n$ need to be used, where $\\epsilon \\to 0$.\nWeisenberg has observed that if $n$ is a weird number with an abundancy index $\\geq 4$ then it is divisible by an odd weird number. In particular, if there are no odd weird numbers (see [470]) then every weird number has abundancy index $<4$. Indeed, if $l(n)$ is the abundancy index and $n=2^km$ with $m$ odd then $l(n)=l(2^k)l(m)$, and $l(2^k)<2$ so if $l(n)\\geq 4$ then $l(m)>2$, and hence $m$ is weird (as a factor of a weird number).\nA similar argument shows that either there are infinitely many primitive weird numbers or there is an upper bound for the abundancy index of all weird numbers.\nSee also [18] and [470].\nThis is part of problem B2 in Guy's collection \\cite{Gu04} (the \\$25 is reported by Guy as offered by Erd\\H{o}s for a solution to this question).\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2325, "problem_number": "EP-826", "title": "Erdős Problem #826", "statement": "Are there infinitely many $n$ such that, for all $k\\geq 1$, $ \\tau(n+k)\\ll k? $ ", "background": "A stronger form of [248].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2326, "problem_number": "EP-827", "title": "Erdős Problem #827", "statement": "Let $n_k$ be minimal such that if $n_k$ points in $\\mathbb{R}^2$ are in general position then there exists a subset of $k$ points such that all $\\binom{k}{3}$ triples determine circles of different radii.\nDetermine $n_k$.", "background": "In \\cite{Er75h} Erd\\H{o}s asks whether $n_k$ exists. In \\cite{Er78c} he gave a simple argument which proves that it does, and in fact $ n_k \\leq k+2\\binom{k-1}{2}\\binom{k-1}{3}, $ but this argument is incorrect, as explained by Martinez and Rold\\'{a}n-Pensado \\cite{MaRo15}.\nMartinez and Rold\\'{a}n-Pensado give a corrected argument that proves $n_k\\ll k^9$.\nReferences\n\n\n[Er75h] Erd\\H{o}s, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3.\n\n[Er78c] Erd\\H{o}s, P., Some more problems on elementary geometry. Austral. Math. Soc. Gaz. (1978), 52-54.\n\n[MaRo15] Mart\\'{I}nez, L. and Rold\\'an-Pensado, E., Points defining triangles with distinct circumradii. Acta Math. Hungar. (2015), 136--141.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2327, "problem_number": "EP-828", "title": "Erdős Problem #828", "statement": "Is it true that, for any $a\\in\\mathbb{Z}$, there are infinitely many $n$ such that $ \\phi(n) \\mid n+a? $ ", "background": "A conjecture of Graham. Lehmer has conjectured that $\\phi(n)\\mid n-1$ if and only if $n$ is prime. It is an easy exercise to show that $\\phi(n) \\mid n$ if and only if $n=2^a3^b$.\nThis is discussed in problem B37 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2328, "problem_number": "EP-829", "title": "Erdős Problem #829", "statement": "Let $A\\subset\\mathbb{N}$ be the set of cubes. Is it true that $ 1_A\\ast 1_A(n) \\ll (\\log n)^{O(1)}? $ ", "background": "Mordell proved that $ \\limsup_{n\\to \\infty} 1_A\\ast 1_A(n)=\\infty $ and Mahler \\cite{Ma35b} proved $ 1_A\\ast 1_A(n) \\gg (\\log n)^{1/4} $ for infinitely many $n$. Stewart \\cite{St08} improved this to $ 1_A\\ast 1_A(n) \\gg (\\log n)^{11/13}. $ \nReferences\n\n\n[Ma35b] Mahler, Kurt, On the Lattice Points on Curves of Genus 1. Proc. London Math. Soc. (2) (1935), 431-466.\n\n[St08] Stewart, Cameron L., Cubic {T}hue equations with many solutions. Int. Math. Res. Not. IMRN (2008), Art. ID rnn040, 11.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2329, "problem_number": "EP-830", "title": "Erdős Problem #830", "statement": "We say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$. Are there infinitely many amicable pairs? If $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$ then is it true that $ A(x)>x^{1-o(1)}? $ ", "background": "For example $220$ and $284$. Erd\\H{o}s \\cite{Er55b} proved that $A(x)=o(x)$, and Pomerance \\cite{Po81} improved this to $ A(x) \\leq x \\exp(-(\\log x)^{1/3}) $ and later \\cite{Po15} to $ A(x) \\leq x \\exp(-(\\tfrac{1}{2}+o(1))(\\log x\\log\\log x)^{1/2}). $ This is problem B4 in Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er55b] Erd\"{o}s, P., On amicable numbers. Publ. Math. Debrecen (1955), 108-111.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Po15] Pomerance, Carl, On amicable numbers. (2015), 321-327.\n\n[Po81] Pomerance, Carl, On the distribution of amicable numbers. {II}. J. Reine Angew. Math. (1981), 183-188.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2330, "problem_number": "EP-831", "title": "Erdős Problem #831", "statement": "Let $h(n)$ be maximal such that in any $n$ points in $\\mathbb{R}^2$ (with no three on a line and no four on a circle) there are at least $h(n)$ many circles of different radii passing through three points. Estimate $h(n)$.\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2331, "problem_number": "EP-836", "title": "Erdős Problem #836", "statement": "Let $r\\geq 2$ and $G$ be a $r$-uniform hypergraph with chromatic number $3$ (that is, there is a $3$-colouring of the vertices of $G$ such that no edge is monochromatic).\nSuppose any two edges of $G$ have a non-empty intersection. Must $G$ contain $O(r^2)$ many vertices? Must there be two edges which meet in $\\gg r$ many vertices?", "background": "A problem of Erd\\H{o}s and Shelah. The Fano geometry gives an example where there are no two edges which meet in $r-1$ vertices. Are there any other examples?\nErd\\H{o}s and Lov\\'{a}sz \\cite{ErLo75} proved that there must be two edges which meet in $\\gg \\frac{r}{\\log r}$ many vertices.\nAlon has provided the following counterexample to the first question: as vertices take two sets $X$ and $Y$ of sizes $2r-2$ and $\\frac{1}{2}\\binom{2r-2}{r-1}$ respectively, where $Y$ corresponds to all partitions of $X$ into two equal parts. The edges are all subsets of $X$ of size $r$, and also all sets consisting of a subset of $X$ of size $r-1$ together with the unique element of $Y$ corresponding to the induced partition of $X$.\nThis hypergraph is intersecting, its chromatic number is $3$, and it has $\\asymp 4^r/\\sqrt{r}$ many vertices.\nReferences\n\n\n[ErLo75] Erd\\H{o}s, P. and Lov\\'{a}sz, L., Problems and results on {$3$}-chromatic hypergraphs and some\nrelated questions. (1975), 609--627.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2332, "problem_number": "EP-837", "title": "Erdős Problem #837", "statement": "Let $k\\geq 2$ and $A_k\\subseteq [0,1]$ be the set of $\\alpha$ such that there exists some $\\beta(\\alpha)>\\alpha$ with the property that, if $G_1,G_2,\\ldots$ is a sequence of $k$-uniform hypergraphs with $ \\liminf \\frac{e(G_n)}{\\binom{\\lvert G_n\\rvert}{k}} >\\alpha $ then there exist subgraphs $H_n\\subseteq G_n$ such that $\\lvert H_n\\rvert \\to \\infty$ and $ \\liminf \\frac{e(H_n)}{\\binom{\\lvert H_n\\rvert}{k}} >\\beta, $ and further that this property does not necessarily hold if $>\\alpha$ is replaced by $\\geq \\alpha$.\nWhat is $A_3$?", "background": "A problem of Erd\\H{o}s and Simonovits. It is known that $ A_2 = \\left\\{ 1-\\frac{1}{k} : k\\geq 1\\right\\}. $ \n\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2333, "problem_number": "EP-838", "title": "Erdős Problem #838", "statement": "Let $f(n)$ be maximal such that any $n$ points in $\\mathbb{R}^2$, with no three on a line, determine at least $f(n)$ different convex subsets. Estimate $f(n)$ - in particular, does there exist a constant $c$ such that $ \\lim \\frac{\\log f(n)}{(\\log n)^2}=c? $ ", "background": "A question of Erd\\H{o}s and Hammer. Erd\\H{o}s proved in \\cite{Er78c} that there exist constants $c_1,c_2>0$ such that $ n^{c_1\\log n}1/2$. In fact this is false - Freud \\cite{Fr93} constructed a sequence with upper density $19/36$.\nSee also [359] and [867].\nReferences\n\n\n[Fr93] R. Freud, Adding numbers - on a problem of P. Erd\\H{o}s. James Cook Mathematical Notes (1993), 6199-6202.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2335, "problem_number": "EP-840", "title": "Erdős Problem #840", "statement": "Let $f(N)$ be the size of the largest quasi-Sidon subset $A\\subset\\{1,\\ldots,N\\}$, where we say that $A$ is quasi-Sidon if $ \\lvert A+A\\rvert=(1+o(1))\\binom{\\lvert A\\rvert}{2}. $ How does $f(N)$ grow?", "background": "Considered by Erd\\H{o}s and Freud \\cite{ErFr91}, who proved $ \\left(\\frac{2}{\\sqrt{3}}+o(1)\\right)N^{1/2} \\leq f(N) \\leq \\left(2+o(1)\\right)N^{1/2}. $ (Although both bounds were already given by Erd\\H{o}s in \\cite{Er81h}.) Note that $2/\\sqrt{3}=1.15\\cdots$. The lower bound is taking a genuine Sidon set $B\\subset [1,N/3]$ of size $\\sim N^{1/2}/\\sqrt{3}$ and taking the union with $\\{N-b : b\\in B\\}$. The upper bound was improved by Pikhurko \\cite{Pi06} to $ f(N) \\leq \\left(\\left(\\frac{1}{4}+\\frac{1}{(\\pi+2)^2}\\right)^{-1/2}+o(1)\\right)N^{1/2} $ (the constant here is $=1.863\\cdots$).\nThe analogous question with $A-A$ in place of $A+A$ is simpler, and there the maximal size is $\\sim N^{1/2}$, as proved by Cilleruelo.\nSee also [30], [819], and [864].\nReferences\n\n\n[Er81h] Erd\\H{o}s, P., Some problems and results on additive and multiplicative\nnumber theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.\n\n[ErFr91] Erd\\H{o}s, P. and Freud, R., On sums of a {S}idon-sequence. J. Number Theory (1991), 196--205.\n\n[Pi06] Pikhurko, Oleg, Dense edge-magic graphs and thin additive bases. Discrete Math. (2006), 2097--2107.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2336, "problem_number": "EP-846", "title": "Erdős Problem #846", "statement": "Let $A\\subset \\mathbb{R}^2$ be an infinite set for which there exists some $\\epsilon>0$ such that in any subset of $A$ of size $n$ there are always at least $\\epsilon n$ with no three on a line.\nIs it true that $A$ is the union of a finite number of sets where no three are on a line?", "background": "A problem of Erd\\H{o}s, Ne\\v{s}et\\v{r}il, and R\"{o}dl.\nSee also [774] and [847].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2337, "problem_number": "EP-847", "title": "Erdős Problem #847", "statement": "Let $A\\subset \\mathbb{N}$ be an infinite set for which there exists some $\\epsilon>0$ such that in any subset of $A$ of size $n$ there is a subset of size at least $\\epsilon n$ which contains no three-term arithmetic progression.\nIs it true that $A$ is the union of a finite number of sets which contain no three-term arithmetic progression?", "background": "A problem of Erd\\H{o}s, Ne\\v{s}et\\v{r}il, and R\"{o}dl.\nSee also [774] and [846].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2338, "problem_number": "EP-849", "title": "Erdős Problem #849", "statement": "Is it true that, for every integer $t\\geq 1$, there is some integer $a$ such that $ \\binom{n}{k}=a $ (with $1\\leq k\\leq n/2$) has exactly $t$ solutions?", "background": "Erd\\H{o}s \\cite{Er96b} credits this to himself and Gordon 'many years ago', but it is more commonly known as Singmaster's conjecture. For $t=3$ one could take $a=120$, and for $t=4$ one could take $a=3003$. There are no known examples for $t\\geq 5$.\nBoth Erd\\H{o}s and Singmaster believed the answer to this question is no, and in fact that there exists an absolute upper bound on the number of solutions.\nMatomäki, Radziwill, Shao, Tao, and Teräväinen \\cite{MRSTT22} have proved that there are always at most two solutions if we restrict $k$ to $ k\\geq \\exp((\\log n)^{2/3+\\epsilon}), $ assuming $a$ is sufficiently large depending on $\\epsilon>0$.\nReferences\n\n\n[Er96b] Erd\"{o}s, Paul, Some problems I presented or planned to present in my short\ntalk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335.\n\n[MRSTT22] Matom\"{a}ki, Kaisa and Radziwi\\l\\l, Maksym and Shao, Xuancheng\nand Tao, Terence and Ter\"{a}v\"{a}inen, Joni, Singmaster's conjecture in the interior of {P}ascal's\ntriangle. Q. J. Math. (2022), 1137--1177.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2339, "problem_number": "EP-850", "title": "Erdős Problem #850", "statement": "Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?", "background": "This is sometimes known as the Erd\\H{o}s-Woods conjecture.\nFor just $x,y$ and $x+1,y+1$ one can take $ x=2(2^r-1) $ and $ y = x(x+2). $ Erd\\H{o}s also asked whether there are any other examples. Makowski \\cite{Ma68} observed that $x=75$ and $y=1215$ is another example, since $ 75 = 3\\cdot 5^2 \\textrm{ and }1215 = 3^5\\cdot 5 $ while $ 76 = 2^2\\cdot 19\\textrm{ and }1216 = 2^6\\cdot 19. $ (This example was also found independently by Matthew Bolan, and by Dubickas, who posed it as part of the 2024 team selection test in Lithuania.) No other examples are known. This sequence is listed as A343101 at the OEIS.\nShorey and Tijdeman \\cite{ShTi16} have shown that, assuming a strong form of the ABC conjecture due to Baker, then the answer to the original problem is no.\nSee also [677].\nThe case of $x,y$ and $x+1,y+1$ appeared as Problem 1 in the Third Benelux Mathematical Olympiad 2011.\nThis problem is discussed in problem B19 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ma68] Makowski, Andrzej, On a problem of {E}rd\\H{o}s. Enseign. Math. (2) (1968), 193.\n\n[ShTi16] Shorey, Tarlok N. and Tijdeman, Rob, Arithmetic properties of blocks of consecutive integers. (2016), 455--471.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2340, "problem_number": "EP-851", "title": "Erdős Problem #851", "statement": "Let $\\epsilon>0$. Is there some $r\\ll_\\epsilon 1$ such that the density of integers of the form $2^k+n$, where $k\\geq 0$ and $n$ has at most $r$ prime divisors, is at least $1-\\epsilon$?", "background": "Romanoff \\cite{Ro34} proved that the set of integers of the form $2^k+p$ (where $p$ is prime) has positive lower density.\nSee also [205].\nReferences\n\n\n[Ro34] Romanoff, N. P., \"{U}ber einige S\"Atze der additiven Zahlentheorie. Math. Ann. (1934), 668-678.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2341, "problem_number": "EP-852", "title": "Erdős Problem #852", "statement": "Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be maximal such that for some $n(\\log x)^c $ for some constant $c>0$, and $ h(x)=o(\\log x)? $ ", "background": "Brun's sieve implies $h(x) \\to \\infty$ as $x\\to \\infty$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2342, "problem_number": "EP-853", "title": "Erdős Problem #853", "statement": "Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n=t$ has no solutions for $n\\leq x$.\nIs it true that $r(x)\\to \\infty$? Or even $r(x)/\\log x \\to \\infty$?", "background": "In \\cite{Er85c} Erd\\H{o}s omits the condition that $t$ be even, but this is clearly necessary.\nReferences\n\n\n[Er85c] Erd\\H{o}s, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2343, "problem_number": "EP-854", "title": "Erdős Problem #854", "statement": "Let $n_k$ denote the $k$th primorial, i.e. the product of the first $k$ primes.\nIf $1=a_1a$. Estimate the maximum of $ \\frac{1}{\\log N}\\sum_{n\\in A}\\frac{1}{n}. $ ", "background": "Alexander \\cite{Al66} and Erd\\H{o}s, S\\'{a}rk\"{o}zi, and Szemer\\'{e}di \\cite{ESS68} proved that this maximum is $o(1)$ (as $N\\to \\infty$). This condition on $A$ is a weaker form of the usual primitive condition. If $A$ is primitive then Behrend \\cite{Be35} proved $ \\frac{1}{\\log N}\\sum_{n\\in A}\\frac{1}{n}\\ll \\frac{1}{\\sqrt{\\log\\log N}}. $ An example of such a set $A$ is the set of all integers in $[N^{1/2},N]$ divisible by some prime $>N^{1/2}$.\nSee also [143].\nReferences\n\n\n[Al66] Alexander, Ralph, Density and multiplicative structure of sets of integers. Acta Arith. (1966/67), 321--332.\n\n[Be35] Behrend, F., On sequences of numbers not divisible by another. London Math. Soc. Journal (1935), 42-45.\n\n[ESS68] Erd\\H{o}s, P. and S\\'{a}rk\"ozi, A. and Szemer\\'{e}di, E., On the solvability of certain equations in sequences of\npositive upper logarithmic density. J. London Math. Soc. (1968), 71--78.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2347, "problem_number": "EP-859", "title": "Erdős Problem #859", "statement": "Let $t\\geq 1$ and let $d_t$ be the density of the set of integers $n\\in\\mathbb{N}$ for which $t$ can be represented as the sum of distinct divisors of $n$.\nDo there exist constants $c_1,c_2>0$ such that $ d_t \\sim \\frac{c_1}{(\\log t)^{c_2}} $ as $t\\to \\infty$?", "background": "Erd\\H{o}s \\cite{Er70} proved that $d_t$ always exists, and that there exist some constants $c_3,c_4>0$ such that $ \\frac{1}{(\\log t)^{c_3}} < d_t < \\frac{1}{(\\log t)^{c_4}}. $ \nReferences\n\n\n[Er70] Erd\\H{o}s, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2348, "problem_number": "EP-860", "title": "Erdős Problem #860", "statement": "Let $h(n)$ be such that, for any $m\\geq 1$, in the interval $(m,m+h(n))$ there exist distinct integers $a_i$ for $1\\leq i\\leq \\pi(n)$ such that $p_i\\mid a_i$, where $p_i$ denotes the $i$th prime.\nEstimate $h(n)$.", "background": "A problem of Erd\\H{o}s and Pomerance \\cite{ErPo80}, who proved that $ h(n) \\ll \\frac{n^{3/2}}{(\\log n)^{1/2}}. $ Erd\\H{o}s and Selfridge proved $h(n)>(3-o(1))n$, and Ruzsa proved $h(n)/n\\to \\infty$.\nThis is discussed in problem B32 of Guy's collection \\cite{Gu04}.\nSee also [375].\nReferences\n\n\n[ErPo80] P. Erd\\H{o}s and C. Pomerance, Matching the natural numbers up to $n$ with distinct multiples of another interval. Indigationes Math. (1980), 147-151.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2349, "problem_number": "EP-863", "title": "Erdős Problem #863", "statement": "Let $r\\geq 2$ and let $A\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\\leq b$ for any $n$. (That is, $A$ is a $B_2[r]$ set.)\nSimilarly, let $B\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a-b$ for any $n$.\nIf $\\lvert A\\rvert\\sim c_rN^{1/2}$ as $N\\to \\infty$ and $\\lvert B\\rvert \\sim c_r'N^{1/2}$ as $N\\to \\infty$ then is it true that $c_r\neq c_r'$ for $r\\geq 2$? Is it true that $c_r'0$ such that, for all large $N$, if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $\\frac{5}{8}N+C$ then there are distinct $a,b,c\\in A$ such that $a+b,a+c,b+c\\in A$.", "background": "A problem of Erd\\H{o}s and S\\'{o}s (also earlier considered by Choi, Erd\\H{o}s, and Szemer\\'{e}di \\cite{CES75}, but Erd\\H{o}s had forgotten this). Taking all integers in $[N/8,N/4]$ and $[N/2,N]$ shows that $\\frac{5}{8}$ would be best possible here.\nIt is a classical folklore fact that if $A\\subseteq \\{1,\\ldots,2N\\}$ has size $\\geq N+2$ then there are distinct $a,b\\in A$ such that $a+b\\in A$, which establishes the $k=2$ case.\nIn general, one can define $f_k(N)$ to be minimal such that if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $f_k(N)$ then there are $k$ distinct $a_i\\in A$ such that all $\\binom{k}{2}$ pairwise sums are elements of $A$. Erd\\H{o}s and S\\'{o}s conjectured that $ f_k(N)\\sim \\frac{1}{2}\\left(1+\\sum_{1\\leq r\\leq k-2}\\frac{1}{4^r}\\right) N, $ and a similar example shows that this would be best possible.\nChoi, Erd\\H{o}s, and Szemer\\'{e}di \\cite{CES75} have proved that, for all $k\\geq 3$, there exists $\\epsilon_k>0$ such that (for large enough $N$) $ f_k(N)\\leq \\left(\\frac{2}{3}-\\epsilon_k\\right)N. $ \nReferences\n\n\n[CES75] Choi, S. L. G. and Erd\\H{o}s, P. and Szemer\\'{e}di, E., Some additive and multiplicative problems in number theory. Acta Arith. (1975), 37--50.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2352, "problem_number": "EP-866", "title": "Erdős Problem #866", "statement": "Let $k\\geq 3$ and $g_k(N)$ be minimal such that if $A\\subseteq \\{1,\\ldots,2N\\}$ has $\\lvert A\\rvert \\geq N+g_k(N)$ then there exist integers $b_1,\\ldots,b_k$ such that all $\\binom{k}{2}$ pairwise sums are in $A$ (but the $b_i$ themselves need not be in $A$).\nEstimate $g_k(N)$.", "background": "A problem of Choi, Erd\\H{o}s, and Szemer\\'{e}di. It is clear that, for the set of odd numbers in $\\{1,\\ldots,2N\\}$, no such $b_i$ exist, whence $g_k(N)\\geq 0$ always. Choi, Erd\\H{o}s, and Szemer\\'{e}di proved that $g_3(N)=2$ and $g_4(N) \\ll 1$. van Doorn has shown that $g_4(N)\\leq 2032$.\nChoi, Erd\\H{o}s, and Szemer\\'{e}di also proved that $ g_5(N)\\asymp \\log N $ and $ g_6(N)\\asymp N^{1/2}. $ In general they proved that $ g_k(N) \\ll_k N^{1-2^{-k}} $ and for every $\\epsilon>0$ if $k$ is sufficiently large then $ g_k(N) > N^{1-\\epsilon}. $ As an example, taking $A$ to be the set of all odd integers and the powers of $2$ shows that $g_5(N)\\gg \\log N$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2353, "problem_number": "EP-869", "title": "Erdős Problem #869", "statement": "If $A_1,A_2$ are disjoint additive bases of order $2$ (i.e. $A_i+A_i$ contains all large integers) then must $A=A_1\\cup A_2$ contain a minimal additive basis of order $2$ (one such that deleting any element creates infinitely many $n\not\\in A+A$)?", "background": "A question of Erd\\H{o}s and Nathanson \\cite{ErNa88}.\nHärtter \\cite{Ha56} and Nathanson \\cite{Na74} proved that there exist additive bases which do not contain any minimal additive bases.\nReferences\n\n\n[ErNa88] Erd\\H{o}s, Paul and Nathanson, Melvyn B., Partitions of bases into disjoint unions of bases. J. Number Theory (1988), 1--9.\n\n[Ha56] H\"{a}rtter, Erich, Ein Beitrag zur {T}heorie der {M}inimalbasen. J. Reine Angew. Math. (1956), 170--204.\n\n[Na74] Nathanson, Melvyn B., Minimal bases and maximal nonbases in additive number theory. J. Number Theory (1974), 324--333.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2354, "problem_number": "EP-870", "title": "Erdős Problem #870", "statement": "Let $k\\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\\geq c\\log n$ for all large $n$ then $A$ must contain a minimal basis of order $k$? (Here $r(n)$ counts the number of representations of $n$ as the sum of at most $k$ elements from $A$.)", "background": "A question of Erd\\H{o}s and Nathanson \\cite{ErNa79}, who proved that this is true for $k=2$ if $1_A\\ast 1_A(n) > (\\log \\frac{4}{3})^{-1}\\log n$ for all large $n$.\nHärtter \\cite{Ha56} and Nathanson \\cite{Na74} proved that there exist additive bases which do not contain any minimal additive bases.\nSee also [868].\nReferences\n\n\n[ErNa79] Erd\\H{o}s, Paul and Nathanson, Melvyn B., Systems of distinct representatives and minimal bases in\nadditive number theory. (1979), 89--107.\n\n[Ha56] H\"{a}rtter, Erich, Ein Beitrag zur {T}heorie der {M}inimalbasen. J. Reine Angew. Math. (1956), 170--204.\n\n[Na74] Nathanson, Melvyn B., Minimal bases and maximal nonbases in additive number theory. J. Number Theory (1974), 324--333.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2355, "problem_number": "EP-872", "title": "Erdős Problem #872", "statement": "Consider the two-player game in which players alternately choose integers from $\\{2,3,\\ldots,n\\}$ to be included in some set $A$ (the same set for both players) such that no $a\\mid b$ for $a\neq b\\in A$.\nThe game ends when no legal move is possible. One player wants the game to last as long as possible, the other wants the game to end quickly. How long can the game be guaranteed to last for?\nAt least $\\epsilon n$ moves? (For $\\epsilon>0$ and $n$ sufficiently large.) At least $(1-\\epsilon)\\frac{n}{2}$ moves?", "background": "A number theoretic variant of a combinatorial game of Hajnal, in which players alternately add edges to a graph while keeping it triangle-free. This game must trivially end in at most $n^2/4$ moves, and F\"{u}redi and Seress \\cite{FuSe91} proved that it can be guaranteed to last for $\\gg n\\log n$ moves. Bir\\'{o}, Horn, and Wildstrom \\cite{BPW16} proved that it must end in at most $(\\frac{26}{121}+o(1))n^2$ moves.\nThis type of game is known as a saturation game.\nErd\\H{o}s does not specify which player goes first, which may result in different answers.\nReferences\n\n\n[BPW16] Bir\\'{o}, Csaba and Horn, Paul and Wildstrom, D. Jacob, An upper bound on the extremal version of Hajnal's\ntriangle-free game. Discrete Appl. Math. (2016), 20--28.\n\n[FuSe91] F\"{u}redi, Zolt\\'{a}n and Reimer, Dave and Seress, \\'{A}kos, Hajnal's triangle-free game and extremal graph problems. Congr. Numer. (1991), 123--128.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2356, "problem_number": "EP-873", "title": "Erdős Problem #873", "statement": "Let $A=\\{a_10$, there exists some $k$ such that $ F(A,X,k)H(n)-n^{1+o(1)}? $ Is it true that, for every $k\\geq 2$, if $n$ is sufficiently large then the admissible set which maximises $G(n)$ contains at least one integer with at least $k$ prime factors?", "background": "Erd\\H{o}s and Van Lint proved that $ H(n)-n^{3/2-o(1)}H(n)-n^{1+o(1)}$ assuming 'plausible (but hopeless) assumptions about the distribution of primes'. They also prove the second claim when $k=2$.\nSee also [878].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2361, "problem_number": "EP-881", "title": "Erdős Problem #881", "statement": "Let $A\\subset\\mathbb{N}$ be an additive basis of order $k$ which is minimal, in the sense that if $B\\subset A$ is any infinite set then $A\\backslash B$ is not a basis of order $k$.\nMust there exist an infinite $B\\subset A$ such that $A\\backslash B$ is a basis of order $k+1$?\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2362, "problem_number": "EP-883", "title": "Erdős Problem #883", "statement": "For $A\\subseteq \\{1,\\ldots,n\\}$ let $G(A)$ be the graph with vertex set $A$, where two integers are joined by an edge if they are coprime.\nIs it true that if $ \\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor $ then $G(A)$ contains all odd cycles of length $\\leq \\frac{n}{3}+1$?\nIs it true that, for every $\\ell\\geq 1$, if $n$ is sufficiently large and $ \\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor $ then $G(A)$ must contain a complete $(1,\\ell,\\ell)$ triparite graph on $2\\ell+1$ vertices?", "background": "A problem of Erd\\H{o}s and S\\'{a}rk\\H{o}zy \\cite{ErSa97}, who prove that if $ \\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor $ then $G(A)$ contains all odd cycles of length $\\leq cn$ for some constant $c>0$.\nThis threshold is the best possible, since one could take $A$ to be the set of $m\\leq n$ which are divisible by either $2$ or $3$, in which case $G(A)$ contains no triangles.\nThe second question was solved by S\\'{a}rk\"{o}zy \\cite{Sa99} who proved that, for large $n$, if $\\lvert A\\rvert$ exceeds the given threshold then $G(A)$ contains a complete $(1,\\ell,\\ell)$ triparite graph with $ \\ell \\gg \\frac{\\log n}{\\log\\log n}. $ \nReferences\n\n\n[ErSa97] Erd\\H{o}s, Paul and Sarkozy, Gabor N., On cycles in the coprime graph of integers. Electron. J. Combin. (1997), Research Paper 8, approx. 11.\n\n[Sa99] S\\'ark\"ozy, G\\'abor N., Complete tripartite subgraphs in the coprime graph of\nintegers. Discrete Math. (1999), 227--238.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2363, "problem_number": "EP-884", "title": "Erdős Problem #884", "statement": "Is it true that, for any $n$, if $d_1<\\cdots 0$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\\epsilon})$ is $O_\\epsilon(1)$?", "background": "Erd\\H{o}s attributes this conjecture to Ruzsa. Erd\\H{o}s and Rosenfeld \\cite{ErRo97} proved that there are infinitely many $n$ such that there are four divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/4})$.\nSee also [887].\nReferences\n\n\n[ErRo97] Erd\\H{o}s, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2366, "problem_number": "EP-887", "title": "Erdős Problem #887", "statement": "Is there an absolute constant $K$ such that, for every $C>0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{1/2},n^{1/2}+C n^{1/4})$.", "background": "A question of Erd\\H{o}s and Rosenfeld \\cite{ErRo97}, who proved that there are infinitely many $n$ with $4$ divisors in $(n^{1/2},n^{1/2}+n^{1/4})$, and ask whether $4$ is best possible here.\nReferences\n\n\n[ErRo97] Erd\\H{o}s, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359.\n\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2367, "problem_number": "EP-888", "title": "Erdős Problem #888", "statement": "What is the size of the largest $A\\subseteq \\{1,\\ldots,n\\}$ such that if $a\\leq b\\leq c\\leq d\\in A$ are such that $abcd$ is a square then $ad=bc$?", "background": "A question of Erd\\H{o}s, S\\'{a}rk\"{o}zy, and S\\'{o}s. Erd\\H{o}s claims that S\\'{a}rk\"{o}zy proved that $\\lvert A\\rvert =o(n)$ (a proof of this bound is provided by Tao in the comments).\nThe primes show that $\\lvert A\\rvert \\gg n/\\log n$ is possible. Cambie and Weisenberg have noted in the comments that the set of semiprimes also works, showing $ (1+o(1))\\frac{\\log\\log n}{\\log n}n \\leq \\lvert A\\rvert $ is achievable.\nSee also [121].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2368, "problem_number": "EP-889", "title": "Erdős Problem #889", "statement": "For $k\\geq 0$ and $n\\geq 1$ let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\\leq ik$.\nIs it true that $ v_0(n)=\\max_{k\\geq 0}v(n,k)\\to \\infty $ as $n\\to \\infty$?", "background": "A question of Erd\\H{o}s and Selfridge \\cite{ErSe67}, who could only show that $v_0(n)\\geq 2$ for $n\\geq 17$. More generally, they conjecture that $ v_l(n)=\\max_{k\\geq l}v(n,k)\\to \\infty $ as $n\\to \\infty$, for every fixed $l$, but could not even prove that $v_1(n)\\geq 2$ for all large $n$.\nThis is problem B27 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[ErSe67] Erd\\H{o}s, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2369, "problem_number": "EP-890", "title": "Erdős Problem #890", "statement": "If $\\omega(n)$ counts the number of distinct prime factors of $n$, then is it true that, for every $k\\geq 1$, $ \\liminf_{n\\to \\infty}\\sum_{0\\leq ik$ by P\\'{o}lya's theorem.\nIt is a classical fact that $ \\limsup_{n\\to \\infty}\\omega(n)\\frac{\\log\\log n}{\\log n}=1. $ \nReferences\n\n\n[ErSe67] Erd\\H{o}s, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2370, "problem_number": "EP-891", "title": "Erdős Problem #891", "statement": "Let $2=p_1k$ many prime factors?", "background": "Schinzel deduced from P\\'{o}lya's theorem \\cite{Po18} (that the sequence of $k$-smooth integers has unbounded gaps) that this is true with $p_1\\cdots p_k$ replaced by $p_1\\cdots p_{k-1}p_{k+1}$.\nThis is unknown even for $k=2$ - that is, is it true that in every interval of $6$ (sufficiently large) consecutive integers there must exist one with at least $3$ prime factors?\nWeisenberg has observed that Dickson's conjecture implies the answer is no if we replace $p_1\\cdots p_k$ with $p_1\\cdots p_k-1$. Indeed, let $L_k$ be the lowest common multiple of all integers at most $p_1\\cdots p_k$. By Dickson's conjecture there are infinitely many $n'$ such that $\\frac{L_k}{m}n'+1$ is prime for all $1\\leq m f(C) e? $ \",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2378, "problem_number": "EP-912", "title": "Erdős Problem #912", "statement": "If $ n! = \\prod_i p_i^{k_i} $ is the factorisation into distinct primes then let $h(n)$ count the number of distinct exponents $k_i$.\nProve that there exists some $c>0$ such that $ h(n) \\sim c \\left(\\frac{n}{\\log n}\\right)^{1/2} $ as $n\\to \\infty$.", "background": "A problem of Erd\\H{o}s and Selfridge, who proved (see \\cite{Er82c}) $ h(n) \\asymp \\left(\\frac{n}{\\log n}\\right)^{1/2}. $ A heuristic of Tao using the Cram\\'{e}r model for the primes (detailed in the comments) suggests this is true with $ c=\\sqrt{2\\pi}=2.506\\cdots. $ \nReferences\n\n\n[Er82c] Erd\\H{o}s, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2379, "problem_number": "EP-913", "title": "Erdős Problem #913", "statement": "Are there infinitely many $n$ such that if $ n(n+1) = \\prod_i p_i^{k_i} $ is the factorisation into distinct primes then all exponents $k_i$ are distinct?", "background": "It is likely that there are infinitely many primes $p$ such that $8p^2-1$ is also prime, in which case this is true with exponents $\\{1,2,3\\}$, letting $n=8p^2-1$.\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2380, "problem_number": "EP-917", "title": "Erdős Problem #917", "statement": "Let $k\\geq 4$ and $f_k(n)$ be the largest number of edges in a graph on $n$ vertices which has chromatic number $k$ and is critical (i.e. deleting any edge reduces the chromatic number).\nIs it true that $ f_k(n) \\gg_k n^2? $ Is it true that $ f_6(n)\\sim n^2/4? $ More generally, is it true that, for $k\\geq 6$, $ f_k(n) \\sim \\frac{1}{2}\\left(1-\\frac{1}{\\lfloor k/3\\rfloor}\\right)n^2? $ ", "background": "Erd\\H{o}s \\cite{Er93} wrote 'I learned of this definition from Dirac in 1949 and immediately asked whether $f_k(n)=o(n^2)$. To my great surprise Dirac constructed a $6$ critical graph on $n$ vertices with more than $\\frac{n^2}{4}$ edges.' In fact Dirac \\cite{Di52} proved $ f_6(4n+2) \\geq 4n^2+8n+3, $ as witnessed by taking two disjoint copies of $C_{2n+1}$ and adding all edges between them.\nErd\\H{o}s \\cite{Er69b} observed that Dirac's construction generalises to show that, if $3\\mid k$, there are infinitely many values of $n$ (those of the shape $mk/3$ where $m$ is odd) such that $ f_k(n) \\geq \\frac{1}{2}\\left(1-\\frac{1}{k/3}\\right)n^2 + n. $ Toft \\cite{To70} proved that $f_k(n)\\gg_k n^2$ for $k\\geq 4$.\nConstructions of Stiebitz \\cite{St87} show that, for $k\\geq 6$, there exist infinitely many values of $n$ such that $ f_k(n) \\geq \\frac{1}{2}\\left(1-\\frac{1}{\\lfloor k/3\\rfloor+\\delta_k}\\right)n^2 $ where $\\delta_k=0$ if $k\\equiv 0\\pmod{3}$, $\\delta_k=1/7$ if $k\\equiv 1\\pmod{3}$, and $\\delta_k\\equiv 24/69$ if $k\\equiv 2\\pmod{3}$, which disproves Erd\\H{o}s' conjectured asympotic for $k\not\\equiv 0\\pmod{3}$.\nStiebitz also proved the general upper bound $ f_k(n) < \\mathrm{ex}(n;K_{k-1})\\sim \\frac{1}{2}\\left(1-\\frac{1}{k-2}\\right)n^2 $ for large $n$. Luo, Ma, and Yang \\cite{LMY23} have improved this upper bound to $ f_k(n) \\leq \\frac{1}{2}\\left(1-\\frac{1}{k-2}-\\frac{1}{36(k-1)^2}+o(1)\\right)n^2 $ See also [944] and [1032].\nReferences\n\n\n[Di52] Dirac, G. A., A property of {$4$}-chromatic graphs and some remarks on\ncritical graphs. J. London Math. Soc. (1952), 85-92.\n\n[Er69b] Erd\\H{o}s, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[LMY23] Luo, Cong and Ma, Jie and Yang, Tianchi, On the maximum number of edges in {$k$}-critical graphs. Combin. Probab. Comput. (2023), 900--911.\n\n[St87] Stiebitz, M., Subgraphs of colour-critical graphs. Combinatorica (1987), 303--312.\n\n[To70] Toft, B., On the maximal number of edges of critical {$k$}-chromatic\ngraphs. Studia Sci. Math. Hungar. (1970), 461--470.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2381, "problem_number": "EP-918", "title": "Erdős Problem #918", "statement": "Is there a graph with $\\aleph_2$ vertices and chromatic number $\\aleph_2$ such that every subgraph on $\\aleph_1$ vertices has chromatic number $\\leq\\aleph_0$?\nIs there a graph with $\\aleph_{\\omega+1}$ vertices and chromatic number $\\aleph_1$ such that every subgraph on $\\aleph_\\omega$ vertices has chromatic number $\\leq\\aleph_0$?", "background": "A question of Erd\\H{o}s and Hajnal \\cite{ErHa68b}, who proved that for every finite $k$ there is a graph with chromatic number $\\aleph_1$ where each subgraph on less than $\\aleph_k$ vertices has chromatic number $\\leq \\aleph_0$.\nIn \\cite{Er69b} it is asked with chromatic number $=\\aleph_0$, but in the comments louisd observes this is (assuming subgraph and not induced subgraph was intended) trivially impossible, and hence presumably the problem was intended as written here (which is how it is posed in \\cite{ErHa68b}).\nReferences\n\n\n[Er69b] Erd\\H{o}s, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\n\n[ErHa68b] Erd\\H{o}s, P. and Hajnal, A., On chromatic number of infinite graphs. (1968), 83--98.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2382, "problem_number": "EP-919", "title": "Erdős Problem #919", "statement": "Is there a graph $G$ with vertex set $\\omega_2^2$ and chromatic number $\\aleph_2$ such that every subgraph whose vertices have a lesser type has chromatic number $\\leq \\aleph_0$?\nWhat if instead we ask for $G$ to have chromatic number $\\aleph_1$?", "background": "This question was inspired by a theorem of Babai, that if $G$ is a graph on a well-ordered set with chromatic number $\\geq \\aleph_0$ there is a subgraph on vertices with order-type $\\omega$ with chromatic number $\\aleph_0$.\nErd\\H{o}s and Hajnal showed this does not generalise to higher cardinals - they (see \\cite{Er69b}) constructed a set on $\\omega_1^2$ with chromatic number $\\aleph_1$ such that every strictly smaller subgraph has chromatic number $\\leq \\aleph_0$ as follows: the vertices of $G$ are the pairs $(x_\\alpha,y_\\beta)$ for $1\\leq \\alpha,\\beta <\\omega_1$, ordered lexicographically. We connect $(x_{\\alpha_1},y_{\\beta_1})$ and $(x_{\\alpha_2},y_{\\beta_2})$ if and only if $\\alpha_1<\\alpha_2$ and $\\beta_1<\\beta_2$.\nA similar construction produces a graph on $\\omega_2^2$ with chromatic number $\\aleph_2$ such that every smaller subgraph has chromatic number $\\leq \\aleph_1$.\nReferences\n\n\n[Er69b] Erd\\H{o}s, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2383, "problem_number": "EP-920", "title": "Erdős Problem #920", "statement": "Let $f_k(n)$ be the maximum possible chromatic number of a graph with $n$ vertices which contains no $K_k$.\nIs it true that, for $k\\geq 4$, $ f_k(n) \\gg \\frac{n^{1-\\frac{1}{k-1}}}{(\\log n)^{c_k}} $ for some constant $c_k>0$?", "background": "Graver and Yackel \\cite{GrYa68} proved that $ f_k(n) \\ll \\left(n\\frac{\\log\\log n}{\\log n}\\right)^{1-\\frac{1}{k-1}}. $ It is known that $f_3(n)\\asymp (n/\\log n)^{1/2}$ (see [1104]).\nThe lower bound $R(4,m) \\gg m^3/(\\log m)^4$ of Mattheus and Verstraete \\cite{MaVe23} (see [166]) implies $ f_4(n) \\gg \\frac{n^{2/3}}{(\\log n)^{4/3}}. $ A positive answer to this question would follow from [986]. The known bounds for that problem imply $ f_k(n) \\gg \\frac{n^{1-\\frac{2}{k+1}}}{(\\log n)^{c_k}}. $ See [1104] (and also [1013]) for the case $k=3$.\nReferences\n\n\n[GrYa68] Graver, Jack E. and Yackel, James, Some graph theoretic results associated with Ramsey's theorem. J. Combinatorial Theory (1968), 125--175.\n\n[MaVe23] Mattheus, S. and Verstraete, J., The asymptotics of $r(4,t)$. arXiv:2306.04007 (2023).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2384, "problem_number": "EP-928", "title": "Erdős Problem #928", "statement": "Let $\\alpha,\\beta\\in (0,1)$ and let $P(n)$ denote the largest prime divisor of $n$. Does the density of integers $n$ such that $P(n) k^{1/2-o(1)}$.\nIt is trivial that $S(k)\\leq k+1$ since, for example, one can take $n\\equiv 1\\pmod{(k+1)!}$. The best bound on large gaps between primes due to Ford, Green, Konyagin, Maynard, and Tao \\cite{FGKMT18} (see [4]) implies $ S(k) \\ll k \\frac{\\log\\log\\log k}{\\log\\log k\\log\\log\\log\\log k}. $ \nReferences\n\n\n[FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence, Long gaps between primes. J. Amer. Math. Soc. (2018), 65-105.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2386, "problem_number": "EP-930", "title": "Erdős Problem #930", "statement": "Is it true that, for every $r$, there is a $k$ such that if $I_1,\\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$, then $ \\prod_{1\\leq i\\leq r}\\prod_{m\\in I_i}m $ is not a perfect power?", "background": "Erd\\H{o}s and Selfridge \\cite{ErSe75} proved that the product of consecutive integers is never a power (establishing the case $r=1$). The condition that the intervals be large in terms of $r$ is necessary for $r=2$ - see the constructions in [363].\nSee also [363] for the case of squares.\nReferences\n\n\n[ErSe75] Erd\\H{o}s, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2387, "problem_number": "EP-931", "title": "Erdős Problem #931", "statement": "Let $k_1\\geq k_2\\geq 3$. Are there only finitely many $n_2\\geq n_1+k_1$ such that $ \\prod_{1\\leq i\\leq k_1}(n_1+i)\\textrm{ and }\\prod_{1\\leq j\\leq k_2}(n_2+j) $ have the same prime factors?", "background": "Tijdeman gave the example $ 19,20,21,22\\textrm{ and }54,55,56,57. $ Erd\\H{o}s \\cite{Er76d} was unsure of this conjecture, and thought perhaps if the two products have the same prime factors then $n_2>2(n_1+k_1)$. It is not clear but it is possible that he meant to ask this question also permitting finitely many counterexamples. Indeed, without this caveat it is false - AlphaProof has found the counterexample $ 10! = 2^8\\cdot 3^4\\cdot 5^2\\cdot 7 $ and $ 14\\cdot 15\\cdot 16 = 2^5\\cdot 3\\cdot 5\\cdot 7, $ so that $n_1=0$, $k_1=10$, $n_2=13$, and $k_2=3$.\nSee also [388].\nThis is discussed in problem B35 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er76d] Erd\\H{o}s, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2388, "problem_number": "EP-932", "title": "Erdős Problem #932", "statement": "Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two integers $p_rn\\log n. $ Steinerberger has noted a simple proof of this fact follows from taking $n=2^{3^r}$ for any integer $r\\geq 1$, when $k=3^r$ and $l=r+1$.\nReferences\n\n\n[Er76d] Erd\\H{o}s, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2390, "problem_number": "EP-934", "title": "Erdős Problem #934", "statement": "Let $h_t(d)$ be minimal such that every graph $G$ with $h_t(d)$ edges and maximal degree $\\leq d$ contains two edges whose shortest path between them has length $\\geq t$.\nEstimate $h_t(d)$.", "background": "A problem of Erd\\H{o}s and Ne\\v{s}et\\v{r}il. Erd\\H{o}s \\cite{Er88} wrote 'This problem seems to be interesting only if there is a nice expression for $h_t(d)$.'\nIt is easy to see that $h_t(d)\\leq 2d^t$ always and $h_1(d)=d+1$.\nErd\\H{o}s and Ne\\v{s}et\\v{r}il and Bermond, Bond, Paoli, and Peyrat \\cite{BBPP83} independently conjectured that $h_2(d) \\leq \\tfrac{5}{4}d^2+1$, with equality for even $d$ (see [149]). This was proved by Chung, Gy\\'{a}rf\\'{a}s, Tuza, and Trotter \\cite{CGTT90}.\nCambie, Cames van Batenburg, de Joannis de Verclos, and Kang \\cite{CCJK22} conjectured that $ h_3(d) \\leq d^3-d^2+d+2, $ with equality if and only if $d=p^k+1$ for some prime power $p^k$, and proved that $h_3(3)=23$. They also conjecture that, for all $t\\geq 3$, $h_t(d)\\geq (1-o(1))d^t$ for infinitely many $d$ and $h_t(d)\\leq (1+o(1))d^t$ for all $d$ (where the $o(1)$ term $\\to 0$ as $d\\to \\infty$).\nThe same authors prove that, if $t$ is large, then there are infinitely many $d$ such that $h_t(d) \\geq 0.629^td^t$, and that for all $t\\geq 1$ we have $ h_t(d) \\leq \\tfrac{3}{2}d^t+1. $ \nReferences\n\n\n[BBPP83] Bermond, J.-C. and Bond, J. and Paoli, M. and Peyrat, C., Graphs and interconnection networks: diameter and\nvulnerability. (1983), 1--30.\n\n[CCJK22] Cambie, Stijn and Cames van Batenburg, Wouter and de Joannis\nde Verclos, R\\'{e}mi and Kang, Ross J., Maximizing line subgraphs of diameter at most {$t$}. SIAM J. Discrete Math. (2022), 939--950.\n\n[CGTT90] Chung, F. R. K. and Gy\\'arf\\'as, A. and Tuza, Z. and Trotter,\nW. T., The maximum number of edges in {$2K_2$}-free graphs of bounded\ndegree. Discrete Math. (1990), 129--135.\n\n[Er88] Erd\\H{o}s, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2391, "problem_number": "EP-935", "title": "Erdős Problem #935", "statement": "For any integer $n=\\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that $ Q_2(n) = \\prod_{\\substack{p\\\\ k_p\\geq 2}}p^{k_p}. $ Is it true that, for every $\\epsilon>0$ and $\\ell\\geq 1$, if $n$ is sufficiently large then $ Q_2(n(n+1)\\cdots(n+\\ell))2$, only keeping those prime powers with exponent $\\geq r$.\nReferences\n\n\n[Er76d] Erd\\H{o}s, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2392, "problem_number": "EP-936", "title": "Erdős Problem #936", "statement": "Are $ 2^n\\pm 1 $ and $ n!\\pm 1 $ powerful (i.e. if $p\\mid m$ then $p^2\\mid m$) for only finitely many $n$?", "background": "Cushing and Pascoe \\cite{CuPa16} have shown the answer to the second question is yes assuming the abc conjecture - in fact, for any fixed $k\\geq 0$, there are only finitely many $n$ and powerful $x$ such that $\\lvert x-n!\\rvert \\leq k$.\nCrowdMath \\cite{Cr20} has shown that the answer to the first question is yes, again assuming the abc conjecture.\nReferences\n\n\n[Cr20] P. A. CrowdMath, Applications of the abc conjecture to powerful numbers. arXiv:2005.07321 (2020).\n\n[CuPa16] D. Cushing and J. E. Pascoe, Powerful numbers and the ABC-conjecture. arXiv:1611.01192 (2016).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2393, "problem_number": "EP-938", "title": "Erdős Problem #938", "statement": "Let $A=\\{n_10$ such that $ h(n) < (\\log n)^{c+o(1)} $ and, for infinitely many $n$, $ h(n) >(\\log n)^{c-o(1)}? $ ", "background": "Erd\\H{o}s writes it is not hard to prove that $\\limsup h(n)=\\infty$, and that the density $\\delta_l$ of integers for which $h(n)=l$ exists and $\\sum \\delta_l=1$.\nA proof that $h(n)$ is unbounded is provided by van Doorn in the comments.\nDe Koninck and Luca \\cite{DeLu04} have proved, for infinitely many $n$, $ h(n) \\gg \\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/3}. $ They also give the density ($\\approx 0.275$) of those $n$ such that $h(n)=1$.\nReferences\n\n\n[DeLu04] De Koninck, Jean-Marie and Luca, Florian, Sur la proximit\\'{e} des nombres puissants. Acta Arith. (2004), 149--157.\n\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2397, "problem_number": "EP-943", "title": "Erdős Problem #943", "statement": "Let $A$ be the set of powerful numbers (if $p\\mid n$ then $p^2\\mid n$). Is it true that $ 1_A\\ast 1_A(n)=n^{o(1)} $ for every $n$?\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2398, "problem_number": "EP-944", "title": "Erdős Problem #944", "statement": "A critical vertex, edge, or set of edges, is one whose deletion lowers the chromatic number.\nLet $k\\geq 4$ and $r\\geq 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$?", "background": "A graph $G$ with chromatic number $k$ in which every vertex is critical is called $k$-vertex-critical.\nThis was conjectured by Dirac in 1970 for $k\\geq 4$ and $r=1$. Dirac's conjecture was proved, for $k=5$, by Brown \\cite{Br92}. Lattanzio \\cite{La02} proved there exist such graphs for all $k$ such that $k-1$ is not prime. Independently, Jensen \\cite{Je02} gave an alternative construction for all $k\\geq 5$. The case $k=4$ and $r=1$ remains open.\nMartinsson and Steiner \\cite{MaSt25} proved this is true for every $r\\geq 1$ if $k$ is sufficiently large, depending on $r$. Skottova and Steiner \\cite{SkSt25} have improved this, proving that such graphs exist for all $k\\geq 5$ and $r\\geq 1$. The only remaining open case is $k=4$ (even the case $k=4$ and $r=1$ remains open).\nErd\\H{o}s also asked a stronger quantitative form of this question: let $f_k(n)$ denote the largest $r\\geq 1$ such that there exists a $k$-vertex-critical graph on $n$ vertices such that no set of at most $r$ edges is critical. He then asks whether $f_k(n)\\to \\infty$ as $n\\to \\infty$. Skottova and Steiner \\cite{SkSt25} have proved this for $k\\geq 5$, establishing the bounds $ n^{1/3}\\ll_k f_k(n) \\ll_k \\frac{n}{(\\log n)^C} $ for all $k\\geq 5$, where $C>0$ is an absolute constant.\nThis is Problem 91 in the graph problems collection. See also [917] and [1032].\nReferences\n\n\n[Br92] Brown, Jason I., A vertex critical graph without critical edges. Discrete Math. (1992), 99--101.\n\n[Je02] Jensen, Tommy R., Dense critical and vertex-critical graphs. Discrete Math. (2002), 63--84.\n\n[La02] Lattanzio, John J., A note on a conjecture of {D}irac. Discrete Math. (2002), 323--330.\n\n[MaSt25] Martinsson, Anders and Steiner, Raphael, Vertex-critical graphs far from edge-criticality. Combin. Probab. Comput. (2025), 151--157.\n\n[SkSt25] E. Skottova and R. Steiner, Critical edge sets in vertex-critical graphs. arXiv:2508.08703 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2399, "problem_number": "EP-945", "title": "Erdős Problem #945", "statement": "Let $F(x)$ be the maximal $k$ such that there exist $n+1,\\ldots,n+k\\leq x$ with $\\tau(n+1),\\ldots,\\tau(n+k)$ all distinct (where $\\tau(m)$ counts the divisors of $m$). Estimate $F(x)$. In particular, is it true that $ F(x) \\leq (\\log x)^{O(1)}? $ In other words, is there a constant $C>0$ such that, for all large $x$, every interval $[x,x+(\\log x)^C]$ contains two integers with the same number of divisors?", "background": "A problem of Erd\\H{o}s and Mirsky \\cite{ErMi52}, who proved that $ \\frac{(\\log x)^{1/2}}{\\log\\log x}\\ll F(x) \\ll \\exp\\left(O\\left(\\frac{(\\log x)^{1/2}}{\\log\\log x}\\right)\\right). $ Erd\\H{o}s \\cite{Er85e} claimed that the lower bound could be improved via their method 'with some more work' to $(\\log x)^{1-o(1)}$. Beker has improved the upper bound to $ F(x) \\ll \\exp\\left(O\\left((\\log x)^{1/3+o(1)}\\right)\\right). $ Cambie has observed that Cram\\'{er's conjecture} implies that $F(x) \\ll (\\log x)^2$, and furthermore if every interval in $[x,2x]$ of length $\\gg \\log x$ contains a squarefree number (see [208]) then every interval of length $\\gg (\\log x)^2$ contains two numbers with the same number of divisors, whence $ F(x) \\ll (\\log x)^2. $ See [1004] for the analogous problem with the Euler totient function.\nThis problem is discussed in problem B18 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er85e] Erd\\H{o}s, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo,\nOkayama and Kyoto, 1984) (1985), 65-87.\n\n[ErMi52] Erd\\H{o}s, P. and Mirsky, L., The distribution of values of the divisor function {$d(n)$}. Proc. London Math. Soc. (3) (1952), 257--271.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2400, "problem_number": "EP-948", "title": "Erdős Problem #948", "statement": "Is there a function $f(n)$ and a $k$ such that in any $k$-colouring of the integers there exists a sequence $a_1<\\cdots$ such that $a_n0$ such that there are $\\gg n^c/\\log n$ many primes in $[n,n+n^c]$ implies that $\\liminf f(n)>0$.\nErd\\H{o}s writes that a 'weaker conjecture which is perhaps not quite inaccessible' is that, for every $\\epsilon>0$, if $x$ is sufficiently large there exists $y0$ then $f(n)\\ll \\log\\log\\log n$.\nThe study of $f(p)$ is even harder, and Erd\\H{o}s could not prove that $ \\sum_{p0$ such that $h(n)>n^{1+c}$ for all large $n$.", "background": "A problem of Erd\\H{o}s and Pach \\cite{ErPa90}, who proved that $h(n) \\ll n^{4/3}$. They also consider the related function where we consider $n$ disjoint convex sets (not necessarily translates), for which they give an upper bound of $\\ll n^{7/5}$.\nIt is trivial that $h(n)\\geq f(n)$, where $f(n)$ is the maximal number of unit distances determined by $n$ points in $\\mathbb{R}^2$ (see [90]).\nReferences\n\n\n[ErPa90] Erd\\H{o}s, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2409, "problem_number": "EP-959", "title": "Erdős Problem #959", "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of size $n$ and let $\\{d_1,\\ldots,d_k\\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the number of times the distance $d$ is determined, and suppose the $d_i$ are ordered such that $ f(d_1)\\geq f(d_2)\\geq \\cdots \\geq f(d_k). $ Estimate $ \\max (f(d_1)-f(d_2)), $ where the maximum is taken over all $A$ of size $n$.", "background": "More generally, one can ask about $ \\max (f(d_r)-f(d_{r+1})). $ Clemen, Dumitrescu, and Liu \\cite{CDL25}, have shown that $ \\max (f(d_1)-f(d_2))\\gg n\\log n. $ More generally, for any $1\\leq k\\leq \\log n$, there exists a set $A$ of $n$ points such that $ f(d_r)-f(d_{r+1})\\gg \\frac{n\\log n}{r}. $ They conjecture that $n\\log n$ can be improved to $n^{1+c/\\log\\log n}$ for some constant $c>0$.\nReferences\n\n\n[CDL25] F. Clemen, A. Dumitrescu, and D. Liu, On multiplicities of interpoint distances. arXiv:2505.04283 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2410, "problem_number": "EP-960", "title": "Erdős Problem #960", "statement": "Let $r,k\\geq 2$ be fixed. Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such that if there are at least $f_{r,k}(n)$ many ordinary lines (lines containing exactly two points) then there is a set $A'\\subseteq A$ of $r$ points such that all $\\binom{r}{2}$ many lines determined by $A'$ are ordinary.\nIs it true that $f_{r,k}(n)=o(n^2)$, or perhaps even $\\ll n$?", "background": "Tur\\'{a}n's theorem implies $ f_{r,k}(n) \\leq \\left(1-\\frac{1}{r-1}\\right)\\frac{n^2}{2}+1. $ See also [209].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2411, "problem_number": "EP-961", "title": "Erdős Problem #961", "statement": "Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$. Estimate $f(k)$.", "background": "In other words, how large can a consecutive set of $k$-smooth integers be? Sylvester and Schur (see \\cite{Er34}) proved $f(k)\\leq k$ and Erd\\H{o}s \\cite{Er55d} proved $ f(k)<3\\frac{k}{\\log k}. $ Jutila \\cite{Ju74} and Ramachandra, and Shorey \\cite{RaSh73} proved $ f(k) \\ll \\frac{\\log\\log\\log k}{\\log \\log k}\\frac{k}{\\log k}. $ It is likely that $f(k) \\ll (\\log k)^{O(1)}$.\nThis is essentially equivalent to [683].\nReferences\n\n\n[Er34] Erd\\H{o}s, Paul, A {T}heorem of {S}ylvester and {S}chur. J. London Math. Soc. (1934), 282--288.\n\n[Er55d] Erd\\H{o}s, P., On consecutive integers. Nieuw Arch. Wisk. (3) (1955), 124--128.\n\n[Ju74] Jutila, Matti, On numbers with a large prime factor. {II}. J. Indian Math. Soc. (N.S.) (1974), 125--130.\n\n[RaSh73] Ramachandra, K. and Shorey, T. N., On gaps between numbers with a large prime factor. Acta Arith. (1973), 99--111.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2412, "problem_number": "EP-962", "title": "Erdős Problem #962", "statement": "Let $k(n)$ be the maximal $k$ such that there exists $m\\leq n$ such that each of the integers $ m+1,\\ldots,m+k $ are divisible by at least one prime $>k$. Estimate $k(n)$.", "background": "Erd\\H{o}s \\cite{Er65} wrote it is 'not hard to prove' that $ k(n)\\gg_\\epsilon \\exp((\\log n)^{1/2-\\epsilon}) $ and it 'seems likely' that $k(n)=o(n^\\epsilon)$, but had no non-trivial upper bound for $k(n)$.\nIt is not clear what he meant by a non-trivial bound for this problem, but Tao in the comments has given a simple argument proving $k(n) \\leq (1+o(1))n^{1/2}$.\nTang has proved a lower bound of $ k(n)\\geq \\exp\\left(\\left(\\frac{1}{\\sqrt{2}}-o(1)\\right)\\sqrt{\\log n\\log\\log n}\\right). $ \nReferences\n\n\n[Er65] Erd\\H{o}s, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2413, "problem_number": "EP-963", "title": "Erdős Problem #963", "statement": "Let $f(n)$ be the maximal $k$ such that in any set $A\\subset \\mathbb{R}$ of size $n$ there is a subset $B\\subseteq A$ of size $\\lvert B\\rvert\\geq k$ which is dissociated that is, the sums $\\sum_{b\\in S}b$ are distinct for all $S\\subseteq B$. Estimate $f(n)$ - in particular, is it true that $ f(n)\\geq \\lfloor \\log_2 n\\rfloor? $ ", "background": "Erd\\H{o}s noted that the greedy algorithm showed $f(n)\\geq \\lfloor \\log_3 n\\rfloor$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2414, "problem_number": "EP-968", "title": "Erdős Problem #968", "statement": "Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_nu_{n+1}$ has positive density.\nErd\\H{o}s also asks whether $ u_nu_{n+1}>u_{n+2} $ have infinitely many solutions.\nReferences\n\n\n[ErPr61] Erd\\H{o}s, P. and Prachar, K., S\"{a}tze und {P}robleme \"{u}ber {$p\\sb{k}/k$}. Abh. Math. Sem. Univ. Hamburg (1961/62), 251--256.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2415, "problem_number": "EP-969", "title": "Erdős Problem #969", "statement": "Let $Q(x)$ count the number of squarefree integers in $[1,x]$. Determine the order of magnitude in the error term in the asymptotic $ Q(x)=\\frac{6}{\\pi^2}x+E(x). $ ", "background": "It is elementary to prove $E(x)\\ll x^{1/2}$, and the prime number theorem implies $o(x^{1/2})$. The best known unconditional upper bound is of the shape $x^{1/2-o(1)}$, due to Walfisz \\cite{Wa63}. Evelyn and Linfoot \\cite{EvLi31} proved that $ E(x) \\gg x^{1/4}, $ and this is likely the true order of magnitude. The Riemann Hypothesis would follow from $E(x)\\ll x^{1/4}$.\nThe true order of magnitude is unknown even assuming the Riemann Hypothesis. Conditional on this assumption, the best known upper bound is $ E(x)\\ll x^{\\frac{11}{35}+o(1)}, $ due to Liu \\cite{Li16}.\nReferences\n\n\n[EvLi31] Evelyn, C. J. A. and Linfoot, E. H., On a problem in the additive theory of numbers. Ann. of Math. (2) (1931), 261--270.\n\n[Li16] Liu, H.-Q., On the distribution of squarefree numbers. J. Number Theory (2016), 202--222.\n\n[Wa63] Walfisz, Arnold, Weylsche {E}xponentialsummen in der neueren {Z}ahlentheorie. (1963), 231.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2416, "problem_number": "EP-970", "title": "Erdős Problem #970", "statement": "Let $h(k)$ be Jacobsthal's function, defined to as the minimal $m$ such that, if $n$ has at most $k$ prime factors, then in any set of $m$ consecutive integers there exists an integer coprime to $n$. Determine the order of magnitude of $h(k)$. In particular, is it true that $ h(k) \\ll k^2? $ ", "background": "That $h(k)\\ll k^2$ is a conjecture of Jacobsthal. Iwaniec \\cite{Iw78} proved $ h(k) \\ll (k\\log k)^2. $ The best lower bound known is $ h(k) \\gg \\frac{(\\log k)(\\log\\log\\log k)}{(\\log\\log k)^2}k, $ due to Ford, Green, Konyagin, Maynard, and Tao \\cite{FGKMT18}.\nThis is a more general form of the function considered in [687].\nReferences\n\n\n[FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence, Long gaps between primes. J. Amer. Math. Soc. (2018), 65-105.\n\n[Iw78] Iwaniec, Henryk, On the problem of {J}acobsthal. Demonstratio Math. (1978), 225--231.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2417, "problem_number": "EP-971", "title": "Erdős Problem #971", "statement": "Let $p(a,d)$ be the least prime congruent to $a\\pmod{d}$. Does there exist a constant $c>0$ such that, for all large $d$, $ p(a,d) > (1+c)\\phi(d)\\log d $ for $\\gg \\phi(d)$ many values of $a$?", "background": "Erd\\H{o}s \\cite{Er49c} could prove this is true for an infinite sequence of $d$. He also proved that, for any $\\epsilon>0$, $ p(a,d)< \\epsilon \\phi(d)\\log d $ for $\\gg_\\epsilon \\phi(d)$ many values of $a$.\nReferences\n\n\n[Er49c] Erd\\H{o}s, P., On some applications of {B}run's method. Acta Univ. Szeged. Sect. Sci. Math. (1949), 57--63.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2418, "problem_number": "EP-972", "title": "Erdős Problem #972", "statement": "Let $\\alpha>1$ be irrational. Are there infinitely many primes $p$ such that $\\lfloor p\\alpha\\rfloor$ is also prime?", "background": "Vinogradov \\cite{Vi48} proved that the sequence $\\{p\\alpha\\}$ is uniformly distributed for every irrational $\\alpha$, and hence there are infinitely many primes $p$ of the shape $p=\\lfloor n\\alpha\\rfloor$ for every irrational $\\alpha>1$. Indeed, this occurs if and only if $ \\frac{p}{\\alpha}\\leq n<\\frac{p+1}{\\alpha}, $ which is true if and only if $\\{p\\alpha^{-1}\\}>1-\\alpha^{-1}$, which happens infinitely often by the uniform distribution of $\\{p\\alpha^{-1}\\}$.\nReferences\n\n\n[Vi48] Vinogradov, I. M., On an estimate of trigonometric sums with prime numbers. Izv. Akad. Nauk SSSR Ser. Mat. (1948), 225--248.\",\n \"difficulty\": \"L3\"\n},{", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2419, "problem_number": "EP-973", "title": "Erdős Problem #973", "statement": "Does there exist a constant $C>1$ such that, for every $n\\geq 2$, there exists a sequence $z_i\\in \\mathbb{C}$ with $z_1=1$ and $\\lvert z_i\\rvert \\geq 1$ for all $1\\leq i\\leq n$ with $ \\max_{2\\leq k\\leq n+1}\\left\\lvert \\sum_{1\\leq i\\leq n}z_i^k\\right\\rvert < C^{-n}? $ ", "background": "This is Problem 7.3 in \\cite{Ha74}, where it is attributed to Erd\\H{o}s.\nErd\\H{o}s proved (as described on p.35 of \\cite{Tu84b}) that such a sequence does exist with $\\lvert z_i\\rvert\\leq 1$. Indeed, Erd\\H{o}s' construction gives a value of $C\\approx 1.32$.\nIn \\cite{Er92f} (a different) Erd\\H{o}s refines this analysis, proving that if $ M_2=\\min_{z_j} \\max_{2\\leq k\\leq n+1} \\left\\lvert \\sum_{1\\leq j\\leq n}z_j^k\\right\\rvert, $ where the minimum is take over all $z_j\\in \\mathbb{C}$ with $\\max \\lvert z_j\\rvert=1$, then $ (1.746)^{-n} < M_2 < (1.745)^{-n}. $ Tang notes in the comments that Theorem 6.1 of \\cite{Tu84b} implies that, if $\\lvert z_i\\rvert \\geq 1$ for all $i$, then $ \\max_{2\\leq k\\leq n+1}\\left\\lvert \\sum_{1\\leq i\\leq n}z_i^k\\right\\rvert \\geq (2e)^{-(1+o(1))n}. $ See also [519].\nReferences\n\n\n[Er92f] Erd\\H{o}s, L., On some problems of {P}. {T}ur\\'an concerning power sums of\ncomplex numbers. Acta Math. Hungar. (1992), 11--24.\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[Tu84b] Tur\\'an, Paul, On a new method of analysis and its applications. (1984), xvi+584.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2420, "problem_number": "EP-975", "title": "Erdős Problem #975", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible non-constant polynomial such that $f(n)\\geq 1$ for all large $n\\in\\mathbb{N}$. Does there exist a constant $c=c(f)>0$ such that $ \\sum_{n\\leq X} \\tau(f(n))\\sim cX\\log X, $ where $\\tau$ is the divisor function?", "background": "Van der Corput \\cite{Va39} proved that $ \\sum_{n\\leq X} \\tau(f(n))\\gg_f X\\log X. $ Erd\\H{o}s \\cite{Er52b} proved using elementary methods that $ \\sum_{n\\leq X} \\tau(f(n))\\ll_f X\\log X. $ Such an asymptotic formula is known whenever $f$ is an irreducible quadratic, as proved by Hooley \\cite{Ho63}. The form of $c$ depends on $f$ in a complicated fashion (see the work of McKee \\cite{Mc95}, \\cite{Mc97}, and \\cite{Mc99} for expressions for various types of quadratic $f$). For example, $ \\sum_{n\\leq x}\\tau(n^2+1)=\\frac{3}{\\pi}x\\log x+O(x). $ Tao has a blog post on this topic.\nReferences\n\n\n[Er52b] Erd\\H{o}s, P., On the sum {$\\sum^x_{k=1} d(f(k))$}. J. London Math. Soc. (1952), 7--15.\n\n[Ho63] Hooley, Christopher, On the number of divisors of a quadratic polynomial. Acta Math. (1963), 97--114.\n\n[Mc95] McKee, James, On the average number of divisors of quadratic polynomials. Math. Proc. Cambridge Philos. Soc. (1995), 389--392.\n\n[Mc97] McKee, James, A note on the number of divisors of quadratic polynomials. (1997), 275--281.\n\n[Mc99] McKee, James, The average number of divisors of an irreducible quadratic\npolynomial. Math. Proc. Cambridge Philos. Soc. (1999), 17--22.\n\n[Va39] van der Corput, J. G., Une in\\'{e}galit\\'{e}{} relative au nombre des diviseurs. Nederl. Akad. Wetensch., Proc. (1939), 547--553.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2421, "problem_number": "EP-976", "title": "Erdős Problem #976", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $d\\geq 2$. Let $F_f(n)$ be maximal such that there exists $1\\leq m\\leq n$ with $f(m)$ is divisible by a prime $\\geq F_f(n)$. Equivalently, $F_f(n)$ is the greatest prime divisor of $ \\prod_{1\\leq m\\leq n}f(m). $ Estimate $F_f(n)$. In particular, is it true that $F_f(n)\\gg n^{1+c}$ for some constant $c>0$? Or even $\\gg n^d$?", "background": "Nagell and Ricci \\cite{Na22} proved that $ F_f(n) \\gg n\\log n, $ which Erd\\H{o}s \\cite{Er52c} improved to $ F_f(n) \\gg n(\\log n)^{\\log\\log\\log n}. $ In \\cite{Er65b} he claimed a proof of $ F_f(n) \\gg n\\exp((\\log n)^c) $ for some constant $c>0$, but said he had never published the proof, which was 'fairly complicated'. This seems to have been flawed, since Erd\\H{o}s and Schinzel \\cite{ErSc90} later published a weaker bound. A proof of the stronger bound above was finally provided by Tenenbaum \\cite{Te90}.\nReferences\n\n\n[Er52c] Erd\\H{o}s, P., On the greatest prime factor of {$\\prod^x_{k=1}f(k)$}. J. London Math. Soc. (1952), 379--384.\n\n[Er65b] Erd\\H{o}s, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[ErSc90] Erd\\H{o}s, P. and Schinzel, A., On the greatest prime factor of {$\\prod^x_{k=1}f(k)$}. Acta Arith. (1990), 191--200.\n\n[Na22] No reference found.\n\n\n[Te90] Tenenbaum, G\\'{e}rald, Sur une question d'{E}rd\\H{o}s et {S}chinzel. (1990), 405--443.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2422, "problem_number": "EP-978", "title": "Erdős Problem #978", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\neq 2^l$ for any $l\\geq 1$) such that the leading coefficient of $f$ is positive.\nDoes the set of integers $n\\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density?\nAre there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free?\nIn particular, does $ n^4+2 $ represent infinitely many squarefree numbers?", "background": "Erd\\H{o}s \\cite{Er53} proved there are infinitely many $n$ for which $f(n)$ is $(k-1)$-power-free, except for possibly when $k=2^l$, when it may happen that $2^{l-1}\\mid f(n)$ for all $n$.\nHooley \\cite{Ho67} settled the first question, in fact providing a precise asymptotic for the number of such $n\\leq x$.\nHeath-Brown \\cite{He06} proved the answer to the second question is yes when $k\\geq 10$, and Browning \\cite{Br11} extended this to $k\\geq 9$ (in fact establishing an asymptotic formula for the number of such $n$).\nIn \\cite{Er65b} Erd\\H{o}s mentions the question of whether $2^n\\pm 1$ represents infinitely many $k$th power-free integers, or $n!\\pm 1$, but that these are 'intractable at present'. (See also [936].)\nReferences\n\n\n[Br11] Browning, T. D., Power-free values of polynomials. Arch. Math. (Basel) (2011), 139--150.\n\n[Er53] Erd\\H{o}s, P., Arithmetical properties of polynomials. J. London Math. Soc. (1953), 416--425.\n\n[Er65b] Erd\\H{o}s, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[He06] Heath-Brown, D. R., Counting rational points on algebraic varieties. (2006), 51--95.\n\n[Ho67] Hooley, C., On the power free values of polynomials. Mathematika (1967), 21--26.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2423, "problem_number": "EP-979", "title": "Erdős Problem #979", "statement": "Let $k\\geq 2$, and let $f_k(n)$ count the number of solutions to $ n=p_1^k+\\cdots+p_k^k, $ where the $p_i$ are prime numbers. Is it true that $\\limsup f_k(n)=\\infty$?", "background": "Erd\\H{o}s \\cite{Er37b} proved this is true when $k=2$, and also when $k=3$ (but this proof appears to be unpublished).\nReferences\n\n\n[Er37b] Erd\\H{o}s, Paul, On the {S}um and {D}ifference of {S}quares of {P}rimes. J. London Math. Soc. (1937), 133--136.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2424, "problem_number": "EP-983", "title": "Erdős Problem #983", "statement": "Let $n\\geq 2$ and $\\pi(n)r$ many $a\\in A$ are only divisible by primes from $\\{p_1,\\ldots,p_r\\}$.\nIs it true that $ 2\\pi(n^{1/2})-f(\\pi(n)+1,n)\\to \\infty $ as $n\\to \\infty$?\nIn general, estimate $f(k,n)$, particularly when $\\pi(n)+10$ and, for any constant $1>c>0$, $ f(cn,n)=\\log\\log n+(c_1+o(1))\\sqrt{2\\log\\log n}, $ where $ c=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{c_1}e^{-x^2/2}\\mathrm{d}x. $ \nReferences\n\n\n[Er70b] Erd\\H{o}s, P., Some applications of graph theory to number theory. Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications (Univ. North Carolina, Chapel Hill, N.C., 1970) (1970), 136-145.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2425, "problem_number": "EP-985", "title": "Erdős Problem #985", "statement": "Is it true that, for every prime $p$, there is a prime $q0$.", "background": "Spencer \\cite{Sp77} proved this for $k=3$ and Mattheus and Verstraete \\cite{MaVe23} proved this for $k=4$.\nThe best general bounds available are $ \\frac{n^{\\frac{k+1}{2}}}{(\\log n)^{\\frac{1}{k-2}-\\frac{k+1}{2}}}\\ll_k R(k,n) \\ll_k \\frac{n^{k-1}}{(\\log n)^{k-2}}. $ The lower bound was proved by Bohman and Keevash \\cite{BoKe10}. The upper bound was proved by Ajtai, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AKS80}. Li, Rousseau, and Zang \\cite{LRZ01} have shown that $\\ll_k$ in the upper bound can be improved to $\\leq (1+o(1))$.\nThe special case $k=3$ is the topic of [165] and $k=4$ is the topic of [166].\nThis problem is #6 in Ramsey Theory in the graphs problem collection.\nSee also [920].\nReferences\n\n\n[AKS80] Ajtai, Mikl\\'{o}s and Koml\\'{o}s, J\\'{a}nos and Szemer\\'{e}di, Endre, A note on Ramsey numbers. J. Combin. Theory Ser. A (1980), 354-360.\n\n[BoKe10] Bohman, Tom and Keevash, Peter, The early evolution of the {$H$}-free process. Invent. Math. (2010), 291--336.\n\n[LRZ01] Li, Yusheng and Rousseau, Cecil C. and Zang, Wenan, Asymptotic upper bounds for {R}amsey functions. Graphs Combin. (2001), 123--128.\n\n[MaVe23] Mattheus, S. and Verstraete, J., The asymptotics of $r(4,t)$. arXiv:2306.04007 (2023).\n\n[Sp77] Spencer, J., Asymptotic lower bounds for Ramsey functions. Discrete Math. (1977), 69-76.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2427, "problem_number": "EP-987", "title": "Erdős Problem #987", "statement": "Let $x_1,x_2,\\ldots \\in (0,1)$ be an infinite sequence and let $ A_k=\\limsup_{n\\to \\infty}\\left\\lvert \\sum_{j\\leq n} e(kx_j)\\right\\rvert, $ where $e(x)=e^{2\\pi ix}$.\nIs it true that $ \\limsup_{k\\to \\infty} A_k=\\infty? $ Is it possible for $A_k=o(k)$?", "background": "This is Problem 7.21 in \\cite{Ha74}, where it is attributed to Erd\\H{o}s.\nErd\\H{o}s \\cite{Er64b} remarks it is 'easy to see' that $ \\limsup_{k\\to \\infty}\\left(\\sup_n\\left\\lvert \\sum_{j\\leq n} e(kx_j)\\right\\rvert\\right)=\\infty. $ Erd\\H{o}s \\cite{Er65b} later found a 'very easy' proof that $A_k\\gg \\log k$ for infinitely many $k$. Clunie \\cite{Cl67} proved that $A_k\\gg k^{1/2}$ infinitely often, and that there exist sequences with $A_k\\leq k$ for all $k$. Tao has independently found a proof that $A_k\\gg k^{1/2}$ infinitely often (see the comment section).\nLiu \\cite{Li69} showed that, for any $\\epsilon>0$, $A_k\\gg k^{1-\\epsilon}$ infinitely often, under the additional assumption that there are only a finite number of distinct points. Clunie observed in the Mathscinet review of \\cite{Li69}, however, that under this assumption in fact $A_k=\\infty$ infinitely often.\nThe question of whether $A_k=o(k)$ is possible (repeated in \\cite{Er65b} and \\cite{Ha74}) seems to still be open.\nReferences\n\n\n[Cl67] Clunie, J., On a problem of {E}rd\\H{o}s. J. London Math. Soc. (1967), 133--136.\n\n[Er64b] Erd\\H{o}s, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.\n\n[Er65b] Erd\\H{o}s, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[Li69] Lindstr\"{o}m, B., An inequality for $B_2$-sequences. J. Combinatorial Theory (1969), 211-212.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2428, "problem_number": "EP-990", "title": "Erdős Problem #990", "statement": "Let $f=a_0+\\cdots+a_dx^d\\in \\mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\\ldots,z_d$ with corresponding arguments $\\theta_1,\\ldots,\\theta_d\\in [0,2\\pi]$, then for all intervals $I\\subseteq [0,2\\pi]$ $ \\left\\lvert (\\# \\theta_i \\in I) - \\frac{\\lvert I\\rvert}{2\\pi}d\\right\\rvert \\ll \\left(n\\log M\\right)^{1/2}, $ where $n$ is the number of non-zero coefficients of $f$ and $ M=\\frac{\\lvert a_0\\rvert+\\cdots +\\lvert a_d\\rvert}{(\\lvert a_0\\rvert\\lvert a_d\\rvert)^{1/2}}. $ ", "background": "Erd\\H{o}s and Tur\\'{a}n \\cite{ErTu50} proved such an upper bound with $n$ replaced by $d$.\nReferences\n\n\n[ErTu50] Erd\\H{o}s, P. and Tur\\'an, P., On the distribution of roots of polynomials. Ann. of Math. (2) (1950), 105--119.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2429, "problem_number": "EP-992", "title": "Erdős Problem #992", "statement": "Let $x_1 \\lambda>1$ for all $i$.\nReferences\n\n\n[Ba81] No reference found.\n\n\n[Ca50] Cassels, J. W. S., Some metrical theorems of {D}iophantine approximation. {III}. Proc. Cambridge Philos. Soc. (1950), 219--225.\n\n[ErKo49] Erd\\H{o}s, P. and Koksma, J. F., On the uniform distribution modulo {$1$} of sequences\n{$(f(n,\\theta))$}. Nederl. Akad. Wetensch., Proc. (1949), 851--854 = Indagationes Math. 11, 299--302.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2430, "problem_number": "EP-995", "title": "Erdős Problem #995", "statement": "Let $n_10$, for almost all $\\alpha$ $ \\limsup_{N\\to \\infty}\\frac{1}{N(\\log\\log N)^{\\frac{1}{2}-\\epsilon}}\\sum_{1\\leq k\\leq N}f(\\{\\alpha n_k\\})=\\infty. $ Erd\\H{o}s also proved that, for every lacunary sequence and $f\\in L^2$, for every $\\epsilon>0$, for almost all $\\alpha$, $ \\sum_{1\\leq k\\leq N}\\sum_{1\\leq k\\leq N}f(\\{\\alpha n_k\\})=o( N(\\log N)^{\\frac{1}{2}+\\epsilon}). $ Erd\\H{o}s \\cite{Er64b} thought that his lower bound was closer to the truth.\nReferences\n\n\n[Er49d] Erd\\H{o}s, P., On the strong law of large numbers. Trans. Amer. Math. Soc. (1949), 51--56.\n\n[Er64b] Erd\\H{o}s, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2431, "problem_number": "EP-996", "title": "Erdős Problem #996", "statement": "Let $n_10$ such that, if $ \\| f-f_n\\|_2 \\ll \\frac{1}{(\\log\\log\\log n)^{C}} $ then $ \\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{k\\leq N}f(\\{\\alpha n_k\\})=\\int_0^1 f(x)\\mathrm{d}x $ for almost every $\\alpha$?", "background": "Raikov proved the conclusion always holds (for every $f\\in L^2([0,1])$, with no assumption on $\\| f-f_n\\|_2$) if $n_k=a^k$ for some integer $a\\geq 2$. Erd\\H{o}s \\cite{Er64b} also asked whether this is true for $n_k=\\lfloor a^k\\rfloor$ for some $a>1$.\nKac, Salem, and Zygmund \\cite{KSZ48} proved that the conclusion holds if $ \\| f-f_n\\|_2 \\ll \\frac{1}{(\\log n)^{c}} $ for some constant $c>1$. Erd\\H{o}s \\cite{Er49d} proved that the conclusion holds if $ \\| f-f_n\\|_2 \\ll \\frac{1}{(\\log\\log n)^{c}} $ for some constant $c>1$. Matsuyama \\cite{Ma66} improved this to $c>1/2$.\nIn \\cite{Er64b} Erd\\H{o}s asked whether the conclusion holds for all bounded functions $f$ and lacunary sequences $n_k$.\nReferences\n\n\n[Er49d] Erd\\H{o}s, P., On the strong law of large numbers. Trans. Amer. Math. Soc. (1949), 51--56.\n\n[Er64b] Erd\\H{o}s, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.\n\n[KSZ48] Kac, M. and Salem, R. and Zygmund, A., A gap theorem. Trans. Amer. Math. Soc. (1948), 235--243.\n\n[Ma66] Matsuyama, Noboru, On the strong law of large numbers. Tohoku Math. J. (2) (1966), 259--269.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2432, "problem_number": "EP-997", "title": "Erdős Problem #997", "statement": "Call $x_1,x_2,\\ldots \\in (0,1)$ well-distributed if, for every $\\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\\subseteq [0,1]$, $ \\lvert \\# \\{ n0$ is an explicit constant.\nReferences\n\n\n[Ke60] Kesten, Harry, Uniform distribution {${\\rm mod}\\,1$}. Ann. of Math. (2) (1960), 445--471.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2434, "problem_number": "EP-1003", "title": "Erdős Problem #1003", "statement": "Are there infinitely many solutions to $\\phi(n)=\\phi(n+1)$, where $\\phi$ is the Euler totient function?", "background": "Erd\\H{o}s \\cite{Er85e} says that, presumably, for every $k\\geq 1$ the equation $ \\phi(n)=\\phi(n+1)=\\cdots=\\phi(n+k) $ has infinitely many solutions.\nErd\\H{o}s, Pomerance, and S\\'{a}rk\"{o}zy \\cite{EPS87} proved that the number of $n\\leq x$ with $\\phi(n)=\\phi(n+1)$ is at most $ \\frac{x}{\\exp((\\log x)^{1/3})}. $ See [946] for the analogous question with the divisor function.\nReferences\n\n\n[EPS87] Erd\\H{o}s, Paul and Pomerance, Carl and S\\'ark\"ozy, Andr\\'as, On locally repeated values of certain arithmetic functions.\n{III}. Proc. Amer. Math. Soc. (1987), 1--7.\n\n[Er85e] Erd\\H{o}s, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo,\nOkayama and Kyoto, 1984) (1985), 65-87.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2435, "problem_number": "EP-1004", "title": "Erdős Problem #1004", "statement": "Let $c>0$. If $x$ is sufficiently large then does there exist $n\\leq x$ such that the values of $\\phi(n+k)$ are all distinct for $1\\leq k\\leq (\\log x)^c$, where $\\phi$ is the Euler totient function?", "background": "Erd\\H{o}s, Pomerenace, and S\\'{a}rk\"{o}zy \\cite{EPS87} proved that if $\\phi(n+k)$ are all distinct for $1\\leq k\\leq K$ then $ K \\leq \\frac{n}{\\exp(c(\\log n)^{1/3})} $ for some constant $c>0$.\nSee [945] for the analogous problem with the divisor function.\nReferences\n\n\n[EPS87] Erd\\H{o}s, Paul and Pomerance, Carl and S\\'ark\"ozy, Andr\\'as, On locally repeated values of certain arithmetic functions.\n{III}. Proc. Amer. Math. Soc. (1987), 1--7.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2436, "problem_number": "EP-1005", "title": "Erdős Problem #1005", "statement": "Let $\\frac{a_1}{b_1},\\frac{a_2}{b_2},\\ldots$ be the Farey fractions of order $n\\geq 4$. Let $f(n)$ be the largest integer such that if $1\\leq k0$ such that $f(n)=(c+o(1))n$ for all large $n$?", "background": "The function $f(n)$ was first considered by Mayer \\cite{Ma42}, who proved $f(n)\\to \\infty$ as $n\\to \\infty$. Erd\\H{o}s \\cite{Er43} proved $f(n)\\gg n$.\nvan Doorn \\cite{vD25b} has proved that $ \\left(\\frac{1}{12}-o(1)\\right)n\\leq f(n) \\leq \\frac{1}{4}n+O(1), $ and conjectures that the upper bound is optimal.\nReferences\n\n\n[Er43] Erd\\H{o}s, P., A note on {F}arey series. Quart. J. Math. Oxford Ser. (1943), 82--85.\n\n[Ma42] Mayer, A. E., A mean value theorem concerning {F}arey series. Quart. J. Math. Oxford Ser. (1942), 48--57.\n\n[vD25b] W. van Doorn, Improved bounds for the Mayer-Erd\\H{o}s phenomenon on similarly ordered Farey fractions. arXiv:2509.00121 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2437, "problem_number": "EP-1011", "title": "Erdős Problem #1011", "statement": "Let $f_r(n)$ be minimal such that every graph on $n$ vertices with $\\geq f_r(n)$ edges and chromatic number $\\geq r$ contains a triangle. Determine $f_r(n)$.", "background": "Tur\\'{a}n's theorem implies $f_2(n)=\\lfloor n^2/4\\rfloor+1$. Erd\\H{o}s and Gallai \\cite{Er62d} proved $f_3(n)=\\lfloor \\frac{1}{4}(n-1)^2\\rfloor+2$.\nSimonovits showed in his PhD thesis (see the discussion on p. 358 of \\cite{Si74}) that $ f_r(n)=\\frac{n^2}{4}-\\frac{g(r)}{2}{n}+O(1), $ where $g(r)$ is the largest $m$ such that, for any triangle-free graph with chromatic number $\\geq r$, at least $m$ vertices of $G$ need to be removed to obtain a bipartite graph. Simonovits \\cite{Si74} notes $ \\frac{\\log r}{\\log\\log r}r^2 \\ll g(r) \\ll (\\log r)^2r^2. $ Hunter in the comments has noted that other results imply $g(r)\\asymp r^2\\log r$ - in fact $ (1/2-o(1))r^2\\log r\\leq g(r)\\leq (2+o(1))r^2\\log r. $ The lower bound follows from work of Davies and Illingworth \\cite{DaIl22} (see [1104]). The upper bound follows from work of Hefty, Horn, King, and Pfender \\cite{HHKP25} on $R(3,k)$.\nRen, Wang, Wang, and Yang \\cite{RWWY24} showed that, for $n\\geq 150$, $ f_4(n)=\\left\\lfloor\\frac{(n-3)^2}{4}\\right\\rfloor+6. $ \nReferences\n\n\n[DaIl22] Davies, Ewan and Illingworth, Freddie, The {$\\chi$}-{R}amsey problem for triangle-free graphs. SIAM J. Discrete Math. (2022), 1124--1134.\n\n[Er62d] Erd\\H{o}s, P., On a theorem of {R}ademacher-{T}ur\\'an. Illinois J. Math. (1962), 122--127.\n\n[HHKP25] Z. Hefty, P. Horn, D. King, and F. Pfender, Improving $R(3,k)$ in just two bites. arXiv:2510.19718 (2025).\n\n[RWWY24] S. Ren, J. Wang, S. Wang, and W. Yang, Extremal triangle-free graphs with chromatic number at least four. arXiv:2404.07486 (2024).\n\n[Si74] Simonovits, M., Extermal graph problems with symmetrical extremal graphs.\n{A}dditional chromatic conditions. Discrete Math. (1974), 349--376.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2438, "problem_number": "EP-1013", "title": "Erdős Problem #1013", "statement": "Let $h_3(k)$ be the minimal $n$ such that there exists a triangle-free graph on $n$ vertices with chromatic number $k$. Find an asymptotic for $h_3(k)$, and also prove $ \\lim_{k\\to \\infty}\\frac{h_3(k+1)}{h_3(k)}=1. $ ", "background": "The function $h_3(k)$ is dual to the function $f(n)$ considered in [1104], in that $h_3(k)= n$ if and only if $n$ is minimal such that $f(n)=k$.\nGraver and Yackel \\cite{GrYa68} proved $ h_3(k)\\gg \\frac{\\log k}{\\log\\log k}k^2. $ The bounds for $f(n)$ from [1104] imply $ \\left(\\frac{1}{2}-o(1)\\right)k^2\\log k\\leq h_3(k) \\leq (1+o(1))k^2\\log k. $ See also [920] for a generalisation to $K_r$-free graphs.\nReferences\n\n\n[GrYa68] Graver, Jack E. and Yackel, James, Some graph theoretic results associated with Ramsey's theorem. J. Combinatorial Theory (1968), 125--175.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2439, "problem_number": "EP-1014", "title": "Erdős Problem #1014", "statement": "Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on $l$ vertices.\nProve, for fixed $k\\geq 3$, that $ \\lim_{l\\to \\infty}\\frac{R(k,l+1)}{R(k,l)}=1. $ ", "background": "This is open even for $k=3$.\nSee also [544] for other behaviour of $R(3,k)$, and [1030] for the diagonal version of this question.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2440, "problem_number": "EP-1016", "title": "Erdős Problem #1016", "statement": "Let $h(n)$ be minimal such that there is a graph on $n$ vertices with $n+h(n)$ edges which contains a cycle on $k$ vertices, for all $3\\leq k\\leq n$. Estimate $h(n)$. In particular, is it true that $ h(n) \\geq \\log_2n+\\log_*n-O(1), $ where $\\log_*n$ is the iterated logarithmic function?", "background": "Such graphs are called pancyclic. A problem of Bondy \\cite{Bo71}, who claimed a proof (without details) of $ \\log_2(n-1)-1\\leq h(n) \\leq \\log_2n+\\log_*n+O(1). $ Erd\\H{o}s \\cite{Er71} believed the upper bound is closer to the truth, but could not even prove $h(n)-\\log_2n\\to \\infty$.\nA proof of the above lower bound is provided by Griffin \\cite{Gr13}. The first published proof of the upper bound appears to be in Chapter 4.5 of George, Khodkar, and Wallis \\cite{GKW16}.\nReferences\n\n\n[Bo71] Bondy, J. A., Pancyclic graphs. {I}. J. Combinatorial Theory Ser. B (1971), 80--84.\n\n[Er71] Erd\\H{o}s, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc.\nConf., Oxford, 1969) (1971), 97-109.\n\n[GKW16] George, John C. and Khodkar, Abdollah and Wallis, W. D., Pancyclic and bipancyclic graphs. (2016), xii+108.\n\n[Gr13] S. Griffin, Minimal Pancyclicity. arXiv:1312.0274 (2013).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2441, "problem_number": "EP-1017", "title": "Erdős Problem #1017", "statement": "Let $f(n,k)$ be such that every graph on $n$ vertices and $k$ edges can be partitioned into at most $f(n,k)$ edge-disjoint complete graphs. Estimate $f(n,k)$ for $k>n^2/4$.", "background": "The function $f(n,k)$ is sometimes called the clique partition number.\nErd\\H{o}s, Goodman, and P\\'{o}sa \\cite{EGP66} proved that $f(n,k)\\leq n^2/4$ for all $k$ (and in fact the complete graphs can be taken to be edges and triangles), which is best possible in general, as witnessed for example by a complete bipartite graph. In \\cite{Er71} Erd\\H{o} asks vaguely whether this result can be 'sharpened' for $k>n^2/4$.\nLov\\'{a}sz \\cite{Lo68} proved that every graph on $n$ vertices and $k$ edges is the union of $\\binom{n}{2}-k+t$ complete graphs, where $t$ is maximal such that $t^2-t\\leq \\binom{n}{2}-k$, but without the assumption that the complete graphs are edge disjoint. Lov\\'{a}sz's result is sharp in many cases.\nIf $k>n^2/4$ and the graph contains no $K_4$ then this is equivalent to finding the minimum number of edge disjoint triangles. This special case was also asked about by Erd\\H{o}s. A complete answer was provided by Gy\"{o}ri and Keszegh \\cite{GyKe17}, who proved that every $K_4$-free graph with $n$ vertices and $\\lfloor n^2/4\\rfloor+m$ edges contins $m$ pairwise edge disjoint triangles.\nSee also [184] for an analogous problem decomposing into edges and cycles, and [583] for decomposing into paths. The clique partition problem for chordal graphs is the subject of [81].\nReferences\n\n\n[EGP66] Erd\\H{o}s, Paul and Goodman, A. W. and P\\'{o}sa, Lajos, The representation of a graph by set intersections. Canadian J. Math. (1966), 106-112.\n\n[Er71] Erd\\H{o}s, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc.\nConf., Oxford, 1969) (1971), 97-109.\n\n[GyKe17] Gy\\H{o}ri, Ervin and Keszegh, Bal\\'azs, On the number of edge-disjoint triangles in {$K_4$}-free\ngraphs. Combinatorica (2017), 1113--1124.\n\n[Lo68] Lov\\'{a}sz, L., On covering of graphs. Theory of Graphs (Proc. Colloq., Tihany, 1966) (1968), 231-236.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2442, "problem_number": "EP-1021", "title": "Erdős Problem #1021", "statement": "Is it true that, for every $k\\geq 3$, there is a constant $c_k>0$ such that $ \\mathrm{ex}(n,G_k) \\ll n^{3/2-c_k}, $ where $G_k$ is the bipartite graph between $\\{y_1,\\ldots,y_k\\}$ and $\\{z_1,\\ldots,z_{\\binom{k}{2}}\\}$, with each $z_j$ joined to a unique pair of $y_i$?", "background": "A conjecture of Erd\\H{o}s and Simonovits, who proved (in unpublished work) that in such a result one must have $c_k\\to 0$ as $k\\to \\infty$. Erd\\H{o}s \\cite{Er71} could not even prove whether $\\mathrm{ex}(n,G_k)=o(n^{3/2})$.\nWhen $k=3$ the graph $G_3$ is the $6$-cycle $C_6$, for which Erd\\H{o}s \\cite{Er64c} and Bondy and Simonovits \\cite{BoSi74} proved $\\mathrm{ex}(n,C_6)\\ll n^{7/6}$ (see [572]).\nThe graph $G_k$ is the graph $H_k$ of [926] with the vertex $x$ omitted, and can also be described as the $1$-subdivision of $K_k$.\nThis was proved by Conlon and Lee \\cite{CoLe21}, with a value of $c_k=6^{-k}$. This was improved to $c_k=\\frac{1}{4k-6}$ by Janzer \\cite{Ja19}.\nReferences\n\n\n[BoSi74] Bondy, J. A. and Simonovits, M., Cycles of even length in graphs. J. Combinatorial Theory Ser. B (1974), 97-105.\n\n[CoLe21] Conlon, David and Lee, Joonkyung, On the extremal number of subdivisions. Int. Math. Res. Not. IMRN (2021), 9122--9145.\n\n[Er64c] Erd\\H{o}s, P., Extremal problems in graph theory. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963) (1964), 29-36.\n\n[Er71] Erd\\H{o}s, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc.\nConf., Oxford, 1969) (1971), 97-109.\n\n[Ja19] Janzer, Oliver, Improved bounds for the extremal number of subdivisions. Electron. J. Combin. (2019), Paper No. 3.3, 6.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2443, "problem_number": "EP-1022", "title": "Erdős Problem #1022", "statement": "Is there a constant $c_t$, where $c_t\\to \\infty$ as $t\\to \\infty$, such that if $\\mathcal{F}$ is a finite family of finite sets, all of size at least $t$, and for every set $X$ there are $0$ such that $ \\lim_k \\frac{R(k+1,k)}{R(k,k)}> 1+c. $ ", "background": "A problem of Erd\\H{o}s and S\\'{o}s, who could not even prove whether $R(k+1,k)-R(k,k)>k^c$ for any $c>1$.\nIt is trivial that $R(k+1,k)-R(k,k)\\geq k-2$. Burr, Erd\\H{o}s, Faudree, and Schelp \\cite{BEFS89} proved $ R(k+1,k)-R(k,k)\\geq 2k-5. $ See also [544] for a similar question concerning $R(3,k)$, and [1014] for the general off-diagonal case.\nReferences\n\n\n[BEFS89] Burr, S. A. and Erd\\H{o}s, P. and Faudree, R. J. and Schelp, R.\nH., On the difference between consecutive {R}amsey numbers. Utilitas Math. (1989), 115--118.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2446, "problem_number": "EP-1032", "title": "Erdős Problem #1032", "statement": "We say that a graph is $4$-chromatic critical if it has chromatic number $4$, and removing any edge decreases the chromatic number to $3$.\nIs there, for arbitrarily large $n$, a $4$-chromatic critical graph on $n$ vertices with minimum degree $\\gg n$?", "background": "In \\cite{Er93} Erd\\H{o}s said he asked this 'more than 20 years ago'.\nDirac gave an example of a $6$-chromatic critical graph with minimum degree $>n/2$. This problem is also open for $5$-chromatic critical graphs.\nSimonovits \\cite{Si72} and Toft \\cite{To72} independently constructed $4$-chromatic critical graphs with minimum degree $\\gg n^{1/3}$. Toft conjectured that a $4$-chromatic critical graph on $n$ vertices has at least $(\\frac{5}{3}+o(1))n$ vertices, and has examples to show this would be the best possible.\nSee also [917] and [944].\nReferences\n\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[Si72] Simonovits, M., On colour-critical graphs. Studia Sci. Math. Hungar. (1972), 67--81.\n\n[To72] Toft, B., Two theorems on critical {$4$}-chromatic graphs. Studia Sci. Math. Hungar. (1972), 83--89.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2447, "problem_number": "EP-1033", "title": "Erdős Problem #1033", "statement": "Let $h(n)$ be such that every graph on $n$ vertices with $>n^2/4$ many edges contains a triangle whose vertices have degrees summing to at least $h(n)$. Estimate $h(n)$. In particular, is it true that $ h(n)\\geq (2(\\sqrt{3}-1)-o(1))n? $ ", "background": "Erd\\H{o}s and Laskar \\cite{ErLa85} proved $ 2(\\sqrt{3}-1)n \\geq h(n) \\geq (1+c)n $ for some $c>0$. The lower bound was improved to $\\frac{21}{16}n$ by Fan \\cite{Fa88}.\nReferences\n\n\n[ErLa85] Erd\\H{o}s, Paul and Laskar, Renu, A note on the size of a chordal subgraph. Congr. Numer. (1985), 81--86.\n\n[Fa88] Fan, Genghua, Degree sum for a triangle in a graph. J. Graph Theory (1988), 249--263.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2448, "problem_number": "EP-1035", "title": "Erdős Problem #1035", "statement": "Is there a constant $c>0$ such that every graph on $2^n$ vertices with minimum degree $>(1-c)2^n$ contains the $n$-dimensional hypercube $Q_n$?", "background": "Erd\\H{o}s \\cite{Er93} says 'if the conjecture is false, two related problems could be asked':\n{UL}\n{LI}Determine or estimate the smallest $m>2^n$ such that every graph on $m$ vertices with minimum degree $>(1-c)2^n$ contains a $Q_n$, and {/LI}\n{LI}For which $u_n$ is it true that every graph on $2^n$ vertices with minimum degree $>2^n-u_n$ contains a $Q_n$.{/LI}\n{/UL}\nSee also [576] for the extremal number of edges that guarantee a $Q_n$.\nReferences\n\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2449, "problem_number": "EP-1038", "title": "Erdős Problem #1038", "statement": "Determine the infimum and supremum of $ \\lvert \\{ x\\in \\mathbb{R} : \\lvert f(x)\\rvert < 1\\}\\rvert $ as $f\\in \\mathbb{R}[x]$ ranges over all non-constant monic polynomials, all of whose roots are real and in the interval $[-1,1]$.", "background": "A problem of Erd\\H{o}s, Herzog, and Piranian \\cite{EHP58}, who proved that the measure of the set in question is always at most $2\\sqrt{2}$ under the assumption that all the roots are in $\\{-1,1\\}$, and conjecture this is the best possible upper bound.\nThey also note that the infimum of the set in question is less than $2$, as witnessed by $f(x)=(x+1)(x-1)^m$ for $m\\geq 3$. They further note that if the roots are restricted to $[-2,2]$ then the infimum is zero, as witnessed by a small perturbation of the Chebyshev polynomials.\nThey further conjectured that, if the roots are restricted to $[-2,2]$, then $ \\lvert \\{ x\\in \\mathbb{R} : \\lvert f(x)\\rvert < 1\\}\\rvert\\geq n^{-c} $ for an absolute constant $c>0$. This was proved by Pommerenke \\cite{Po61}, who in fact showed that this set must contain an interval of width $\\gg n^{-4}$.\nThe current best known bounds (see the discussion in the comments) are $ 1.519\\approx 2^{4/3}-1\\leq \\inf \\leq 1.835\\cdots $ and $ \\sup = 2\\sqrt{2}\\approx 2.828. $ \nReferences\n\n\n[EHP58] Erd\\H{o}s, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.\n\n[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2450, "problem_number": "EP-1039", "title": "Erdős Problem #1039", "statement": "Let $f(z)=\\prod_{i=1}^n(z-z_i)\\in \\mathbb{C}[z]$ with $\\lvert z_i\\rvert \\leq 1$ for all $i$. Let $\\rho(f)$ be the radius of the largest disc which is contained in $\\{z: \\lvert f(z)\\rvert< 1\\}$.\nDetermine the behaviour of $\\rho(f)$. In particular, is it always true that $\\rho(f)\\gg 1/n$?", "background": "A problem of Erd\\H{o}s, Herzog, and Piranian, who note that $f(z)=z^n-1$ has $\\rho(f) \\leq \\frac{\\pi/2}{n}$.\nPommerenke \\cite{Po61} proved that $ \\rho(f) \\geq \\frac{1}{2en^2}. $ Krishnapur, Lundberg, and Ramachandran \\cite{KLR25} proved $ \\rho(f) \\gg \\frac{1}{n\\sqrt{\\log n}}. $ \nReferences\n\n\n[KLR25] M. Krishnapur, E. Lundberg, and K. Ramachandran, On the area of polynomial lemniscates. arXiv:2503.18270 (2025).\n\n[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2451, "problem_number": "EP-1040", "title": "Erdős Problem #1040", "statement": "Let $F\\subseteq \\mathbb{C}$ be a closed infinite set, and let $\\mu(F)$ be the infimum of $ \\lvert \\{ z: \\lvert f(z)\\rvert < 1\\}\\rvert, $ as $f$ ranges over all polynomials of the shape $\\prod (z-z_i)$ with $z_i\\in F$.\nIs $\\mu(F)$ determined by the transfinite diameter of $F$? In particular, is $\\mu(F)=0$ whenever the transfinite diameter of $F$ is $\\geq 1$?", "background": "A problem of Erd\\H{o}s, Herzog, and Piranian \\cite{EHP58}, who show that the answer is yes if $F$ is a line segment or disc, and that if the transfinite diameter is $<1$ then $\\{ z: \\lvert f(z)\\rvert < 1\\}$ always contains a disc of radius $\\gg_F 1$.\nErd\\H{o}s and Netanyahu \\cite{ErNe73} proved that if $F$ is also bounded and connected, with transfinite diameter $01$ be a rational number. Is $ \\sum_{n=1}^\\infty\\frac{1}{t^n-1}=\\sum_{n=1}^\\infty \\frac{\\tau(n)}{t^n} $ irrational, where $\\tau(n)$ counts the divisors of $n$?", "background": "A conjecture of Chowla. Erd\\H{o}s \\cite{Er48} proved that this is true if $t\\geq 2$ is an integer.\nReferences\n\n\n[Er48] Erd\\H{o}s, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.) (1948), 63-66.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2453, "problem_number": "EP-1051", "title": "Erdős Problem #1051", "statement": "Is it true that if $a_11 $ then $ \\sum_{n=1}^\\infty \\frac{1}{a_na_{n+1}} $ is irrational?", "background": "In \\cite{Er88c} Erd\\H{o}s notes this is true if $a_n\\to \\infty$ 'rapidly'.\nReferences\n\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2454, "problem_number": "EP-1052", "title": "Erdős Problem #1052", "statement": "A unitary divisor of $n$ is $d\\mid n$ such that $(d,n/d)=1$. A number $n\\geq 1$ is a unitary perfect number if it is the sum of its unitary divisors (aside from $n$ itself).\nAre there only finite many unitary perfect numbers?", "background": "Guy \\cite{Gu04} reports that Carlitz, Erd\\H{o}s, and Subbarao offer \\$10 for settling this question, and that Subbarao offers 10 cents for each new example.\nThere are no odd unitary perfect numbers. There are five known unitary perfect numbers (A002827 in the OEIS): $ 6, 60, 90, 87360, 146361946186458562560000. $ This is problem B3 in Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2455, "problem_number": "EP-1053", "title": "Erdős Problem #1053", "statement": "Call a number $k$-perfect if $\\sigma(n)=kn$, where $\\sigma(n)$ is the sum of the divisors of $n$. Must $k=o(\\log\\log n)$?", "background": "A question of Erd\\H{o}s, as reported in problem B2 of Guy's collection \\cite{Gu04}. Guy further writes 'It has even been suggested that there may be only finitely many $k$-perfect numbers with $k\\geq 3$.' The largest $k$ for which a $k$-perfect number has been found is $k=11$ - see this page for more information.\nThese are known as multiply perfect numbers. When $k=2$ this is the definition of a perfect number.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2456, "problem_number": "EP-1054", "title": "Erdős Problem #1054", "statement": "Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\\geq 1$.\nIs it true that $f(n)=o(n)$? Or is this true only for almost all $n$, and $\\limsup f(n)/n=\\infty$?", "background": "A question of Erd\\H{o}s reported in problem B2 of Guy's collection \\cite{Gu04}. The function $f(n)$ is undefined for $n=2$ and $n=5$, but is likely well-defined for all $n\\geq 6$ (which would follow from a strong form of Goldbach's conjecture).\nThe sequence of values of $f(n)$ is given by A167485 in the OEIS.\nSee also [468].\nThe strong claim that $f(n)=o(n)$ was disproved by Tao in the comments to [468], in which he proves that the upper density of $\\{ n : f(n)\\leq \\delta n\\}$ is $\\ll \\delta^2$.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2457, "problem_number": "EP-1055", "title": "Erdős Problem #1055", "statement": "A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.\nAre there infinitely many primes in each class? If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave?", "background": "A classification due to Erd\\H{o}s and Selfridge. It is easy to prove that the number of primes $\\leq n$ in class $r$ is at most $n^{o(1)}$.\nThe sequence $p_r$ begins $2,13,37,73,1021$ (A005113 in the OEIS). Erd\\H{o}s thought $p_r^{1/r}\\to \\infty$, while Selfridge thought it quite likely to be bounded.\nA similar question can be asked replacing $p+1$ with $p-1$.\nThis is problem A18 in Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2458, "problem_number": "EP-1056", "title": "Erdős Problem #1056", "statement": "Let $k\\geq 2$. Does there exist a prime $p$ and consecutive intervals $I_1,\\ldots,I_k$ such that $ \\prod_{n\\in I_i}n \\equiv 1\\pmod{p} $ for all $1\\leq i\\leq k$?", "background": "This is problem A15 in Guy's collection \\cite{Gu04}, where he reports that in a letter in 1979 Erd\\H{o}s observed that $ 3\\cdot 4\\equiv 5\\cdot 6\\cdot 7\\equiv 1\\pmod{11}, $ establishing the case $k=2$. Makowski \\cite{Ma83} found, for $k=3$, $ 2\\cdot 3\\cdot 4\\cdot 5\\equiv 6\\cdot 7\\cdot 8\\cdot 9\\cdot 10\\cdot 11\\equiv 12\\cdot 13\\cdot 14\\cdot 15\\equiv 1\\pmod{17}. $ Noll and Simmons asked, more generally, whether there are solutions to $q_1!\\equiv\\cdots \\equiv q_k!\\pmod{p}$ for arbitrarily large $k$ (with $q_1<\\cdots0$. Pomerance \\cite{Po89} gave a heuristic suggesting that this is the true order of growth, and in fact $ C(x)= x \\exp\\left(-(1+o(1))\\frac{\\log x\\log\\log\\log x}{\\log\\log x}\\right). $ Alford, Granville, and Pomerance \\cite{AGP94} proved that $C(x)\\to \\infty$, and in fact $C(x)>x^{2/7}$ for large $x$. The lower bound $ C(x)> x^{0.33336704} $ was proved by Harman \\cite{Ha08}. This exponent was improved to $0.3389$ by Lichtman \\cite{Li22}.\nKorselt observed that $n$ being a Carmichael number is equivalent to $n$ being squarefree and $p-1\\mid n-1$ for all primes $p\\mid n$.\nThis is discussed in problem A13 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[AGP94] Alford, W. R. and Granville, Andrew and Pomerance, Carl, There are infinitely many {C}armichael numbers. Ann. of Math. (2) (1994), 703--722.\n\n[Er56c] Erd\\H{o}s, P., On pseudoprimes and {C}armichael numbers. Publ. Math. Debrecen (1956), 201--206.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ha08] Harman, Glyn, Watt's mean value theorem and {C}armichael numbers. Int. J. Number Theory (2008), 241--248.\n\n[Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022).\n\n[Po89] Pomerance, Carl, Two methods in elementary analytic number theory. (1989), 135--161.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2460, "problem_number": "EP-1059", "title": "Erdős Problem #1059", "statement": "Are there infinitely many primes $p$ such that $p-k!$ is composite for each $k$ such that $1\\leq k!l$, and all the numbers $n-k!$ are composite for $1\\leq k\\leq l$.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2461, "problem_number": "EP-1060", "title": "Erdős Problem #1060", "statement": "Let $f(n)$ count the number of solutions to $k\\sigma(k)=n$, where $\\sigma(k)$ is the sum of divisors of $k$. Is it true that $f(n)\\leq n^{o(\\frac{1}{\\log\\log n})}$? Perhaps even $\\leq (\\log n)^{O(1)}$?", "background": "This is discussed in problem B11 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2462, "problem_number": "EP-1061", "title": "Erdős Problem #1061", "statement": "How many solutions are there to $ \\sigma(a)+\\sigma(b)=\\sigma(a+b) $ with $a+b\\leq x$, where $\\sigma$ is the sum of divisors function? Is it $\\sim cx$ for some constant $c>0$?", "background": "A question of Erd\\H{o}s reported in problem B15 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2463, "problem_number": "EP-1062", "title": "Erdős Problem #1062", "statement": "Let $f(n)$ be the size of the largest subset $A\\subseteq \\{1,\\ldots,n\\}$ such that there are no three distinct elements $a,b,c\\in A$ such that $a\\mid b$ and $a\\mid c$. How large can $f(n)$ be? Is $\\lim f(n)/n$ irrational?", "background": "The example $[m+1,3m+2]$ shows that $f(n)\\geq\\lceil \\frac{2}{3}n\\rceil$. Lebensold \\cite{Le76} has shown that, for large $n$, $ 0.6725 n \\leq f(n) \\leq 0.6736 n. $ This is problem B24 in Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Le76] Lebensold, Kenneth, A divisibility problem. Studies in Appl. Math. (1976/77), 291--294.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2464, "problem_number": "EP-1063", "title": "Erdős Problem #1063", "statement": "Let $k\\geq 2$ and define $n_k\\geq 2k$ to be the least value of $n$ such that $n-i$ divides $\\binom{n}{k}$ for all but one $0\\leq i1$. Is it true that $g_d(n) \\gg n/d$ in general? The upper bound $g_d(n) \\ll n/d$ is trivial, considering widely spaced unit simplices.\nSee [1070] for the general estimate of independence number of unit distance graphs.\nReferences\n\n\n[Cs98] Csizmadia, G., On the independence number of minimum distance graphs. Discrete Comput. Geom. (1998), 179--187.\n\n[PaTo96] Pach, J\\'anos and T\\'oth, G\\'{e}za, On the independence number of coin graphs. Geombinatorics (1996), 30--33.\n\n[Po85] Pollack, R., Increasing the minimum distance of a set of points. J. Combin. Theory Ser. A (1985), 450.\n\n[Sw02] Swanepoel, Konrad J., Independence numbers of planar contact graphs. Discrete Comput. Geom. (2002), 649--670.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2467, "problem_number": "EP-1068", "title": "Erdős Problem #1068", "statement": "Does every graph with chromatic number $\\aleph_1$ contain a countable subgraph which is infinitely vertex-connected?", "background": "I do not think this was originally a question of Erd\\H{o}s - it appears in \\cite{BoPi24} as a 'version of the Erd\\H{o}s-Hajnal problem' (which is [1067]).\nI could not in fact find this in the paper of Erd\\H{o}s and Hajnal \\cite{ErHa66}, however, and hence the first place it appears may in fact be in \\cite{BoPi24}. In hindsight this should not have been included as a separate problem, but this has been discovered too late, and so we must leave it here.\nWe say a graph is infinitely (vertex) connected if any two vertices are connected by infinitely many pairwise vertex-disjoint paths.\nSoukup \\cite{So15} constructed a graph with uncountable chromatic number in which every uncountable set is finitely vertex-connected. A simpler construction was given by Bowler and Pitz \\cite{BoPi24}.\nSee also [1067].\nReferences\n\n\n[BoPi24] N. Bowler and M. Pitz, A note on uncountably chromatic graphs. arXiv:2402.05984 (2024).\n\n[ErHa66] Erd\\H{o}s, P. and Hajnal, A., On chromatic number of graphs and set-systems. Acta Math. Acad. Sci. Hungar. (1966), 61-99.\n\n[So15] Soukup, D\\'aniel T., Trees, ladders and graphs. J. Combin. Theory Ser. B (2015), 96--116.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2468, "problem_number": "EP-1070", "title": "Erdős Problem #1070", "statement": "Let $f(n)$ be maximal such that, given any $n$ points in $\\mathbb{R}^2$, there exist $f(n)$ points such that no two are distance $1$ apart. Estimate $f(n)$. In particular, is it true that $f(n)\\geq n/4$?", "background": "In other words, estimate the minimal independence number of a unit distance graph with $n$ vertices. If $\\omega$ is the independence number and $\\chi$ is the chromatic number then $\\omega \\chi\\geq n$, and hence $f(n)\\geq n/\\chi$, where $\\chi$ is the answer to the Hadwiger-Nelson problem [508].\nThe Moser spindle shows $f(n)\\leq \\frac{2}{7}n\\approx 0.285n$. Larman and Rogers \\cite{LaRo72} noted that if $m_1$ is the supremum of the upper densities of measurable subsets of $\\mathbb{R}^2$ which have no unit distance pairs then $ f(n)\\geq m_1n. $ Croft \\cite{Cr67} gave the best-known lower bound of $m_1\\geq 0.22936$ and hence $ 0.22936n \\leq f(n) \\leq \\frac{2}{7}n\\approx 0.285n. $ Ambrus, Csisz\\'{a}rik, Matolcsi, Varga, and Zs\\'{a}mboki \\cite{ACMVZ23} have proved that $m_1\\leq 0.247$, and hence this approach cannot achieve $f(n)\\geq n/4$. See [232] for more on $m_1$.\nMatolcsi, Ruzsa, Varga, and Zs\\'{a}mboki \\cite{MRVZ23} have improved the upper bound to $ f(n) \\leq \\left(\\frac{1}{4}+o(1)\\right)n. $ They conjecture that $m_1=0.22936\\cdots$ (the lower bound of Croft mentioned above) and $f(n)=(1/4+o(1))n$.\nIf we also insist that no two points are distance $<1$ apart then this is problem becomes [1066].\nReferences\n\n\n[ACMVZ23] G. Ambrus, A. Csisz\\'{a}rik, M. Matolcsi, D. Varga, and P. Zs\\'{a}mboki, The density of planar sets avoiding unit distances. arXiv:2207.14179 (2023).\n\n[Cr67] H. T. Croft, Incidence incidents. Eureka (1967), 22-26.\n\n[LaRo72] Larman, D. G. and Rogers, C. A., The realization of distances within sets in Euclidean space. Mathematika (1972), 1-24.\n\n[MRVZ23] M. Matolcsi, I. Z. Ruzsa, D. Varga, and P. Zs\\'{a}mboki, The fractional chromatic number of the plane is at least $4$. arXiv:2311.10069 (2023).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2469, "problem_number": "EP-1071", "title": "Erdős Problem #1071", "statement": "Is there a finite set of unit line segments (rotated and translated copies of $(0,1)$) in the unit square, no two of which intersect, which are maximal with respect to this property?\nIs there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite?", "background": "A question of Erd\\H{o}s and T\\'{o}th. The answer to the first question is yes (which Erd\\H{o}s gave Danzer \\$10 for). There is no prize mentioned in \\cite{Er87b} for the (still open) second question.\nThere are two examples Erd\\H{o}s gives in \\cite{Er87b}, the {IMAGE=1071-one,first} by Danzer, the {IMAGE=1071-two,second} by an unnamed participant.\nIn \\cite{Er87b} he further asks what happens if the unit line segments are rotated/translated copies of $[0,1]$ that are allowed to intersect only at their endpoints.\nReferences\n\n\n[Er87b] Erd\\H{o}s, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Si\\'{o}fok, 1985) (1987), 167-177.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2470, "problem_number": "EP-1072", "title": "Erdős Problem #1072", "statement": "For any prime $p$, let $f(p)$ be the least integer such that $f(p)!+1\\equiv 0\\pmod{p}$.\nIs it true that there are infinitely many $p$ for which $f(p)=p-1$?\nIs it true that $f(p)/p\\to 0$ for almost all $p$?", "background": "Questions formulated by Erd\\H{o}s, Hardy, and Subbarao \\cite{HaSu02}, who believed that the number of $p\\leq x$ for which $f(p)=p-1$ is $o(x/\\log x)$.\nThese are mentioned in problem A2 of Guy's collection.\nReferences\n\n\n[HaSu02] Hardy, G. E. and Subbarao, M. V., A modified problem of Pillai and some related questions. Amer. Math. Monthly (2002), 554--559.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2471, "problem_number": "EP-1073", "title": "Erdős Problem #1073", "statement": "Let $A(x)$ count the number of composite $ur^{-r}$ such that, for any $\\epsilon>0$, if $n$ is sufficiently large, the following holds.\nAny $r$-uniform hypergraph on $n$ vertices with at least $(1+\\epsilon)(n/r)^r$ many edges contains a subgraph on $m$ vertices with at least $c_rm^r$ edges, where $m=m(n)\\to \\infty$ as $n\\to \\infty$.", "background": "Erd\\H{o}s \\cite{Er64f} proved that this is true with $c_r=r^{-r}$ whenever the graph has at least $\\epsilon n^r$ many edges.\nReferences\n\n\n[Er64f] Erd\\H{o}s, P., On extremal problems of graphs and generalized graphs. Israel J. Math. (1964), 183--190.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2474, "problem_number": "EP-1083", "title": "Erdős Problem #1083", "statement": "Let $d\\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\\mathbb{R}^d$ determines at least $m$ distinct distances. Estimate $f_d(n)$ - in particular, is it true that $ f_d(n)=n^{\\frac{2}{d}-o(1)}? $ ", "background": "A generalisation of the distinct distance problem [89] to higher dimensions. Erd\\H{o}s \\cite{Er46b} proved $ n^{1/d}\\ll_d f_d(n)\\ll_d n^{2/d}, $ the upper bound construction being given by a set of lattice points.\n{UL}\n{LI} Clarkson, Edelsbrunner, Gubias, Sharir, and Welzl \\cite{CEGSW90} proved $f_3(n)\\gg n^{1/2}$.{/LI}\n{LI}Aronov, Pach, Sharir, and Tardos \\cite{APST04} proved $f_d(n)\\gg n^{\\frac{1}{d-90/77}-o(1)}$ for any $d\\geq 3$ (for example, $f_3(n)\\gg n^{0.546}$).{/LI}\n{LI}Solymosi and Vu \\cite{SoVu08} proved $f_3(n) \\gg n^{3/5}$ and $ f_d(n)\\gg_d n^{\\frac{2}{d}-\\frac{c}{d^2}} $ for all $d\\geq 4$ for some constant $c>0$. (The result in their paper for $d=3$ is slightly weaker than stated here, but uses as a black box the bound for distinct distances in $2$ dimensions; we have recorded the consequence of combining their method with the work of Guth and Katz on [89].){/LI}\n{/UL}\nThe function $f_d(n)$ is essentially the inverse of the function $g_d(n)$ considered in [1089] - with our definitions, $g_d(n)>m$ if and only if $f_d(m)0$, which the triangular lattice shows is the best possible up to the value of $c$. In \\cite{Er75f} he speculated that the triangular lattice is exactly the best possible, and in particular $ f_2(3n^2+3n+1)=9n^2+6n. $ Harborth \\cite{Ha74b} proved that $ f_2(n)=\\lfloor 3n-\\sqrt{12n-3}\\rfloor $ for all $n\\geq 2$.\nIn \\cite{Er75f} he claims the existence of $c_1,c_2>0$ such that $ 6n-c_1n^{2/3}< f_3(n) < 6n-c_2n^{2/3}. $ An upper bound of $ f_3(n) < 6n-0.926n^{2/3} $ for all $n\\geq 2$ was proved by Bezdek and Reid \\cite{BeRe13}.\nIn general, it is known that $ (d-o(1))n \\leq f_d(n) \\leq 2^{O(d)}n, $ the lower bound coming from points arranged in an integer grid and the upper bound from the fact that $2^{O(d)}$ many non-intersecting congruent balls can touch a fixed ball (the kissing number problem).\nA recent survey on contact numbers for sphere packings is by Bezdek and Khan \\cite{BeKa18}.\nSee [223] for the analogous problem with maximal distance $1$.\nReferences\n\n\n[BeKa18] No reference found.\n\n\n[BeRe13] Bezdek, K\\'aroly and Reid, Samuel, Contact graphs of unit sphere packings revisited. J. Geom. (2013), 57--83.\n\n[Er46b] Erd\\H{o}s, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250.\n\n[Er75f] Erd\\H{o}s, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\n\n[Ha74b] Harborth, Heiko, L\"{o}sung zu Problem 664A. Elem. Math. (1974), 14-15.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2476, "problem_number": "EP-1085", "title": "Erdős Problem #1085", "statement": "Let $f_d(n)$ be minimal such that, in any set of $n$ points in $\\mathbb{R}^d$, there exist at most $f_d(n)$ pairs of points which distance $1$ apart. Estimate $f_d(n)$.", "background": "The most difficult cases are $d=2$ and $d=3$. When $d=2$ this is the unit distance problem [90], and the best known bounds are $ n^{1+\\frac{c}{\\log\\log n}}< f_2(n) \\ll n^{4/3} $ for some constant $c>0$, the lower bound by Erd\\H{o}s \\cite{Er46b} and the upper bound by Spencer, Szemer\\'{e}di, and Trotter \\cite{SST84}.\nWhen $d=3$ the best known bounds are $ n^{4/3}\\log\\log n \\ll f_3(n) \\ll n^{3/2}\\beta(n) $ where $\\beta(n)$ is a very slowly growing function, the lower bound by Erd\\H{o}s \\cite{Er60b} and the upper bound by Clarkson, Edelsbrunner, Guibas, Sharir, and Welzl \\cite{CEGSW90}.\nA construction of Lenz (taking points on orthogonal circles) shows that, for $d\\geq 4$, $ f_d(n)\\geq \\frac{p-1}{2p}n^2-O(1) $ with $p=\\lfloor d/2\\rfloor$. Erd\\H{o}s \\cite{Er60b} showed that the Erd\\H{o}s-Stone theorem implies $ f_d(n) \\leq \\left(\\frac{p-1}{2p}+o(1)\\right)n^2 $ for $d\\geq 4$.\nErd\\H{o}s \\cite{Er67e} determined $f_d(n)$ up to $O(1)$ for all even $d\\geq 4$. Brass \\cite{Br97} determined $f_4(n)$ exactly. Swanepoel \\cite{Sw09} determined $f_d(n)$ exactly for even $d\\geq 6$. For odd $d\\geq 5$ Erd\\H{o}s and Pach \\cite{ErPa90} proved that there exist constants $c_1(d),c_2(d)>0$ such that $ \\frac{p-1}{2p}n^2 +c_1n^{4/3}\\leq f_d(n) \\leq \\frac{p-1}{2p}n^2 +c_2n^{4/3}. $ \nReferences\n\n\n[Br97] Brass, P., On the maximum number of unit distances among {$n$} points in\ndimension four. (1997), 277--290.\n\n[CEGSW90] Clarkson, Kenneth L. and Edelsbrunner, Herbert and Guibas,\nLeonidas J. and Sharir, Micha and Welzl, Emo, Combinatorial complexity bounds for arrangements of curves and\nspheres. Discrete Comput. Geom. (1990), 99--160.\n\n[Er46b] Erd\\H{o}s, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250.\n\n[Er60b] Erd\\H{o}s, P., On sets of distances of {$n$} points in {E}uclidean space. Magyar Tud. Akad. Mat. Kutat\\'o{} Int. K\"ozl. (1960), 165--169.\n\n[Er67e] Erd\\H{o}s, P., On some applications of graph theory to geometry. Canadian J. Math. (1967), 968--971.\n\n[ErPa90] Erd\\H{o}s, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269.\n\n[SST84] Spencer, J. and Szemer\\'{e}di, E. and Trotter, Jr., W., Unit distances in the Euclidean plane. Graph theory and combinatorics (Cambridge, 1983) (1984), 293-303.\n\n[Sw09] Swanepoel, Konrad J., Unit distances and diameters in {E}uclidean spaces. Discrete Comput. Geom. (2009), 1--27.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2477, "problem_number": "EP-1086", "title": "Erdős Problem #1086", "statement": "Let $g(n)$ be minimal such that any set of $n$ points in $\\mathbb{R}^2$ contains the vertices of at most $g(n)$ many triangles with the same area. Estimate $g(n)$.", "background": "Equivalently, how many triangles of area $1$ can a set of $n$ points in $\\mathbb{R}^2$ determine? Erd\\H{o}s and Purdy attribute this question to Oppenheim. Erd\\H{o}s and Purdy \\cite{ErPu71} proved $ n^2\\log\\log n \\ll g(n) \\ll n^{5/2}, $ and believed the lower bound to be closer to the truth. The upper bound has been improved a number of times - by Pach and Sharir \\cite{PaSh92}, Dumitrescu, Sharir, and T\\'{o}th \\cite{DST09}, Apfelbaum and Sharir \\cite{ApSh10}, and Apfaulbaum \\cite{Ap13}. The best known bound is $ g(n) \\ll n^{20/9} $ by Raz and Sharir \\cite{RaSh17}.\nErd\\H{o}s and Purdy also ask a similar question about the higher-dimensional generalisations - more generally, let $g_d^{r}(n)$ be minimal such that any set of $n$ points in $\\mathbb{R}^d$ contains the vertices of at most $g_d^{r}(n)$ many $r$-dimensional simplices with the same volume.\nErd\\H{o}s and Purdy \\cite{ErPu71} proved $g_3^2(n) \\ll n^{8/3}$, and Dumitrescu, Sharir, and T\\'{o}th \\cite{DST09} improved this to $g_3^2(n) \\ll n^{2.4286}$.\nErd\\H{o}s and Purdy \\cite{ErPu71} proved $g_6^2(n)\\gg n^3$. Purdy \\cite{Pu74} proved $ g_4^2(n)\\leq g^2_5(n) \\ll n^{3-c} $ for some constant $c>0$. An observation of Oppenheim (using a construction of Lenz) detailed in \\cite{ErPu71} shows that $ g_{2k+2}^k(n)\\geq \\left(\\frac{1}{(k+1)^{k+1}}+o(1)\\right)n^{k+1} $ and Erd\\H{o}s and Purdy conjecture this is the best possible.\nSee also [90] and [755].\nReferences\n\n\n[Ap13] R. Apfelbaum, Geometric Incidences and Repeated Configurations. Ph.D. Dissertation, School of Computer Science, Tel Aviv University (2013).\n\n[ApSh10] Apfelbaum, Roel and Sharir, Micha, An improved bound on the number of unit area triangles. Discrete Comput. Geom. (2010), 753--761.\n\n[DST09] Dumitrescu, Adrian and Sharir, Micha and T\\'oth, Csaba D., Extremal problems on triangle areas in two and three\ndimensions. J. Combin. Theory Ser. A (2009), 1177--1198.\n\n[ErPu71] Erd\\H{o}s, Paul and Purdy, George, Some extremal problems in geometry. J. Combinatorial Theory Ser. A (1971), 246--252.\n\n[PaSh92] Pach, J\\'anos and Sharir, Micha, Repeated angles in the plane and related problems. J. Combin. Theory Ser. A (1992), 12--22.\n\n[Pu74] Purdy, George, Some extremal problems in geometry. Discrete Math. (1974), 305--315.\n\n[RaSh17] Raz, Orit E. and Sharir, Micha, The number of unit-area triangles in the plane: theme and\nvariation. Combinatorica (2017), 1221--1240.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2478, "problem_number": "EP-1087", "title": "Erdős Problem #1087", "statement": "Let $f(n)$ be minimal such that every set of $n$ points in $\\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate $f(n)$ - in particular, is it true that $f(n)\\leq n^{3+o(1)}$?", "background": "A question of Erd\\H{o}s and Purdy \\cite{ErPu71}, who proved $ n^3\\log n \\ll f(n) \\ll n^{7/2}. $ \nReferences\n\n\n[ErPu71] Erd\\H{o}s, Paul and Purdy, George, Some extremal problems in geometry. J. Combinatorial Theory Ser. A (1971), 246--252.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2479, "problem_number": "EP-1088", "title": "Erdős Problem #1088", "statement": "Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\\mathbb{R}^d$ contains a set of $n$ points such that any two determined distances are distinct. Estimate $f_d(n)$. In particular, is it true that, for fixed $n\\geq 3$, $ f_d(n)=2^{o(d)}? $ ", "background": "It is easy to prove that $f_d(n) \\leq n^{O_d(1)}$. Erd\\H{o}s \\cite{Er75f} claimed that he and Straus proved $f_d(n)\\leq c_n^d$ for some constant $c_n>0$.\nWhen $d=1$ this is the subject of [530], and $f_1(n)\\asymp n^2$.\nWhen $n=3$ this is the subject of [503]. Erd\\H{o}s could prove $f_2(3)=7$ and Croft \\cite{Cr62} proved $f_3(3)=9$. The results described at [503] demonstrate that $f_d(3)=d^2/2+O(d)$.\nReferences\n\n\n[Cr62] Croft, H. T., $9$-point and $7$-point configurations in $3$-space. Proc. London Math. Soc. (3) (1962), 400-424.\n\n[Er75f] Erd\\H{o}s, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2480, "problem_number": "EP-1089", "title": "Erdős Problem #1089", "statement": "Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $\\mathbb{R}^d$ determines at least $n$ many distinct distances. Estimate $g_d(n)$. In particular, does $ \\lim_{d\\to \\infty}\\frac{g_d(n)}{d^{n-1}} $ exist?", "background": "A question of Kelly. Erd\\H{o}s \\cite{Er75f} writes it is 'easy' to see that $g_d(n)\\gg d^{n-1}$. Erd\\H{o}s and Straus proved (in unpublished work mentioned in \\cite{Er75f}) that $ g_d(n) \\leq c^{d^{1-b_n}} $ for some constants $c>0$ and $b_n>0$.\nIt is trivial that $g_1(3)=4$, and easy to see that $g_2(3)=6$. Croft \\cite{Cr62} proved $g_3(3)=7$. The vertices of a $d$-dimensional cube demonstrate that $ g_d(d+1)>2^d. $ The function $g_d(n)$ is essentially the inverse of the function $f_d(n)$ considered in [1083] - with our definitions, $g_d(n)>m$ if and only if $f_d(m)1$?", "background": "A problem of Erd\\H{o}s, Lacampagne, and Selfridge \\cite{ELS88}, that was also asked in the 1986 problem session of West Coast Number Theory (as reported here).\nIn \\cite{ELS93} they prove that if the deficiency exists and is $\\geq 1$ then $n\\ll 2^k\\sqrt{k}$.\nThe following examples are either from \\cite{ELS88} or here. The following have deficiency $1$ (there are $58$ examples with $n\\leq 10^5$): $ \\binom{7}{3},\\binom{13}{4},\\binom{14}{4},\\binom{23}{5},\\binom{62}{6},\\binom{94}{10},\\binom{95}{10}. $ The examples which follow are the only known examples with deficiency $>1$. The following have deficiency $2$: $ \\binom{44}{8},\\binom{74}{10},\\binom{174}{12},\\binom{239}{14},\\binom{5179}{27},\\binom{8413}{28},\\binom{8414}{28},\\binom{96622}{42}. $ The following have deficiency $3$: $ \\binom{46}{10},\\binom{47}{10},\\binom{241}{16},\\binom{2105}{25},\\binom{1119}{27},\\binom{6459}{33}. $ The following has deficiency $4$: $ \\binom{47}{11}. $ The following has deficiency $9$: $ \\binom{284}{28}. $ See also [384] and [1094].\nBarreto in the comments has given a positive answer to the second question, conditional on two (strong) conjectures.\nReferences\n\n\n[ELS88] Erd\\H{o}s, P. and Lacampagne, C. B. and Selfridge, J. L., Prime factors of binomial coefficients and related problems. Acta Arith. (1988), 507--523.\n\n[ELS93] Erd\\H{o}s, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215--224.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2484, "problem_number": "EP-1094", "title": "Erdős Problem #1094", "statement": "For all $n\\geq 2k$ the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$, with only finitely many exceptions.", "background": "A stronger form of [384] that appears in a paper of Erd\\H{o}s, Lacampagne, and Selfridge \\cite{ELS88}. Erd\\H{o}s observed that the least prime factor is always $\\leq n/k$ provided $n$ is sufficiently large depending on $k$. Selfridge \\cite{Se77} further conjectured that this always happens if $n\\geq k^2-1$, except $\\binom{62}{6}$.\nThe threshold $g(k)$ below which $\\binom{n}{k}$ is guaranteed to be divisible by a prime $\\leq k$ is the subject of [1095].\nMore precisely, in \\cite{ELS88} they conjecture that if $n\\geq 2k$ then the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$ with the following $14$ exceptions: $ \\binom{7}{3},\\binom{13}{4},\\binom{23}{5},\\binom{14}{4},\\binom{44}{8},\\binom{46}{10},\\binom{47}{10}, $ $ \\binom{47}{11},\\binom{62}{6},\\binom{74}{10},\\binom{94}{10},\\binom{95}{10},\\binom{241}{16},\\binom{284}{28}. $ They also suggest the stronger conjecture that, with a finite number of exceptions, the least prime factor is $\\leq \\max(n/k,\\sqrt{k})$, or perhaps even $\\leq \\max(n/k,O(\\log k))$. Indeed, in \\cite{ELS93} they provide some further computational evidence, and point out it is consistent with what they know that in fact this holds with $\\leq \\max(n/k,13)$, with only $12$ exceptions.\nDiscussed in problem B31 and B33 of Guy's collection \\cite{Gu04} - there Guy credits Selfridge with the conjecture that if $n> 17.125k$ then $\\binom{n}{k}$ has a prime factor $p\\leq n/k$.\nThis is related to [1093], in that the only counterexamples to this conjecture can occur from $\\binom{n}{k}$ with deficiency $\\geq 1$.\nThere is an interesting discussion about this problem on MathOverflow.\nReferences\n\n\n[ELS88] Erd\\H{o}s, P. and Lacampagne, C. B. and Selfridge, J. L., Prime factors of binomial coefficients and related problems. Acta Arith. (1988), 507--523.\n\n[ELS93] Erd\\H{o}s, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215--224.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Se77] J. L. Selfridge, Some problems on the prime factors of consecutive integers. Notices Amer. Math. Soc. (1977), A456-457.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2485, "problem_number": "EP-1095", "title": "Erdős Problem #1095", "statement": "Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\\binom{n}{k}$ are $>k$. Estimate $g(k)$.", "background": "A question of Ecklund, Erd\\H{o}s, and Selfridge \\cite{EES74}, who proved $ k^{1+c}0$, and conjectured $g(k)0$, due to Konyagin \\cite{Ko99b}.\nErd\\H{o}s, Lacampagne, and Selfridge \\cite{ELS93} write 'it is clear to every right-thinking person' that $g(k)\\geq\\exp(c\\frac{k}{\\log k})$ for some constant $c>0$.\nSorenson, Sorenson, and Webster \\cite{SSW20} give heuristic evidence that $ \\log g(k) \\asymp \\frac{k}{\\log k}. $ See also [1094].\nReferences\n\n\n[EES74] Ecklund, Jr., E. F. and Erd\\H{o}s, P. and Selfridge, J. L., A new function associated with the prime factors of\n{$(\\sp{n}\\sb{k})$}. Math. Comp. (1974), 647--649.\n\n[ELS93] Erd\\H{o}s, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215--224.\n\n[GrRa96] Granville, Andrew and Ramar\\'{e}, Olivier, Explicit bounds on exponential sums and the scarcity of\nsquarefree binomial coefficients. Mathematika (1996), 73--107.\n\n[Ko99b] Konyagin, S. V., Estimates of the least prime factor of a binomial coefficient. Mathematika (1999), 41--55.\n\n[SSW20] Sorenson, Brianna and Sorenson, Jonathan and Webster,\nJonathan, An algorithm and estimates for the {E}rd\\H{o}s-{S}elfridge\nfunction. (2020), 371--385.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2486, "problem_number": "EP-1096", "title": "Erdős Problem #1096", "statement": "Let $10$ is sufficiently small, $x_{k+1}-x_k \\to 0$?", "background": "A problem of Erd\\H{o}s and Jo\\'{o} posed in the 1991 problem session of Great Western Number Theory.\nThey speculate that the threshold may be $q_0$, where $q_0\\approx 1.3247$ is the real root of $x^3=x+1$, and is the smallest Pisot-Vijayaraghavan number.\nIn \\cite{EJK90} Erd\\H{o}, Jo\\'{o}, and Komornik prove that any Pisot-Vijayaraghavan number cannot have this property, and also prove that, for any $10 $ for all $m\\geq 1$, where $x_k^m$ is the set of those numbers which can be written as a finite sum $\\sum_{n\\geq 0}c_nq^n$ for some $c_n\\in \\{0,\\ldots,m\\}$ (so that the sequence in the question is $x_k^1$). Erd\\H{o}s, Jo\\'{o}, and Schnitzer \\cite{EJS96} improved this to show that, if $10. $ \nReferences\n\n\n[Bu96] Bugeaud, Y., On a property of {P}isot numbers and related questions. Acta Math. Hungar. (1996), 33--39.\n\n[EJK90] Erd\\H{o}s, P\\'al and Jo\\'o, Istv\\'an and Komornik, Vilmos, Characterization of the unique expansions\n{$1=\\sum^\\infty_{i=1}q^{-n_i}$} and related problems. Bull. Soc. Math. France (1990), 377--390.\n\n[EJS96] Erd\\H{o}s, P. and Jo\\'o, I. and Schnitzer, F. J., On {P}isot numbers. Ann. Univ. Sci. Budapest. E\"otv\"os Sect. Math. (1996), 95--99.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2487, "problem_number": "EP-1097", "title": "Erdős Problem #1097", "statement": "Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$? Are there always $O(n^{3/2})$ many such $d$?", "background": "A problem Erd\\H{o}s posed in the 1989 problem session of Great Western Number Theory.\nHe states that Erd\\H{o}s and Ruzsa gave an explicit construction which achieved $n^{1+c}$ for some $c>0$, and Erd\\H{o}s and Spencer gave a probabilistic proof which achieved $n^{3/2}$, and speculated this may be the best possible.\nIn the comment section, Chan has noticed that this problem is exactly equivalent to a sums-differences question of Bourgain \\cite{Bo99}, introduced as an arithmetic path towards the Kakeya conjecture: find the smallest $c\\in [1,2]$ such that, for any finite sets of integers $A$ and $B$ and $G\\subseteq A\\times B$ we have $ \\lvert A\\overset{G}{-}B\\rvert \\ll \\max(\\lvert A\\rvert,\\lvert B\\rvert, \\lvert A\\overset{G}{+}B\\rvert)^c $ (where, for example, $A\\overset{G}{+}B$ denotes the set of $a+b$ with $(a,b)\\in G$).\nThis is equivalent in the sense that the greatest exponent $c$ achievable for the main problem here is equal to the smallest constant achievable for the sums-differences question. The current best bounds known are thus $ 1.77898\\cdots \\leq c \\leq 11/6 \\approx 1.833. $ The upper bound is due to Katz and Tao \\cite{KaTa99}. The lower bound is due to Lemm \\cite{Le15} (with a very small improvement found by AlphaEvolve \\cite{GGTW25}).\nReferences\n\n\n[Bo99] Bourgain, J., On the dimension of {K}akeya sets and related maximal\ninequalities. Geom. Funct. Anal. (1999), 256--282.\n\n[GGTW25] B. Georgiev, J. G\\'{o}mez-Serrano, T. Tao, and A. Wagner, Mathematical exploration and discovery at scale. arXiv:2511.02864 (2025).\n\n[KaTa99] Katz, Nets Hawk and Tao, Terence, Bounds on arithmetic projections, and applications to the\n{K}akeya conjecture. Math. Res. Lett. (1999), 625--630.\n\n[Le15] Lemm, Marius, New counterexamples for sums-differences. Proc. Amer. Math. Soc. (2015), 3863--3868.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2488, "problem_number": "EP-1100", "title": "Erdős Problem #1100", "statement": "If $1=d_1<\\cdots0$ and sufficiently large $x$, $ \\max_{n \\exp((\\log\\log x)^{2-\\epsilon}). $ Erd\\H{o}s and Simonovits (see \\cite{Er81h}) proved $ (2^{1/2}+o(1))^k < g(k) < (2-c)^k $ for some constant $c>0$.\nReferences\n\n\n[Er81h] Erd\\H{o}s, P., Some problems and results on additive and multiplicative\nnumber theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.\n\n[ErHa78] Erd\\H{o}s, P. and Hall, R. R., On some unconventional problems on the divisors of integers. J. Austral. Math. Soc. Ser. A (1978), 479--485.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2489, "problem_number": "EP-1101", "title": "Erdős Problem #1101", "statement": "If $u=\\{u_10$, if $x$ is sufficiently large then $ \\max_{a_k (1+o(1))t_x \\prod_{i}\\left(1-\\frac{1}{u_i}\\right)^{-1}. $ The strong form of [208] is asking whether if $u_i=p_i^2$, the sequence of prime squares, is good.\nReferences\n\n\n[Er81h] Erd\\H{o}s, P., Some problems and results on additive and multiplicative\nnumber theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2490, "problem_number": "EP-1103", "title": "Erdős Problem #1103", "statement": "Let $A$ be an infinite sequence of integers such that every $n\\in A+A$ is squarefree. How fast must $A$ grow?", "background": "Erd\\H{o}s notes there exists such a sequence which grows exponentially, but does not expect such a sequence of polynomial growth.\nIn \\cite{Er81h} he asked whether there is an infinite sequence of integers $A$ such that, for every $a\\in A$ and prime $p$, if $ a\\equiv t\\pmod{p^2} $ then $1\\leq t 0.24j^{4/3}$ for all $j$, and further that there exists such a sequence (furthermore with squarefree terms) such that $ a_j < \\exp(5j/\\log j) $ for all large $j$. A superior lower bound of $a_j \\gg j^{15/11-o(1)}$ had earlier been found by Konyagin \\cite{Ko04} when considering the finite case [1109].\nThey also obtain further results for the generalisation from squarefree to $k$-free integers, and also replacing $A+A$ with $A\\cup (A+A)\\cup(A+A+A)$.\nSee also [1109] for the finite analogue of this problem.\nReferences\n\n\n[Er81h] Erd\\H{o}s, P., Some problems and results on additive and multiplicative\nnumber theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.\n\n[Ko04] Konyagin, S. V., Problems of the set of square-free numbers. Izv. Ross. Akad. Nauk Ser. Mat. (2004), 63--90.\n\n[vDTa25] W. van Doorn and T. Tao, Growth rates of sequences governed by the squarefree properties of its translates. arXiv:2512.01087 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2491, "problem_number": "EP-1104", "title": "Erdős Problem #1104", "statement": "Let $f(n)$ be the maximum possible chromatic number of a triangle-free graph on $n$ vertices. Estimate $f(n)$.", "background": "The best bounds available are $ (1-o(1))(n/\\log n)^{1/2}\\leq f(n) \\leq (2+o(1))(n/\\log n)^{1/2}. $ The upper bound is due to Davies and Illingworth \\cite{DaIl22}, the lower bound follows from a construction of Hefty, Horn, King, and Pfender \\cite{HHKP25}.\nOne can ask a similar question for the maximum possible chromatic number of a triangle-free graph on $m$ edges. Let this be $g(m)$. Davies and Illingworth \\cite{DaIl22} prove $ g(m) \\leq (3^{5/3}+o(1))\\left(\\frac{m}{(\\log m)^2}\\right)^{1/3}. $ Kim \\cite{Ki95} gave a construction which implies $g(m) \\gg (m/(\\log m)^2)^{1/3}$.\nThe function $f(n)$ is the inverse to the function $h_3(k)$ considered in [1013].\nA generalisation of $f(n)$ is considered in [920].\nReferences\n\n\n[DaIl22] Davies, Ewan and Illingworth, Freddie, The {$\\chi$}-{R}amsey problem for triangle-free graphs. SIAM J. Discrete Math. (2022), 1124--1134.\n\n[HHKP25] Z. Hefty, P. Horn, D. King, and F. Pfender, Improving $R(3,k)$ in just two bites. arXiv:2510.19718 (2025).\n\n[Ki95] Kim, J. H., The Ramsey number $R(3,t)$ has order of magnitude $t^2/\\log t$. Random Structures and Algorithms (1995), 173-207.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2492, "problem_number": "EP-1105", "title": "Erdős Problem #1105", "statement": "The anti-Ramsey number $\\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the edges of $K_n$ can be coloured without creating a rainbow copy of $G$ (i.e. one in which all edges have different colours).\nLet $C_k$ be the cycle on $k$ vertices. Is it true that $ \\mathrm{AR}(n,C_k)=\\left(\\frac{k-2}{2}+\\frac{1}{k-1}\\right)n+O(1)? $ Let $P_k$ be the path on $k$ vertices and $\\ell=\\lfloor\\frac{k-1}{2}\\rfloor$. If $n\\geq k\\geq 5$ then is $\\mathrm{AR}(n,P_k)$ equal to $ \\max\\left(\\binom{k-2}{2}+1, \\binom{\\ell-1}{2}+(\\ell-1)(n-\\ell+1)+\\epsilon\\right) $ where $\\epsilon=1$ if $k$ is odd and $\\epsilon=2$ otherwise?", "background": "A conjecture of Erd\\H{o}s, Simonovits, and S\\'{o}s \\cite{ESS75}, who gave a simple proof that $\\mathrm{AR}(n,C_3)=n-1$. In this paper they announced proofs of the claimed formula for $\\mathrm{AR}(n,P_k)$ for $n\\geq \\frac{5}{4}k+C$ for some large constant $C$, and also for all $n\\geq k$ if $k$ is sufficiently large, but these never appeared.\nSimonovits and S\\'{o}s \\cite{SiSo84} published a proof that the claimed formula for $\\mathrm{AR}(n,P_k)$ is true for $n\\geq ck^2$ for some constant $c>0$.\nA proof of the formula for $\\mathrm{AR}(n,P_k)$ for all $n\\geq k\\geq 5$ has been announced by Yuan \\cite{Yu21}\nReferences\n\n\n[ESS75] Erd\\H{o}s, P. and Simonovits, M. and S\\'os, V. T., Anti-{R}amsey theorems. (1975), 633--643.\n\n[SiSo84] Simonovits, Mikl\\'os and S\\'os, Vera T., On restricted colourings of {$K_n$}. Combinatorica (1984), 101--110.\n\n[Yu21] L.-T. Yuan, The anti-Ramsey number for paths. arXiv:2102.00807 (2021).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2493, "problem_number": "EP-1106", "title": "Erdős Problem #1106", "statement": "Let $p(n)$ denote the partition function of $n$ and let $F(n)$ count the number of distinct prime factors of $ \\prod_{1\\leq k\\leq n}p(k). $ Does $F(n)\\to \\infty$ with $n$? Is $F(n)>n$ for all sufficiently large $n$?", "background": "Asked by Erd\\H{o}s at Oberwolfach in 1986. Schinzel noted in the Oberwolfach problem book that $F(n)\\to \\infty$ follows from the asymptotic formula for $p(n)$ and a result of Tijdeman \\cite{Ti73}. This is not obvious; details are given in a paper of Erd\\H{o}s and Ivi\\'{c} (see page 69 of \\cite{ErIv90}).\nSchinzel and Wirsing \\cite{ScWi87} have proved $F(n) \\gg \\log n$.\nOno \\cite{On00} has proved that every prime divides $p(n)$ for some $n\\geq 1$ (indeed this holds, for any fixed prime, for a positive density set of $n$).\nReferences\n\n\n[ErIv90] Erd\\H{o}s, Paul and Ivi\\'c, Aleksandar, The distribution of values of a certain class of arithmetic\nfunctions at consecutive integers. (1990), 45--91.\n\n[On00] Ono, Ken, Distribution of the partition function modulo {$m$}. Ann. of Math. (2) (2000), 293--307.\n\n[ScWi87] Schinzel, A. and Wirsing, E., Multiplicative properties of the partition function. Proc. Indian Acad. Sci. Math. Sci. (1987), 297--303.\n\n[Ti73] Tijdeman, R., On integers with many small prime factors. Compositio Math. (1973), 319--330.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2494, "problem_number": "EP-1107", "title": "Erdős Problem #1107", "statement": "Let $r\\geq 2$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\\mid n$. Is every large integer the sum of at most $r+1$ many $r$-powerful numbers?", "background": "Given in the 1986 Oberwolfach problem book as a problem of Erd\\H{o}s and Ivi\\'{c}.\nThis is true when $r=2$, as proved by Heath-Brown \\cite{He88} (see [941]).\nSee [940] for the problem of which integers are the sum of at most $r$ many $r$-powerful numbers.\nReferences\n\n\n[He88] Heath-Brown, D. R., Ternary quadratic forms and sums of three square-full numbers. (1988), 137--163.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2495, "problem_number": "EP-1108", "title": "Erdős Problem #1108", "statement": "Let $ A = \\left\\{ \\sum_{n\\in S}n! : S\\subset \\mathbb{N}\\textrm{ finite}\\right\\}. $ If $k\\geq 2$, then does $A$ contain only finitely many $k$th powers? Does it contain only finitely many powerful numbers?", "background": "Asked by Erd\\H{o}s at Oberwolfach in 1988. It is open even whether there are infinitely many squares of the form $1+n!$ (see [398]).\nThis was motivated in part by a problem of Mahler which he discussed with Erd\\H{o}s a few days before his death in 1988: if $k\\geq 5$ and $ A_k= \\left\\{ \\sum_{n\\in S}k^n : S\\subset \\mathbb{N}\\textrm{ finite}\\right\\} $ then does $A_k$ contain only finitely many squares? Mahler showed that there are infinitely many squares in $A_k$ for $k\\leq 4$, and found only one square for $k\\geq 5$, namely $ 1+7+7^2+7^3=400. $ Brindza and Erd\\H{o}s \\cite{BrEr91} proved that, for any $r$, if $n_1!+\\cdots+n_r!$ is powerful then $n_1\\ll_r 1$.\nReferences\n\n\n[BrEr91] Brindza, B. and Erd\\H{o}s, P., On some {D}iophantine problems involving powers and\nfactorials. J. Austral. Math. Soc. Ser. A (1991), 1--7.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2496, "problem_number": "EP-1109", "title": "Erdős Problem #1109", "statement": "Let $f(N)$ be the size of the largest subset $A\\subseteq \\{1,\\ldots,N\\}$ such that every $n\\in A+A$ is squarefree. Estimate $f(N)$. In particular, is it true that $f(N)\\leq N^{o(1)}$, or even $f(N) \\leq (\\log N)^{O(1)}$?", "background": "First studied by Erd\\H{o}s and S\\'{a}rk\"{o}zy \\cite{ErSa87}, who proved $ \\log N \\ll f(N) \\ll N^{3/4}\\log N, $ and guessed the lower bound is nearer the truth. S\\'{a}rk\"{o}zy \\cite{Sa92c} extended this to consider the case of $A+B$ and also looking for sumsets which are $k$-power-free.\nGyarmati \\cite{Gy01} gave an alternative proof of $f(N)\\gg \\log N$, and also gave new bounds for the case of $A+B$. Konyagin \\cite{Ko04} improved this to $ \\log\\log N(\\log N)^2\\ll f(N) \\ll N^{11/15+o(1)}. $ The infinite analogue of this problem is [1103]. (In particular upper bounds for this $f(N)$ directly imply lower bounds for the size of the $a_j$ considered there.)\nReferences\n\n\n[ErSa87] Erd\\H{o}s, P. and S\\'ark\"ozy, A., On divisibility properties of integers of the form {$a+a'$}. Acta Math. Hungar. (1987), 117--122.\n\n[Gy01] Gyarmati, Katalin, On divisibility properties of integers of the form {$ab+1$}. Period. Math. Hungar. (2001), 71--79.\n\n[Ko04] Konyagin, S. V., Problems of the set of square-free numbers. Izv. Ross. Akad. Nauk Ser. Mat. (2004), 63--90.\n\n[Sa92c] S\\'ark\"ozy, G. N., On a problem of {P}. {E}rd\\H{o}s. Acta Math. Hungar. (1992), 271--282.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2497, "problem_number": "EP-1110", "title": "Erdős Problem #1110", "statement": "Let $p>q\\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divide each other.\nIf $\\{p,q\\}\neq \\{2,3\\}$ then what can be said about the density of non-representable numbers? Are there infinitely many coprime non-representable numbers?", "background": "A problem of Erd\\H{o}s and Lewin \\cite{ErLe96}, who proved that there are finitely many non-representable numbers if and only if $\\{p,q\\}=\\{2,3\\}$.\nIndeed, in \\cite{Er92b} Erd\\H{o}s wrote 'last year I made the following silly conjecture': every integer $n$ can be written as the sum of distinct integers of the form $2^k3^l$, none of which divide any other. He wrote 'I mistakenly thought that this was a nice and difficult conjecture but Jansen and several others found a simple proof by induction.'\nThis simple proof is as follows: one proves the stronger fact that such a representation always exists, and moreover if $n$ is even then all the summands can be taken to be even: if $n=2m$ we are done applying the inductive hypothesis to $m$. Otherwise if $n$ is odd then let $3^k$ be the largest power of $3$ which is $\\leq n$ and apply the inductive hypothesis to $n-3^k$ (which is even).\nYu and Chen \\cite{YuCh22} prove that the set of non-representable numbers has density zero whenever $q>3$ or $q=3$ and $p>6$ or $q=2$ and $p>10$. They also prove that there are infinitely many coprime non-representable numbers if $q>3$ or $q=3$ and $p\neq 5$ or $q=2$ and $p\not\\in \\{3,5,9\\}$.\nErd\\H{o}s and Lewin \\cite{ErLe96} also asked whether all large integers $n$ can be written as a sum of $2^k3^l$, none of which divide another, each of which is $>f(n)$ for some $f(n)\\to \\infty$. Let $f(n)$ be the fastest growing such $f(n)$. Yu and Chen \\cite{YuCh22} proved $ \\frac{n}{(\\log n)^{\\log_23}}\\ll f(n) \\ll \\frac{n}{\\log n}. $ Yang and Zhao \\cite{YaZh25} improved the lower bound to $f(n)\\gg n/\\log n$.\nThe case of three powers is the subject of [123], and see also [845] for more on the case $\\{p,q\\}=\\{2,3\\}$. The problem [246] addresses the topic without the non-divisibility condition.\nReferences\n\n\n[Er92b] Erd\\H{o}s, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240.\n\n[ErLe96] Erd\\H{o}s, P. and Lewin, Mordechai, $d$-complete sequences of integers. Math. Comp. (1996), 837-840.\n\n[YaZh25] Yang, Quan-Hui and Zhao, Lilu, A conjecture of {Y}u and {C}hen related to the {E}rd\\H\nos-{L}ewin theorem. Acta Arith. (2025), 277--286.\n\n[YuCh22] Yu, Wang-Xing and Chen, Yong-Gao, On a conjecture of {E}rd\\H{o}s and {L}ewin. J. Number Theory (2022), 763--778.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2498, "problem_number": "EP-1111", "title": "Erdős Problem #1111", "statement": "If $G$ is a finite graph and $A,B$ are disjoint sets of vertices then we call $A,B$ anticomplete if there are no edges between $A$ and $B$.\nIf $t,c\\geq 1$ then there exists $d\\geq 1$ such that if $\\chi(G)\\geq d$ and $\\omega(G)3$.\nNguyen, Scott, and Seymour \\cite{NSS24} prove that if $t,c\\geq 1$ then there exists $d\\geq 1$ such that if $\\chi(G)\\geq d$ and $\\omega(G)0$, and if $g(p)$ does not have very large values (in a certain technical sense).\nSee also [491].\nReferences\n\n\n[Ma22] Mangerel, Alexander P., Additive functions in short intervals, gaps and a conjecture\nof {E}rd\\H{o}s. Ramanujan J. (2022), 1023--1090.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2504, "problem_number": "EP-1129", "title": "Erdős Problem #1129", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let $ l_k(x)=\\frac{\\prod_{i\neq k}(x-x_i)}{\\prod_{i\neq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.\nDescribe which choice of $x_i$ minimise $ \\Lambda(x_1,\\ldots,x_n)=\\max_{x\\in [-1,1]} \\sum_k \\lvert l_k(x)\\rvert. $ ", "background": "The functions $l_k(x)$ are sometimes called the fundamental functions of Lagrange interpolation, and $\\Lambda$ is sometimes called the Lebesgue constant.\nFaber \\cite{Fa14} proved $ \\Lambda(x_1,\\ldots,x_n)\\gg \\log n $ for all choices of $x_i$, and Bernstein \\cite{Be31} proved it is $>(\\frac{2}{\\pi}-o(1))\\log n$. Erd\\H{o}s \\cite{Er61c} improved this to $ \\Lambda(x_1,\\ldots,x_n)> \\frac{2}{\\pi}\\log n-O(1). $ This is best possible, since taking the $x_i$ as the roots of the $n$th Chebyshev polynomial yields $ \\Lambda(x_1,\\ldots,x_n)< \\frac{2}{\\pi}\\log n+O(1). $ Erd\\H{o}s thought that the minimising choice is characterised by the property that the sums $ \\lambda_i=\\max_{x\\in [x_i,x_{i+1}]}\\sum_k \\lvert l_k(x)\\rvert $ are all equal for $0\\leq i\\leq n$ (where $x_0=-1$ and $x_{n+1}=1$). This conjecture was also made by Bernstein \\cite{Be31}. Kilgore and Cheney \\cite{KiCh76} proved that there exists $x_i$ for which all $\\lambda_i$ are equal. Kilgore \\cite{Ki77} proved that $\\Lambda$ is minimised only when all $\\lambda_i$ are equal. Finally, de Boor and Pinkus \\cite{dBPi78} proved that there exists a unique minimising choice of $x_i$.\nIf $x_1=-1$ and $x_n=1$ then there is a unique minimising set of $x_i$, which are symmetric around $0$. (Such a choice is called canonical.) The exact minimising canonical choice is known only for $n\\leq 4$. For $n=2$ the points are $-1,1$ (with $\\Lambda=1$). For $n=3$ the points are $-1,0,1$ (with $\\Lambda=1.25$), as shown by Bernstein \\cite{Be31}. Rack and Vajda \\cite{RaVa15} have shown that for $n=4$ the points are $-1,-t,t,1$ where $t\\approx 0.4177$ is an explicit algebraic constant (with $\\Lambda \\approx 1.4229$).\nIn \\cite{Er67} Erd\\H{o}s suggests that an easier variant might be to have the $x_i\\in \\mathbb{C}$ with $\\lvert x_i\\rvert=1$, and seek to minimise $\\max_{\\lvert z\\rvert=1}\\sum_{k}\\lvert l_k(z)\\rvert$, adding it 'seems certain' that the minimising $x_i$ are the $n$th roots of unity. This was proved by Brutman \\cite{Br80} for odd $n$ and by Brutman and Pinkus \\cite{BrPi80} for even $n$.\nSee also [671], [1130], and [1132].\nReferences\n\n\n[Be31] S. Bernstein, Sur la limitation des valeurs d'un polynome $P_n(x)$ de degr\\'{e} n sur tout un segment par ses valeurs en $(n+1)$ points du segment. Izv. Akad. Nauk. SSSR (1931), 1025-1050.\n\n[Br80] Brutman, L., On the polynomial and rational projections in the complex\nplane. SIAM J. Numer. Anal. (1980), 366--372.\n\n[BrPi80] Brutman, L. and Pinkus, A., On the {E}rd\\H{o}s conjecture concerning minimal norm\ninterpolation on the unit circle. SIAM J. Numer. Anal. (1980), 373--375.\n\n[Er61c] Erd\\H{o}s, P., Problems and results on the theory of interpolation. II. Acta Math. Acad. Sci. Hungar. (1961), 235-244.\n\n[Er67] Erd\\H{o}s, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) (1967), 65-73.\n\n[Fa14] G. Faber, \"{U}ber die interpolatorische Darstellung stetiger Funktionen. Jahresb. der Deutschen Math. Ver. (1914), 190-210.\n\n[Ki77] Kilgore, T. A., Optimization of the norm of the {L}agrange interpolation\noperator. Bull. Amer. Math. Soc. (1977), 1069--1071.\n\n[KiCh76] Kilgore, T. A. and Cheney, E. W., A theorem on interpolation in {H}aar subspaces. Aequationes Math. (1976), 391--400.\n\n[RaVa15] Rack, Heinz-Joachim and Vajda, Robert, Optimal cubic {L}agrange interpolation: extremal node systems\nwith minimal {L}ebesgue constant. Stud. Univ. Babe\\c s-Bolyai Math. (2015), 151--171.\n\n[dBPi78] de Boor, Carl and Pinkus, Allan, Proof of the conjectures of {B}ernstein and {E}rd\\H os\nconcerning the optimal nodes for polynomial interpolation. J. Approx. Theory (1978), 289--303.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2505, "problem_number": "EP-1130", "title": "Erdős Problem #1130", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let $ l_k(x)=\\frac{\\prod_{i\neq k}(x-x_i)}{\\prod_{i\neq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.\nLet $x_0=-1$ and $x_{n+1}=1$ and $ \\Upsilon(x_1,\\ldots,x_n)=\\min_{0\\leq i\\leq n}\\max_{x\\in[x_i,x_{i+1}]} \\sum_k \\lvert l_k(x)\\rvert. $ Is it true that $ \\Upsilon(x_1,\\ldots,x_n)\\ll \\log n? $ Describe which choice of $x_i$ maximise $\\Upsilon(x_1,\\ldots,x_n)$.", "background": "The functions $l_k(x)$ are sometimes called the fundamental functions of Lagrange interpolation.\nErd\\H{o}s \\cite{Er47} could prove $ \\Upsilon(x_1,\\ldots,x_n)< \\sqrt{n}. $ Erd\\H{o}s thought that the maximising choice is characterised by the property that the sums $ \\lambda_i=\\max_{x\\in [x_i,x_{i+1}]}\\sum_k \\lvert l_k(x)\\rvert $ are all equal for $0\\leq i\\leq n$ (where $x_0=-1$ and $x_{n+1}=1$), which would be the same characterisation as [1129].\nThis is true, and was proved by de Boor and Pinkus \\cite{dBPi78}. It follows by the bounds discussed in [1129] that $ \\Upsilon(x_1,\\ldots,x_n)\\leq \\frac{2}{\\pi}\\log n+O(1). $ See also [1129].\nReferences\n\n\n[Er47] Erd\\H{o}s, P., Some remarks on polynomials. Bull. Amer. Math. Soc. (1947), 1169--1176.\n\n[dBPi78] de Boor, Carl and Pinkus, Allan, Proof of the conjectures of {B}ernstein and {E}rd\\H os\nconcerning the optimal nodes for polynomial interpolation. J. Approx. Theory (1978), 289--303.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2506, "problem_number": "EP-1131", "title": "Erdős Problem #1131", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let $ l_k(x)=\\frac{\\prod_{i\neq k}(x-x_i)}{\\prod_{i\neq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.\nWhat is the minimal value of $ I(x_1,\\ldots,x_n)=\\int_{-1}^1 \\sum_k \\lvert l_k(x)\\rvert^2\\mathrm{d}x? $ In particular, is it true that $ \\min I =2-(1+o(1))\\frac{1}{n}? $ ", "background": "Erd\\H{o}s first conjectured this minimum was achieved by taking the $x_i$ to be the roots of the integral of the Legendre polynomial, since Fejer \\cite{Fe32} had earlier shown these to be minimisers of $ \\max_{x\\in [-1,1]}\\sum_k \\lvert l_k(x)\\rvert^2. $ This was disproved by Szabados \\cite{Sz66} for every $n>3$.\nErd\\H{o}s, Szabados, Varma, and V\\'{e}rtesi \\cite{ESVV94} proved that $ 2-O\\left(\\frac{(\\log n)^2}{n}\\right)\\leq \\min I\\leq 2-\\frac{2}{2n-1} $ where the upper bound is witnessed by the roots of the integral of the Legendre polynomial as above.\nReferences\n\n\n[ESVV94] Erd\\H{o}s, P. and Szabados, J. and Varma, A. K. and V\\'{e}rtesi,\nP., On an interpolation theoretical extremal problem. Studia Sci. Math. Hungar. (1994), 55--60.\n\n[Fe32] Fej\\'{e}r, Leopold, Bestimmung derjenigen {A}bszissen eines {I}ntervalles, f\"ur\nwelche die {Q}uadratsumme der {G}rundfunktionen der\n{L}agrangeschen {I}nterpolation im {I}ntervalle ein\n{M}\"oglichst kleines {M}aximum {B}esitzt. Ann. Scuola Norm. Super. Pisa Cl. Sci. (2) (1932), 263--276.\n\n[Sz66] Szabados, J., On a problem of {P}. {E}rd\\H{o}s. Acta Math. Acad. Sci. Hungar. (1966), 155--157.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2507, "problem_number": "EP-1132", "title": "Erdős Problem #1132", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let $ l_k(x)=\\frac{\\prod_{i\neq k}(x-x_i)}{\\prod_{i\neq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.\nLet $x_1,x_2,\\ldots\\in [-1,1]$ be an infinite sequence, and let $ L_n(x) = \\sum_{1\\leq k\\leq n}\\lvert l_k(x)\\rvert, $ where each $l_k(x)$ is defined above with respect to $x_1,\\ldots,x_n$.\nMust there exist $x\\in (-1,1)$ such that $ L_n(x) >\\frac{2}{\\pi}\\log n-O(1) $ for infinitely many $n$?\nIs it true that $ \\limsup_{n\\to \\infty}\\frac{L_n(x)}{\\log n}\\geq \\frac{2}{\\pi} $ for almost all $x\\in (-1,1)$?", "background": "A result of Bernstein \\cite{Be31} implies that the set of $x\\in(-1,1)$ for which $ \\limsup_{n\\to \\infty}\\frac{L_n(x)}{\\log n}\\geq \\frac{2}{\\pi} $ is everywhere dense.\nErd\\H{o}s \\cite{Er61c} proved that, for any fixed $x_1,\\ldots,x_n\\in [-1,1]$, $ \\max_{x\\in [-1,1]}\\sum_{1\\leq k\\leq n}\\lvert l_k(x)\\rvert>\\frac{2}{\\pi}\\log n-O(1). $ See also [1129] for more on $L_n(x)$.\nReferences\n\n\n[Be31] S. Bernstein, Sur la limitation des valeurs d'un polynome $P_n(x)$ de degr\\'{e} n sur tout un segment par ses valeurs en $(n+1)$ points du segment. Izv. Akad. Nauk. SSSR (1931), 1025-1050.\n\n[Er61c] Erd\\H{o}s, P., Problems and results on the theory of interpolation. II. Acta Math. Acad. Sci. Hungar. (1961), 235-244.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2508, "problem_number": "EP-1133", "title": "Erdős Problem #1133", "statement": "Let $C>0$. There exists $\\epsilon>0$ such that if $n$ is sufficiently large the following holds.\nFor any $x_1,\\ldots,x_n\\in [-1,1]$ there exist $y_1,\\ldots,y_n\\in [-1,1]$ such that, if $P$ is a polynomial of degree $m<(1+\\epsilon)n$ with $P(x_i)=y_i$ for at least $(1-\\epsilon)n$ many $1\\leq i\\leq n$, then $ \\max_{x\\in [-1,1]}\\lvert P(x)\\rvert >C. $ ", "background": "Erd\\H{o}s proved that, for any $C>0$, there exists $\\epsilon>0$ such that if $n$ is sufficiently large and $m=\\lfloor (1+\\epsilon)n\\rfloor$ then for any $x_1,\\ldots,x_m\\in [-1,1]$ there is a polynomial $P$ of degree $n$ such that $\\lvert P(x_i)\\rvert\\leq 1$ for $1\\leq i\\leq m$ and $ \\max_{x\\in [-1,1]}\\lvert P(x)\\rvert>C. $ The conjectured statement would also imply this, but Erd\\H{o}s in \\cite{Er67} says he could not even prove it for $m=n$.\nReferences\n\n\n[Er67] Erd\\H{o}s, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) (1967), 65-73.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2509, "problem_number": "EP-1135", "title": "Erdős Problem #1135", "statement": "Define $f:\\mathbb{N}\\to \\mathbb{N}$ by $f(n)=n/2$ if $n$ is even and $f(n)=\\frac{3n+1}{2}$ if $n$ is odd.\nGiven any integer $m\\geq 1$ does there exist $k\\geq 1$ such that $f^{(k)}(m)=1$?", "background": "The infamous Collatz conjecture. For a detailed discussion of the history and theory surrounding this problem we refer to the overview by Lagarias \\cite{La10}.\nThis is not a problem due to Erd\\H{o}s; it was first devised by Collatz before 1952. Erd\\H{o}s referred to this problem on several occasions as 'hopeless'. As Lagarias \\cite{La16} notes, the closest Erd\\H{o}s ever came to working on problems of this nature is the theorem described in the remarks to [1134].\nIt is often claimed that Erd\\H{o}s offered \\$500 for a solution to this problem; this claim originated in a survey article by Lagarias \\cite{La85}.\nLagarias reported, in personal communication, that this came from a conversation he had with Erd\\H{o}s and Graham around 1983, in which Graham asked Erd\\H{o}s to make an estimate of what value Erd\\H{o}s would put the problem on his prize scale, to which Erd\\H{o}s replied \\$500. Therefore, strictly speaking, Erd\\H{o}s never offered \\$500 specifically as a prize, but we include this prize value here for comparing those problems which Erd\\H{o}s rated as 'prize problems'.\nThis is Problem E16 in Guy's collection \\cite{Gu04}, in which Guy quotes Erd\\H{o}s as saying \"Mathematics may not be ready for such problems\".\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[La10] Lagarias, Jeffrey C., The {$3x+1$} problem: an overview. (2010), 3--29.\n\n[La16] Lagarias, Jeffrey C., Erd\\H os, {K}larner, and the {$3x+1$} problem. Amer. Math. Monthly (2016), 753--776.\n\n[La85] Lagarias, Jeffrey C., The {$3x+1$} problem and its generalizations. Amer. Math. Monthly (1985), 3--23.\",\n \"difficulty\": \"L5\"\n}", "difficulty_level_id": 5, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2510, "problem_number": "KOU-21.1", "title": "Kourovka Notebook Problem 21.1", "statement": "Let $n$ be a positive integer. For a finite group $K$ and an automorphism $\\varphi$ of $K$ of order dividing $n$, let\n\n$$\nX_{n,\\varphi}(K):=\\{x\\in K\\mid x x^{\\varphi}\\cdots x^{\\varphi^{n-1}}=1\\}.\n$$\n\nLet $c_n$ be the supremum of the ratios $|X_{n,\\varphi}(H)|/|H|$ over all finite groups $H$ and their automorphisms $\\varphi\\in\\operatorname{Aut}(H)$ such that $\\varphi^n=\\operatorname{id}$ and $X_{n,\\varphi}(H)\\ne H$.\n\n(a) Let $n>1$ be a positive integer such that $c_d<1$ for all prime power divisors $d$ of $n$. Is it true that $c_n<1$?\n\n(b) For a finite group $G$ and a positive integer $n$, the generalized Hughes--Thompson subgroup is defined as $H_n(G)=\\langle x\\in G\\mid x^n\\ne 1\\rangle$. Suppose that $n$ is a positive integer for which there is a positive integer $k_n$ depending only on $n$ such that $|G:H_n(G)|\\leqslant k_n$ for all finite groups $G$ with $H_n(G)\\ne 1$. Is it true that then $c_n<1$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.1.\n\nDiscussion and literature:\nThis question is open even when $n\\geqslant 5$ is prime. A. Abdollahi, M. S. Malekan", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2511, "problem_number": "KOU-21.2", "title": "Kourovka Notebook Problem 21.2", "statement": "Let $S$ be a finite simple group, and let $G$ be a finite group for which there exists a bijection $f:G\\to S$ such that $|x|$ divides $|f(x)|$ for all $x\\in G$. Must $G$ necessarily be simple?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.2.\n\nDiscussion and literature:\nM. Amiri", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2512, "problem_number": "KOU-21.3", "title": "Kourovka Notebook Problem 21.3", "statement": "Let $G=A_n$ or $S_n$ and let $H,K$ be soluble subgroups of $G$. For all sufficiently large $n$, can we always find an element $x\\in G$ such that $H\\cap K^x=1$? Does this hold for all $n\\geqslant 21$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.3.\n\nDiscussion and literature:\nNote that the conclusion is false when $G=S_{20}$ and $H=K=(S_4\\wr S_4)\\times S_4$. M. Anagnostopoulou-Merkouri, T. C. Burness", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2513, "problem_number": "KOU-21.4", "title": "Kourovka Notebook Problem 21.4", "statement": "Let $G$ be a finite group with trivial solvable radical and let $H_1,\\ldots,H_5$ be solvable subgroups of $G$. Then do there always exist elements $x_i\\in G$ such that $\\bigcap_i H_i^{x_i}=1$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.4.\n\nDiscussion and literature:\nCf. 17.41(b). M. Anagnostopoulou-Merkouri, T. C. Burness", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2514, "problem_number": "KOU-21.5", "title": "Kourovka Notebook Problem 21.5", "statement": "Let $p$ be a prime. Let $G$ be a transitive subgroup of the group of finitary permutations $\\operatorname{FSym}(\\Omega)$ of a set $\\Omega$, let $N$ be a normal subgroup of $G$, and let $S$ be a transitive Sylow $p$-subgroup of $G$.\n\n(a) Is it true that $S\\cap N$ is a Sylow $p$-subgroup of $N$?\n\n(b) Is it true that $SN/N$ is a Sylow $p$-subgroup of $G/N$?\n\n(c) Are any two transitive Sylow $p$-subgroups of $G$ locally conjugate in $G$?\n\nTwo subgroups $X,Y$ of a group $G$ are said to be locally conjugate if there is a locally inner automorphism $\\phi$ of $G$ such that $X^\\phi=Y$. An automorphism $\\phi$ of $G$ is said to be locally inner if for every finite subset $A\\subseteq G$ there is an element $g=g(A)\\in G$ such that $a^\\phi=g^{-1}ag$ for all $a\\in A$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.5.\n\nDiscussion and literature:\nA. O. Asar", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2515, "problem_number": "KOU-21.6", "title": "Kourovka Notebook Problem 21.6", "statement": "Let p be a prime. A totally imprimitive p-group H of finitary permutations is said to have the cyclic-block property if in the cycle decomposition of every element the support of every cycle is a block for H. Let G be a transitive subgroup of the group of finitary permutations $\\operatorname{FSym}(\\Omega)$ of a set $\\Omega$. Does every transitive Sylow p-subgroup of G contain a transitive subgroup which has the cyclic-block property?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.6.\n\nDiscussion and literature:\nA. O. Asar", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2516, "problem_number": "KOU-21.7", "title": "Kourovka Notebook Problem 21.7", "statement": "(Well-known problem). A finite group G is called an IYB-group if it is isomorphic to the permutation group of a finite involutive non-degenerate set-theoretic solution of the Yang--Baxter equation, or equivalently, G is isomorphic to the multiplicative group of a finite left brace. Assume that the Sylow subgroups of a finite soluble group G are IYB-groups. Is G an IYB-group?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.7.\n\nDiscussion and literature:\nA. Ballester-Bolinches", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2517, "problem_number": "KOU-21.8", "title": "Kourovka Notebook Problem 21.8", "statement": "As in 17.57, let $r(m)=\\{r+km\\mid k\\in\\mathbb Z\\}$ for integers $0\\leqslant r3$ the group $\\operatorname{CT}(k)$ is isomorphic to the symmetric group $S_N$, where N is the least common multiple of the numbers $2,3,\\ldots,k$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.8.\n\nDiscussion and literature:\nV. G. Bardakov, A. L. Iskra\n\nYes, it is true (Junyao Pan, Preprint of 14 April 2026, https://arxiv.org/abs/2604.12553; P. Monticone, Preprint of 30 April 2026, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/05/21_8-1.pdf).", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2518, "problem_number": "KOU-21.9", "title": "Kourovka Notebook Problem 21.9", "statement": "Let $F$ be a non-abelian free pro-$p$ group of finite rank. Can one find a finite collection $U_1,\\ldots,U_n$ of open subgroups of $F$, including $F$ itself, such that the only subgroup of $F$ which is contained in $U_i$ and is characteristic in $U_i$ for every $i$ is the trivial subgroup?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.9.\n\nDiscussion and literature:\nY. Barnea", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2519, "problem_number": "KOU-21.10", "title": "Kourovka Notebook Problem 21.10", "statement": "We call a group presentation finite if it represents a finite group. We say that a presentation is just finite if it is finite and is no longer finite on removal of any relation from it. Is it true that every finite group has a just finite presentation?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.10.\n\nDiscussion and literature:\nNote that if a group has a balanced presentation, then it is just finite. A similar argument can be applied for some p-groups using the Golod--Shafarevich inequality. Y. Barnea", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2520, "problem_number": "KOU-21.11", "title": "Kourovka Notebook Problem 21.11", "statement": "Can some or all groups of the following sorts be written as homomorphic images of nonprincipal ultraproducts of countable families of groups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.11.\n\nDiscussion and literature:\nThis is Question 18 in (G. M. Bergman, Pacific J. Math., 274 (2015) 451--495).\n\n(a) Infinite finitely generated groups of finite exponent.\n\n(b) For an infinite set X, the group of those permutations of X that move only finitely many elements.\n\nIt is known that no group of permutations containing an element with exactly one infinite orbit can be written as an image of such an ultraproduct (ibid.). G. M. Bergman", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2521, "problem_number": "KOU-21.12", "title": "Kourovka Notebook Problem 21.12", "statement": "Suppose that $U$ is a nonprincipal ultrafilter on $\\omega$, and $B$ is a group such that every element $b\\in B$ belongs to a subgroup of $B$ that is a homomorphic image of $\\mathbb Z^\\omega/U$. Must $B$ then be a homomorphic image of an ultraproduct group $(\\prod_{i\\in\\omega}G_i)/U$ for some groups $G_i$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.12.\n\nDiscussion and literature:\nThis is Question 19 in (G. M. Bergman, Pacific J. Math., 274 (2015) 451--495).\n\nAn affirmative answer would imply that every torsion group was such a homomorphic image for every $U$, and so would give positive answers to both parts of 21.11. G. M. Bergman\n\nNo, it need not (S. M. Corson, J. Algebra, 681 (2025), 306--317).", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2522, "problem_number": "KOU-21.13", "title": "Kourovka Notebook Problem 21.13", "statement": "Does $\\mathbb Z^\\omega$ have a subgroup whose dual is free abelian of still larger rank (the largest possible being $2^{2^{\\aleph_0}}$)?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.13.\n\nDiscussion and literature:\nIt is known that the group $\\mathbb Z^\\omega$ has a subgroup whose dual is free abelian of rank $2^{\\aleph_0}$ (see 17.24 in Archive).\n\nThis is Question 11 in (G. M. Bergman, Portugaliae Math., 69 (2012) 69--84). G. M. Bergman", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2523, "problem_number": "KOU-21.14", "title": "Kourovka Notebook Problem 21.14", "statement": "Suppose $\\alpha$ is an endomorphism of a group G such that for every group H and every homomorphism $f:G\\to H$, there exists an endomorphism $\\beta_f$ of $H$ such that $\\beta_f f=f\\alpha$. Must $\\alpha$ then be either an inner automorphism of G or the trivial endomorphism?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.14.\n\nDiscussion and literature:\nThis is Question 5 in (G. M. Bergman, Publ. Matem., 56 (2012), 91--126). G. M. Bergman\n\nYes, it must (F. Fournier-Facio, Preprint, 2026, https://arxiv.org/abs/2604.05728)", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2524, "problem_number": "KOU-21.15", "title": "Kourovka Notebook Problem 21.15", "statement": "Suppose B is a subgroup of the symmetric group $S_\\Omega$ on an infinite set $\\Omega$. Will the amalgamated free product $S_\\Omega *_B S_\\Omega$ of two copies of $S_\\Omega$ with amalgamation of B be embeddable in $S_\\Omega$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.15.\n\nDiscussion and literature:\nThis is a weakened form of the group case of Question 4.4 in (G. M. Bergman, Indag. Math., 18 (2007), 349--403).\n\nIt is known that $S_\\Omega *_B S_\\Omega$ need not be so embeddable by a map respecting B (Algebra Number Theory, 3 (2009), 847--879, \\S{} 10). G. M. Bergman\n\nNo, not necessarily: take $\\Omega$ countably infinite, $B$ a subgroup of $S_\\Omega$ which is not Borel, and $\\varphi:S_\\Omega *_B S_\\Omega\\to S_\\Omega$ an injective group homomorphism. Let $\\varphi_1$ be the restriction of $\\varphi$ to the first copy of $S_\\Omega$ and $\\varphi_2$ that to the second. Each $\\varphi_i$ is continuous (A. S. Kechris, C. Rosendal, Proc. Lond. Math. Soc., 94, no. 2 (2007), 302--350), so $\\operatorname{im}(\\varphi_i)$ is Borel in $S_\\Omega$ (as a continuous injective image of a Polish space). Now $B=\\varphi_1^{-1}(\\varphi(B))=\\varphi_1^{-1}(\\operatorname{im}(\\varphi_1)\\cap\\operatorname{im}(\\varphi_2))$ is Borel, a contradiction. (S. M. Corson, Letter of 26 January 2026; see also https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/03/solution-of-21.15.pdf.)", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2525, "problem_number": "KOU-21.16", "title": "Kourovka Notebook Problem 21.16", "statement": "Let the width of a group (respectively, a monoid) H with respect to a generating set X mean the supremum over h $\\in$ H of the least length of a group word (respectively, a monoid word) in elements of X expressing h. A group (or monoid) is said to have finite width if its width with respect to every generating set is finite. (A common finite bound for these widths is not required.) Do there exist groups G having finite width as groups, but not as monoids?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.16.\n\nDiscussion and literature:\nThis is Question 9 in (G. M. Bergman, Bull. London Math. Soc., 38 (2006), 429--440). G. M. Bergman", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2526, "problem_number": "KOU-21.17", "title": "Kourovka Notebook Problem 21.17", "statement": "If $X$ is a class of groups, let $H(X)$ denote the class of homomorphic images of groups in $X$, let $S(X)$ denote the class of groups isomorphic to subgroups of groups in $X$, let $P(X)$ denote the class of groups isomorphic to (unrestricted) direct products of families of groups in $X$, and let $P_f(X)$ denote the class of groups isomorphic to direct products of finite families of groups in $X$. By Birkhoff's theorem, $H(S(P(X)))$ is the variety of groups generated by $X$. If $M$ is a class of metabelian groups, must $H(S(P_f(M)))\\subseteq S(H(P(S(M))))$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.17.\n\nDiscussion and literature:\nThis is Question 27 in (G. M. Bergman, Algebra Universalis, 26 (1989), 267--283). G. M. Bergman", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2527, "problem_number": "KOU-21.18", "title": "Kourovka Notebook Problem 21.18", "statement": "Suppose that G is a finite group, and $A_1,A_2,A_3$ are subsets of G such that the multiplication map $A_1\\times A_2\\times A_3\\to G$ is bijective. Must the subgroup $\\langle A_2\\rangle$ generated by $A_2$ have order divisible by the cardinality $|A_2|$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.18.\n\nDiscussion and literature:\nThis is Question 8 in (G. M. Bergman, J. Iranian Math. Soc., 1 (2020), 157--161).\n\nIt is known (ibid.) that the corresponding statement is true for the subgroups $\\langle A_1\\rangle$ and $\\langle A_3\\rangle$. Moreover, $|A_2|$ will at least divide the order of the least subgroup containing $A_2$ and closed under conjugation by members of $A_1$, and similarly of the least subgroup containing $A_2$ and closed under conjugation by members of $A_3$. G. M. Bergman\n\nNo, it need not (M. I. Kabenyuk, Preprint, 2021, https://arxiv.org/abs/2102.08605).", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2528, "problem_number": "KOU-21.19", "title": "Kourovka Notebook Problem 21.19", "statement": "Suppose that S and M are groups of finite Morley rank, S is an infinite group, and M is a non-trivial connected group definably and faithfully acting on S. This action is said to be irreducible if M does not leave invariant any definable non-trivial proper subgroup of S. Prove that if S is a simple group such that every proper definable subgroup of S is nilpotent, and the action of M on S is irreducible, then this action is equivalent to the action of S on itself by conjugation.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.19.\n\nDiscussion and literature:\nA. V. Borovik", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2529, "problem_number": "KOU-21.20", "title": "Kourovka Notebook Problem 21.20", "statement": "Prove that a simple group of finite Morley rank without involutions cannot act definably, faithfully, and irreducibly on a connected group other than on itself acting by conjugation.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.20.\n\nDiscussion and literature:\nA. V. Borovik", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2530, "problem_number": "KOU-21.21", "title": "Kourovka Notebook Problem 21.21", "statement": "Prove that a simple (that is, without proper non-trivial connected normal subgroups) algebraic group M over an algebraically closed field cannot act definably, faithfully, and irreducibly on a simple group of finite Morley rank other than M/Z(M).", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.21.\n\nDiscussion and literature:\nA. V. Borovik", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2531, "problem_number": "KOU-21.22", "title": "Kourovka Notebook Problem 21.22", "statement": "Is the (standard, restricted) wreath product $G\\wr H$ of two finitely generated Hopfian groups Hopfian?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.22.\n\nDiscussion and literature:\nThe same question where G is assumed to be abelian or nilpotent is equivalent to Kaplansky's direct finiteness conjecture; see (H. Bradford, F. Fournier-Facio, Math. Z., 308, no. 4 (2024), Paper no. 58). H. Bradford, F. Fournier-Facio", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2532, "problem_number": "KOU-21.23", "title": "Kourovka Notebook Problem 21.23", "statement": "A graph is called a cograph if it has no induced subgraph isomorphic to a path with 4 vertices. A graph is said to be chordal if it has no induced cycles with n vertices for every $n\\geqslant 4$. For a finite group G, the enhanced power graph E(G) is the graph with vertex set G and edges \\{x, y\\} for all $x\\ne y\\in G$ such that $\\langle x,y\\rangle$ is cyclic.\n\n(a) For a given integer $n\\geqslant 4$, determine the set of all finite nonabelian simple groups G such that E(G) has no induced cycles with n vertices.\n\n(b) Determine the set of all finite nonabelian simple groups G such that E(G) is chordal.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.23.\n\nDiscussion and literature:\nIn (Preprint, 2025, https://arxiv.org/abs/2510.18073) we proved that if the enhanced power graph of a given finite group is a cograph, then it is also chordal. Also the finite nonabelian simple groups whose enhanced power graph is a cograph are described, and additional information is obtained on finite nonabelian simple groups whose enhanced power graph has no induced cycles with 4 vertices. D. Bubboloni, F. Fumagalli, C. E. Praeger", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2533, "problem_number": "KOU-21.24", "title": "Kourovka Notebook Problem 21.24", "statement": "For a finite group G, the power graph P(G) is the graph with vertex set G and edges \\{x, y\\} for all $x\\ne y\\in G$ such that either $x\\in\\langle y\\rangle$ or $y\\in\\langle x\\rangle$. Is it true that, for every finite group G, if P(G) is a cograph, then P(G) is chordal?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.24.\n\nDiscussion and literature:\nCf. 21.23. This holds if every element of G has prime power order (D. Bubboloni, F. Fumagalli, C. E. Praeger, Preprint, 2025, https://arxiv.org/abs/2510.18073) and if G is a nonabelian simple group (J. Cameron, P. Manna, R. Mehatari, J. Algebra, 591 (2022), 59--74; J. Brachter, E. Kaja, J. Algebr. Comb., 58 (2023), 1095--1124). D. Bubboloni, F. Fumagalli, C. E. Praeger\n\nYes, it is true (M. Rundstr\\\"om, Preprint of 30 January 2026, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/04/21.24-runds.pdf; P. Monticone, Preprint of 29 March 2026, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/04/21_24-1.pdf)", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2534, "problem_number": "KOU-21.25", "title": "Kourovka Notebook Problem 21.25", "statement": "Let $G$ be a finite simple group and let $p_1,p_2$ be any (not necessarily distinct) prime divisors of $|G|$. Then can we always find Sylow $p_i$-subgroups $H_i$ such that $G=\\langle H_1,H_2\\rangle$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.25.\n\nDiscussion and literature:\n(T. Breuer, R. M. Guralnick).\n\nT. C. Burness", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2535, "problem_number": "KOU-21.26", "title": "Kourovka Notebook Problem 21.26", "statement": "Let $G$ be a non-trivial finite group and let $p_1,\\ldots,p_k$ be the distinct prime divisors of $|G|$. For each $i$, let $H_i$ be a Sylow $p_i$-subgroup of $G$. Is it true that there exists an element $x\\in G$ such that for all $i$ the subgroup $H_i\\cap H_i^x$ is inclusion-minimal in $\\{H_i\\cap H_i^g:g\\in G\\}$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.26.\n\nDiscussion and literature:\n(F. Lisi, L. Sabatini).\n\nT. C. Burness", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2536, "problem_number": "KOU-21.27", "title": "Kourovka Notebook Problem 21.27", "statement": "A permutation on a set $\\Omega$ is called a derangement if it has no fixed points in $\\Omega$. Let G be a finite simple transitive permutation group. Is it true that every element in G is the product of two derangements?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.27.\n\nDiscussion and literature:\n(M. Larsen, A. Shalev, P. H. Tiep).\n\nT. C. Burness", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2537, "problem_number": "KOU-21.28", "title": "Kourovka Notebook Problem 21.28", "statement": "Let $G$ be a finite simple transitive permutation group, and let $\\delta(G)$ be the proportion of derangements in $G$. Is it true that $\\delta(G)\\geqslant 89/325$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.28.\n\nDiscussion and literature:\nNote that $\\delta(G)=89/325$ for the action of the Tits group $G={}^2F_4(2)'$ on the cosets of a maximal parabolic subgroup of the form $2^2.[2^8].S_3$ (Forum Math. Sigma, 13 (2025), paper no. e98, 62 pp.). T. C. Burness, M. Fusari", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2538, "problem_number": "KOU-21.29", "title": "Kourovka Notebook Problem 21.29", "statement": "Let $G\\leqslant\\operatorname{Sym}(\\Omega)$ be a finite primitive permutation group with a regular suborbit (that is, $G$ has a trivial 2-point stabiliser). Then is it true that for all $\\alpha,\\beta\\in\\Omega$, there exists $\\gamma\\in\\Omega$ such that the 2-point stabilisers $G_{\\alpha,\\gamma}$ and $G_{\\beta,\\gamma}$ are both trivial?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.29.\n\nDiscussion and literature:\nT. C. Burness, M. Giudici", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2539, "problem_number": "KOU-21.30", "title": "Kourovka Notebook Problem 21.30", "statement": "(Well-known question). A discrete group G is said to have the Haagerup property (also known as Gromov's a-T-menability property) if there exists a metrically proper isometric action of G on a (possibly infinite-dimensional) Hilbert space. Are all 1-relator groups Haagerup groups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.30.\n\nDiscussion and literature:\nJ. O. Button", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2540, "problem_number": "KOU-21.31", "title": "Kourovka Notebook Problem 21.31", "statement": "Conjecture: If N is a finite soluble group, then any regular subgroup in the holomorph Hol(N) of N is also soluble.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.31.\n\nDiscussion and literature:\nN. Byott", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2541, "problem_number": "KOU-21.32", "title": "Kourovka Notebook Problem 21.32", "statement": "Is the following problem decidable, and if so, what is its complexity? Given a finite group G, is there a finite group H such that the derived subgroup of H is isomorphic to G?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.32.\n\nDiscussion and literature:\nP. J. Cameron", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2542, "problem_number": "KOU-21.33", "title": "Kourovka Notebook Problem 21.33", "statement": "Does an analogue of Dunwoody's theorem hold for totally disconnected locally compact groups, that is, must a tdlc group of rational discrete cohomological dimension at most 1 be topologically isomorphic to the fundamental group of a graph of profinite groups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.33.\n\nDiscussion and literature:\nI. Castellano", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2543, "problem_number": "KOU-21.34", "title": "Kourovka Notebook Problem 21.34", "statement": "(Well-known problem). A group $G$ is a unique product group if, for any nonempty finite subsets $A,B$ of $G$, there exists an element of $G$ which can be written uniquely as $ab$ with $a\\in A$ and $b\\in B$. A group $G$ is locally invariant orderable if $G$ admits a partial order $<$ such that for all $g,h\\in G$ with $h\\ne 1$, we have either $gh>g$ or $gh^{-1}>g$. Does there exist a unique product group which is not locally invariant orderable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.34.\n\nDiscussion and literature:\nA. Clay", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2544, "problem_number": "KOU-21.35", "title": "Kourovka Notebook Problem 21.35", "statement": "Let $G$ be a finite group, $w$ a multilinear commutator group-word, and $p$ a prime. Suppose that $p$ divides the order $|xy|$ whenever $x$ is a $w$-value of $p'$-order in $G$ and $y$ is a $w$-value in $G$ of order divisible by $p$. Is it true that then the verbal subgroup $w(G)$ must be $p$-nilpotent?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.35.\n\nDiscussion and literature:\nWithout the assumption that $w$ be multilinear, the answer is negative. An affirmative answer has been obtained in several special cases (J. Algebra, 609 (2022), 926--936). Y. Contreras Rojas, V. Grazian, C. Monetta", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2545, "problem_number": "KOU-21.36", "title": "Kourovka Notebook Problem 21.36", "statement": "Let a hierarchy of tdlc groups $\\mathbf H\\mathcal K$ be defined analogously to Kropholler's hierarchy in 15.45, with $\\mathcal K$ being the class of profinite groups and with the cell stabilisers of the admissible action required to be open. Is it true that $\\mathbf H\\mathcal K$ is closed under profinite extensions, that is, $(\\mathbf H_\\alpha\\mathcal K)\\mathcal K\\subseteq \\mathbf H_\\alpha\\mathcal K$ for every $\\alpha$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.36.\n\nDiscussion and literature:\nKropholler's hierarchy (see 15.45) is closed under finite extensions, that is, $(\\mathbf H_\\alpha\\mathcal F)\\mathcal F\\subseteq \\mathbf H_\\alpha\\mathcal F$ for every $\\alpha$ (P. Kropholler, J. Pure Appl. Algebra, 90 (1993), 55--67).\n\nG. C. Cook", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2546, "problem_number": "KOU-21.37", "title": "Kourovka Notebook Problem 21.37", "statement": "By definition, a constructible totally disconnected, locally compact (tdlc) group is the result of a sequence of profinite extensions and ascending HNN-extensions starting from the trivial group. Are soluble tdlc groups of type $FP_\\infty$ constructible?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.37.\n\nDiscussion and literature:\nAs in the discrete case, soluble constructible tdlc groups have type $FP_\\infty$ (G. C. Cook, I. Castellano, J. Algebra, 543 (2020), 54--97).\n\nG. C. Cook", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2547, "problem_number": "KOU-21.38", "title": "Kourovka Notebook Problem 21.38", "statement": "The spread of a group $G$ is the greatest nonnegative integer $k$ such that for all nontrivial elements $x_1,\\ldots,x_k\\in G$ there exists $y\\in G$ such that $\\langle x_1,y\\rangle=\\cdots=\\langle x_k,y\\rangle=G$, or is $\\infty$ in case there is no such maximum. Does there exist a group with spread equal to 1?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.38.\n\nDiscussion and literature:\n(S. Harper, C. Donoven).\n\nSuch a group must be infinite if it exists (T. C. Burness, R. M. Guralnick, S. Harper, Ann. Math., 193 (2021), 619--687). S. Corson", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2548, "problem_number": "KOU-21.39", "title": "Kourovka Notebook Problem 21.39", "statement": "Are there any locally finite, characteristically simple groups with finitely many orbits under automorphisms that are not residually finite?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.39.\n\nDiscussion and literature:\nIt is known that there exist residually finite, locally finite, characteristically simple groups with finitely many orbits under automorphisms (A. B. Apps, J. Algebra, 81 (1983), 320--339).\n\nA. Dantas, E. de Melo", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2549, "problem_number": "KOU-21.40", "title": "Kourovka Notebook Problem 21.40", "statement": "Let G be a subgroup of GL(n, Q) with finitely many orbits under automorphisms. Is G a virtually soluble group?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.40.\n\nDiscussion and literature:\nA. Dantas, E. de Melo", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2550, "problem_number": "KOU-21.41", "title": "Kourovka Notebook Problem 21.41", "statement": "A group is said to be self-similar if it admits a faithful state-closed representation by automorphisms of a regular one-rooted m-tree for some m. Can a torsion-free finitely presented metabelian group which is self-similar contain a subgroup isomorphic to the restricted wreath product H = $\\mathbb Z\\wr\\mathbb Z$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.41.\n\nDiscussion and literature:\nIt is known that $\\mathbb Z\\wr\\mathbb Z$ itself is self-similar (A. C. Dantas, T. M. G. Santos, S. N. Sidki, J. Algebra, 567 (2021), 564--581). A. Dantas, S. Sidki", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2551, "problem_number": "KOU-21.42", "title": "Kourovka Notebook Problem 21.42", "statement": "Let $\\mathcal T_{d,c}$ denote the class of $d$-generated, torsion-free nilpotent groups having class $c$. Are there $\\mathcal T_{3,3}$-groups that are not self-similar?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.42.\n\nDiscussion and literature:\nIt is known that $\\mathcal T_{d,2}$-groups are self-similar for all $d$, that $\\mathcal T_{2,3}$-groups are self-similar, and that there are $\\mathcal T_{4,3}$-groups that are not self-similar (A. Berlatto, T. Santos, Preprint, 2025, https://arxiv.org/abs/2509.16947).\n\nA. Dantas, S. Sidki", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2552, "problem_number": "KOU-21.43", "title": "Kourovka Notebook Problem 21.43", "statement": "Conjecture: Suppose that for a fixed positive integer $k$ at least half of the elements of a finite group $G$ have order $k$. Then $G$ is solvable.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.43.\n\nDiscussion and literature:\nM. Deaconescu\n\nThis is not always true. In a direct product of $N$ copies of $A_5$ there are $60^N$ elements, while the number of elements of order exactly 30 is $60^N-45^N-40^N-36^N+25^N+21^N+16^N-1$. As $N\\to\\infty$, the ratio of elements of order 30 converges to 1. (L. Tae Young, Letter of 9 January 2026).", "difficulty_level_id": 3, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2553, "problem_number": "KOU-21.44", "title": "Kourovka Notebook Problem 21.44", "statement": "Let $W_n=A_5\\wr\\cdots\\wr A_5$ be the $n$-times iterated permutational wreath product of $A_5$ in its natural action (so $W_n$ acts on $5^n$ points), and let $W=\\varprojlim W_n$ be the inverse limit (infinite iterated wreath product of $A_5$). Does $W$ contain a finitely generated dense subgroup of subexponential growth?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.44.\n\nDiscussion and literature:\nS. Eberhard", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2554, "problem_number": "KOU-21.45", "title": "Kourovka Notebook Problem 21.45", "statement": "(Well-known problem). Does there exist a finitely presented (infinite) simple group requiring more than two generators?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.45.\n\nDiscussion and literature:\nCf. 6.44 in Archive. F. Fournier-Facio", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2555, "problem_number": "KOU-21.46", "title": "Kourovka Notebook Problem 21.46", "statement": "(Well-known problem). Does there exist a finitely presented (infinite) simple group of finite cohomological dimension greater than 2?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.46.\n\nDiscussion and literature:\nF. Fournier-Facio", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2556, "problem_number": "KOU-21.47", "title": "Kourovka Notebook Problem 21.47", "statement": "(Well-known problem). Does there exist a finitely presented group $G$ such that $G\\cong G\\times H$ for some non-trivial group $H$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.47.\n\nDiscussion and literature:\nThe first finitely generated example was constructed in (J. M. Tyrer Jones, J. Austral. Math. Soc., 17 (1974), 174--196).\n\nA finitely presented group that surjects onto its own direct square was constructed in (G. Baumslag, C. F. Miller, III, Bull. London Math. Soc., 20, no. 3 (1988), 239--244). F. Fournier-Facio", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2557, "problem_number": "KOU-21.48", "title": "Kourovka Notebook Problem 21.48", "statement": "A quasimorphism on a group $G$ is a function $f:G\\to\\mathbb R$ such that the quantity $\\sup_{g,h}|f(g)+f(h)-f(gh)|$ is finite. A quasimorphism is homogeneous if it restricts to a homomorphism on every cyclic subgroup of $G$. Let $G$ be a group admitting an unbounded homogeneous quasimorphism $G\\to\\mathbb R$ that is not a homomorphism. Must $G$ contain a non-abelian free subgroup?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.48.\n\nDiscussion and literature:\nF. Fournier-Facio", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2558, "problem_number": "KOU-21.49", "title": "Kourovka Notebook Problem 21.49", "statement": "An isometric action of a group G on a metric space S is called acylindrical if for every $\\varepsilon>0$ there exist R, N > 0 such that for every two points x, y with $d(x,y)\\geqslant R$, there are at most N elements $g\\in G$ satisfying $d(x,gx)\\leqslant\\varepsilon$ and $d(y,gy)\\leqslant\\varepsilon$. A group is said to be acylindrically hyperbolic if it is not virtually cyclic and admits an acylindrical action on a hyperbolic space with unbounded orbits. Is the automorphism group of a finitely generated acylindrically hyperbolic group also acylindrically hyperbolic?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.49.\n\nDiscussion and literature:\nA. Genevois", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2559, "problem_number": "KOU-21.50", "title": "Kourovka Notebook Problem 21.50", "statement": "Does every finite 3-group $T$ have a nontrivial characteristic subgroup $C$ such that if $T$ is a Sylow 3-subgroup of a finite group $G$, then $T\\cap G'=T\\cap H'$, where $H=N_G(C)$?\n\nThe group $S_4$ shows that no such characteristic subgroups can be found in some Sylow 2-subgroups.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.50.\n\nDiscussion and literature:\nSuch a characteristic subgroup is known to exist in $p$-groups for $p\\geqslant 5$ (G. Glauberman, Math. Z., 117 (1970), 46--56), and for $p=3$ there are two characteristic subgroups $K_1,K_2$ such that $T\\cap G'=(T\\cap H_1')(S\\cap H_2')$, where $H_i=N_G(K_i)$ (G. Glauberman, J. Algebra, 648 (2024), 62--86).\n\nG. Glauberman", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2560, "problem_number": "KOU-21.51", "title": "Kourovka Notebook Problem 21.51", "statement": "Let $p$ be a prime, and $P$ a finite $p$-group.\n\n(a) Suppose that $P$ has an abelian subgroup of order $p^n$. For which $n$ does $P$ necessarily have a normal abelian subgroup of order $p^n$?\n\n(b) Suppose that $P$ has an elementary abelian subgroup of order $p^n$. For which $n$ does $P$ necessarily have a normal elementary abelian subgroup of order $p^n$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.51.\n\nDiscussion and literature:\nIt is easy to see that for $p=2$, the answer to (b) is \"yes\" only for $n=1$. The answer to both questions is \"yes\" for $n<(p+2)/2$ (G. G. Glauberman, J. Algebra, 319, no. 2 (2008), 800--805), as well as for $n\\leqslant 5$ when $p\\ne 2$ (M. Konvisser, D. Jonah, J. Algebra, 34 (1975), 309--330). The answer to both questions is \"no\" for $n\\geqslant (p+9)/2$ when $p\\geqslant 5$ (for $p\\geqslant 7$ due to G. Glauberman, Contemp. Math., 524 (2010), 61--65; for $p=3,5$ due to Ya. G. Berkovich, J. Algebra, 248, no. 2 (2002), 472--553). Thus, the only open cases for $p\\geqslant 5$ are $n=6$ for $p=5$, and $n=(p+3)/2,(p+5)/2,(p+7)/2$ for $p>5$. G. Glauberman", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2561, "problem_number": "KOU-21.52", "title": "Kourovka Notebook Problem 21.52", "statement": "Let $L$ be a finite non-abelian simple group, and let $D$ be a conjugacy class of involutions in $L$. Consider the complete graph $\\Gamma$ with vertex set $D$. Define an equivalence relation $\\sim$ (graph coloring) on the set of edges as follows: $(a,b)\\sim(c,d)$ if and only if $|ab|=|cd|$. An automorphism of the coloured graph $\\Gamma$ is a permutation $\\tau\\in S_D$ such that $(a,b)\\sim(a^\\tau,b^\\tau)$ for every edge $(a,b)$. Is it true that the automorphism group of $\\Gamma$ is a subgroup of $\\operatorname{Aut}(L)$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.52.\n\nDiscussion and literature:\nI. B. Gorshkov", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2562, "problem_number": "KOU-21.53", "title": "Kourovka Notebook Problem 21.53", "statement": "In the notation of 21.52, let $\\operatorname{Aut}_t(\\Gamma)$ be the set of permutations $\\tau\\in S_D$ such that $(a,b)\\sim(a^\\tau,b^\\tau)$ whenever $|ab|=t$ for $a,b\\in D$. Clearly, $\\operatorname{Aut}(\\Gamma)=\\bigcap_t\\operatorname{Aut}_t(\\Gamma)$. Is it true that for every finite simple group $G$ we have $\\operatorname{Aut}(\\Gamma)=\\operatorname{Aut}_2(\\Gamma)\\cap\\operatorname{Aut}_p(\\Gamma)$, where $\\{2,p\\}$ are the two minimal prime divisors of $|G|$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.53.\n\nDiscussion and literature:\nI. B. Gorshkov", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2563, "problem_number": "KOU-21.54", "title": "Kourovka Notebook Problem 21.54", "statement": "Let $G$ be a finite soluble group with triality, which means that $G$ admits a group of automorphisms $S$ isomorphic to the symmetric group of degree 3 given by the presentation $S=\\langle\\sigma,\\rho\\mid \\sigma^2=\\rho^3=1;\\ \\sigma\\rho\\sigma=\\rho^2\\rangle$ such that $m\\cdot m^\\rho\\cdot m^{\\rho^2}=1$ for all $m$ in the set of commutators $M(G):=\\{[g,\\sigma]\\mid g\\in G\\}$. Suppose in addition that $G=[G,S]$, the group $G$ is generated by $d$ elements of $M(G)$ and their images under $S$, and $x^n=1$ for all $x\\in M(G)$. Is it true that the Fitting height of $G$ is bounded in terms of $d$ and $n$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.54.\n\nDiscussion and literature:\nAn affirmative answer would provide a reduction of the analogue of the Restricted Burnside Problem for Moufang loops to the nilpotent case. A. N. Grishkov, A. V. Zavarnitsine", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2564, "problem_number": "KOU-21.55", "title": "Kourovka Notebook Problem 21.55", "statement": "Let $q$ be a power of a prime $p$, and let $m_n(q)$ be the maximum $p$-length of $p$-solvable subgroups of $\\operatorname{GL}(n,q)$. Is it true that $\\lim_{n\\to\\infty}m_n(q)/\\log_2 n=1$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.55.\n\nDiscussion and literature:\nI. G\\\"ulo\\u{g}lu", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2565, "problem_number": "KOU-21.56", "title": "Kourovka Notebook Problem 21.56", "statement": "Let $\\ell(X)$ denote the composition length of a finite group $X$. Let $A$ be a finite nilpotent group acting by automorphisms on a finite soluble group $G$. Let $c(G,A)$ be the number of trivial $A$-modules in a given $A$-composition series of $G$. (Note that $c(G,A)=\\ell(C_G(A))$ if $(|A|,|G|)=1$.)\n\nConjecture: there are absolute constants $C_1$ and $C_2$ such that the Fitting height of $G$ is at most $C_1\\ell(A)+C_2c(G,A)$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.56.\n\nDiscussion and literature:\nI. G\\\"ulo\\u{g}lu", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2566, "problem_number": "KOU-21.57", "title": "Kourovka Notebook Problem 21.57", "statement": "Let X be a non-empty class of finite groups of odd order closed under taking subgroups, homomorphic images, and extensions. Let H be an X-maximal subgroup of a finite group G, and N a normal subgroup of G. Must H $\\cap$ N be an X-maximal subgroup of N?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.57.\n\nDiscussion and literature:\nW. Guo, D. O. Revin", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2567, "problem_number": "KOU-21.58", "title": "Kourovka Notebook Problem 21.58", "statement": "We say that a product $XY=\\{xy\\mid x\\in X,\\ y\\in Y\\}$ of two subsets $X,Y$ of a group $G$ is direct if for every $z\\in XY$ there are unique $x\\in X$, $y\\in Y$ such that $z=xy$. Is there an infinite group $G$ such that every subset $A\\subseteq G$ satisfies the following property: all the maximal subsets $B$ for which the product $AB$ is direct have the same cardinality?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.58.\n\nDiscussion and literature:\nNote that for checking the property for a given infinite group $G$, it suffices to consider only those subsets $A\\subseteq G$ for which $|A|=|G\\setminus A|$. Indeed, the property is equivalent to $A^{-1}A\\cap BB^{-1}=\\{1\\}$ and $A^{-1}AB=G$, and these imply $|G|=|A||B|$, since $G$ is infinite. Now, if $|A|<|G\\setminus A|$, then $|A|<|G|$, and so $|B|=|G|$; and if $|A|>|G\\setminus A|$, then $A^{-1}A=G$, and so $|B|=1$, for all $B$ satisfying the property. M. H. Hooshmand\n\nNo, there are no such groups (M. I. Kabenyuk, Preprint, 2026, https://arxiv.org/abs/2602.22876)", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2568, "problem_number": "KOU-21.59", "title": "Kourovka Notebook Problem 21.59", "statement": "For a finite group $G$, let $\\chi_1(G)$ denote the totality of the degrees of all irreducible complex characters of $G$ with allowance for their multiplicities. Suppose that $H$ is a finite group with $\\chi_1(H)=\\chi_1(G)$.\n\na) If $G$ is an almost simple group, must $H$ be isomorphic to $G$?\n\nb) If $G$ is a quasisimple group, must $H$ be isomorphic to $G$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.59.\n\nDiscussion and literature:\nThis is true if $G$ is a simple group (see 11.8(a) in Archive). A. Iranmanesh, F. Shirjian", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2569, "problem_number": "KOU-21.60", "title": "Kourovka Notebook Problem 21.60", "statement": "Let $G$ be a finite group, $\\mathbb Z_{(p)}$ the localization at $p$, and $\\mathbb F_p$ the field of $p$ elements. Let $\\mathcal X$ be the class of $\\mathbb F_pG$-modules obtained by reduction of simple $\\mathbb QG$-modules. Is it true that $\\mathbb Z_{(p)}G$ is semiperfect if and only if each projective indecomposable $\\mathbb F_pG$-module can be written as an $\\mathbb N$-linear combination of modules in $\\mathcal X$ inside the Grothendieck group of $\\mathbb F_pG$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.60.\n\nDiscussion and literature:\nThe \"only if\" direction is proved, and the affirmative answer is obtained if $p$ does not divide $|G|$ (D. Johnston, D. Rumynin, J. Algebra, 687, no. 1 (2026), 776--791). D. Johnston, D. Rumynin", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2570, "problem_number": "KOU-21.61", "title": "Kourovka Notebook Problem 21.61", "statement": "For a fixed (finitely generated free)-by-cyclic group $G=F_n\\rtimes\\mathbb Z$, is there an algorithm that, given a finite subset $S$ of $G$, finds a finite presentation for the subgroup $H=\\langle S\\rangle$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.61.\n\nDiscussion and literature:\nCf. 4.8 in Archive. I. Kapovich", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2571, "problem_number": "KOU-21.62", "title": "Kourovka Notebook Problem 21.62", "statement": "Is the uniform subgroup membership problem decidable for (finitely generated free)-by-cyclic groups? That is, for a fixed group $G=F_n\\rtimes\\mathbb Z$, is there an algorithm that, given elements $w,h_1,\\ldots,h_k\\in G$, decides whether or not $w$ belongs to the subgroup $H=\\langle h_1,\\ldots,h_k\\rangle$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.62.\n\nDiscussion and literature:\nI. Kapovich", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2572, "problem_number": "KOU-21.63", "title": "Kourovka Notebook Problem 21.63", "statement": "Let $F$ be a field of characteristic $p>0$, and let $\\Gamma$ be the principal congruence subgroup of $\\operatorname{Aut}(F[x_1,\\ldots,x_n])$ consisting of all automorphisms that send each variable $x_i$ to $x_i$ modulo terms of higher degree. Then $\\Gamma$ is a residually-$p$ group. Does $\\Gamma$ satisfy a pro-$p$ identity?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.63.\n\nDiscussion and literature:\n(E. Zelmanov).\n\nE. I. Khukhro", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2573, "problem_number": "KOU-21.64", "title": "Kourovka Notebook Problem 21.64", "statement": "Is it true that if a normal subgroup $A$ of a Sylow $p$-subgroup of a $p$-soluble finite group $G$ has exponent $p^e$, then the normal closure of $A$ in $G$ has $(p,e)$-bounded (or even $e$-bounded) $p$-length?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.64.\n\nDiscussion and literature:\nE. I. Khukhro", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2574, "problem_number": "KOU-21.65", "title": "Kourovka Notebook Problem 21.65", "statement": "Suppose that $\\phi$ is an automorphism of a finite soluble group $G$. Must $G$ contain a subgroup of index bounded in terms of $|\\phi|$ and $|C_G(\\phi)|$ whose Fitting height is bounded\n\n(a) in terms of $|\\phi|$?\n\n(b) or even in terms of the composition length of $\\langle\\phi\\rangle$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.65.\n\nDiscussion and literature:\nAn affirmative answer to part (b) is known when $|\\phi|$ is a prime power (B. Hartley--V. Turau), or when $(|G|,|\\phi|)=1$ (A. Turull, B. Hartley--I. M. Isaacs). Also cf. 19.43. E. I. Khukhro", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2575, "problem_number": "KOU-21.66", "title": "Kourovka Notebook Problem 21.66", "statement": "Suppose that A is a nilpotent group of automorphisms of a finite soluble group G. Is the Fitting height of G bounded in terms of |A| and |CG(A)|?\n\nIsaacs), when A is cyclic (see 19.43), or when CG(A) = 1 (E. C. Dade).", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.66.\n\nDiscussion and literature:\nAn affirmative answer is known when (|G|, |A|) = 1 (J. G. Thompson, even for soluble A, with improved bounds in subsequent papers of H. Kurzweil, A. Turull, B. Hartley--I. M.\n\nNote that for any non-nilpotent finite group A there are finite soluble groups G of unbounded Fitting height with CG(A) = 1 (S. D. Bell--B. Hartley). E. I. Khukhro", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2576, "problem_number": "KOU-21.67", "title": "Kourovka Notebook Problem 21.67", "statement": "Suppose that $\\phi$ is an automorphism of a finite soluble group $G$, and let $r$ be the (Pr\\\"ufer) rank of the fixed-point subgroup $C_G(\\phi)$. Is the Fitting height of $G$ bounded in terms of $|\\phi|$ and $r$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.67.\n\nDiscussion and literature:\nThis is known to be true in the case where $|\\phi|$ is a product of at most two prime powers (B. Hartley, Preprint, 1994, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2025/08/hartley94-prepr-mims.pdf). An affirmative answer to this question would imply an affirmative answer to 13.8(b). E. I. Khukhro", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2577, "problem_number": "KOU-21.68", "title": "Kourovka Notebook Problem 21.68", "statement": "A finite group $G$ is said to be semi-abelian if it has a sequence of subgroups $1=G_0\\leqslant G_1\\leqslant\\cdots\\leqslant G_n=G$ such that for every $i$ the subgroup $G_{i+1}$ is isomorphic to a quotient of a semidirect product $A_i\\rtimes G_i$ for some abelian group $A_i$.\n\nConjecture: Semi-abelian finite groups are monomial.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.68.\n\nDiscussion and literature:\nM. Kida", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2578, "problem_number": "KOU-21.69", "title": "Kourovka Notebook Problem 21.69", "statement": "Is there an algorithm deciding if a given one-relator group is hyperbolic?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.69.\n\nDiscussion and literature:\nD. Kielak", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2579, "problem_number": "KOU-21.70", "title": "Kourovka Notebook Problem 21.70", "statement": "A group $G$ is called an orientable Poincar\\'e duality group of dimension $n$ over a ring $R$ if it is of type $FP$ over $R$ and $H^i(G;RG)=0$ for $i\\ne n$, while $H^n(G;RG)=R$ as an $RG$-module, where the action on $R$ is trivial. (Note that $G$ is not required to be finitely presented.) If $G$ is an orientable Poincar\\'e duality group of dimension $n$ over all fields, is it an orientable Poincar\\'e duality group over the integers?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.70.\n\nDiscussion and literature:\nD. Kielak", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2580, "problem_number": "KOU-21.71", "title": "Kourovka Notebook Problem 21.71", "statement": "For a ring $R$, we say that a group $G$ is of type $FL(R)$ if the trivial $RG$-module $R$ admits a finite resolution by finitely generated free modules. If $R$ is a field, we define the Euler characteristic of $G$ over $R$ to be the alternating sum of $R$-ranks of homology groups $H_i(G;R)$. Does there exist a group of type $FL(F_i)$ for two fields $F_1$ and $F_2$ such that the Euler characteristics of the group over the fields $F_1$ and $F_2$ differ?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.71.\n\nDiscussion and literature:\nD. Kielak", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2581, "problem_number": "KOU-21.72", "title": "Kourovka Notebook Problem 21.72", "statement": "We say that a group is a Tarski monster if it is finitely generated, not cyclic, and all of its proper non-trivial subgroups are isomorphic to each other.\n\na) Do there exist amenable torsion-free Tarski monsters?\n\nb) Do there exist amenable Tarski monsters of prime exponent?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.72.\n\nDiscussion and literature:\nD. Kielak", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2582, "problem_number": "KOU-21.73", "title": "Kourovka Notebook Problem 21.73", "statement": "Is the conjugacy problem in $\\operatorname{CT}(\\mathbb Z)$ algorithmically decidable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.73.\n\nDiscussion and literature:\nSee the definition of $\\operatorname{CT}(\\mathbb Z)$ in 17.57. S. Kohl", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2583, "problem_number": "KOU-21.74", "title": "Kourovka Notebook Problem 21.74", "statement": "Is it algorithmically decidable whether a given element $g\\in\\operatorname{CT}(\\mathbb Z)$\n\n(a) permutes a nontrivial partition of $\\mathbb Z$ into residue classes?\n\n(b) has only finite cycles?\n\n(c) has no finite cycles?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.74.\n\nDiscussion and literature:\nS. Kohl", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2584, "problem_number": "KOU-21.75", "title": "Kourovka Notebook Problem 21.75", "statement": "Given two distinct sets $P_1$ and $P_2$ of odd primes none of which is a subset of the other, is it true that $\\langle \\operatorname{CT}_{P_1}(\\mathbb Z),\\operatorname{CT}_{P_2}(\\mathbb Z)\\rangle \\lneq \\operatorname{CT}_{P_1\\cup P_2}(\\mathbb Z)$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.75.\n\nDiscussion and literature:\nSee the definition of $\\operatorname{CT}_P(\\mathbb Z)$ in 17.60. S. Kohl", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2585, "problem_number": "KOU-21.76", "title": "Kourovka Notebook Problem 21.76", "statement": "Let $\\sigma=(\\sigma_{ij})$, $1\\leqslant i\\ne j\\leqslant n$, be an irreducible elementary net (carpet) of order $n\\geqslant 3$ over a field $K$ (see 19.48). The net $\\sigma$ is said to be closed if the elementary net subgroup $E(\\sigma)$ does not contain new elementary transvections. The net $\\sigma$ is said to be completable if its diagonal can be supplemented with subgroups to a complete net. Do there exist irreducible closed elementary nets of order $n\\geqslant 3$ over a field of odd characteristic that are not completable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.76.\n\nDiscussion and literature:\nCompletable elementary nets are closed. It is known that over fields of characteristic 0 and 2 there exist irreducible closed elementary nets that are not completable (V. A. Koibaev, Trudy Inst. Mat. Mekh. Ural Div. Ross. Akad. Nauk, 17, no. 4 (2011), 134--141 (Russian); V. A. Koibaev, Siberian Math. J., 62, no. 2 (2021), 262--266).\n\nV. A. Koibaev", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2586, "problem_number": "KOU-21.77", "title": "Kourovka Notebook Problem 21.77", "statement": "Let $d$ be an integer that is not divisible by $n$-th powers of primes, let $x^n-d$ be an irreducible polynomial over $\\mathbb Q$, let $\\theta=\\sqrt[n]{d}$, and let $K=\\mathbb Q(\\theta)$ be the radical extension of degree $n$ of the field $\\mathbb Q$. The multiplicative group $K^*$ of the field $K$ is canonically embedded into the group $\\operatorname{Aut}_{\\mathbb Q}(K)$ of all invertible $\\mathbb Q$-linear mappings of the $\\mathbb Q$-space $K$; let $T$ be the image of $K^*$ under this embedding. In the natural basis $1,\\theta,\\theta^2,\\ldots,\\theta^{n-1}$ of the $\\mathbb Q$-space $K$ the group $\\operatorname{Aut}_{\\mathbb Q}(K)$ corresponds to $G=\\operatorname{GL}(n,\\mathbb Q)$, and the subgroup $T$ to a subgroup $T(d)$ (unsplit maximal torus). Every subgroup $H$ of $G$ containing $T(d)$ and some one-dimensional transformation is rich in elementary transvections and thus defines a net $\\sigma=\\sigma(H)$. Let $E(\\sigma)$ denote the subgroup generated by all transvections in the net group $G(\\sigma)$. Is it true that $H\\leqslant N_G(E(\\sigma))$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.77.\n\nDiscussion and literature:\nBackground on rich elementary transvections and the associated net: (V. A. Koibaev, St. Petersbg. Math. J., 21, no. 5 (2010), 731--742); (V. A. Koibaev, A. V. Shilov, J. Math. Sci. New York 171, no. 3 (2010), 380--385).\n\nThis inclusion was proved in the case $n=2$ (V. A. Koibaev, Dokl. Math., 41, no. 3 (1990), 414--416). V. A. Koibaev", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2587, "problem_number": "KOU-21.78", "title": "Kourovka Notebook Problem 21.78", "statement": "Let $p$ be a prime and let $G$ be a pro-$p$ group. Suppose that all of the (continuous Galois) cohomology groups $H^n(G,\\mathbb F_p)$ of $G$ with coefficients in the field of $p$ elements are finite. Does it necessarily follow that the cohomology ring $H^*(G,\\mathbb F_p)$ is finitely generated?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.78.\n\nDiscussion and literature:\nThe answer is known to be `yes' if $G$ is abelian-by-($p$-adic analytic), as follows from (J. King, Commun. Algebra, 27, no. 10 (1999), 4969--4991). P. Kropholler", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2588, "problem_number": "KOU-21.79", "title": "Kourovka Notebook Problem 21.79", "statement": "Let $G$ be a finitely generated group with a fixed finite generating set $S$ and the corresponding word metric $L_S(*)$. An element $g$ is said to be distorted in $G$ if $L_S(g^n)/n\\to 0$ as $n\\to\\infty$; this notion is independent of the choice of the generating set $S$. For any, not necessarily finitely generated, group $H$, an element $g\\in H$ is said to be distorted if there is a finitely generated subgroup $G$ of $H$ containing $g$ in which $g$ is distorted. Do there exist finitely generated left-orderable groups in which every nontrivial element is distorted?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.79.\n\nDiscussion and literature:\nNote that it is straightforward to construct countable (not finitely generated) left-orderable groups with this property using HNN-extensions and applying results of V. V. Bludov and A. M. W. Glass. Y. Lodha, A. Navas", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2589, "problem_number": "KOU-21.80", "title": "Kourovka Notebook Problem 21.80", "statement": "Do there exist finitely generated left-orderable groups with only one nontrivial conjugacy class?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.80.\n\nDiscussion and literature:\nA positive answer to this question implies a positive answer to 21.78. Note that D. Osin constructed torsion-free finitely generated groups with only one nontrivial conjugacy class; see 9.10 in Archive. Y. Lodha, A. Navas", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2590, "problem_number": "KOU-21.81", "title": "Kourovka Notebook Problem 21.81", "statement": "Let $\\Gamma$ be a finite simple group and let $N_n(\\Gamma)$ denote the set of normal subgroups of the free group $F_n$ of rank $n$ whose quotient is isomorphic to $\\Gamma$.\n\nConjecture: $\\operatorname{Aut}(F_n)$ acts transitively on $N_n(\\Gamma)$ for $n\\geqslant 3$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.81.\n\nDiscussion and literature:\n(J. Wiegold).\n\nThis is not true for $n=2$ (B. H. Neumann, H. Neumann, Math. Nachr., 4 (1951), 106--125). A. Lubotzky", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2591, "problem_number": "KOU-21.82", "title": "Kourovka Notebook Problem 21.82", "statement": "Conjecture: For $n\\geqslant 3$, there are no finite simple characteristic quotients of the free group $F_n$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.82.\n\nDiscussion and literature:\nThis is not true for $n=2$ (W. Y. Chen, A. Lubotzky, P. H. Tiep, to appear in Comment. Math. Helvetici, 2025). A. Lubotzky", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2592, "problem_number": "KOU-21.83", "title": "Kourovka Notebook Problem 21.83", "statement": "Conjecture: Metabelian groups are permutation-stable.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.83.\n\nDiscussion and literature:\nThe function $d_n(\\sigma,\\tau)=(1/n)\\cdot|\\{x\\in\\{1,\\ldots,n\\}\\mid \\sigma(x)\\ne\\tau(x)\\}|$ is a distance on the symmetric group $S_n$. For a finitely generated group $G$, an almost-homomorphism is a sequence of set-theoretic maps $f_n:G\\to S_n$ satisfying $d_n(f_n(g)f_n(h),f_n(gh))\\to 0$ as $n\\to\\infty$ for all $g,h\\in G$. An almost-homomorphism $\\{f_n\\}$ is said to be close to a homomorphism if there is a sequence of group homomorphisms $\\rho_n:G\\to S_n$ such that $d_n(\\rho_n(g),f_n(g))\\to 0$ as $n\\to\\infty$ for all $g\\in G$. The group $G$ is said to be permutation stable if every almost-homomorphism of $G$ is close to a homomorphism.\n\nA. Lubotzky", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2593, "problem_number": "KOU-21.84", "title": "Kourovka Notebook Problem 21.84", "statement": "For $\\sigma\\in S_n$ and $\\tau\\in S_m$, where $n\\leqslant m$, let $d_n^{\\mathrm{flex}}(\\sigma,\\tau)=(1/n)\\cdot(|\\{x\\in\\{1,\\ldots,n\\}\\mid \\sigma(x)\\ne\\tau(x)\\}|+(m-n))$. An almost-homomorphism $\\{f_n\\}$ is said to be flexibly close to a homomorphism if there is a sequence of group homomorphisms $\\rho_n:G\\to S_{m_n}$ with $n\\leqslant m_n$ such that $d_n^{\\mathrm{flex}}(\\rho_n(g),f_n(g))\\to 0$ as $n\\to\\infty$ for all $g\\in G$. The group $G$ is said to be flexibly permutation stable if every almost-homomorphism of $G$ is flexibly close to a homomorphism. Is $\\operatorname{SL}_n(\\mathbb Z)$ flexibly permutation-stable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.84.\n\nDiscussion and literature:\nA. Lubotzky", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2594, "problem_number": "KOU-21.85", "title": "Kourovka Notebook Problem 21.85", "statement": "Is a flexibly permutation-stable group always permutation-stable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.85.\n\nDiscussion and literature:\nA. Lubotzky", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2595, "problem_number": "KOU-21.86", "title": "Kourovka Notebook Problem 21.86", "statement": "A group $G$ is said to be sofic if for every finite set $F\\subseteq G$ containing $1$ and every $\\varepsilon>0$ there exist $n\\in\\mathbb N$ and a map $\\phi:F\\to S_n$ such that $\\phi(1)=1$, $d(\\phi(gh),\\phi(g)\\phi(h))<\\varepsilon$ for all $g,h$ such that $gh\\in F$, and $\\phi(g)$ does not have fixed points for every $g\\in F\\setminus\\{1\\}$. Is every group sofic?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.86.\n\nDiscussion and literature:\n(M. Gromov, B. Weiss).\n\nA. Lubotzky", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2596, "problem_number": "KOU-21.87", "title": "Kourovka Notebook Problem 21.87", "statement": "Assume that a finite group G has a family of d-generator subgroups whose indices have no common divisor. Is it true that G can be generated by d+1 elements?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.87.\n\nDiscussion and literature:\nThe answer is positive if G is solvable (L. G. Kov\\'acs, H.-S. Sim, Indag. Math., 2 (1991), 229--232). For an arbitrary finite group G it is proved that G can be generated by d + 2 elements (A. Lucchini, Commun. Algebra, 28, no. 4 (2000), 1875--1880). A. Lucchini", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2597, "problem_number": "KOU-21.88", "title": "Kourovka Notebook Problem 21.88", "statement": "Is there a finite non-abelian group $G$ of odd order, with $k(G)$ conjugacy classes, such that $k(G)/|G|=1/17$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.88.\n\nDiscussion and literature:\nD. MacHale", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2598, "problem_number": "KOU-21.89", "title": "Kourovka Notebook Problem 21.89", "statement": "For $n>39$, is it true that the number of conjugacy classes in the symmetric group $S_n$ of degree $n$ is never a divisor of the order of $S_n$? In other words, is it true that, for $n>39$, the number $p(n)$ of integer partitions of $n$ is never a divisor of $n!$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.89.\n\nDiscussion and literature:\nD. MacHale", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2599, "problem_number": "KOU-21.90", "title": "Kourovka Notebook Problem 21.90", "statement": "Let $\\Gamma$ be a graph of diameter $d$. For $i\\in\\{1,2,\\ldots,d\\}$, let $\\Gamma_i$ be the graph on the same vertex set as $\\Gamma$ with vertices $u,w$ adjacent in $\\Gamma_i$ if and only if $d_\\Gamma(u,w)=i$. Does there exist a $Q$-polynomial distance-regular graph $\\Gamma$ of diameter 3 such that $\\Gamma_2$ and $\\Gamma_3$ are strongly regular?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.90.\n\nDiscussion and literature:\nA. A. Makhn\\\"ev", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2600, "problem_number": "KOU-21.91", "title": "Kourovka Notebook Problem 21.91", "statement": "Conjecture: The sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$ is at least the $p'$-part of $|G|$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.91.\n\nDiscussion and literature:\n(W. Willems).\n\nThis is known to be true for $p=2$ (G. Malle, Adv. Math., 380 (2021), Paper no. 107609, 15 pp.). G. Malle", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2601, "problem_number": "KOU-21.92", "title": "Kourovka Notebook Problem 21.92", "statement": "Conjecture: The number of irreducible $p$-Brauer characters of a finite group $G$ is bounded above by the maximum of the number of conjugacy classes $k(H)$ in $p'$-subgroups $H$ of $G$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.92.\n\nDiscussion and literature:\nG. Malle, G. Navarro, G. Robinson", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2602, "problem_number": "KOU-21.93", "title": "Kourovka Notebook Problem 21.93", "statement": "Let $G$ be a group and let $k\\geqslant 2$. Let $H_1,\\ldots,H_k$ be subgroups of $G$, and $g_1,\\ldots,g_k$ elements of $G$ such that the cosets $g_1H_1,\\ldots,g_kH_k$ form a partition of $G$. Is it true that $|G:H_i|=|G:H_j|$ for some $i\\ne j$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.93.\n\nDiscussion and literature:\n(M. Herzog, J. Sch\\\"onheim).\n\nThis is known to be true for groups with a Sylow tower (M. A. Berger, A. Felzenbaum, A. Fraenkel, Fund. Math., 128, no. 3 (1987), 139--144). Also cf. 20.99. L. Margolis", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2603, "problem_number": "KOU-21.94", "title": "Kourovka Notebook Problem 21.94", "statement": "The Gruenberg--Kegel graph (or the prime graph) GK(G) of a finite group G is a labelled graph with vertex set consisting of all prime divisors of the order of G in which different vertices p and q are adjacent if and only if G contains an element of order pq. Let GK(G) denote the abstract graph obtained from GK(G) by removing all labels. A finite group G is said to be recognizable by the isomorphism type of its Gruenberg--Kegel graph if there are no finite groups H $\\ncong$ G with GK(H) isomorphic to GK(G). Are there infinitely many (pairwise non-isomorphic) finite groups which are recognizable by the isomorphism type of the Gruenberg--Kegel graph?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.94.\n\nDiscussion and literature:\nN. V. Maslova", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2604, "problem_number": "KOU-21.95", "title": "Kourovka Notebook Problem 21.95", "statement": "Is there an almost simple but not simple group which is recognizable by the isomorphism type of its Gruenberg--Kegel graph?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.95.\n\nDiscussion and literature:\nNote that if a group G is recognizable by the isomorphism type of its Gruenberg-- Kegel graph, then G is recognizable by its Gruenberg--Kegel graph, and therefore G is almost simple (P. J. Cameron, N. V. Maslova, J. Algebra, 607 (2022), 186--213). N. V. Maslova", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2605, "problem_number": "KOU-21.96", "title": "Kourovka Notebook Problem 21.96", "statement": "Is it true that a periodic group containing an involution is locally finite if the centralizer of every element of even order is locally finite?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.96.\n\nDiscussion and literature:\nV. D. Mazurov", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2606, "problem_number": "KOU-21.97", "title": "Kourovka Notebook Problem 21.97", "statement": "Is it true that for every positive rational number $r$ there exists a finite group $G$ such that $|\\operatorname{Aut}(G)|/|G|=r$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.97.\n\nDiscussion and literature:\n(M. T\\u{a}rn\\u{a}uceanu).\n\nA similar question is answered in the positive for graphs, monoids, partial groups, and posets (R. Molinier, Preprint, 2025, https://arxiv.org/abs/2504.21059). It is also known that the set $\\{|\\operatorname{Aut}(G)|/|G|\\mid G\\text{ is a finite abelian group}\\}$ is dense in $[0,+\\infty)$ (M. T\\u{a}rn\\u{a}uceanu, Elemente Math. (2025), https://ems.press/journals/em/articles/14298544). R. Molinier", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2607, "problem_number": "KOU-21.98", "title": "Kourovka Notebook Problem 21.98", "statement": "Let w be a multilinear commutator word, and assume that G is a group where the set of w-values is covered by finitely many cyclic subgroups. Is it true that the verbal subgroup w(G) is finite-by-cyclic?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.98.\n\nDiscussion and literature:\nThis is true for lower central words (G. Cutolo, C. Nicotera, J. Algebra, 324, no. 7 (2010), 1616--1624). M. Morigi", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2608, "problem_number": "KOU-21.99", "title": "Kourovka Notebook Problem 21.99", "statement": "Conjecture: If $G$ is a transitive permutation group on a finite set $\\Omega$, then for any distinct $\\alpha,\\beta\\in\\Omega$ there is an element $g\\in G$ with $\\alpha^g=\\beta$ whose number of fixed points is different from 1.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.99.\n\nDiscussion and literature:\nP. M\\\"uller", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2609, "problem_number": "KOU-21.100", "title": "Kourovka Notebook Problem 21.100", "statement": "Suppose that $A$ and $G$ are finite groups such that $A$ acts coprimely on $G$ by automorphisms. Let $C=C_G(A)$ be the fixed-point subgroup, and let $C'$ denote its derived subgroup. Is it true that the number of $A$-invariant irreducible characters $\\chi$ of $G$ whose restriction $\\chi_C$ is never zero is exactly $|C/C'|$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.100.\n\nDiscussion and literature:\nThis would follow if one could show that $\\chi_C$ is never zero if and only if the Glauberman--Isaacs correspondent $\\chi^*$ of $\\chi$ is linear. G. Navarro", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2610, "problem_number": "KOU-21.101", "title": "Kourovka Notebook Problem 21.101", "statement": "Which finite almost simple groups are the automorphism groups of regular polytopes of rank 3? In other words, which finite almost simple groups are generated by three involutions two of which commute?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.101.\n\nDiscussion and literature:\nThis question has been answered for finite simple groups; see 7.30 in Archive. Ya. N. Nuzhin", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2611, "problem_number": "KOU-21.102", "title": "Kourovka Notebook Problem 21.102", "statement": "Let $V$ be a variety generated by a finite group, and let $f(n)$ be the order of the free group in $V$ on $n$ generators. Is it true that the sequence $\\sqrt[n]{\\log f(n)}$ has a limit as $n\\to\\infty$, and this limit is an integer?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.102.\n\nDiscussion and literature:\nA. Yu. Olshanskii", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2612, "problem_number": "KOU-21.103", "title": "Kourovka Notebook Problem 21.103", "statement": "A Hausdorff topological group G is called minimal if it does not admit a strictly coarser Hausdorff group topology. A topological group is called Raikov complete if its two-sided uniform structure is complete. Is it true that an arbitrary Cartesian product of Raikov complete minimal topological groups remains minimal?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.103.\n\nDiscussion and literature:\n(V. V. Uspenskii).\n\nIt is known that a finite direct product of Raikov complete minimal topological groups is again minimal.\n\nIt is known that an arbitrary Cartesian product of centre-free minimal topological groups is minimal (M. Megrelishvili, Topology Appl., 62, no. 1 (1995), 1--19). D. Peng", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2613, "problem_number": "KOU-21.104", "title": "Kourovka Notebook Problem 21.104", "statement": "For a group word $w(x_1,\\ldots,x_n)$ on $n$ letters, define $e_0(x_1,\\ldots,x_n)=x_1$ and $e_{k+1}(x_1,\\ldots,x_n)=w(e_k(x_1,\\ldots,x_n),\\ldots,x_n)$ for all $k\\in\\mathbb N$. A group $G$ is said to satisfy the Engel type iterated identity $w$ if for all $x_1,\\ldots,x_n\\in G$ there exists $m\\in\\mathbb N$ such that $e_m(x_1,\\ldots,x_n)=1$.\n\nConjecture: For every non-trivial word $w$, if a finitely generated branch group $G$ (see 15.12) satisfies the iterated identity $w$, then $G$ is a torsion group.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.104.\n\nDiscussion and literature:\nThis is true in the case of the commutator word $w=[x_1,x_2]$ (G. Fern\\'andez-Alcober, M. Noce, G. Tracey, J. Algebra, 554 (2020), 54--77). M. Petschick", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2614, "problem_number": "KOU-21.105", "title": "Kourovka Notebook Problem 21.105", "statement": "A group word $w$ is said to be concise in a class $\\mathcal C$ of groups if for every group $G$ in $\\mathcal C$ such that the set $G_w$ of word values of $w$ in $G$ is finite, the verbal subgroup $w(G)=\\langle G_w\\rangle$ is also finite. Is every word concise in the class of residually finite groups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.105.\n\nDiscussion and literature:\n(D. Segal).\n\nSee also Archive 2.45.\n\nSee (D. Segal, Words. Notes on verbal width in groups, Cambridge Univ. Press, 2009) for a partial solution. M. Petschick", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2615, "problem_number": "KOU-21.106", "title": "Kourovka Notebook Problem 21.106", "statement": "A first order formula $\\phi(x)$ in the group language with one free variable is said to be concise in a class $\\mathcal C$ of groups if for every group $G$ in $\\mathcal C$ such that the set $G_\\phi$ of elements in $G$ satisfying $\\phi$ is finite, the subgroup $\\phi(G)$ generated by $G_\\phi$ is also finite. Is every formula with one free variable concise in the class of residually finite groups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.106.\n\nDiscussion and literature:\nCf. 21.105 and Archive 2.45. M. Petschick", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2616, "problem_number": "KOU-21.107", "title": "Kourovka Notebook Problem 21.107", "statement": "A sequence $\\{F_n\\}$ of pairwise disjoint finite subsets of a topological group is called expansive if for every open subset $U$ there is a number $m$ such that $F_n\\cap U\\ne\\emptyset$ for all $n>m$. Suppose that a countable group $G$ can be partitioned into countably many dense subsets. Is it true that in $G$ there exists an expansive sequence?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.107.\n\nDiscussion and literature:\nCf. 15.80. I. V. Protasov", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2617, "problem_number": "KOU-21.108", "title": "Kourovka Notebook Problem 21.108", "statement": "For a finite group $G$ let $\\operatorname{Cod}(G)$ denote the set of irreducible character codegrees of $G$ (see 20.78). Define $\\sigma(G)=\\max\\{|\\pi(m)|:m\\in\\operatorname{Cod}(G)\\}$, where $\\pi(m)$ denotes the set of prime divisors of an integer $m$. Can the constant $k$ be taken as 4?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.108.\n\nDiscussion and literature:\nIt is proved that there exists a constant $k$ such that $|\\pi(G)|\\leqslant k\\cdot\\sigma(G)$ for every finite group $G$ (Y. Yang, G. Qian, J. Algebra, 478 (2017), 215--219), but the estimate provided for $k$ is very crude.\n\nG. Qian", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2618, "problem_number": "KOU-21.109", "title": "Kourovka Notebook Problem 21.109", "statement": "Conjecture: The derived length of a finite solvable group $G$ does not exceed $|\\operatorname{Cod}(G)|-1$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.109.\n\nDiscussion and literature:\nThe Fitting height of $G$ is known to be at most $\\min\\{|\\operatorname{Cod}(G)|-1,\\ |\\operatorname{Cod}(G)|/2+1\\}$ (G. Qian, Y. Zeng, J. Group Theory, to appear). G. Qian", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2619, "problem_number": "KOU-21.110", "title": "Kourovka Notebook Problem 21.110", "statement": "Let $S$ be a nonabelian finite simple group, and $x$ a nonidentity automorphism of $S$. Let $\\alpha(x)$ be the smallest number of conjugates of $x$ in $G=\\langle x,\\operatorname{Inn}S\\rangle$ that generate $G$.\n\n(a) Conjecture: If $S$ is an exceptional group of Lie type, then $\\alpha(x)\\leqslant 5$ for every nonidentity automorphism $x$ of $S$.\n\n(b) For each exceptional group $S$ of Lie type, find the largest value of $\\alpha(x)$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.110.\n\nDiscussion and literature:\nThe values of $\\alpha(x)$ had been studied in (R. Guralnick, J. Saxl, J. Algebra, 268, no. 2 (2003), 519--571).\n\nPart (a) is attributed to R. Guralnick and J. Saxl. D. O. Revin", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2620, "problem_number": "KOU-21.111", "title": "Kourovka Notebook Problem 21.111", "statement": "Let $S$ be a finite simple nonabelian group that is not isomorphic to any group ${}^2B_2(q)$. A nonidentity automorphism $x$ of $S$ is called a $\\tau$-automorphism if every two conjugates of $x$ in $\\langle x,\\operatorname{Inn}(S)\\rangle$ generate a subgroup of order not divisible by 3. If $S$ admits a $\\tau$-automorphism, we call $S$ a $\\tau$-group.\n\n(a) List all $\\tau$-groups up to isomorphism.\n\n(b) Do $\\tau$-automorphisms of odd order exist?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.111.\n\nDiscussion and literature:\nD. O. Revin, N. Y. Yang", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2621, "problem_number": "KOU-21.112", "title": "Kourovka Notebook Problem 21.112", "statement": "A nonempty class $\\mathcal X$ of finite groups is said to be complete if $\\mathcal X$ is closed under taking subgroups, homomorphic images, and extensions. The symmetric boundary of a complete class $\\mathcal X$ other than the class of all finite groups is defined as the largest integer $n$ such that $S_n\\in\\mathcal X$. For every positive integer $n\\ne 3$, let $f_+(n)$ and $f_-(n)$ be respectively the maximum and the minimum of $\\operatorname{BS}(\\mathcal X)$, where $\\mathcal X$ runs over all complete classes of symmetric boundary $n$.\n\n(a) Find $f_+(n)$ for $n=4,5,6$.\n\n(b) Is it true that $f_-(n)=n$ for all $n\\ne 3$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.112.\n\nDiscussion and literature:\nEvery positive integer $n\\ne 3$ coincides with the symmetric boundary of some complete class. It is proved (mod CFSG, D. O. Revin, Algebra i Analiz, 37, no. 1 (2025), 141--176 (Russian)) that, for every complete class $\\mathcal X$, there exists a nonnegative integer $m$ with the following property: for every finite group $G$ and each conjugacy class $D$ of $G$, if every $m$ elements of $D$ generate a subgroup belonging to $\\mathcal X$, then $\\langle D\\rangle\\in\\mathcal X$. The smallest such $m$ is called the Baer--Suzuki width of $\\mathcal X$, denoted by $\\operatorname{BS}(\\mathcal X)$. It is also proved (mod CFSG, ibid.) that, for a complete class $\\mathcal X$ of symmetric boundary $n$, the value of $\\operatorname{BS}(\\mathcal X)$ is at least $n$ and is bounded above in terms of $n$.\n\nIt is known that $f_+(1)=2$, $f_+(2)=3$, and $f_+(n)=2(n-1)$ for $n\\geqslant 7$.\n\nThis is known to be true for $n=1,2,4$. D. O. Revin, N. Y. Yang", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2622, "problem_number": "KOU-21.113", "title": "Kourovka Notebook Problem 21.113", "statement": "Let $G$ be a finite group and $p$ be a prime. Let $\\Psi_{p,G}$ be the class function of $G$ which vanishes on all $p$-singular elements of $G$ and whose value at each $p$-regular element $x$ of $G$ is the number of $p$-elements of $C_G(x)$.\n\n(a) Is it true that $\\Psi_{p,G}$ is a character of $G$?\n\n(b) If yes, can $\\Psi_{p,G}$ be afforded by a projective $RG$-module, where $R$ is a complete discrete valuation ring of characteristic zero such that the field of fractions of $R$ is a splitting field for $G$ and its subgroups, and the residue field $R/J(R)$ is a splitting field of characteristic $p$ for $G$ and its subgroups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.113.\n\nDiscussion and literature:\nIt is known that $\\Psi_{p,G}$ is a character when $G\\cong S_n$ for any positive integer $n$ and any prime $p$ (T. Scharf, J. Algebra, 139, no. 2 (1991), 446--457). G. Robinson", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2623, "problem_number": "KOU-21.114", "title": "Kourovka Notebook Problem 21.114", "statement": "A finite group G is called weakly ab-maximal if |H : [H, H]| $\\leqslant$ |G : [G, G]| for all H $\\leqslant$ G. Do weakly ab-maximal groups have bounded derived length?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.114.\n\nDiscussion and literature:\nIt is known that weakly ab-maximal groups are direct products of weakly ab-maximal p-groups (F. Lisi, L. Sabatini, J. Group Theory, 27 (2024), 1203--1217). L. Sabatini", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2624, "problem_number": "KOU-21.115", "title": "Kourovka Notebook Problem 21.115", "statement": "Let $C_1,\\ldots,C_n$ be (left or right) cosets of a finite group $G$ such that $U:=C_1\\cup\\cdots\\cup C_n$ is not $G$. Is it always true that $|G\\setminus U|\\geqslant |G|/2^n$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.115.\n\nDiscussion and literature:\nAffirmative answers are known in some special cases (B. Sambale, M. T\\u{a}rn\\u{a}uceanu, J. Algebraic Combin., 55 (2022), 979--987). B. Sambale", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2625, "problem_number": "KOU-21.116", "title": "Kourovka Notebook Problem 21.116", "statement": "A group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Is every branch group boundedly acyclic?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.116.\n\nDiscussion and literature:\nE. Schesler", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2626, "problem_number": "KOU-21.117", "title": "Kourovka Notebook Problem 21.117", "statement": "(a) Does there exist a finitely generated simple group that is of exponential growth but not of uniformly exponential growth?\n\n(b) Does there exist a finitely generated hereditarily just-infinite group that is of exponential growth but not of uniformly exponential growth?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.117.\n\nDiscussion and literature:\nE. Schesler", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2627, "problem_number": "KOU-21.118", "title": "Kourovka Notebook Problem 21.118", "statement": "Is there any group which is not isomorphic to the quotient of a residually finite group by an amenable normal subgroup?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.118.\n\nDiscussion and literature:\n(A. Thom).\n\nE. Schesler", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2628, "problem_number": "KOU-21.119", "title": "Kourovka Notebook Problem 21.119", "statement": "Does there exist a group $G$ that contains a family $(G_n)_{n\\in\\mathbb N}$ of finite-index subgroups such that for every $n$ there is a homomorphism $f_n:G_n\\to\\mathbb Z$ whose kernel is of type $F_n$, but not of type $F_{n+1}$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.119.\n\nDiscussion and literature:\nE. Schesler", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2629, "problem_number": "KOU-21.120", "title": "Kourovka Notebook Problem 21.120", "statement": "A pro-p group is (relatively) strictly finitely presented if it is the pro-p completion of a group that is finitely presented (respectively, finitely presented in some finitely-based variety of groups). A pro-p group is finitely axiomatizable if it is determined up to isomorphism by a single sentence in the first-order language of group theory.\n\n(a) Does there exist a (relatively) strictly finitely presented pro-p group that is not finitely axiomatizable in the class of all pro-p groups?\n\n(b) In particular, is every finitely generated free pro-p group finitely axiomatizable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.120.\n\nDiscussion and literature:\nSee (A. Nies, K. Tent, D. Segal, Proc. London Math. Soc. (3), 123 (2021), 597-- 635; D. Segal, Preprint, 2025, https://arxiv.org/abs/2505.04816). D. Segal", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2630, "problem_number": "KOU-21.121", "title": "Kourovka Notebook Problem 21.121", "statement": "Let $p$ be a prime number. A group $\\Gamma$ is called $p$-Jordan if there exist constants $J$ and $e$ such that any finite subgroup $G\\subset\\Gamma$ contains a normal abelian subgroup of order coprime to $p$ and of index at most $J\\cdot |G(p)|^e$. (For example by the results of Brauer--Feit and Larsen--Pink, for any field $K$ of characteristic $p$ the group $\\operatorname{GL}_n(K)$ is $p$-Jordan with $e=3$.) Let the $p$-Jordan exponent $e(\\Gamma)$ of the group $\\Gamma$ be the infimum of all constants $e$ for which the above bound holds for some constant $J=J(e)$.\n\na) Is it true that this infimum is always attained?\n\nb) Is it true that $e(\\Gamma)\\leqslant 3$ for any $p$-Jordan group $\\Gamma$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.121.\n\nDiscussion and literature:\nC. Shramov", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2631, "problem_number": "KOU-21.122", "title": "Kourovka Notebook Problem 21.122", "statement": "Let w be a group word, and G a profinite group. Is it true that the cardinality of the set of w-values in G is either finite or at least continuum?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.122.\n\nDiscussion and literature:\nAn affirmative answer is known for several important words; see (E. Detomi, B. Klopsch, P. Shumyatsky, J. London Math. Soc. (2), 102 (2020), 977--993). P. Shumyatsky", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2632, "problem_number": "KOU-21.123", "title": "Kourovka Notebook Problem 21.123", "statement": "Is it true that the extension of the A. Agrachev--R. Gamkrelidze construction of groups from pre-Lie rings suggested in Definition 66 produces groups from pre-Lie rings? If this extended construction does give a group, then it also gives a brace, and so an affirmative answer to this question would have consequences for the theory of set-theoretic solutions of the Yang--Baxter equation and for the theory of braces.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.123.\n\nDiscussion and literature:\nA. Agrachev--R. Gamkrelidze construction: (J. Soviet Math., 17 (1981), 1650--1675). Definition 66: (A. Smoktunowicz, J. Pure Appl. Algebra, 229, no. 12 (2025), 108128). A. Smoktunowicz", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2633, "problem_number": "KOU-21.124", "title": "Kourovka Notebook Problem 21.124", "statement": "A group G is said to be virtually special if G has a finite-index subgroup isomorphic to the fundamental group of a special complex. A group G is called a CAT(0) group if it acts properly discontinuously and cocompactly by isometries on a CAT(0) metric space.\n\na) Is every CAT(0) free-by-cyclic group virtually special?\n\nb) A weaker question: does every CAT(0) free-by-cyclic group virtually embed into a right-angled Artin group?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.124.\n\nDiscussion and literature:\nSpecial complex is meant in the sense of F. Haglund, D. T. Wise, Geom. Funct. Anal., 17, no. 5 (2008), 1551--1620; cf 20.60. I. Soroko", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2634, "problem_number": "KOU-21.125", "title": "Kourovka Notebook Problem 21.125", "statement": "Let $F_m$ be a free group of rank $m$ and let $\\varphi\\in\\operatorname{Aut}(F_m)$ be a polynomially growing automorphism of maximal degree $m-1$, which means that for some (equivalently, any) free basis $\\{x_1,\\ldots,x_m\\}$ of $F_m$, the sequence $\\max_i |\\varphi^n(x_i)|$ grows at the rate of $n^{m-1}$, where $|g|$ denotes the minimal length of $g$ in the $x_i$ and their inverses.\n\na) Is the free-by-cyclic group $F_m\\rtimes_\\varphi\\mathbb Z$ virtually special?\n\nb) In particular, are the Hydra groups $G_m=F_m\\rtimes\\mathbb Z=\\langle a_1,\\ldots,a_m,t\\mid t^{-1}a_1t=a_1,\\ t^{-1}a_it=a_ia_{i-1}\\text{ for all }i>1\\rangle$ virtually special?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.125.\n\nDiscussion and literature:\n(M. Bridson).\n\nI. Soroko", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2635, "problem_number": "KOU-21.126", "title": "Kourovka Notebook Problem 21.126", "statement": "Do there exist finitely presented subgroups of right-angled Artin groups whose Dehn functions are super-exponential, or sub-exponential but not polynomial?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.126.\n\nDiscussion and literature:\n(N. Brady).\n\nSuch subgroups are known to exist in general CAT(0) groups, whereas the only Dehn functions currently realized for subgroups of right-angled Artin groups are exponential and polynomial of arbitrary degree. I. Soroko", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2636, "problem_number": "KOU-21.127", "title": "Kourovka Notebook Problem 21.127", "statement": "Let G be a right-angled Artin group. Is the stable commutator length scl(g) a rational number for every g $\\in$ [G, G]?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.127.\n\nDiscussion and literature:\nFor free groups this is true by Calegari's Rationality Theorem. (See 18.40 for the definition of scl(g).) I. Soroko", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2637, "problem_number": "KOU-21.128", "title": "Kourovka Notebook Problem 21.128", "statement": "Two groups $G_1$ and $G_2$ are said to be commensurable if there exist finite-index subgroups $H_1\\leqslant G_1$ and $H_2\\leqslant G_2$ (not necessarily of the same index) such that $H_1\\cong H_2$. Let $A[F_4]$ and $A[H_4]$ denote the Artin groups of spherical types $F_4$ and $H_4$, respectively. Are these two groups commensurable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.128.\n\nDiscussion and literature:\nThis is the most difficult case in the classification of Artin groups of spherical type up to commensurability. I. Soroko", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2638, "problem_number": "KOU-21.129", "title": "Kourovka Notebook Problem 21.129", "statement": "If two Artin groups of spherical type are quasi-isometric, must they be commensurable? (This is not true for right-angled Artin groups.)", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.129.\n\nDiscussion and literature:\nI. Soroko", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2639, "problem_number": "KOU-21.130", "title": "Kourovka Notebook Problem 21.130", "statement": "Conjecture: Let $G$ be a finite additive abelian group with $|G|$ odd. Then any subset $A$ of $G$ with $|A|=n>2$ can be written as $\\{a_1,\\ldots,a_n\\}$ in such a way that all the sums $a_1+a_2,a_2+a_3,\\ldots,a_{n-1}+a_n,a_n+a_1$ are distinct.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.130.\n\nDiscussion and literature:\nZ. W. Sun", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2640, "problem_number": "KOU-21.131", "title": "Kourovka Notebook Problem 21.131", "statement": "Construct a homomorphism of a subgroup of a Golod group onto an infinite AT-group.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.131.\n\nDiscussion and literature:\nGolod group: see 9.76. AT-group: (A. V. Rozhkov, Math. Notes, 40, no. 5 (1986), 827--836). Cf. 13.55 A. V. Timofeenko", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2641, "problem_number": "KOU-21.132", "title": "Kourovka Notebook Problem 21.132", "statement": "Based on the development of E. S. Golod's construction, for each prime number p, construct a finitely generated residually finite p-group with a non-trivial finite centre.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.132.\n\nDiscussion and literature:\nDevelopment of E. S. Golod's construction: see, for example, Discrete Math. Appl., 23, no. 5--6 (2013), 491--501.\n\nSuch groups with infinite and trivial centres are known (see 9.76 and Archive 11.101). A. V. Timofeenko", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2642, "problem_number": "KOU-21.133", "title": "Kourovka Notebook Problem 21.133", "statement": "Does a group need to have a subnormal abelian series if every countable subgroup of it has such a series?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.133.\n\nDiscussion and literature:\nM. Trombetti", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2643, "problem_number": "KOU-21.134", "title": "Kourovka Notebook Problem 21.134", "statement": "For a finite group $G$, let the type of $G$ be the function on positive integers whose value at $n$ is the number of solutions of the equation $x^n=1$ in $G$.\n\na) Is it true that a group having the same type as a group with trivial solvable radical must also have trivial solvable radical?\n\nb) Is it true that a group having the same type as an almost simple group must be isomorphic to it?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.134.\n\nDiscussion and literature:\nNote that there are solvable and nonsolvable groups with the same type (see 12.37).\n\nThis is true for a group having the same type as a simple group, as follows from the affirmative answer to 12.39 (in Archive). A. V. Vasil'ev", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2644, "problem_number": "KOU-21.135", "title": "Kourovka Notebook Problem 21.135", "statement": "For a finite group $G$, let $\\chi_1(G)$ denote the totality of the degrees of all irreducible complex characters of $G$ with allowance for their multiplicities. Suppose that $H$ is a finite group with $\\chi_1(H)=\\chi_1(G)$. If $G$ has trivial solvable radical, must $H$ also have trivial solvable radical?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.135.\n\nDiscussion and literature:\nA. V. Vasil'ev", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2645, "problem_number": "KOU-21.136", "title": "Kourovka Notebook Problem 21.136", "statement": "Let G be a profinite group with fewer than $2^{\\aleph_0}$ conjugacy classes of elements of infinite order. Must G be a torsion group?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.136.\n\nDiscussion and literature:\nThis holds in the case when G is finitely generated (J. S. Wilson, Arch. Math., 120 (2023), 557--563). John S. Wilson", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2646, "problem_number": "KOU-21.137", "title": "Kourovka Notebook Problem 21.137", "statement": "If the $p$-th powers in a finite $p$-group form a subgroup, must that subgroup be powerful? That is, for $p\\ne 2$, if the $p$-th powers in a $p$-group of exponent $p^2$ form a subgroup, must that subgroup be abelian? For a 2-group of exponent 8, if the squares form a subgroup, must that subgroup be abelian?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.137.\n\nDiscussion and literature:\nL. Wilson", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2647, "problem_number": "KOU-21.138", "title": "Kourovka Notebook Problem 21.138", "statement": "Let G be an infinite finitely presented group such that every subgroup of infinite index is free. Must G be isomorphic to either a free group or a surface group?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.138.\n\nDiscussion and literature:\nH. Wilton", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2648, "problem_number": "KOU-21.139", "title": "Kourovka Notebook Problem 21.139", "statement": "Let G be a hyperbolic group which is virtually compact special in the sense of Haglund--Wise. Suppose that the set of second Betti numbers of the finite-index subgroups of G is bounded. Must G be virtually either a free group or a surface group?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.139.\n\nDiscussion and literature:\nH. Wilton", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2649, "problem_number": "KOU-21.140", "title": "Kourovka Notebook Problem 21.140", "statement": "Let G be a torsion-free group of type $F_\\infty$ of infinite cohomological dimension. Must G contain a copy of Thompson's group F?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.140.\n\nDiscussion and literature:\nS. Witzel, M. C. B. Zaremsky", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2650, "problem_number": "KOU-21.141", "title": "Kourovka Notebook Problem 21.141", "statement": "Let $G=G_1\\amalg_H G_2$ be a free pro-$p$ product of coherent pro-$p$ groups with polycyclic amalgamation. Is $G$ coherent?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.141.\n\nDiscussion and literature:\nFor abstract groups this is known to be true. A group is said to be coherent if each of its finitely generated subgroups is finitely presented, and in the question the coherency is used in the pro-$p$ sense. P. A. Zalesskii", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2651, "problem_number": "KOU-21.142", "title": "Kourovka Notebook Problem 21.142", "statement": "A group $G$ is said to be invariably generated by $a$ and $b$ if $G$ is generated by the conjugates $a^g,b^h$ for every $g,h$. Let $p\\ne q$ be fixed primes. Does every finite group embed into a finite group invariably generated by an element of order $p$ and an element of order $q$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.142.\n\nDiscussion and literature:\nP. A. Zalesskii", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2652, "problem_number": "KOU-21.143", "title": "Kourovka Notebook Problem 21.143", "statement": "(Well-known problem). Is Thompson's group F automatic?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.143.\n\nDiscussion and literature:\nM. C. B. Zaremsky", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2653, "problem_number": "KOU-21.144", "title": "Kourovka Notebook Problem 21.144", "statement": "Conjecture: Every subgroup of Thompson's group F is either elementary amenable or else contains a subgroup isomorphic to F.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.144.\n\nDiscussion and literature:\n(M. Brin, M. Sapir).\n\nM. C. B. Zaremsky", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2654, "problem_number": "KOU-21.145", "title": "Kourovka Notebook Problem 21.145", "statement": "Is Thompson's group F quasi-isometric\n\n(a) to F $\\times$ Z?\n\n(b) to F $\\times$ F?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.145.\n\nDiscussion and literature:\n(M. Bridson).\n\nM. C. B. Zaremsky", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2655, "problem_number": "KOU-21.146", "title": "Kourovka Notebook Problem 21.146", "statement": "(Well-known problem). A classifying space for a group $G$ is a connected CW-complex with fundamental group $G$ and all higher homotopy groups trivial. A group is of type $F_n$ if it has a classifying space with finite $n$-skeleton. For example, type $F_1$ is equivalent to finite generation, and type $F_2$ is equivalent to finite presentability. Type $F_\\infty$ means type $F_n$ for all $n$. For $n\\geqslant 3$, does every group of type $F_{n-1}$ embed as a subgroup of a group of type $F_n$? Or even in a group of type $F_\\infty$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.146.\n\nDiscussion and literature:\nM. C. B. Zaremsky", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2656, "problem_number": "KOU-21.147", "title": "Kourovka Notebook Problem 21.147", "statement": "A subgroup H of a right-orderable group G is said to be right-relatively convex if it is convex under some right ordering on G. Is the lattice of right-relatively convex subgroups of a right-orderable group always a sublattice of the lattice of its subgroups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.147.\n\nDiscussion and literature:\n(V. M. Kopytov, N. Ya. Medvedev).\n\nA. V. Zenkov", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2657, "problem_number": "KOU-21.148", "title": "Kourovka Notebook Problem 21.148", "statement": "Is it true that the lattice of right-relatively convex subgroups of a right-orderable group is distributive if and only if it is a chain?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.148.\n\nDiscussion and literature:\n(V. M. Kopytov, N. Ya. Medvedev).\n\nA. V. Zenkov", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2658, "problem_number": "KOU-21.149", "title": "Kourovka Notebook Problem 21.149", "statement": "Are there order automorphisms of Dlab groups that are not inner automorphisms?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.149.\n\nDiscussion and literature:\n(V. M. Kopytov, N. Ya. Medvedev).\n\nA. V. Zenkov", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2659, "problem_number": "KOU-21.150", "title": "Kourovka Notebook Problem 21.150", "statement": "Let $G$ be an extension of a normal elementary abelian subgroup $A$ by an elementary abelian group $B\\cong G/A$ such that $A$ contains an element $a$ with $C_B(a)=1$. Is it true that the rank of the subgroup $Z(\\langle a,B\\rangle)\\cap\\langle a,B\\rangle'$ is at most the rank of $B$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.150.\n\nDiscussion and literature:\nV. I. Zenkov\n\nNo, not always. Let $A=\\mathbb F_3[x,y]/I^3$ be the additive group of the quotient of the polynomial algebra $\\mathbb F_3[x,y]$ by the ideal $I^3$, where $I$ is the ideal generated by $x,y$. Let $B$ be the group of automorphisms of $A$ generated by multiplication by $1+x$ and $1+y$. Let $G=A\\rtimes B$. For $a=1\\in A$ we have $C_B(a)=1_B$. For $H=\\langle a,B\\rangle$ we have $Z(H)\\cap H'=\\langle x^2,xy,y^2\\rangle$, which has rank 3, while the rank of $B$ is 2. (P. Monticone, Letter of 21 March 2026; see also https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/03/solution_21_150-3.pdf).", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2660, "problem_number": "KP-1.1", "title": "Kirby Problem 1.1", "statement": "Is the crossing number additive under connected sum:\n$c(K_{1}\\#K_{2}) = c(K_{1}) + c(K_{2})$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.1.\n\nLiterature notes:\nThis is Problem 1.65 in [Kir97]. Many special cases are known.\nThe most famous case is due to Kauffman [Kau87], Murasugi [Mur87], and\nThistlewaite [Thi87], who showed the conjecture for alternating knots (and more\ngenerally adequate knots, defined in [LT88]). Lackenby [Lac09] has shown in the\ngeneral case that if $K_{1},..., K_{n}$ are knots, then\n\n$$\n\\frac{c(K_{1}) + \\cdots + c(K_{n})}{152} \\leq c(K_{1}\\# \\cdots \\#K_{n}).\n$$\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Kau87] Louis H. Kauffman. State models and the Jones polynomial. Topology, 26(3):395– 407, 1987. doi:10.1016/0040-9383(87)90009-7.\n- [Mur87] Kunio Murasugi. Jones polynomials and classical conjectures in knot theory. Topology, 26(2):187–194, 1987. doi:10.1016/0040-9383(87)90058-9.\n- [Thi87] Morwen B. Thistlethwaite. A spanning tree expansion of the Jones polynomial. Topology, 26(3):297–309, 1987. doi:10.1016/0040-9383(87)90003-6.\n- [LT88] W. B. R. Lickorish and M. B. Thistlethwaite. Some links with nontrivial polynomials and their crossing-numbers. Comment. Math. Helv., 63(4):527–539, 1988. doi:10.1007/BF02566777.\n- [Lac09] Marc Lackenby. The crossing number of composite knots. J. Topol., 2(4):747–768, 2009. doi:10.1112/jtopol/jtp028.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2661, "problem_number": "KP-1.2", "title": "Kirby Problem 1.2", "statement": "(a) Show that if $P$ is a nontrivial satellite operator and $K_{P}$ is a nontrivial\nsatellite of a knot $K$, then\n\n$$\nc(K_{P}) \\geq c(K),\n$$\n\nwhere $c$ denotes the crossing number.\n(b) More generally, show that if $P$ is a satellite operator then\n\n$$\nc(K_{P}) \\geq c(K) \\cdot w(P)^{2},\n$$\n\nwhere $w(P)$ denotes the geometric winding number.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.2.\n\nLiterature notes:\n(1) Part (a) is Problem 1.67 in [Kir97]. Part (b) appears as the Satellite\nCrossing Number Conjecture in [BM24a].\n(2) Note that a special case of this problem is whether\n\n$$\nc(K_{1}\\#K_{2}) \\geq c(K_{1}),\n$$\n\nwhich is a weaker form of Problem 1.1.\nIt is reasonable to expect equality in part (a) only when $P$ is the core\nof the solid torus.\nLackenby [Lac14] has shown the inequality\n\n$$\nc(K_{P}) \\geq c(K)/10^{13}.\n$$\n\nKalfagianni-Lee [KL23a] have shown that if $K$ is a nontrivial adequate\nknot that has an adequate diagram with writhe 0, then\n\n$$\nc(\\operatorname{Wh}_{\\pm}(K)) = 4c(K) + 2,\n$$\n\nwhere $\\operatorname{Wh}_{\\pm}(K)$ denote the Whitehead doubles of $K$. Kalfagianni-McConkey\n[KM24] have shown that if $K$ is adequate, then the cable knot $K_{p,q}$ sat-\nisfies\n\n$$\nc(K_{p,q}) \\geq q^{2} \\cdot c(K) + 1\n$$\n\nfor any coprime $p, q$.\nHere $q$, denotes the longitudinal winding of the\ncabling pattern. A recent preprint of Baker–Motegi claims that the more\ngeneral conjecture is true if one puts sufficiently many full twists in a\npattern with geometric winding number at least 2 [BM24a].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [BM24a] Kenneth L. Baker and Kimihiko Motegi. The stable crossing number of a twist family of knots and the satellite crossing number conjecture, 2024. arXiv:2404.05308.\n- [Lac14] Marc Lackenby. The crossing number of satellite knots. Algebr. Geom. Topol., 14(4):2379–2409, 2014. doi:10.2140/agt.2014.14.2379.\n- [KL23a] Efstratia Kalfagianni and Christine Ruey Shan Lee. Jones diameter and crossing number of knots. Adv. Math., 417:Paper No. 108937, 35, 2023. doi:10.1016/j.aim.2023.108937.\n- [KM24] Efstratia Kalfagianni and Rob Mcconkey. Crossing numbers of cable knots. Bulletin of the London Mathematical Society, 56(11):3400–3411, 2024. doi:10.1112/blms.13140.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2662, "problem_number": "KP-1.3", "title": "Kirby Problem 1.3", "statement": "How does unknotting number behave under connected sum and\nmutation?\n(a) Does the connected sum of $n$ nontrivial knots have unknotting number at\nleast $n$?\n(b) Is unknotting number invariant under mutation?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.3.\n\nLiterature notes:\n(1) Problem 1.69 (B) from [Kir97] asks if unknotting number is additive; a\nrecent preprint [BH25] claims an example where it fails to be additive,\nwhere the knots in question are the $(2, 7)$ torus knot and its mirror.\n(2) Question (a) is [Kir97, Problem 1.69 (A)], and is still open. It has a\npositive answer for $n = 2$ by Scharlemann [Sch85b].\n(3) With regard to Question (b), Gordon–Luecke [GL06] show that having\nunknotting number one is invariant under mutation.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [BH25] Mark Brittenham and Susan Hermiller. Unknotting number is not additive under connected sum, 2025. arXiv:2506.24088.\n- [Sch85b] Martin G. Scharlemann. Unknotting number one knots are prime. Invent. Math., 82(1):37–55, 1985. doi:10.1007/BF01394778.\n- [GL06] C. McA. Gordon and John Luecke. Knots with unknotting number 1 and essential Conway spheres. Algebr. Geom. Topol., 6:2051–2116, 2006. doi:10.2140/agt.2006.6.2051.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2663, "problem_number": "KP-1.4", "title": "Kirby Problem 1.4", "statement": "Let $P$ be a nontrivial satellite pattern with winding number\n$w(P) \\neq 0$. Then for any nontrivial knot $K$ and its satellite $K_{P}$ , one has\n\n$$\nu(K_{P}) \\geq w(P) + 1.\n$$", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.4.\n\nLiterature notes:\n(1) This problem appears as [HLP22, Conjecture 1.10].\n(2) One might guess that $w(P)^{2}$ is a lower bound. However, it can be shown\nthat\n\n$$\nu((T_{2,3})_{2,1}) \\leq 3,\n$$\n\nwhere $K_{p,q}$ denotes the $(p, q)$ cable of the knot $K$.\n(3) It follows from [ST88] that if $w(P) \\neq 0$, then\n\n$$\nu(K_{P}) \\geq 2.\n$$\n\nProblem 1.4 would be a generalization of this result. More recently, Hom,\nLidman and Park [HLP22] have claimed that if $K$ is a nontrivial knot,\nthen\n\n$$\nu(K_{p,q}) \\geq p.\n$$\n\nIn the above, $p$ indicates the winding number of the satellite operation.\n\nReferences cited:\n- [HLP22] Jennifer Hom, Tye Lidman, and JungHwan Park. Unknotting number and cabling, 2022. arXiv:2206.04196.\n- [ST88] Martin Scharlemann and Abigail Thompson. Unknotting number, genus, and companion tori. Math. Ann., 280(2):191–205, 1988. doi:10.1007/BF01456051.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2664, "problem_number": "KP-1.5", "title": "Kirby Problem 1.5", "statement": "Is there a relationship between genus and unknotting number\nfor specific classes of knots? Here are two instances of classes of knots for which\nthere might be an interesting answer.\n(a) Conjecture (Murasugi-Przytycki). For every positive fibered knot $K$, the\ngenus of $K$ equals its unknotting number.\n(b) Question (Stoimenow). For every alternating fibered knot, is the genus\ngreater than or equal to the unknotting number?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.5.\n\nLiterature notes:\n(1) There are examples of knots where the genus of a knot is greater than the\nunknotting number, and vice versa. However, there could be an inequality\nin some particular classes of knots. In part (a) there is evidence for an\ninequality one direction; in a different setting, part (b), the evidence points\nto an inequality in the other direction.\n(2) The conjecture in (a) appears at the end of [MP89]. By work of Rudolph\n[Rud93, Rud99] it is now known that genus $\\leq$ unknotting number for\na positive knot. For a positive braid closure, manipulations with braid\ngroup relations [BW84] allow one to construct an unknotting sequence of\nlength = genus, so that genus = unknotting number for a positive braid\nknot. However, such arguments are not available for the slightly more\ngeneral case of a positive fibered knot.\n(3) A Knotinfo [LM23b] search shows that this holds for alternating fibered\nknots of at most 13 crossings, with strict inequality holding in all but 13\nof the 2106 such knots.\n\nReferences cited:\n- [MP89] Kunio Murasugi and Józef H. Przytycki. The skein polynomial of a planar star product of two links. Math. Proc. Cambridge Philos. Soc., 106(2):273–276, 1989. doi:10.1017/S0305004100078099.\n- [Rud93] Lee Rudolph. Quasipositivity as an obstruction to sliceness. Bull. Amer. Math. Soc. (N.S.), 29(1):51–59, 1993. doi:10.1090/S0273-0979-1993-00397-5.\n- [Rud99] Lee Rudolph. Positive links are strongly quasipositive. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 555–562. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.555.\n- [BW84] Michel Boileau and Claude Weber. Le problème de J. Milnor sur le nombre gordien des nœuds algébriques. Enseign. Math. (2), 30(3-4):173–222, 1984.\n- [LM23b] Charles Livingston and Allison H. Moore. Knotinfo: Table of knot invariants, November 2023. https://knotinfo.org.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2665, "problem_number": "KP-1.6", "title": "Kirby Problem 1.6", "statement": "Suppose that $V_{1}$ and $V_{2}$ are $S$–equivalent Seifert forms. Does\nthere exist a fixed knot $K$ bounding Seifert surfaces $F_{1}$ and $F_{2}$ for which the as-\nsociated Seifert forms are $V_{1}$ and $V_{2}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.6.\n\nLiterature notes:\n(1) A Seifert form of a knot is a bilinear integral form $V$ for which $V - V^{*}$ is\nunimodular (with $V^{*}(\\alpha, \\beta) = V (\\beta, \\alpha)$). Every Seifert form $V$ arises as the\nSeifert form of a Seifert surface $F$ for some knot $K$. If $K$ bounds Seifert\nsurfaces $F_{1}$ and $F_{2}$, then the associated Seifert forms are $S$–equivalent.\nSee [Tro73] or [Kaw96, Chapter 5] for more details.\n(2) One way to think about this problem is as a comparison of two flavors\nof $S$–equivalence: that of $S$–equivalence of Seifert forms, versus that of\n$S$–equivalence of knots, with two knots being $S$–equivalent if they possess\nSeifert forms that are $S$–equivalent [Kea04, NS03]. It is shown in [NS03]\nthat two knots are $S$–equivalent if and only if they are related by a finite\nnumber of doubled-delta moves. These moves do not preserve the knot\ntype in general: for example, it is shown in [NS03] that a knot is $S$–\nequivalent to the unknot if and only if it has Alexander polynomial is\nequal to 1.\n\n(3) Consider the notation, defined for a Seifert form $V$ and for a knot $K$:\n$[V]_{S} = \\{$ $\\widetilde{V} |$ $\\widetilde{V}$ is a Seifert form $S$–equivalent to $V\\},$\n$[K]_{S} = \\{$ $\\widetilde{K} |$ $\\widetilde{K}$ is a knot $S$–equivalent to $K\\}.$\nSince for a given knot $K$, any two of its Seifert forms $V_{1}$ and $V_{2}$ are $S$–\nequivalent, it follows that $V_{1}, V_{2} \\in [V]_{S}$, where $V$ is any Seifert form of\n$K$. Thus, the set of all Seifert forms of $K$ maps to a subset of the $S$–\nequivalence class of any one of its Seifert forms. Given this, the question\nasked in this problem, can be rephrased as follows.\nQuestion. For a given $S$–equivalence class $[V]_{S}$ of a Seifert form $V$ ,\ndoes there exist a fixed knot $K$ such that the obvious map\n$\\{$ $\\widetilde{V} |$ $\\widetilde{V}$ is a Seifert form of $K\\} \\longrightarrow [V]_{S}$\nis surjective?\nThe key here being the existence of a single knot $K$ for which the\nindicated map is surjective, as every element of $[V]_{S}$ arises as the Seifert\nform of some knot; cf. [AFMW23, page 20].\n(4) In [AFMW23], the authors pose the genus-one version of this problem as\nProblem 7.7, and claim a complete answer under the additional hypothesis\nthat Seifert surfaces be disjoint.\nThey find a necessary and sufficient\ncondition for a pair of $2\\times2$ matrices to be Seifert forms for a pair of disjoint\nSeifert surfaces of the same knot, and they give an explicit construction\nrealizing infinitely many pairs of disjoint Seifert surfaces with the same\nboundary for any pair satisfying the condition.\n\nReferences cited:\n- [Tro73] H. F. Trotter. On S-equivalence of Seifert matrices. Invent. Math., 20:173–207, 1973. doi:10.1007/BF01394094.\n- [Kaw96] Akio Kawauchi. A survey of knot theory. Birkhäuser Verlag, Basel, 1996. Translated and revised from the 1990 Japanese original by the author.\n- [Kea04] C. Kearton. S-equivalence of knots. J. Knot Theory Ramifications, 13(6):709–717, 2004. doi:10.1142/S0218216504003408.\n- [NS03] Swatee Naik and Theodore Stanford. A move on diagrams that generates S-equivalence of knots. J. Knot Theory Ramifications, 12(5):717–724, 2003. doi: 10.1142/S0218216503002639.\n- [AFMW23] Menny Aka, Peter Feller, Alison Beth Miller, and Andreas Wieser. Seifert surfaces in the four-ball and composition of binary quadratic forms, 2023. arXiv:2311.17746.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2666, "problem_number": "KP-1.7", "title": "Kirby Problem 1.7", "statement": "Show that the sequence of absolute values of the coefficients of\nthe Alexander polynomial of a link are:\n(a) concave $($ Fox’s trapezoidal conjecture $)$, or\n(b) for an alternating link, log concave $($ Stoimenow’s strong Fox conjecture $)$.\n(c) For a positive link, is the sequence of the coefficients of the Conway poly-\nnomial strongly log concave?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.7.\n\nLiterature notes:\n(1) Let $K$ be a link with Alexander polynomial (up to a factor of $t^{r}$, $r \\in \\frac{1}{2}\\mathbb{Z}$)\n\n$$\n\\Delta_{K}(t)=\\sum_{i=0}^{n} a_i t^i,\n$$\n\nwith $a_{0}, a_{n} \\neq 0$, and Conway polynomial\n\n$$\n\\nabla_{K}(z)=z^{c(K)-1}\\sum_{i=0}^{m} c_i z^{2i}\n$$\n\ndefined by $\\nabla_{K}(t - t^{-1}) = t^{-n}\\Delta_{K}(t^{2})$, where $c(K)$ denotes the number\nof components of $K$. Recall that $a_{i} = (-1)^{n}a_{n-i}$; that for a non-split\nalternating link, $a_{i}a_{i+1} < 0$, $0 \\leq i < n$; and that for a positive link,\n$c_{i} > 0$, $0 \\leq i \\leq m$.\nFor the Strong Fox Conjecture (3) below, recall that a sequence of\npositive numbers $(a_{i})_{0\\leq i\\leq n}$ is called log concave if $a_{i-1}a_{i+1} \\leq a_i^{2}$\nfor all $i$ with $0 < i < n$. That is, the sequence $(\\log(a_{i}))$ is concave. For the final\nconjecture (4), recall that a sequence of positive numbers $(c_{i})_{0\\leq i\\leq m}$ is\ncalled strongly log concave if $c_{i-1}c_{i+1} < c_i^{2}$ for all $i$ with $0 < i < m$.\n(2) (Fox’s trapezoidal conjecture [Fox62, Problem 12]) Let $K$ be an alternating knot\nwith Alexander polynomial as above. Then\n\n$$\n|a_{0}| \\leq |a_{1}| \\leq \\cdots \\leq |a_{\\lfloor n/2\\rfloor}|\n\\geq |a_{\\lfloor n/2\\rfloor+1}| \\geq \\cdots \\geq |a_{n}|.\n$$\n\nMoreover, if $|a_{i}| = |a_{i+1}|$ for some $i < n/2$, then\n\n$$\n|a_{i}| = |a_{i+1}| = \\cdots = |a_{n-i}|.\n$$\n\n(1)\nSee for example [Mur85, AJK24] for some progress on Fox’s conjecture.\n(3) (Strong Fox Conjecture) Let $K$ be an alternating link with Alexander\npolynomial as above. Then the sequence of absolute values of the coeffi-\ncients $(|a_{i}|)$ is log concave. The equality $a_{i-1}a_{i+1} = a_i^{2}$ for some $i < n/2$\nholds only if $a_{i-1} = -a_{i} = a_{i+1}$. Moreover, the length of this ‘trapezoid\ntop’, i.e. the length of the longest string of equalities as in Equation (1) is\nbounded above by the absolute value of the signature. Conjecture (3) first\nappeared in [Sto05]. It is called the Strong Fox Conjecture in [Ban22].\nLog-concavity is a ubiquitous phenomenon appearing in many fields of\nmathematics, for example, [AHK17] and [Sta89].\nThe Strong Fox Conjecture was proved for 2-bridge knots by Ban-\nfield [Ban22] and for special alternating links by Hafner–Meszaros–Vidinas\n[HMV24]. The conjecture, including the statement regarding equality,\nwas verified by Stoimenow [Sto16, Section 9.2] for alternating knots of\ngenus at most 4 and for certain plumbed links [Sto21, Theorem 5.1]\n(4) Conjecture (Stoimenow): Let $K$ be a positive link with Conway poly-\nnomial as above. Then the sequence of coefficients $(c_{i})$ is strongly log\nconcave.\nConjecture (4) was proved for special alternating links by Stoimenow\n[Sto05, Theorem 1].\n\nReferences cited:\n- [Fox62] Ralph Fox. Some problems in knot theory. In M. K. Fort, Jr., editor, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961). Prentice-Hall, Englewood Cliffs, N.J., 1962.\n- [Mur85] Kunio Murasugi. On the Alexander polynomial of alternating algebraic knots. J. Austral. Math. Soc. Ser. A, 39(3):317–333, 1985.\n- [AJK24] Soheil Azarpendar, András Juhász, and Tamás Kálmán. On Fox’s trapezoidal conjecture, 2024. arXiv:2406.08662.\n- [Sto05] Alexander Stoimenow. Newton-like polynomials of links. Enseign. Math. (2), 51(3-4):211–230, 2005.\n- [Ban22] Ian M. Banfield. Christoffel words and the strong Fox conjecture for two-bridge knots, 2022. arXiv:2212.04561.\n- [AHK17] Karim Adiprasito, June Huh, and Eric Katz. Hodge theory of matroids. Notices Amer. Math. Soc., 64(1):26–30, 2017. doi:10.1090/noti1463.\n- [Sta89] Richard P. Stanley. Log-concave and unimodal sequences in algebra, combinatorics, and geometry. In Graph theory and its applications: East and West (Jinan, 1986), volume 576 of Ann. New York Acad. Sci., pages 500–535. New York Acad. Sci., New York, 1989. doi:10.1111/j.1749-6632.1989.tb16434.x.\n- [HMV24] Elena S. Hafner, Karola Mészáros, and Alexander Vidinas. Log-concavity of the Alexander polynomial. Int. Math. Res. Not. IMRN, 2024(13):10273–10284, 2024. doi:10.1093/imrn/rnae058.\n- [Sto16] Alexander Stoimenow. Diagram genus, generators, and applications. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016.\n- [Sto21] A. Stoimenow. Independence polynomials and Alexander-Conway polynomials of plumbing links. J. Combin. Theory Ser. A, 183:Paper No. 105487, 18, 2021. doi: 10.1016/j.jcta.2021.105487.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2667, "problem_number": "KP-1.8", "title": "Kirby Problem 1.8", "statement": "Which multi-variable Laurent polynomials arise as the multi-\nvariable Alexander polynomial of a link in the 3-sphere or, more generally, a ho-\nmology 3-sphere?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.8.\n\nLiterature notes:\n(1) A link in a homology sphere $Y$ has a covering corresponding to the ho-\nmomorphism $\\pi_{1}(Y \\setminus L) \\twoheadrightarrow\\mathbb{Z}^{n}$ arising from the isomorphism $H_{1}(Y \\setminus L) \\cong$\n$\\mathbb{Z}^{n}$, where $n$ is the number of components of the link $L$.\nThe homol-\nogy of this covering gives rise to a multi-variable Alexander polynomial\n\n$$\n\\Delta_{L}(t_{1},\\dots,t_{n}) \\in \\mathbb{Z}[t_{1}^{\\pm 1},\\dots,t_{n}^{\\pm 1}],\n$$\n\nwell defined up to multiplication by units, which are all of the form\n$\\pm t_{1}^{\\nu_{1}}\\cdots t_{n}^{\\nu_{n}}$.\n(2) The problem has been solved for the single-variable Alexander polynomial\nof knots. If $K$ is a knot in a homology sphere, then:\n\n$$\n\\bullet \\Delta_{K}(t) = \\Delta_{K}(t^{-1})\n$$\n\n$\\bullet \\Delta_{K}(1) = 1$.\nMoreover, any polynomial satisfying these two properties is realized by a\nknot in the 3-sphere (or any homology sphere) [Sei35, Lev65, Rol90].\nTorres [Tor53] found analogous (necessary but not sufficient–see [Hil81,\nPla86]) properties of the multi-variable Alexander polynomial of a link.\nIt is perhaps surprising that while there is no known characterization of\nthe Alexander polynomial of two-component links, Bailey found a charac-\nterization [Bai77] (discussed in [Pla86]) of their Alexander modules. A\ngeneral reference for the ideas here is Hillman’s book [Hil12].\n(3) The problem can be refined by focusing on specific families of links.\nQuestion (i). Which polynomials arise for a given family of knots\nor links?\nSome partial results are known.\n$\\bullet$ Torus links. If $T_{p,q}$ is the $(p, q)-$ torus link, let $d = \\gcd(p, q)$. Then\n\n$$\n\\Delta_{Tp,q}(t_{1}, t_{2},..., t_{d}) = [(t_{1}t_{2}... t_{d})^{pq/d} - 1]^{d}[(t_{1}t_{2}... t_{d}) - 1]\n$$\n\n$$\n[(t_{1}t_{2}... t_{d})^{p/d} - 1][(t_{1}t_{2}... t_{d})^{q/d} - 1].\n$$\n\nSee [Rol90] for knots and [EN85, Theorem 12.1]; compare [Ore21,\nCorollary 8.2] for links with $d > 1$.\n$\\bullet$ Two-bridge links. Many authors have found necessary (but still in-\nsufficient) conditions on the Alexander polynomial of a 2-bridge knot\nor link. Results of Koseleff and Pecker [KP15] subsume most earlier\nresults. Given a 2-variable polynomial, Hoste [Hos20], gives an al-\ngorithm to decide if the polynomial is the Alexander polynomial of\na 2-component, 2-bridge link.\n$\\bullet$ Alternating links. The Alexander polynomials of alternating knots\nhave special properties [Cro59, Mur58], and their coefficients are\nconjectured to have various concavity properties. See Problem 1.7\nfor an extensive discussion of these. It would be of interest to extend\nsuch properties to the case of alternating links.\n(4) Similarly, any 3-manifold with $H_{1}(X) \\cong \\mathbb{Z}^{n}$ has an associated multi-\nvariable Alexander polynomial $\\Delta_{X} \\in \\mathbb{Z}[H_{1}(X)] \\cong \\mathbb{Z}[t_{1}^{\\pm1},\\ldots,t_{n}^{\\pm1}]$.\nQuestion (ii). Which Laurent polynomials arise as the multi-variable\nAlexander polynomial of a 3-manifold?\n\nReferences cited:\n- [Sei35] H. Seifert. Über das Geschlecht von Knoten. Math. Ann., 110(1):571–592, 1935. doi:10.1007/BF01448044.\n- [Lev65] J. Levine. A characterization of knot polynomials. Topology, 4:135–141, 1965. doi: 10.1016/0040-9383(65)90061-3.\n- [Rol90] Dale Rolfsen. Knots and links, volume 7 of Mathematics Lecture Series. Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original.\n- [Tor53] Guillermo Torres. On the Alexander polynomial. Ann. of Math. (2), 57:57–89, 1953. doi:10.2307/1969726.\n- [Hil81] Jonathan A. Hillman. The Torres conditions are insufficient. Math. Proc. Cambridge Philos. Soc., 89(1):19–22, 1981. doi:10.1017/S0305004100057893.\n- [Pla86] M. L. Platt. Insufficiency of Torres’ conditions for two-component classical links. Trans. Amer. Math. Soc., 296(1):125–136, 1986. doi:10.2307/2000564.\n- [Bai77] James Leonard Bailey. ALEXANDER INVARIANTS OF LINKS. ProQuest LLC, Ann Arbor, MI, 1977. Thesis (Ph.D.)–The University of British Columbia (Canada). URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\& rft val fmt=info:ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri: pqdiss:NK34756.\n- [Hil12] Jonathan Hillman. Algebraic invariants of links, volume 52 of Series on Knots and Everything. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, second edition, 2012. doi:10.1142/8493.\n- [EN85] David Eisenbud and Walter Neumann. Three-dimensional link theory and invariants of plane curve singularities, volume 110 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1985.\n- [Ore21] S. Yu. Orevkov. Multivariate signatures of iterated torus links. Funktsional. Anal. i Prilozhen., 55(1):73–92, 2021. English translation in Funct. Anal. Appl. 55 (2021), no. 1, 59–74. doi:10.1134/S001626632101007X.\n- [KP15] Pierre-Vincent Koseleff and Daniel Pecker. On Alexander-Conway polynomials of two-bridge links. J. Symbolic Comput., 68:215–229, 2015. doi:10.1016/j.jsc.2014.09.011.\n- [Hos20] Jim Hoste. A note on Alexander polynomials of 2-bridge links. J. Knot Theory Ramifications, 29(8):1971003, 7, 2020. doi:10.1142/S0218216519710032.\n- [Cro59] Richard Crowell. Genus of alternating link types. Ann. of Math. (2), 69:258–275, 1959. doi:10.2307/1970181.\n- [Mur58] Kunio Murasugi. On the genus of the alternating knot. I, II. J. Math. Soc. Japan, 10:94–105, 235–248, 1958. doi:10.2969/jmsj/01010094.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2668, "problem_number": "KP-1.9", "title": "Kirby Problem 1.9", "statement": "If Dehn surgery on a knot $K$ gives a lens space, then $K$ is a\nBerge knot.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.9.\n\nLiterature notes:\n(1) This appears as [Kir97, Problem 1.78]. Let $(\\Sigma; H_{1}, H_{2})$ denote the genus-\ntwo Heegaard splitting of $S^{3}$ (cf. Section 3.3). A knot $K \\subset S^{3}$ is a Berge\nknot if it lies on $\\Sigma$ and there are properly embedded disks $D_{i} \\subset H_{i}$\nsuch that $|K \\cap D_{i}| = 1$. These knots are also called doubly primitive;\nsee [Gor09, Section 3.2].\n(2) The conjecture, often referred to as the Berge Conjecture, is known for\nsmall order lens spaces [Gor09]. For example, if surgery on $K$ gives $L(p, q)$\nwith $|p| \\leq 5$, then $K$ is the unknot or the trefoil [KMOS07]. In general,\nif a lens space were to arise as surgery on a non-Berge knot, that lens\nspace must also arise as surgery on a Berge knot [Gre13b]. The putative\nknot necessarily has the same knot Floer homology as that corresponding\nBerge knot.\n(3) From its description, any Berge knot is necessarily a tunnel number one\nknot. The Berge conjecture has been verified for all tunnel number one\nknots in the preprint [LMP25] and all non-hyperbolic knots [Mos71,\nBL89].\n(4) If surgery on a non-torus knot produces a lens space, then it must be an\nintegral surgery by the Cyclic Surgery Theorem [CGLS87]. The size of\nthe surgery coefficient is also heavily constrained. If $S^{3}_{p}(K) = L(p,q)$ for a\nnontrivial, non-trefoil knot $K$, then\n$2g(K)-1 \\leq |p|-2\\sqrt{4(|p|+1)/5}$ [Gre13b].\nIf $K$ is a counterexample to the Berge conjecture, then $|p| < 4g(K) - 1$\n[Bak06].\n(5) Using Heegaard Floer homology, it is also known that a knot with a lens\nspace surgery must be fibered and, after possibly mirroring, strongly quasi-\npositive [Ni07, Hed10]. The knot Floer homology of the dual knot $\\widetilde{K}$\nin $L(p,q)$ has smallest possible knot Floer homology,\n\n$$\n\\dim \\widehat{\\mathrm{HFK}}(\\widetilde{K}) = \\dim \\widehat{\\mathrm{HF}}(L(p,q))\n$$\n\n[Ras07, Hed11, Gre13b]. Such a knot is called Floer\nsimple, and one proposed strategy is to prove that Floer simple knots are\none-bridge with respect to the standard genus one Heegaard splitting of\n$L(p, q)$ [BGH08], as Berge proved that such knots are necessarily doubly\nprimitive [Ber18].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Gor09] Cameron Gordon. Dehn surgery and 3-manifolds. In Low dimensional topology, volume 15 of IAS/Park City Math. Ser., pages 21–71. Amer. Math. Soc., Providence, RI, 2009.\n- [KMOS07] P. Kronheimer, T. Mrowka, P. Ozsváth, and Z. Szabó. Monopoles and lens space surgeries. Ann. of Math. (2), 165(2):457–546, 2007. doi:10.4007/annals.2007.165.457.\n- [Gre13b] Joshua Evan Greene. The lens space realization problem. Ann. of Math. (2), 177(2):449–511, 2013. doi:10.4007/annals.2013.177.2.3.\n- [LMP25] Tao Li, Yoav Moriah, and Tali Pinsky. Tunnel number one knots satisfy the Berge conjecture. Geom. Topol., 29(6):3271–3343, 2025. doi:10.2140/gt.2025.29.3271.\n- [Mos71] Louise Moser. Elementary surgery along a torus knot. Pacific J. Math., 38:737–745, 1971. http://projecteuclid.org/euclid.pjm/1102969920.\n- [BL89] Steven A. Bleiler and Richard A. Litherland. Lens spaces and Dehn surgery. Proc. Amer. Math. Soc., 107(4):1127–1131, 1989. doi:10.2307/2047677.\n- [CGLS87] Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen. Dehn surgery on knots. Ann. of Math. (2), 125(2):237–300, 1987. doi:10.2307/1971311.\n- [Bak06] Kenneth L. Baker. Small genus knots in lens spaces have small bridge number. Algebr. Geom. Topol., 6:1519–1621, 2006. doi:10.2140/agt.2006.6.1519.\n- [Ni07] Yi Ni. Knot Floer homology detects fibred knots. Invent. Math., 170(3):577–608, 2007. doi:10.1007/s00222-007-0075-9.\n- [Hed10] Matthew Hedden. Notions of positivity and the Ozsváth-Szabó concordance invariant. J. Knot Theory Ramifications, 19(5):617–629, 2010. doi:10.1142/S0218216510008017.\n- [Ras07] Jacob Rasmussen. Lens space surgeries and L-space homology spheres, 2007. arXiv: 0710.2531.\n- [Hed11] Matthew Hedden. On Floer homology and the Berge conjecture on knots admitting lens space surgeries. Trans. Amer. Math. Soc., 363(2):949–968, 2011. doi:10.1090/S0002-9947-2010-05117-7.\n- [BGH08] Kenneth L. Baker, J. Elisenda Grigsby, and Matthew Hedden. Grid diagrams for lens spaces and combinatorial knot Floer homology. Int. Math. Res. Not. IMRN, 2008(10):Art. ID rnm024, 39, 2008. doi:10.1093/imrn/rnn024.\n- [Ber18] John Berge. Some knots with surgeries yielding lens spaces, 2018. arXiv:1802.09722.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2669, "problem_number": "KP-1.10", "title": "Kirby Problem 1.10", "statement": "(Generalized Property R Conjecture). Let $L \\subset S^{3}$ be an $n$-\ncomponent link such that 0-framed Dehn surgery on $L$ results in $\\#^{n}(S^{1} \\times S^{2})$. Is\nthere a sequence of handleslides converting $L$ into an $n$-component unlink?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.10.\n\nLiterature notes:\n(1) An $n$-component link $L \\subset S^{3}$ whose 0-surgery is $\\#^{n}(S^{1} \\times S^{2})$ is called\nan R-link. An R-link is said to have generalized property R if it can be\nconverted by handleslides to an unlink. (Note that all framings and linking\nnumbers are zero.)\nGabai proved [Gab87a] that the only 1-component R-link is the\nunknot, solving the Property R Conjecture. The Generalized Property\nR Conjecture (GPRC) posits that all R-links have generalized property\nR [Kir97, Problem 1.82] and was studied in [GST10].\n(2) There are two ways to weaken the GPRC that are of interest. Let $L \\sqcup L'$\ndenote the split union of links $L$ and $L'$.\nConjecture (Stable GPRC). If $L$ is an R-link, then there is an\nunlink $\\operatorname{U}$ such that $L \\sqcup \\operatorname{U}$ can be converted by handleslides to an unlink.\nMany potential counterexamples to the Stable GPRC are given in\n[GST10, MZ22].\n(3) A Hopf pair is a canceling 1-handle/2-handle pair, which can be repre-\nsented in the Kirby calculus by a Hopf link with a dotted-circle component\nand a 0-framed component [GS99, Section 5.4].\nConjecture (Weak GPRC). If $L$ is an R-link, then there is an un-\nlink $\\operatorname{U}$ and a split union $H$ of Hopf pairs such that $L \\sqcup \\operatorname{U} \\sqcup H$ can be\nconverted by handleslides to the split union $\\operatorname{U}' \\sqcup H'$ of an unlink and a\nsplit union of Hopf pairs.\nA sufficient condition for a 2-component link to satisfy the Weak\nGPRC is given in [MZ22, Theorem 1.1].\n(4) There are connections between these conjectures and other important\nproblems. Given an R-link $L$, one can construct a homotopy 4-ball $B_{L}$\nby attaching 0-framed 2-handles to $S^{3} \\times I$ along $L$ and capping off with\na 4-dimensional 1-handlebody. Turning this handle-decomposition upside\ndown gives a balanced presentation $P_{L}$ of the trivial group.\nAny homotopy 4-sphere that can be built without 1-handles is $B_{L}\\cup B^{4}$\nfor some R-link $L$. If an R-link L satisfies the Weak GPRC, then $B_{L}$ is\ndiffeomorphic to $B^{4}$. So, the Weak GPRC implies the Smooth Poincaré\nConjecture in dimension 4 for homotopy 4-spheres built without 1-handles.\n\nThe Andrews–Curtis Conjecture (ACC–see Problem 5.10) posits that\nany balanced presentation of the trivial group can be converted to the\ntrivial presentation through balanced presentations. If the GPRC is true,\nthen so is the ACC. There is also a stable version of the ACC that is\nimplied by the Stable GPRC [GST10, MZ22].\nIf $L$ is an R-link, then $L$ bounds a collection of homotopy-ribbon disks\nin the homotopy 4-ball $B_{L}$. Thus, such links provide potential counterex-\namples to the Slice-Ribbon Conjecture (Problem 1.50) but are ribbon if\nthey satisfy the Stable GPRC [AT16b]; see [MZ25, Proposition 5.2]. If\n$K$ is a fibered, homotopy-ribbon knot, then the closed monodromy of $K$\nadmits an extension across a handlebody [CG83a]. Any such extension\ngives rise to a link $L$ lying on the fiber of $K$ such that $L$ and $K \\cup L$ are\nboth R-links, and $K \\cup L$ is ribbon if $L$ (equivalently, $K \\cup L$) satisfies the\nStable GPRC.\n\nReferences cited:\n- [Gab87a] David Gabai. Foliations and the topology of 3-manifolds. II. J. Differential Geom., 26(3):461–478, 1987. http://projecteuclid.org/euclid.jdg/1214441487.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [GST10] Robert E. Gompf, Martin Scharlemann, and Abigail Thompson. Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures. Geom. Topol., 14(4):2305–2347, 2010. doi:10.2140/gt.2010.14.2305.\n- [MZ22] Jeffrey Meier and Alexander Zupan. Generalized square knots and homotopy 4-spheres. J. Differential Geom., 122(1):69–129, 2022. doi:10.4310/jdg/1668186788.\n- [GS99] Robert E. Gompf and András I. Stipsicz. 4-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1999. doi:10.1090/gsm/020.\n- [AT16b] Tetsuya Abe and Motoo Tange. A construction of slice knots via annulus twists. Michigan Math. J., 65(3):573–597, 2016. doi:10.1307/mmj/1472066149.\n- [MZ25] Jeffrey Meier and Alexander Zupan. Knots bounding nonisotopic ribbon disks. J. Topol., 18(4):Paper No. e70047, 18, 2025. doi:10.1112/topo.70047.\n- [CG83a] A. J. Casson and C. McA. Gordon. A loop theorem for duality spaces and fibred ribbon knots. Invent. Math., 74(1):119–137, 1983. doi:10.1007/BF01388533.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2670, "problem_number": "KP-1.11", "title": "Kirby Problem 1.11", "statement": "(Cabling conjecture). Let $K \\subset S^{3}$ be a knot and $r \\in \\mathbb{Q}$. If\n$r$-framed Dehn surgery on $K$ is not prime, then $K$ is a nontrivial cable of a knot\n$J \\subset S^{3}$ and $r$ is the slope of the cabling annulus.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.11.\n\nLiterature notes:\n(1) The conjecture is due to González-Acuña and Short [GAnS86]. If $K =$\n$J_{p,q}$ is a cable knot, then there is an annulus in $S^{3}\\setminus\\nu(K)$ that has boundary\nequal to two parallel curves of slope $pq$. This annulus is called the cabling\nannulus and $pq$ is referred to as the slope of the cabling annulus. Using a\ncut and paste argument, one can show that\n\n$$\nS^{3}_{pq}(K)=S^{3}_{q/p}(J)\\#S^{3}_{p/q}(\\operatorname{U}),\n$$\n\nwhere $\\operatorname{U}$ is the unknot. Therefore, if $p \\neq 1$, then $S^{3}_{pq}(K)$ is not prime.\n(2) Many special cases of the conjecture are known.\nGordon and Luecke\n[GL87] prove that if $r$ is a reducible slope, then $r$ is an integer and the\nsurgery contains a lens space summand.\nThe conjecture is known for\nsatellite knots [Sch90], strongly invertible knots [EMn92], alternating\nknots [MT92] and symmetric knots [LZ94, HS98, HM97b].\n(3) Greene [Gre15] has shown that if $r$-framed surgery on $K$ gives a connected\nsum of nontrivial lens spaces, then $K$ is a $(p, q)$-cable of either the unknot\nor a torus knot and that $r = pq$.\n(4) Dunfield [Dun20] observed that a similar conjecture may hold for knots\nin $S^{1} \\times S^{2}$. He conjectures that there are no hyperbolic knots in $S^{1} \\times S^{2}$\nthat have non-prime Dehn surgery.\n(5) A weaker version of the cabling conjecture is the two summands conjecture,\nwhich says that if $S^{3}_{r}(K)$ is non-prime, then it cannot have more than two\nprime summands.\nNote that it follows from work of Howie [How02],\nSayari [Say09] and Valdez-Sánchez [VS99] that $S^{3}_{r}(K)$ can never contain\nmore than three irreducible summands.\n\nReferences cited:\n- [GAnS86] Francisco González-Acuña and Hamish Short. Knot surgery and primeness. Math. Proc. Cambridge Philos. Soc., 99(1):89–102, 1986. doi:10.1017/S0305004100063969.\n- [GL87] C. McA. Gordon and J. Luecke. Only integral Dehn surgeries can yield reducible manifolds. Math. Proc. Cambridge Philos. Soc., 102(1):97–101, 1987. doi:10.1017/S0305004100067086.\n- [Sch90] Martin Scharlemann. Producing reducible 3-manifolds by surgery on a knot. Topology, 29(4):481–500, 1990. doi:10.1016/0040-9383(90)90017-E.\n- [EMn92] Mario Eudave Muñoz. Band sums of links which yield composite links. The cabling conjecture for strongly invertible knots. Trans. Amer. Math. Soc., 330(2):463–501, 1992. doi:10.2307/2153918.\n- [MT92] William W. Menasco and Morwen B. Thistlethwaite. Surfaces with boundary in alternating knot exteriors. J. Reine Angew. Math., 426:47–65, 1992. doi:10.1515/crll.1992.426.47.\n- [LZ94] E. Luft and X. Zhang. Symmetric knots and the cabling conjecture. Math. Ann., 298(3):489–496, 1994. doi:10.1007/BF01459747.\n- [HS98] Chuichiro Hayashi and Koya Shimokawa. Symmetric knots satisfy the cabling conjecture. Math. Proc. Cambridge Philos. Soc., 123(3):501–529, 1998. doi:10.1017/S0305004197002399.\n- [HM97b] Chuichiro Hayashi and Kimihiko Motegi. Dehn surgery on knots in solid tori creating essential annuli. Trans. Amer. Math. Soc., 349(12):4897–4930, 1997. doi: 10.1090/S0002-9947-97-01723-6.\n- [Gre15] Joshua Evan Greene. L-space surgeries, genus bounds, and the cabling conjecture. J. Differential Geom., 100(3):491–506, 2015. http://projecteuclid.org/euclid.jdg/1432842362.\n- [Dun20] Nathan M. Dunfield. A census of exceptional Dehn fillings. In Characters in lowdimensional topology, volume 760 of Contemp. Math., pages 143–155. Amer. Math. Soc., [Providence], RI, [2020] ©2020. doi:10.1090/conm/760/15289.\n- [How02] James Howie. A proof of the Scott-Wiegold conjecture on free products of cyclic groups. J. Pure Appl. Algebra, 173(2):167–176, 2002. doi:10.1016/S0022-4049(02) 00042-7.\n- [Say09] Nabil Sayari. Reducible Dehn surgery and the bridge number of a knot. J. Knot Theory Ramifications, 18(4):493–504, 2009. doi:10.1142/S0218216509007038.\n- [VS99] Luis Gerardo Valdez Sánchez. Dehn fillings of 3-manifolds and non-persistent tori. Topology Appl., 98(1-3):355–370, 1999. II Iberoamerican Conference on Topology and its Applications (Morelia, 1997). doi:10.1016/S0166-8641(99)00038-3.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2671, "problem_number": "KP-1.12", "title": "Kirby Problem 1.12", "statement": "This problem presents several variations on the Cosmetic\nSurgery Conjecture, discussed in turn below.\n(a) $($ Cosmetic Surgery Conjecture $)$ Two surgeries along inequivalent slopes are\nnever purely cosmetic.\n(b) $($ Oriented knot complement conjecture $)$ If $K_{1}$ and $K_{2}$ are knots in a closed,\noriented 3-manifold $N$ whose complements are homeomorphic via an orientation-\npreserving homeomorphism, then there exists a homeomorphism of $N$ tak-\ning $K_{1}$ to $K_{2}$.\n(c) Let $K \\subset S^{3}$ be a nontrivial knot, then $K$ admits no purely cosmetic surg-\neries.\n(d) If $K \\subset S^{3}$ is not the torus knot $T_{2g+1,2}$, then $K$ admits no chirally cos-\nmetic surgeries with inequivalent slopes.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.12.\n\nLiterature notes:\n(1) This is an update to [Kir97, Problem 1.81], contributed by S. Bleiler.\nParts (a) and (b) appear as Conjectures 6.1 and 6.2 of [Gor91] respec-\ntively.\n(2) For the rest of this problem, $M$ denotes an irreducible oriented 3-manifold\nwith torus boundary that is not the solid torus. Two fillings along dis-\ntinct slopes $r_{1}$ and $r_{2}$ are called purely cosmetic if there is an orientation-\npreserving homeomorphism between $M(r_{1})$ and $M(r_{2})$, and chirally cos-\nmetic if there is an orientation-reversing homeomorphism between these\nfillings. (The terminology would be purely or chirally cosmetic surgery if\n$M$ is a knot complement.) Two slopes are said to be equivalent if there\nexists a homeomorphism (either preserving or reversing the orientation)\nof $M$ taking one slope to the other.\n(3) When [Kir97, Problem 1.81] was written, several examples of chirally cos-\nmetic surgeries on inequivalent slopes were known. More recently, Ichihara\nand Jong [IJ18], with help from Masai, found a hyperbolic one-cusped\nmanifold admitting a pair of chirally cosmetic surgeries with inequivalent\nslopes, such that the resulting manifold is also hyperbolic. This gives a\ncounterexample to [Kir97, Problem 1.81B].\nDunfield found that the manifold $o9_{39009}$ from Burton’s census [Bur14]\nhad Dehn fillings $(-1, 3)$ and $(-3, 2)$, which are isometric by an orienta-\ntion reversing isometry, but where the core curves of the Dehn fillings are\ngeodesics of different lengths. This provides a counterexample to [Kir97,\nProblem 1.81C].\n(4) There are different versions of the Cosmetic Surgery Conjecture (a) in\nthe literature. A stronger version of the conjecture, which is still open,\nasserts that if there exist purely cosmetic surgeries on $M$, then there is\nan orientation-preserving homeomorphism of $M$ sending one slope to the\nother. This version would imply that $M$ does not admit purely cosmetic\nsurgeries if $M$ is hyperbolic. If there exist two slopes that are related\nby an orientation-reversing homeomorphism, and the surgeries on these\ntwo slopes are amphichiral, then this would be a counterexample to the\nstronger version of the conjecture, but not to the version of the conjecture\nstated in this problem.\n\n(5) Lackenby [Lac97] proved that, for “most” surgeries on null-homotopic\nknots in manifolds with $b_{1} > 0$, the original manifold, the knot, and the\nslope are determined by the resulting manifold from the surgery.\nWhen $M$ is hyperbolic, there is an explicit constant $C = C(M)$,\nsatisfying that if the normalized lengths of two inequivalent slopes $r_{1}, r_{2}$\nare both greater than $C$, then this pair is not (purely or chirally) cosmetic\n[FPS22, FPS25].\nUsing code that is publicly available on GitHub [FPS24], Conjec-\nture (a) has been verified for all the 59,107 one-cusped manifolds in the\nSnapPy census. This code can also be used to investigate Conjecture (c)\nabove for knots in $S^{3}$.\n(6) If $M$ is the irreducible complement of a null-homologous knot in a 3-\nmanifold of the form $Z\\#S^{1} \\times S^{2}$, then the slopes of any pair of cosmetic\nsurgeries must be $\\{\\pm r\\}$ [Ni13]. A similar result holds when $M$ is the\ncomplement of a knot in a closed 3-manifold whose Thurston norm is\nstrictly smaller than that of $M$ [Ni11].\n(7) The Cosmetic Surgery Conjecture (a) is equivalent to the Oriented Knot\nComplement Conjecture (b), which is an analogue of the Knot Comple-\nment Theorem [GL89].\nIn [Kir97, Problem 1.81D], the homeomorphism of $N$ is required to be\norientation-preserving. Since our definition of equivalent slopes does not\nrequire the homeomorphism to preserve orientation, we do not require\nthe homeomorphism of $N$ to be orientation-preserving in the problem\nstatement here.\nGordon and Luecke [GL89] proved Conjecture (b) for $N = S^{3}$ and\n$S^{1} \\times S^{2}$. It also holds for certain L-spaces [Gai18, Rav22]. Cremaschi–\nYarmola proved it for knots in an orientable circle bundle over a genus\n$g \\geq 2$ surface [CY24].\n(8) When $M$ is the complement of a knot $K \\subset S^{3}$, the Cosmetic Surgery Con-\njecture is equivalent to saying that if $K$ admits purely cosmetic surgeries\nwith slopes $r_{1}, r_{2}$ then $K$ is amphichiral and $r_{1} = -r_{2}$. The conclusion\nthat $r_{1} = -r_{2}$ was proved in [NW15]. Conjecture (c) is stronger than\nthis special case of Conjecture (a), because there may exist an amphichi-\nral knot in $S^{3}$ such that a nonzero surgery on it is also amphichiral. This\nwould be a counterexample to Conjecture (c), but not to Conjecture (a).\nFuter–Purcell–Schleimer [FPS25] verified Conjecture (c) for knots\nwith up to 19 crossings.\nHanselman [Han23] proved that if $K$ admits a pair of purely cosmetic\nsurgeries, then the pair of slopes is either $\\{\\pm2\\}$ or $\\{\\pm 1/q\\}$ for an integer $q$\nthat is explicitly determined from the knot Floer homology of $K$, and the\nformer case happens only when $g(K) = 2$. Daemi-Lidman-Miller Eismeier\n[DLME24] subsequently announced that they have eliminated the case\nof $\\{\\pm 1/q\\}$. As a result, any pair of purely cosmetic slopes must be $\\{\\pm2\\}$,\nand the knot has genus 2 and trivial Alexander polynomial.\n(9) Although [Kir97, Problem 1.81 (B)] has been disproved, no such examples\nhave been found for knots in $S^{3}$. This is the motivation for Conjecture\n(d).\n\n(10) There are many known constraints on chirally cosmetic surgeries; see\n[IIS22] for an overview. Ozsváth and Szabó [OS11] proved that if $K$\nadmits a pair of (purely or chirally) cosmetic surgeries with slopes $r_{1}, r_{2}$,\nthen either $r_{1}, r_{2}$ have opposite signs or the surgery is an L-space. Futer–\nPurcell–Schleimer [FPS25] have shown that Conjecture (d) holds for all\nhyperbolic knots with up to 15 crossings.\n(11) Somewhat weaker versions of Conjecture (b), in which the knots are re-\nquired to be null-homotopic, may be found in [Kir97, Problem 1.80 (B)\nand (C)]. Using standard arguments in Heegaard Floer homology, one can\nverify the conjecture in [Kir97, Problem 1.80(C)] when $Y$ is an L-space.\nSee, for example, [Gai18, Rav22].\n(12) A list of problems about (mostly chirally) cosmetic surgeries on knots in\n$S^{3}$ can be found in [Ito22].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Gor91] Cameron McA. Gordon. Dehn surgery on knots. In Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pages 631–642. Math. Soc. Japan, Tokyo, 1991.\n- [IJ18] Kazuhiro Ichihara and In Dae Jong. Cosmetic banding on knots and links. Osaka J. Math., 55(4):731–745, 2018. With an appendix by Hidetoshi Masai, https://projecteuclid.org/euclid.ojm/1539158668.\n- [Bur14] Benjamin A. Burton. The cusped hyperbolic census is complete, 2014. arXiv:1405.2695.\n- [Lac97] Marc Lackenby. Dehn surgery on knots in 3-manifolds. J. Amer. Math. Soc., 10(4):835–864, 1997. doi:10.1090/S0894-0347-97-00241-5.\n- [FPS22] David Futer, Jessica S. Purcell, and Saul Schleimer. Effective bilipschitz bounds on drilling and filling. Geom. Topol., 26(3):1077–1188, 2022. doi:10.2140/gt.2022.26.1077.\n- [FPS25] David Futer, Jessica S. Purcell, and Saul Schleimer. Excluding cosmetic surgeries on hyperbolic 3-manifolds. Journal of Computational Geometry, 16(1):694–736, Nov. 2025. URL: https://jocg.org/index.php/jocg/article/view/4778, doi:10.20382/jocg.v16i1a19.\n- [FPS24] David Futer, Jessica S. Purcell, and Saul Schleimer. Code for verifying the cosmetic surgery conjecture on a given 3–manifold, 2024. https://github.com/saulsch/Cosmetic.\n- [Ni13] Yi Ni. Nonseparating spheres and twisted Heegaard Floer homology. Algebr. Geom. Topol., 13(2):1143–1159, 2013. doi:10.2140/agt.2013.13.1143.\n- [Ni11] Yi Ni. Thurston norm and cosmetic surgeries. In Low-dimensional and symplectic topology, volume 82 of Proc. Sympos. Pure Math., pages 53–63. Amer. Math. Soc., Providence, RI, 2011. doi:10.1090/pspum/082/2768653.\n- [GL89] C. McA. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc., 2(2):371–415, 1989. doi:10.2307/1990979.\n- [Gai18] Fyodor Gainullin. Heegaard Floer homology and knots determined by their complements. Algebr. Geom. Topol., 18(1):69–109, 2018. doi:10.2140/agt.2018.18.69.\n- [Rav22] Huygens C. Ravelomanana. Knot complement problem for L-space ZH$S^{3}$. J. Knot Theory Ramifications, 31(4):Paper No. 2250028, 7, 2022. doi:10.1142/S0218216522500286.\n- [CY24] Tommaso Cremaschi and Andrew Yarmola. Knots in circle bundles are determined by their complements, 2024. arXiv:2401.02895.\n- [NW15] Yi Ni and Zhongtao Wu. Cosmetic surgeries on knots in $S^{3}$. J. Reine Angew. Math., 706:1–17, 2015. doi:10.1515/crelle-2013-0067.\n- [Han23] Jonathan Hanselman. Heegaard Floer homology and cosmetic surgeries in $S^{3}$. J. Eur. Math. Soc. (JEMS), 25(5):1627–1669, 2023. doi:10.4171/jems/1218.\n- [DLME24] Aliakbar Daemi, Tye Lidman, and Mike Miller Eismeier. Filtered instanton homology and cosmetic surgery, 2024. arXiv:2410.21248.\n- [IIS22] Kazuhiro Ichihara, Tetsuya Ito, and Toshio Saito. On constraints for knots to admit chirally cosmetic surgeries and their calculations. Pacific J. Math., 321(1):167–191, 2022. doi:10.2140/pjm.2022.321.167.\n- [OS11] Peter Ozsváth and Zoltán Szabó. Knot Floer homology and rational surgeries. Algebr. Geom. Topol., 11(1):1–68, 2011. doi:10.2140/agt.2011.11.1.\n- [Ito22] Tetsuya Ito. Cosmetic surgery on knots. In Tomotada Ohtsuki, editor, Problems in low-dimensiontal topology, 2022. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan, 2022. URL: https://www.kurims.kyoto-u.ac.jp/„kyodo/kokyuroku/contents/pdf/2227-12.pdf.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2672, "problem_number": "KP-1.13", "title": "Kirby Problem 1.13", "statement": "Let $K$ be a null-homotopic knot in a 3-manifold $Y$ , and let\n$Y_{0}(K)$ be the manifold obtained by 0-surgery on $K$.\n(a) Conjecture: Let $F$ be a Seifert surface for $K$, and let $\\widetilde{F} \\subset Y_{0}(K)$ be the\nclosed surface obtained by capping off $\\partial F$ with a disk. If $Y_{0}(K)$ is a surface\nbundle over $S^{1}$ with fiber in the homology class $[\\widetilde{F}]$, then $K$ is a fibered\nknot with fiber in the homology class $[F]$.\n(b) Conjecture: If $F$ is taut in the exterior of $K$ in $Y$ , then $\\widetilde{F}$ is also taut in\n$Y_{0}(K)$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.13.\n\nLiterature notes:\n(1) Conjecture (a) is an updating of [Kir97, Problem 1.80C], and is due to\nBoileau. Conjecture (b) is due to Gabai.\n(2) Let $M$ be a compact, oriented, connected 3-manifold, let $F \\subset M$ be a\nproperly embedded connected surface. We say that $F$ is taut if it realizes\nthe Thurston norm in the homology class $[F] \\neq 0 \\in H_{2}(M, \\partial M),$ or if $F$\nis a sphere.\n(3) In the original version of Conjecture (a), the surgery slope can be any\nrational number, and there is no restriction on the homology class of the\nfiber. However, there are simple counterexamples to the original question\n[Ni09]. More precisely, there exist nonfibered null-homotopic knots such\nthat every integral surgery on the knot yields a manifold that fibers over\n$S^{1}$.\n(4) It is known that $Y_{0}(K)$ does not contain a homologically essential 2-sphere\nif $Y \\setminus K$ does not contain such 2-spheres and $K$ is nontrivial [Gab87a,\nLac97, HL22, Ni23b].\nConjecture (a) and Conjecture (b) are known for knots in $S^{3}$ [Gab87b],\nL-spaces [Ni07, AN09], and certain reducible manifolds [Gab87b, Ni13].\nThey are also known for knots obtained from the unknot by crossing\nchanges of the same sign [Ni24].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Ni09] Yi Ni. Dehn surgeries that yield fibred 3-manifolds. Math. Ann., 344(4):863–876, 2009. doi:10.1007/s00208-008-0331-3.\n- [Gab87a] David Gabai. Foliations and the topology of 3-manifolds. II. J. Differential Geom., 26(3):461–478, 1987. http://projecteuclid.org/euclid.jdg/1214441487.\n- [Lac97] Marc Lackenby. Dehn surgery on knots in 3-manifolds. J. Amer. Math. Soc., 10(4):835–864, 1997. doi:10.1090/S0894-0347-97-00241-5.\n- [HL22] Jennifer Hom and Tye Lidman. Dehn surgery and nonseparating two-spheres. In Gauge theory and low-dimensional topology—progress and interaction, volume 5 of Open Book Ser., pages 145–153. Math. Sci. Publ., Berkeley, CA, 2022. https: //msp.org/obs/2022/5-1/p07.xhtml.\n- [Ni23b] Yi Ni. Null-homotopic knots have property R. Math. Proc. Cambridge Philos. Soc., 175(1):217–223, 2023. doi:10.1017/S0305004123000129.\n- [Gab87b] David Gabai. Foliations and the topology of 3-manifolds. III. J. Differential Geom., 26(3):479–536, 1987. http://projecteuclid.org/euclid.jdg/1214441488.\n- [Ni07] Yi Ni. Knot Floer homology detects fibred knots. Invent. Math., 170(3):577–608, 2007. doi:10.1007/s00222-007-0075-9.\n- [AN09] Yinghua Ai and Yi Ni. Two applications of twisted Floer homology. Int. Math. Res. Not. IMRN, 2009(19):3726–3746, 2009. doi:10.1093/imrn/rnp070.\n- [Ni13] Yi Ni. Nonseparating spheres and twisted Heegaard Floer homology. Algebr. Geom. Topol., 13(2):1143–1159, 2013. doi:10.2140/agt.2013.13.1143.\n- [Ni24] Yi Ni. Property G and the 4-genus. Trans. Amer. Math. Soc. Ser. B, 11:120–143, 2024. doi:10.1090/btran/153.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2673, "problem_number": "KP-1.14", "title": "Kirby Problem 1.14", "statement": "For which nonzero $r \\in \\mathbb{Q}$ is it true that for every nontrivial\nknot $K \\subset S^{3}$ there is a homomorphism\n\n$$\n\\pi_{1}(S^{3}_{r}(K)) \\to \\operatorname{SU}(2)\n$$\n\nwith nonabelian image?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.14.\n\nLiterature notes:\n(1) If this statement is true for some $r \\in \\mathbb{Q}$, then it’s true for $-r$; hence,\nwe assume $r > 0$. Kronheimer–Mrowka proved [KM04] that there is a\nhomomorphism with nonabelian image for all $r \\in (0, 2]$. Work of Baldwin–\nSivek [BS23a] together with announced results of Baldwin–Li–Sivek–Ye\n[BLSY24], and Farber–Reinoso–Wang [FRW24] proves the same for all\n$r \\in (2, 5)$ with prime power numerators.\n(2) Let $r = p/q > 0$ with $p$ and $q$ relatively prime, and assume that $\\Delta_{K}(\\zeta^{2}) \\neq$\n0 for any $p$th root of unity $\\zeta$. Then it is further known [BS23a] that if\n$\\pi_{1}(S^{3}_{r}(K))$ only admits abelian $\\operatorname{SU}(2)$-representations, then $K$ is fibered,\nstrongly quasipositive, and $r \\geq 2g(K) - 1.$\n(3) Note that for most $r \\in \\mathbb{Z}$, there is a torus knot whose $r$–surgery is $\\operatorname{SU}(2)$-\nabelian. Problem 3.51 deals with a related problem about the characteri-\nzation of rational homology spheres whose fundamental groups have only\nAbelian $\\operatorname{SU}(2)$ representations.\n\nReferences cited:\n- [KM04] P. B. Kronheimer and T. S. Mrowka. Dehn surgery, the fundamental group and $\\mathrm{SU}(2)$. Math. Res. Lett., 11(5-6):741–754, 2004. doi:10.4310/MRL.2004.v11.n6.a3.\n- [BS23a] John A. Baldwin and Steven Sivek. Instantons and L-space surgeries. J. Eur. Math. Soc. (JEMS), 25(10):4033–4122, 2023. doi:10.4171/jems/1280.\n- [BLSY24] John A. Baldwin, Zhenkun Li, Steven Sivek, and Fan Ye. Small Dehn surgery and $\\mathrm{SU}(2)$. Geom. Topol., 28(4):1891–1922, 2024. doi:10.2140/gt.2024.28.1891.\n- [FRW24] Ethan Farber, Braeden Reinoso, and Luya Wang. Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil, 2024. doi:10.2140/gt.2024.28.4337.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2674, "problem_number": "KP-1.15", "title": "Kirby Problem 1.15", "statement": "(a) Are there integral homology spheres with arbitrarily large (integral) Dehn\nsurgery number? Are there irreducible examples? Does the connected sum\nof $n$ nontrivial homology spheres have Dehn surgery number at least $n$?\n(b) Given a 3-manifold $Y$ with a smooth orientation-preserving $\\mathbb{Z}/n\\mathbb{Z}$ sym-\nmetry $\\varphi$, Sakuma [Sak01] (generalizing [PS01]) proved that $Y$ admits a\nsurgery description compatible with the symmetry. Define the equivariant\n(integral) Dehn surgery number of $(Y, \\varphi)$ as the minimal number of con-\nnected components of any such link. Do there exist $(Y, \\varphi)$ whose equivari-\nant (integral) Dehn surgery number is arbitrarily larger than the (integral)\nDehn surgery number?\n(c) Following the remarks below, we can consider the following specific case:\nDoes there exist a manifold with a 2-fold symmetry whose equivariant\n(integral) Dehn surgery number is larger than the (integral) Dehn surgery\nnumber?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.15.\n\nLiterature notes:\n(1) Any oriented connected closed 3-manifold $Y$ is obtained by (integral) Dehn\nsurgery on a link in the 3-sphere, and the minimal number of connected\ncomponents of any such link is called the (integral) Dehn surgery number\nof $Y$ .\n\n(2) Part (a) is a version of Problem 3.102 from [Kir97], asked by Auckly. The\nintegral Dehn surgery case is interesting to consider from the perspective\nof 4-manifolds bounded by $Y$ , as it is related to the minimum number of\nhandles needed to describe the 4-manifold.\nThere are many examples of integral homology spheres with Dehn\nsurgery number at least 2, the earliest due to [GL87], hyperbolic examples\ndue to [Auc97], and Seifert fibered examples due to [HKL16b].\n(3) Regarding part (b), it should be possible to show that the equivari-\nant surgery number does not agree with the surgery number. Consider\n$Y=S^{3}_{2}(4_{1})$, which is a Seifert fibered space (see, for example, [BW01,\nTheorem 1.1(4))]). Since $Y$ is Seifert fibered, it has a $\\mathbb{Z}/n\\mathbb{Z}$ action, for\nall $n$, while $4_{1}$ does not. It remains to be shown that there is no other\nknot $K$ for which some $2/q$-surgery is $Y$ ; a plausible approach is to use\nthe mapping cone formula for Heegaard Floer homology.\nFinding the examples as in part (b) for which the equivariant surgery\nnumber is arbitrarily larger than the Dehn surgery number is, as in part\n(a), more challenging.\n(4) Lastly, we point out that this problem is closely related to Problem 3.39.\n\nReferences cited:\n- [Sak01] Makoto Sakuma. Surgery description of orientation-preserving periodic maps on compact orientable 3-manfolds. Rend. Istit. Mat. Univ. Trieste, 32:375–396, 2001. Dedicated to the memory of Marco Reni.\n- [PS01] Józef H. Przytycki and Maxim V. Sokolov. Surgeries on periodic links and homology of periodic 3-manifolds. Math. Proc. Cambridge Philos. Soc., 131(2):295–307, 2001. doi:10.1017/S0305004101005308.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [GL87] C. McA. Gordon and J. Luecke. Only integral Dehn surgeries can yield reducible manifolds. Math. Proc. Cambridge Philos. Soc., 102(1):97–101, 1987. doi:10.1017/S0305004100067086.\n- [Auc97] David Auckly. Surgery numbers of 3-manifolds: a hyperbolic example. In Geometric topology (Athens, GA, 1993), volume 2 of AMS/IP Stud. Adv. Math., pages 21–34. Amer. Math. Soc., Providence, RI, 1997. doi:10.1090/amsip/002.1/02.\n- [HKL16b] Jennifer Hom, Çağrı Karakurt, and Tye Lidman. Surgery obstructions and Heegaard Floer homology. Geom. Topol., 20(4):2219–2251, 2016. doi:10.2140/gt.2016.20.2219.\n- [BW01] Mark Brittenham and Ying-Qing Wu. The classification of exceptional Dehn surgeries on 2-bridge knots. Comm. Anal. Geom., 9(1):97–113, 2001. doi:10.4310/CAG.2001.v9.n1.a4.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2675, "problem_number": "KP-1.16", "title": "Kirby Problem 1.16", "statement": "(a) Given a knot $K \\subset S^{3}$ determine all knots $K' \\subset S^{3}$ for which the branched\ndouble covers of $S^{3}$ along $K$ and $K'$ are homeomorphic.\n(b) Find a set of moves on links in $S^{3}$ so that two links are related by these\nmoves if and only if they have the same branched double cover.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.16.\n\nLiterature notes:\n(1) These are Problems 3.25 and 1.22 in [Kir97], respectively.\nSee Prob-\nlem 1.17 for a related problem about alternating links. Let $\\Sigma(K)$ denote\nthe branched double cover of $S^{3}$ along $K$.\n(2) It follows from the Orbifold Geometrization Theorem that there are finitely\nmany $K'$ with $\\Sigma(K') \\cong \\Sigma(K)$. See [Pao05] for a discussion.\n(3) Mecchia and Zimmerman showed that if $\\Sigma(K)$ is hyperbolic, then there are\nat most eight other $K'$ with $\\Sigma(K') \\cong \\Sigma(K)$ [MZ04]. Kawauchi showed\nthat this bound is sharp, exhibiting infinitely many 9-tuples of distinct hy-\nperbolic knots in $S^{3}$ with homeomorphic branched double covers [Kaw06].\nHowever, his construction is very non-explicit, and it would be interest-\ning to have some explicit constructions. Mecchia further described how\nhyperbolic knots with the same double branched cover can be understood\nin terms of isometries of the cover [Mec01]\n(4) If $K$ is a 2-bridge knot and $\\Sigma(K') \\cong \\Sigma(K)$ then $K' = K [\\mathrm{HR}85$].\n(5) If $K$ and $K'$ are related by Conway mutation then $\\Sigma(K') \\cong \\Sigma(K)$. There\nare also non-mutant knots with the same branched double covers, includ-\ning $T(3, 7)$ and $P(-2, 3, 7).$ In this example, the Khovanov homologies\nhave the same total rank, but Watson [Wat10] gives non-mutant knots\n\nwith homeomorphic branched double covers the total rank of whose Kho-\nvanov homologies differ, showing that the rank of Khovanov homology is\nnot an invariant of branched double covers. See also [Pao01].\n(6) Higher order cyclic branched covers are generally much better knot in-\nvariants. For example, work of Zimmerman [Zim98] shows that if $n \\geq 3$\nand $K \\subset S^{3}$ is hyperbolic, then the $n$-fold cyclic branched cover of $K$\ndistinguishes $K$ from other knots so long as $K$ does not admit peri-\nodic symmetries of order $n$, and the latter condition is satisfied whenever\n$n > 2g(K) + 1$. See [Pao05] for further discussion.\n(7) Extending work of Montesinos [Mon85], Piergallini constructed a com-\nplete set of moves relating links with the same simple, irregular 3-fold\nbranched covers [Pie91].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Pao05] Luisa Paoluzzi. Hyperbolic knots and cyclic branched covers. Publ. Mat., 49(2):257– 284, 2005. doi:10.5565/PUBLMAT\\\\_49205\\\\_01.\n- [MZ04] Mattia Mecchia and Bruno Zimmermann. The number of knots and links with the same 2-fold branched covering. Q. J. Math., 55(1):69–76, 2004. doi:10.1093/qjmath/55.1.69.\n- [Kaw06] Akio Kawauchi. Topological imitations and Reni-Mecchia-Zimmermann’s conjecture. Kyungpook Math. J., 46(1):1–9, 2006.\n- [Mec01] Mattia Mecchia. Hyperbolic 2-fold branched coverings. Rend. Istit. Mat. Univ. Trieste, 32:165–180, 2001. Dedicated to the memory of Marco Reni.\n- [HR85] Craig Hodgson and J. H. Rubinstein. Involutions and isotopies of lens spaces. In Knot theory and manifolds (Vancouver, B.C., 1983), volume 1144 of Lecture Notes in Math., pages 60–96. Springer, Berlin, 1985. doi:10.1007/BFb0075012.\n- [Wat10] Liam Watson. A remark on Khovanov homology and two-fold branched covers. Pacific J. Math., 245(2):373–380, 2010. doi:10.2140/pjm.2010.245.373.\n- [Pao01] Luisa Paoluzzi. On hyperbolic type involutions. Rend. Istit. Mat. Univ. Trieste, 32:221–256, 2001. Dedicated to the memory of Marco Reni.\n- [Zim98] Bruno Zimmermann. On hyperbolic knots with homeomorphic cyclic branched coverings. Math. Ann., 311(4):665–673, 1998. doi:10.1007/s002080050205.\n- [Mon85] José Marı́a Montesinos. Lectures on 3-fold simple coverings and 3-manifolds. In Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982), volume 44 of Contemp. Math., pages 157–177. Amer. Math. Soc., Providence, RI, 1985. doi:10.1090/conm/044/813111.\n- [Pie91] R. Piergallini. Covering moves. Trans. Amer. Math. Soc., 325(2):903–920, 1991. doi:10.2307/2001654.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2676, "problem_number": "KP-1.17", "title": "Kirby Problem 1.17", "statement": "Can an alternating link and a non-alternating link have home-\nomorphic branched double covers?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.17.\n\nLiterature notes:\n(1) The answer is conjectured to be no in [Gre13a, Conjecture 1.4] since\nGreene shows that two alternating links have homeomorphic branched\ndouble covers if and only if they are Conway mutants. Note that Viro\nproved that mutant links have homeomorphic branched double-covers\n[Vir76, Theorem 1], and Menasco proved that mutation preserves the\nproperty of being alternating [Men84, Proof of Theorem 3(b)].\n(2) The problem is related to Problems 1.22 and 3.25 of [Kir97] which ask\nfor a kind of classification of pairs of knots with homeomorphic branched\ndouble covers.\n(3) If a Seifert-fibered space is the branched double-cover of a pair of distinct\nlinks, then the pair consists of either a non-alternating torus link and\na non-alternating Montesinos link, or else two mutant Montesinos links\n[MR02, Introduction].\n(4) Dunfield investigated knots with at most 16 crossings whose branched\ndouble-covers are hyperbolic and of small enough volume to appear in the\nHodgson-Weeks census of closed hyperbolic 3-manifolds. He reports 3765\nnon-alternating knots with such branched covers but only 178 alternating\nknots, and no manifold occurs as the branched double-cover of both kinds\nof knots (private correspondence from 2013).\n(5) While there exist many constructions of pairs of non-mutant links with\nhomeomorphic branched double-covers, these constructions typically ap-\npear to produce non-alternating examples (cf. [MW86, MZ04]).\n\nReferences cited:\n- [Gre13a] Joshua Evan Greene. Lattices, graphs, and Conway mutation. Invent. Math., 192(3):717–750, 2013. doi:10.1007/s00222-012-0421-4.\n- [Vir76] O. Ja. Viro. Nonprojecting isotopies and knots with homeomorphic coverings. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 66:133–147, 207–208, 1976. Studies in topology, II.\n- [Men84] W. Menasco. Closed incompressible surfaces in alternating knot and link complements. Topology, 23(1):37–44, 1984. doi:10.1016/0040-9383(84)90023-5.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [MR02] Mattia Mecchia and Marco Reni. Hyperbolic 2-fold branched coverings of links and their quotients. Pacific J. Math., 202(2):429–447, 2002. doi:10.2140/pjm.2002.202.429.\n- [MW86] José M. Montesinos and Wilbur Whitten. Constructions of two-fold branched covering spaces. Pacific J. Math., 125(2):415–446, 1986. http://projecteuclid.org/euclid.pjm/1102700086.\n- [MZ04] Mattia Mecchia and Bruno Zimmermann. The number of knots and links with the same 2-fold branched covering. Q. J. Math., 55(1):69–76, 2004. doi:10.1093/qjmath/55.1.69.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2677, "problem_number": "KP-1.18", "title": "Kirby Problem 1.18", "statement": "(a) $($ Meridional Rank Conjecture $)$ Is the meridional $\\operatorname{rank} \\mu(L)$ of every link\n$L$ equal to its bridge number $b(L)$?\n(b) Given two knots $K_{1}, K_{2} \\subset S^{3}$ such that $\\pi_{1}(S^{3}\\setminus K_{1})$ and $\\pi_{1}(S^{3}\\setminus K_{2})$ have\nthe same finite quotients, does $\\mu(K_{1}) = \\mu(K_{2})$? In particular, is it possible\nto explicitly detect the meridional $\\operatorname{rank} \\mu(K)$ of a knot $K \\subset S^{3}$ from the\nfinite quotients of $\\pi_{1}(S^{3}\\setminus K)$?\n(c) Given two links $L_{1}, L_{2} \\subset S^{3}$, if there exists a surjective homomorphism\n$\\pi_{1}(S^{3}\\setminus L_{1}) \\twoheadrightarrow\\pi_{1}(S^{3}\\setminus L_{2})$ that sends the meridians of $L_{1}$ to the meridians\nof $L_{2}$, do the bridge numbers satisfy $b(L_{1}) \\geq b(L_{2})$?\n(d) Given two knots $K_{1}, K_{2} \\subset S^{3}$, if there exists a surjective homomorphism\n$\\pi_{1}(S^{3}\\setminus K_{1}) \\twoheadrightarrow\\pi_{1}(S^{3}\\setminus K_{2})$ (which may not send meridians to meridians):\n(i) Do the meridional ranks satisfy $\\mu(K_{1}) \\geq \\mu(K_{2})$?\n(ii) Do the bridge numbers satisfy $b(K_{1}) \\geq b(K_{2})$?\n(e) Let $K \\subset S^{3}$ be a hyperbolic knot. Is every subgroup of $\\pi_{1}(S^{3}\\setminus K)$ generated\nby at most $b(K) - 1$ meridians free? See [BJW18].", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.18.\n\nLiterature notes:\n(1) Problem 1.11 of [Kir97] poses Question (a) for knots, and was proposed\nby Cappell and Shaneson.\n(2) The meridional $\\operatorname{rank} \\mu(L)$ of a link $L$ is the minimal number of merid-\nians needed to generate the fundamental group of its complement. It is\nbounded above by the bridge number $b(L)$.\n(3) Question (b) has a negative answer for links: let $W_{n}$ denote the result\nof applying $|n| \\geq 2$ twists to the Whitehead link $W_{0}$. Then $W_{n}$ and $W_{0}$\nhave homeomorphic exteriors, but $\\mu(W_{0}) = b(W_{0}) = 2$, while $\\mu(W_{n}) =$\n$b(W_{n}) = 3$.\nNote that the homeomorphism between the exterior of $W_{n}$ and $W_{0}$ dis-\ncussed above does not send meridians to meridians. Under the meridian-\npreserving hypothesis, the epimorphism of groups in Question (c) implies\n$\\mu(L_{1}) \\geq \\mu(L_{2})$, so a negative answer to Question (c) implies a negative\nanswer to the Meridional Rank Conjecture.\n(4) In the case of knots, Question (d) represents a strengthening of Question\n(c).\n(5) Different answers to the two items under Question (d) would imply a nega-\ntive answer to the Meridional Rank Conjecture. Note that there are many\nepimorphisms between knot groups that are not meridian-preserving [GAn75].\nFor closed orientable manifolds, the answer to the analogous question with\nrespect to the Heegaard genus is negative if one does not assume the epi-\nmorphism to be induced by a non-zero degree map (see [BF18]). The\nnon-zero degree condition, in the knot complement case, implies that the\nepimorphism is meridian preserving.\n(6) In Question 4 the bridge number is defined using bridge trisections (see [MZ17a,\nJMMZ22] and [AAD $^{+}23$, Question 6.1]; cf. Problem 4.111) or, equiva-\nlently, using Morse position.\nBy [JP25], the meridional rank of 2-spheres in $S^{4}$ can collapse un-\nder connected sum; therefore, either additivity of bridge number or the\nequality of bridge number and meridional rank (or both) fails for 2-knots.\n(7) Problem 4.117 describes a four-dimensional analog of the Meridional Rank\nConjecture.\n\nReferences cited:\n- [BJW18] Michel Boileau, Yeonhee Jang, and Richard Weidmann. Meridional rank and bridge number for a class of links. Pacific J. Math., 292(1):61–80, 2018. doi:10.2140/pjm.2018.292.61.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [GAn75] F. González-Acuña. Homomorphs of knot groups. Ann. of Math. (2), 102(2):373– 377, 1975. doi:10.2307/1971036.\n- [BF18] Michel Boileau and Stefan Friedl. Epimorphisms of 3-manifold groups. Q. J. Math., 69(3):931–942, 2018. doi:10.1093/qmath/hay007.\n- [MZ17a] Jeffrey Meier and Alexander Zupan. Bridge trisections of knotted surfaces in $S^{4}$. Trans. Amer. Math. Soc., 369(10):7343–7386, 2017. doi:10.1090/tran/6934.\n- [JMMZ22] Jason Joseph, Jeffrey Meier, Maggie Miller, and Alexander Zupan. Bridge trisections and classical knotted surface theory. Pacific J. Math., 319(2):343–369, 2022. doi:10.2140/pjm.2022.319.343.\n- [AAD+23] Wolfgang Allred, Manuel Aragón, Zack Dooley, Alexander Goldman, Yucong Lei, Isaiah Martinez, Nicholas Meyer, Devon Peters, Scott Warrander, Ana Wright, and Alexander Zupan. Tri-plane diagrams for simple surfaces in $S^{4}$. J. Knot Theory Ramifications, 32(6):Paper No. 2350041, 28, 2023. doi:10.1142/S0218216523500414.\n- [JP25] Jason Joseph and Puttipong Pongtanapaisan. Meridional rank and bridge number of knotted 2-spheres. Canad. J. Math., 77(1):282–299, 2025. doi:10.4153/S0008414X23000883.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2678, "problem_number": "KP-1.19", "title": "Kirby Problem 1.19", "statement": "Let $Y = Y_{1}\\#Y_{2}$ be a connected sum of 3-manifolds with $Y_{i} \\neq$\n$S^{3}$, for $i = 1, 2$. Let $\\Phi: Y \\to Y$ be a Dehn twist around the connected sum separating\n2-sphere, and assume $Y_{1}$ and $Y_{2}$ are such that $\\Phi$ is not isotopic to the identity. Does\nthere exist a knot $K \\subseteq Y$ such that $K$ and $\\Phi(K)$ are not ambiently isotopic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.19.\n\nLiterature notes:\nAceto-Bregman-Davis-Park-Ray [ABD $^{+}20$] studied the analogous\nquestion for knots in prime 3-manifolds. They showed that the Gluck twist on $S^{1} \\times$\n$S^{2}$ (see Section 4.1 for the definition) can be detected by a knot that is equivalent\nbut not isotopic to its image under the Gluck twist. They also showed that for\nirreducible 3-manifolds, diffeomorphisms that preserve free homotopy classes of\nloops are isotopic to the identity. Hence the key remaining question is whether\nDehn twists around connected sum spheres can be detected by knots that are\nequivalent but not isotopic.\n\nReferences cited:\n- [ABD+20] Paolo Aceto, Corey Bregman, Christopher W. Davis, JungHwan Park, and Arunima Ray. Isotopy and equivalence of knots in 3-manifolds, 2020. arXiv:2007.05796.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2679, "problem_number": "KP-1.20", "title": "Kirby Problem 1.20", "statement": "(a) Are there any null-homologous Floer minimal knots with irreducible com-\nplements other than the Borromean knots $B_{g}, g \\geq 0$, in any 3-manifolds?\nIt is reasonable to make the following more concrete conjectures.\n(b) Conjecture (Ni): If $b_{1}(Y) < 2$, then the only null-homologous Floer mini-\nmal knot in $Y$ is the unknot.\n(c) Conjecture (Ni): If the Thurston norm of $Y$ is not identically zero, then $Y$\ndoes not contain any null-homologous Floer minimal knot with irreducible\ncomplement.\n(d) Conjecture (Hedden, Rasmussen): The only Floer minimal knots in lens\nspaces are simple knots.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.20.\n\nLiterature notes:\n(1) Let $Y$ be a closed, oriented, connected 3-manifold, and let $K \\subset Y$ be a\nrationally null-homologous knot. It is known that\n\n$$\n\\operatorname{rank}\\widehat{\\mathrm{HFK}}(Y,K) \\geq \\operatorname{rank}\\widehat{\\mathrm{HF}}(Y).\n$$\n\nWe say $K$ is Floer minimal if equality in the above inequality holds.\nClearly, unknots are Floer minimal. Moreover, the Borromean knot [OS04c,\nFigure 16]$B_{g} \\subset \\#^{2g}(S^{1}\\times S^{2})$ is Floer minimal. If the $p$-surgery on a knot\n$L \\subset S^{3}$ is an L-space and $p \\geq 2g(L)$, then the dual knot in the L-space is\nFloer minimal [Hed11, Ras07].\nIt is known that the only null-homologous Floer minimal knot in $Y$\nis the unknot, when $Y$ is an L-space [NW14], and when $Y$ fibers over $S^{1}$\nwith fiber of genus $> 1$ [Ni14].\n(2) When $Y$ is the lens space $L(p, q)$, there is a standard genus-1 Heegaard\ndiagram with $p$ intersection points. Any two base points $z, w$ will specify a\nFloer minimal knot. Such knots are called simple.\nHedden [Hed11] and\nRasmussen [Ras07] proved that Conjecture (d) would imply the Berge\nConjecture (Problem 1.9 and [Kir97, Problem 1.78]).\n\nReferences cited:\n- [OS04c] Peter Ozsváth and Zoltán Szabó. Holomorphic disks and knot invariants. Adv. Math., 186(1):58–116, 2004. doi:10.1016/j.aim.2003.05.001.\n- [Hed11] Matthew Hedden. On Floer homology and the Berge conjecture on knots admitting lens space surgeries. Trans. Amer. Math. Soc., 363(2):949–968, 2011. doi:10.1090/S0002-9947-2010-05117-7.\n- [Ras07] Jacob Rasmussen. Lens space surgeries and L-space homology spheres, 2007. arXiv: 0710.2531.\n- [NW14] Yi Ni and Zhongtao Wu. Heegaard Floer correction terms and rational genus bounds. Adv. Math., 267:360–380, 2014. doi:10.1016/j.aim.2014.09.006.\n- [Ni14] Yi Ni. Some applications of Gabai’s internal hierarchy. Adv. Math., 250:467–495, 2014. doi:10.1016/j.aim.2013.10.001.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2680, "problem_number": "KP-1.21", "title": "Kirby Problem 1.21", "statement": "(a) For a given positive integer $g$, are there only finitely many L-space knots\nof genus $g$?\nA related but more general question is:\n(b) Question (Hedden): For which knots $L \\subset S^{3}$ are there only finitely many\nknots $K \\subset S^{3}$ with\n$\\widehat{\\mathrm{HFK}}(S^{3},K) \\cong \\widehat{\\mathrm{HFK}}(S^{3},L)$?\n\n(c) In some special cases, one expects stronger results saying that a class of\nknots is characterized by its knot Floer homology. We state 3 conjectures,\nordered by their strength.\n(i) For a knot $K \\subset S^{3}$, if\n$\\widehat{\\mathrm{HFK}}(S^{3},K) \\cong \\widehat{\\mathrm{HFK}}(S^{3},T_{2g+1,2})$, then\n$K = T_{2g+1,2}$.\n(ii) For a knot $K \\subset S^{3}$, if\n$\\widehat{\\mathrm{HFK}}(S^{3},K) \\cong \\widehat{\\mathrm{HFK}}(S^{3},T_{p,q})$, then $K$ is an\niterated torus knot.\n(iii) Conjecture (Ni): For an L-space knot $K \\subset S^{3}$, if all roots of $\\Delta_{K}(t)$\nlie on the unit circle, then $K$ is an iterated torus knot.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.21.\n\nLiterature notes:\n(1) A knot $K \\subset S^{3}$ is called an L-space knot if there exists a positive integer\n$p$ such that $S^{3}_{p}(K)$, the $p$-surgery on $K$ is an L-space (see Problem 3.48).\n(2) The answer to Question (a) is “yes” when $g \\leq 2$ [Ghi08, FRW24].\nIf $K$ is an L-space knot, then $K$ is fibered and strongly quasipositive\n[Ni07, Hed10],\n\n$$\n\\widehat{\\mathrm{HFK}}(S^{3},K,g(K)-1) \\cong \\mathbb{Z}_{(-1)}, \\tag{2}\n$$\n\nwhere $\\mathbb{Z}_{(-1)}$ means a copy of $\\mathbb{Z}$ supported in the Maslov grading $-1$\n[OS05b]. (The condition (2) implies that the monodromy of the fibra-\ntion is freely isotopic to a diffeomorphism without fixed point [BHS25,\nNi23a, GS22a].) However, these properties are not enough to guaran-\ntee the finiteness. For each $g \\geq 2$, Misev [Mis21] constructed infinitely\nmany strongly quasipositive fibered knots with the same Seifert form as\nthe torus knot $T_{2g+1,2}$. By taking cables of these knots, the condition (2)\ncan also be satisfied.\n(3) The finiteness in Question (b) does not hold for all knots $L$: see [HW18,\nAcknowledgements], [MS15], and [Wan22, Theorem 1.3]. In particular,\nHedden and Watson showed that, given any non-trivial band sum of two\nunknots, altering the number of twists in the band gives rise to an infinite\nfamily of distinct knots with the same knot Floer homology [HW18,\nTheorem 1].\n(4) Conjecture (i) in part (c) is known for $g = 1, 2$ [Ghi08, FRW24]. Con-\njecture (iii) would yield a positive answer to Problem 1.22; see [BBG19a].\n\nReferences cited:\n- [Ghi08] Paolo Ghiggini. Knot Floer homology detects genus-one fibred knots. Amer. J. Math., 130(5):1151–1169, 2008. doi:10.1353/ajm.0.0016.\n- [FRW24] Ethan Farber, Braeden Reinoso, and Luya Wang. Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil, 2024. doi:10.2140/gt.2024.28.4337.\n- [Ni07] Yi Ni. Knot Floer homology detects fibred knots. Invent. Math., 170(3):577–608, 2007. doi:10.1007/s00222-007-0075-9.\n- [Hed10] Matthew Hedden. Notions of positivity and the Ozsváth-Szabó concordance invariant. J. Knot Theory Ramifications, 19(5):617–629, 2010. doi:10.1142/S0218216510008017.\n- [OS05b] Peter Ozsváth and Zoltán Szabó. On knot Floer homology and lens space surgeries. Topology, 44(6):1281–1300, 2005. doi:10.1016/j.top.2005.05.001.\n- [BHS25] John A. Baldwin, Ying Hu, and Steven Sivek. Khovanov homology and the cinquefoil. J. Eur. Math. Soc. (JEMS), 27(6):2443–2465, 2025. doi:10.4171/jems/1415.\n- [Ni23a] Yi Ni. A note on knot Floer homology and fixed points of monodromy. Peking Math. J., 6(2):635–643, 2023. doi:10.1007/s42543-022-00051-3.\n- [GS22a] Paolo Ghiggini and Gilberto Spano. Knot Floer homology of fibred knots and Floer homology of surface diffeomorphisms, 2022. arXiv:2201.12411.\n- [Mis21] Filip Misev. On families of fibred knots with equal Seifert forms. Comm. Anal. Geom., 29(2):465–482, 2021. doi:10.4310/CAG.2021.v29.n2.a6.\n- [HW18] Matthew Hedden and Liam Watson. On the geography and botany of knot Floer homology. Selecta Math. (N.S.), 24(2):997–1037, 2018. doi:10.1007/s00029-017-0351-5.\n- [MS15] Allison H. Moore and Laura Starkston. Genus-two mutant knots with the same dimension in knot Floer and Khovanov homologies. Algebr. Geom. Topol., 15(1):43– 63, 2015. doi:10.2140/agt.2015.15.43.\n- [Wan22] Joshua Wang. The cosmetic crossing conjecture for split links. Geom. Topol., 26(7):2941–3053, 2022. doi:10.2140/gt.2022.26.2941.\n- [BBG19a] Michel Boileau, Steven Boyer, and Cameron McA. Gordon. Branched covers of quasi-positive links and L-spaces. J. Topol., 12(2):536–576, 2019. doi:10.1112/topo.12092.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2681, "problem_number": "KP-1.22", "title": "Kirby Problem 1.22", "statement": "If $K$ is a hyperbolic $L$-space knot, show that its branched cover\n$\\Sigma_{2}(K)$ is not an $L$-space.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.22.\n\nLiterature notes:\nThis is a conjecture of A. Moore. The corresponding statement\nfor higher order covers $\\Sigma_{n}(K)$ was shown by Boileau-Boyer-Gordon [BBG19a] for\n$n \\geq 4$ and by [FRW24] for $n = 3$. Boyer-Gordon-Hu [BGH25, Corollary 7.3] have\nannounced that $\\Sigma_{n}(K)$ has a left-orderable group for each $n \\geq 2$, so in particular, a\npositive solution would be in accordance with the $L$-space conjecture, Problem 3.48.\n\nReferences cited:\n- [BBG19a] Michel Boileau, Steven Boyer, and Cameron McA. Gordon. Branched covers of quasi-positive links and L-spaces. J. Topol., 12(2):536–576, 2019. doi:10.1112/topo.12092.\n- [FRW24] Ethan Farber, Braeden Reinoso, and Luya Wang. Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil, 2024. doi:10.2140/gt.2024.28.4337.\n- [BGH25] Steven Boyer, Cameron McA. Gordon, and Ying Hu. Cyclic branched covers of Seifert links and properties related to the ADE link conjecture. J. Lond. Math. Soc. (2), 111(6):Paper No. e70178, 53, 2025. doi:10.1112/jlms.70178.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2682, "problem_number": "KP-1.23", "title": "Kirby Problem 1.23", "statement": "Let $K$ be a cubic graph embedded in the plane, and let $\\mathrm{Tait}(K)$\nbe the number of Tait colorings of $K$.\n(a) Is $\\dim J^{7}(K) = \\mathrm{Tait}(K)$?\n(b) Is $J^{5}(K)$ nonzero when $K$ is bridgeless?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.23.\n\nLiterature notes:\n(1) Part (a) is Conjecture 1.2 in [KM19a].\n(2) A Tait coloring of a cubic (also called trivalent) graph is a labeling of the\nedges of the graph by three colors such that edges of all of the colors are\nincident at each vertex. In [KM19a], Kronheimer–Mrowka used gauge\ntheory to assign to a web $K \\subset \\mathbb{R}^{3}$ a finite-dimensional vector space $J^{7}(K)$\nover $\\mathbb{Z}/2\\mathbb{Z}$, and proved that $J^{7}(K)$ is zero if and only if $K$ has an embedded\nbridge.\nThey also proposed a combinatorial counterpart $J^{5}(K),$ which\nKhovanov–Robert [KR21] showed is well-defined for planar webs $K \\subset \\mathbb{R}^{2}$.\nWork of Kronheimer–Mrowka [KM19b] and Khovanov–Robert [KR21]\nimplies that\n\n$$\n\\dim J^{5}(K) \\leq \\mathrm{Tait}(K) \\leq \\dim J^{7}(K)\n$$\n\nfor any planar web $K$.\nThe Four Color Theorem is equivalent to the\nstatement that every bridgeless cubic planar graph has a Tait coloring,\nand a new proof would therefore follow from an affirmative answer to\neither of the questions in this problem.\n(3) Boozer [Boo23] gave numerical evidence suggesting that the inequality\n$\\dim J^{5}(K) \\leq \\mathrm{Tait}(K)$ may sometimes be strict.\nIn [Boo25], he also\nannounced that $J^{5}$ and $J^{7}$ do not coincide as functors.\n\nReferences cited:\n- [KM19a] P. B. Kronheimer and T. S. Mrowka. Tait colorings, and an instanton homology for webs and foams. J. Eur. Math. Soc. (JEMS), 21(1):55–119, 2019. doi:10.4171/JEMS/831.\n- [KR21] Mikhail Khovanov and Louis-Hadrien Robert. Foam evaluation and KronheimerMrowka theories. Adv. Math., 376:Paper No. 107433, 59, 2021. doi:10.1016/j.aim.2020.107433.\n- [KM19b] Peter B. Kronheimer and Tomasz S. Mrowka. A deformation of instanton homology for webs. Geom. Topol., 23(3):1491–1547, 2019. doi:10.2140/gt.2019.23.1491.\n- [Boo23] David Boozer. Computer bounds for Kronheimer-Mrowka foam evaluation. Exp. Math., 32(4):615–630, 2023. doi:10.1080/10586458.2021.1982078.\n- [Boo25] David Boozer. The combinatorial and gauge-theoretic foam evaluation functors are not the same. Math. Ann., 392(1):47–56, 2025. doi:10.1007/s00208-024-03064-8.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2683, "problem_number": "KP-1.24", "title": "Kirby Problem 1.24", "statement": "(Jones Unknot Detection).\n(a) Is there a nontrivial knot with the same Jones polynomial as the unknot?\n\n(b) Does there exist a nontrivial knot whose colored Jones polynomials are all\ntrivial?\n(c) Is there a nontrivial knot with trivial HOMFLYPT polynomial? (A knot\nwith trivial HOMFLYPT polynomial would have trivial Jones polynomial.)", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.24.\n\nLiterature notes:\n(1) Question (a) was first proposed by V. Jones; see [Kir97, Problems 1.88(a)\nand (c)] and [AALM00].\n(2) Computer calculations by Sikora and Tuzun [TS21] have shown there is\nno nontrivial knot with up to 24 crossings with trivial Jones polynomial.\n(3) As observed in [MM01], a positive answer to Question (b) would follow\nfrom the Volume Conjecture stated in Problem 1.27.\n(4) By tabulating links, Thistlethwaite found links with two or more com-\nponents with trivial Jones polynomials [Thi01]. This was extended in\n[EKT03] to infinite families of links with trivial Jones polynomials. But\nthese examples do not have trivial HOMFLYPT polynomials.\n(5) Kronheimer and Mrowka have proven that a knot has trivial Khovanov\nhomology if and only if it is trivial [KM11].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [AALM00] Vladimir Igorevič Arnol’d, Michael Atiyah, Peter Lax, and Barry Mazur. Mathematics: Frontiers and perspectives. American Mathematical Society Providence, RI, 2000.\n- [TS21] Robert E. Tuzun and Adam S. Sikora. Verification of the Jones unknot conjecture up to 24 crossings. J. Knot Theory Ramifications, 30(3):Paper No. 2150020, 6, 2021. doi:10.1142/S0218216521500206.\n- [MM01] Hitoshi Murakami and Jun Murakami. The colored Jones polynomials and the simplicial volume of a knot. Acta Math., 186(1):85–104, 2001. doi:10.1007/BF02392716.\n- [Thi01] Morwen Thistlethwaite. Links with trivial Jones polynomial. J. Knot Theory Ramifications, 10(4):641–643, 2001. doi:10.1142/S0218216501001050.\n- [EKT03] Shalom Eliahou, Louis H. Kauffman, and Morwen B. Thistlethwaite. Infinite families of links with trivial Jones polynomial. Topology, 42(1):155–169, 2003. doi:10.1016/S0040-9383(02)00012-5.\n- [KM11] P. B. Kronheimer and T. S. Mrowka. Khovanov homology is an unknotdetector. Publ. Math. Inst. Hautes Études Sci., 113:97–208, 2011. doi:10.1007/s10240-010-0030-y.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2684, "problem_number": "KP-1.25", "title": "Kirby Problem 1.25", "statement": "(a) Conjecture: The noncommutative $A$-ideal of a knot $K$ is exactly the an-\nnihilator of the colored Jones polynomial $($ the infinite dimensional vector\nwhose entries are the colored Jones polynomials of $K)$.\n(b) $($ AJ Conjecture $)$ Let $A(K; L, M)$ be the $A$-polynomial of the knot $K$, and\n$\\alpha(K, q; M, L)$ a polynomial of minimal degree in $L$ and co-prime coeffi-\ncients that annihilates the colored Jones polynomial $J_{n}(K; q)$. Then\n\n$$\n\\alpha(K, -1; M, L) = A(K; L, M)\n$$\n\nup to a factor depending on $M.$", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.25.\n\nLiterature notes:\n(1) The version of Conjecture (a) for $\\mathfrak{s}\\mathfrak{l}_{2}$ appears in [FGL02]; the full version\nis stated in [Sik08]. The AJ conjecture (b) is due to Garoufalidis [Gar04].\n(2) The noncommutative $A$-ideal of $S^{3} - K$ is the left ideal in the Kauffman\nbracket skein algebra of the torus that annihilates the empty skein in\nthe knot complement. [FGL02] shows that a series built from the colored\nJones polynomial $J_{n}(K; q)$ annihilates the noncommutative $A$-ideal, which\nleads to recursive relationships satisfied by $J_{n}(K; q)$.\nThe noncommutative $A$-ideal is computed for $3_{1}$ [Gel02], $(2, 2p + 1)$\ntorus knots [GS03], and $4_{1}$ [GS04]. Demonstrating the nontriviality of\nthe noncommutative $A$-ideal remains a significant unresolved problem.\n(3) The $A$-polynomial of a knot $K$ [CL98] is the defining polynomial for\nthe character variety formed by $\\operatorname{SL}_{2}(\\mathbb{C})$-representations of $\\pi_{1}(\\partial(S^{3} - K))$\nthat extend to $\\pi_{1}(S^{3} - K)$.\nSee [GL05] for details on the definition\nof $\\alpha(K, q; M, L)$, which comes from linear recursion relations satisfied\nby $J_{n}(K; q)$.\nThe AJ Conjecture is proved for certain 2-bridge knots\n\n[Lê06, LZ17] and pretzel knots [LT15], and a diagrammatic approach is\nsuggested in [DG20]. See also [Guk05] for the physics context of the AJ\nconjecture.\n(4) The two Conjectures are closely intertwined, and the AJ Conjecture is an\nimportant step towards understanding the relationship between classical\nand quantum invariants.\n\nReferences cited:\n- [FGL02] Charles Frohman, Răzvan Gelca, and Walter Lofaro. The A-polynomial from the noncommutative viewpoint. Trans. Amer. Math. Soc., 354(2):735–747, 2002. doi: 10.1090/S0002-9947-01-02889-6.\n- [Sik08] Adam S. Sikora. Quantizations of character varieties and quantum knot invariants, 2008. arXiv:0807.0943.\n- [Gar04] Stavros Garoufalidis. On the characteristic and deformation varieties of a knot. In Proceedings of the Casson Fest, volume 7 of Geom. Topol. Monogr., pages 291– 309. Geom. Topol. Publ., Coventry, 2004. doi:10.2140/gtm.2004.7.291.\n- [Gel02] Răzvan Gelca. Non-commutative trigonometry and the A-polynomial of the trefoil knot. Math. Proc. Cambridge Philos. Soc., 133(2):311–323, 2002. doi:10.1017/S0305004102006047.\n- [GS03] Răzvan Gelca and Jeremy Sain. The noncommutative A-ideal of a $(2,2p+1)$-torus knot determines its Jones polynomial. J. Knot Theory Ramifications, 12(2):187– 201, 2003. doi:10.1142/S021821650300238X.\n- [GS04] Răzvan Gelca and Jeremy Sain. The computation of the non-commutative generalization of the A-polynomial of the figure-eight knot. J. Knot Theory Ramifications, 13(6):785–808, 2004. doi:10.1142/S0218216504003482.\n- [CL98] D. Cooper and D. D. Long. Representation theory and the A-polynomial of a knot. Chaos Solitons Fractals, 9(4-5):749–763, 1998. Knot theory and its applications. doi:10.1016/S0960-0779(97)00102-1.\n- [GL05] Stavros Garoufalidis and Thang T. Q. Lê. The colored Jones function is qholonomic. Geom. Topol., 9:1253–1293, 2005. doi:10.2140/gt.2005.9.1253.\n- [LZ17] Thang T. Q. Lê and Xingru Zhang. Character varieties, A-polynomials and the AJ conjecture. Algebr. Geom. Topol., 17(1):157–188, 2017. doi:10.2140/agt.2017.17.157.\n- [LT15] Thang T. Q. Le and Anh T. Tran. On the AJ conjecture for knots. Indiana Univ. Math. J., 64(4):1103–1151, 2015. With an appendix written jointly with Vu Q. Huynh. doi:10.1512/iumj.2015.64.5602.\n- [DG20] Renaud Detcherry and Stavros Garoufalidis. A diagrammatic approach to the AJ conjecture. Math. Ann., 378(1-2):447–484, 2020. doi:10.1007/s00208-020-02028-y.\n- [Guk05] Sergei Gukov. Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial. Comm. Math. Phys., 255(3):577–627, 2005. doi:10.1007/s00220-005-1312-y.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2685, "problem_number": "KP-1.26", "title": "Kirby Problem 1.26", "statement": "(Jones Slope Conjecture). For a knot $K$, the Jones slopes $js(K)$\nare the set of cluster points in\n\n$$\n\\left\\{\\frac{4}{n^{2}}\\deg_{+}\\bigl(J_{n}(K;q)\\bigr)\\right\\}_{n\\in\\mathbb{N}}\n\\cup\n\\left\\{\\frac{4}{n^{2}}\\deg_{-}\\bigl(J_{n}(K;q)\\bigr)\\right\\}_{n\\in\\mathbb{N}},\n$$\n\nwhere deg $^{+}$ denotes the highest degree and deg $^{-}$ the lowest degree.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.26.\n\nLiterature notes:\n(1) The Jones slopes of a knot $K$ are a subset of the boundary slopes of\nincompressible surfaces, boundary-incompressible orientable surfaces in\n$S^{3} - K$.\n(2) This conjecture was originally stated by Garoufalidis [Gar11]. There is a\nstronger version of the conjecture due to Kalfagianni and Tran [KT15].\nGaroufalidis proved his conjecture for up to 10 crossings [Gar11], and the\nSlope Conjecture or its stronger form has been proved for various fami-\nlies of knots [FKP11, GvdV16, LvdV16, Lee22, BMT21, BMT20,\nMT17, LYL19, GLVDV20].\n\nReferences cited:\n- [Gar11] Stavros Garoufalidis. The Jones slopes of a knot. Quantum Topol., 2(1):43–69, 2011. doi:10.4171/QT/13.\n- [KT15] Efstratia Kalfagianni and Anh T. Tran. Knot cabling and the degree of the colored Jones polynomial. New York J. Math., 21:905–941, 2015. http://nyjm.albany.edu: 8000/j/2015/21 905.html.\n- [FKP11] David Futer, Efstratia Kalfagianni, and Jessica S. Purcell. Slopes and colored Jones polynomials of adequate knots. Proc. Amer. Math. Soc., 139(5):1889–1896, 2011. doi:10.1090/S0002-9939-2010-10617-2.\n- [GvdV16] Stavros Garoufalidis and Roland van der Veen. Quadratic integer programming and the slope conjecture. New York J. Math., 22:907–932, 2016. http://nyjm.albany.edu: 8000/j/2016/22 907.html.\n- [LvdV16] Christine Ruey Shan Lee and Roland van der Veen. Slopes for pretzel knots. New York J. Math., 22:1339–1364, 2016. http://nyjm.albany.edu:8000/j/2016/22 1339. html.\n- [Lee22] Christine Ruey Shan Lee. Jones slopes and coarse volume of near-alternating knots. Comm. Anal. Geom., 30(4):891–948, 2022.\n- [BMT21] Kenneth L. Baker, Kimihiko Motegi, and Toshie Takata. The strong slope conjecture for cablings and connected sums. New York J. Math., 27:676–704, 2021. https://nyjm.albany.edu/j/2021/27-26v.pdf.\n- [BMT20] Kenneth L. Baker, Kimihiko Motegi, and Toshie Takata. The strong slope conjecture for twisted generalized Whitehead doubles. Quantum Topol., 11(3):545–608, 2020. doi:10.4171/qt/242.\n- [MT17] Kimihiko Motegi and Toshie Takata. The slope conjecture for graph knots. Math. Proc. Cambridge Philos. Soc., 162(3):383–392, 2017. doi:10.1017/S0305004116000566.\n- [LYL19] Xudong Leng, Zhiqing Yang, and Ximin Liu. The slope conjectures for 3-string Montesinos knots. New York J. Math., 25:45–70, 2019. https://nyjm.albany.edu/j/2019/25-2p.pdf.\n- [GLVDV20] Stavros Garoufalidis, Christine Ruey Shan Lee, and Roland Van Der Veen. The slope conjecture for Montesinos knots. Internat. J. Math., 31(7):2050056, 66, 2020. doi:10.1142/S0129167X20500561.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2686, "problem_number": "KP-1.27", "title": "Kirby Problem 1.27", "statement": "(Kashaev–Murakami–Murakami Volume Conjecture). For a\nlink $L \\subset S^{3}$,\n\n$$\n\\frac{1}{2\\pi}\\operatorname{Vol}_{\\mathrm{hyp}}(S^{3}-L)\n=\\lim_{n\\to\\infty}\\frac{1}{n}\\log\\left|J_{n}\\bigl(L,e^{2\\pi i/n}\\bigr)\\right|.\n$$", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.27.\n\nLiterature notes:\nHere, $J_{n}(L, q)$ is the $n^{th}$ colored Jones polynomial of $L$ in the vari-\nable $q$, and $\\operatorname{Vol}_{hyp}(S^{3} - L)$ is the hyperbolic volume of the link complement. This\nproblem was originally proposed for a complex-valued invariant for triangulated\nlinks in $S^{3}$ [Kas97], which was later shown to be determined by the colored Jones\npolynomial of the link complement [MM01]. Note that this conjecture would imply\nthat the collection of colored Jones polynomials would characterize the unknot; see\nProblem 1.24(b). The conjecture has been verified for some knots, including knots\nwith up to seven crossings [Oht16, OY18, Oht17], torus knots [KT00], White-\nhead doubles of $(2, b)$ torus knots or links [Zhe07], Whitehead chains, [vdV08],\nand certain cables of $4_{1}$ [LT10, MT25].\nNote that there is also a stronger complexified version involving the Chern–\nSimons invariant. See [Mur07, MMO $^{+}02$].\n\nReferences cited:\n- [Kas97] Rinat M Kashaev. The hyperbolic volume of knots from the quantum dilogarithm. Letters in mathematical physics, 39(3):269–275, 1997.\n- [MM01] Hitoshi Murakami and Jun Murakami. The colored Jones polynomials and the simplicial volume of a knot. Acta Math., 186(1):85–104, 2001. doi:10.1007/BF02392716.\n- [Oht16] Tomotada Ohtsuki. On the asymptotic expansion of the Kashaev invariant of the 52 knot. Quantum Topol., 7(4):669–735, 2016. doi:10.4171/QT/83.\n- [OY18] Tomotada Ohtsuki and Yoshiyuki Yokota. On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. Math. Proc. Cambridge Philos. Soc., 165(2):287–339, 2018. doi:10.1017/S0305004117000494.\n- [Oht17] Tomotada Ohtsuki. On the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings. Internat. J. Math., 28(13):1750096, 143, 2017. doi:10.1142/S0129167X17500963.\n- [KT00] R. M. Kashaev and O. Tirkkonen. A proof of the volume conjecture on torus knots. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 269:262–268, 370, 2000. doi:10.1023/A:1022608131142.\n- [Zhe07] Hao Zheng. Proof of the volume conjecture for Whitehead doubles of a family of torus knots. Chinese Ann. Math. Ser. B, 28(4):375–388, 2007. doi:10.1007/s11401-006-0373-3.\n- [vdV08] Roland van der Veen. Proof of the volume conjecture for Whitehead chains. Acta Math. Vietnam., 33(3):421–431, 2008.\n- [LT10] Thang T. Q. Le and Anh T. Tran. On the volume conjecture for cables of knots. J. Knot Theory Ramifications, 19(12):1673–1691, 2010. doi:10.1142/S0218216510008534.\n- [MT25] Hitoshi Murakami and Anh T. Tran. The colored Jones polynomial of a cable of the figure-eight knot. J. Knot Theory Ramifications, 34(3):Paper No. 2340019, 24, 2025. doi:10.1142/S0218216523400199.\n- [Mur07] Hitoshi Murakami. Various generalizations of the volume conjecture. In The interaction of analysis and geometry, volume 424 of Contemp. Math., pages 165–186. Amer. Math. Soc., Providence, RI, 2007. doi:10.1090/conm/424/08100.\n- [MMO+02] Hitoshi Murakami, Jun Murakami, Miyuki Okamoto, Toshie Takata, and Yoshiyuki Yokota. Kashaev’s conjecture and the Chern-Simons invariants of knots and links. Experiment. Math., 11(3):427–435, 2002. http://projecteuclid.org/euclid.em/1057777432.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2687, "problem_number": "KP-1.28", "title": "Kirby Problem 1.28", "statement": "Let $L$ be a link in the thickened annulus $S^{1} \\times I \\times I$.\n(a) Wrapping conjecture: $w(L)$ is equal to the maximal nonzero annular de-\ngree of the annular Kauffman bracket of $L$.\n(b) Categorifed wrapping conjecture: $w(L)$ is equal to the maximal nonzero\nannular grading of the annular Khovanov homology of $L$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.28.\n\nLiterature notes:\n(1) The wrapping number $w(L)$ of $L$ is the minimum number of times $L$\nintersects a meridional disk.\n(2) The wrapping number $w(L)$ is less than or equal to the maximal nonzero\nannular degree of the annular Kauffman bracket of $L$, as well as the maxi-\nmal nonzero annular grading of the annular Khovanov homology of $L$. The\nproblem is asking whether these upper bounds are necessarily achieved;\nthat is, whether either of these invariants detects the wrapping number.\n(3) The wrapping conjecture is due to Hoste–Przytycki [HP95a], who proved\nit for any annular link with a $\\pm$-adequately wrapped diagram.\n(4) The categorified wrapping conjecture is due to Grigsby, who proved with\nNi that it holds for string links [GN14]. This conjecture would follow\nfrom the wrapping conjecture since the graded Euler characteristic of an-\nnular Khovanov homology is equal to the annular Kauffman bracket. It is\nknown via spectral sequences relating annular Khovanov homology with\nFloer-theoretic invariants [GW10, Xie21, XZ25] that annular Khovanov\nhomology is nontrivial in the annular grading given by the generalized\nThurston norm of the meridional disk. The difficulty is that the general-\nized meridional Thurston norm can be arbitrarily smaller than the wrap-\nping number; see Martin’s work for this and related progress [Mar23].\n\nReferences cited:\n- [HP95a] Jim Hoste and Józef H. Przytycki. The $(2,8)$-skein module of Whitehead manifolds. J. Knot Theory Ramifications, 4(3):411–427, 1995. doi:10.1142/S021821659500020X.\n- [GN14] J. Elisenda Grigsby and Yi Ni. Sutured Khovanov homology distinguishes braids from other tangles. Math. Res. Lett., 21(6):1263–1275, 2014. doi:10.4310/MRL.2014.v21.n6.a4.\n- [GW10] J. Elisenda Grigsby and Stephan M. Wehrli. Khovanov homology, sutured Floer homology and annular links. Algebr. Geom. Topol., 10(4):2009–2039, 2010. doi: 10.2140/agt.2010.10.2009.\n- [Xie21] Yi Xie. Instantons and annular Khovanov homology. Adv. Math., 388:Paper No. 107864, 51, 2021. doi:10.1016/j.aim.2021.107864.\n- [XZ25] Yi Xie and Boyu Zhang. Instanton Floer homology for sutured manifolds with tangles. J. Differential Geom., 130(3):701–769, 2025. doi:10.4310/jdg/1749496718.\n- [Mar23] Gage Martin. Annular Khovanov homology and meridional disks. J. Knot Theory Ramifications, 32(2):Paper No. 2250088, 14, 2023. doi:10.1142/S0218216522500882.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2688, "problem_number": "KP-1.29", "title": "Kirby Problem 1.29", "statement": "Is the first inequality below true? If so, is the second?\n(a) $(\\operatorname{Vol}$-Det Conjecture $)$ For any alternating hyperbolic knot $K \\subset S^{3}$,\nvol $(K) < 2\\pi \\log \\det(K),$\nwhere vol is the hyperbolic volume of $K$ and det is the determinant.\n(b) For any hyperbolic knot $K \\subset S^{3}$,\nvol $(K) < 2\\pi \\log \\operatorname{rank} \\mathrm{Kh}(K),$\nwhere Kh is the reduced Khovanov homology of $K$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.29.\n\nLiterature notes:\n(1) The proposed inequalities relate quantum topology and hyperbolic geom-\netry in the style of the Volume Conjecture [Kas97, MM01] (see Problem\n1.27), but may be more approachable.\n(2) A correlation between determinant and volume for alternating hyperbolic\nknots was first noticed by Dunfield, and Champanerkar, Kofman and Pur-\ncell conjectured in [CKP16] that (1) holds. They moreover showed that\nthe bound is sharp, and conjectured the second more general inequal-\nity (b).\n(3) Burton verified (a) for several families of knots including 2-bridge knots,\nclosed alternating 3-braids, and an infinite family of alternating 4-braids\n[Bur18].\n(4) Champanerkar, Kofman and Lalin proved (a) for other infinite families of\nalternating knots using number theoretic methods in [CKL19], and gave\nrelated conjectures involving other volume-type quantities and the Mahler\nmeasures of certain graph-theoretic polynomials. See also [CK25].\n\nReferences cited:\n- [Kas97] Rinat M Kashaev. The hyperbolic volume of knots from the quantum dilogarithm. Letters in mathematical physics, 39(3):269–275, 1997.\n- [MM01] Hitoshi Murakami and Jun Murakami. The colored Jones polynomials and the simplicial volume of a knot. Acta Math., 186(1):85–104, 2001. doi:10.1007/BF02392716.\n- [CKP16] Abhijit Champanerkar, Ilya Kofman, and Jessica S. Purcell. Geometrically and diagrammatically maximal knots. J. Lond. Math. Soc. (2), 94(3):883–908, 2016. doi:10.1112/jlms/jdw062.\n- [Bur18] Stephan D. Burton. The determinant and volume of 2-bridge links and alternating 3-braids. New York J. Math., 24:293–316, 2018. http://nyjm.albany.edu:8000/j/2018/24 293.html.\n- [CKL19] Abhijit Champanerkar, Ilya Kofman, and Matilde Lalı́n. Mahler measure and the vol-det conjecture. J. Lond. Math. Soc. (2), 99(3):872–900, 2019. doi:10.1112/jlms.12200.\n- [CK25] Abhijit Champanerkar and Ilya Kofman. Geometric bounds for spanning tree entropy of planar lattice graphs, 2025. arXiv:2505.05688.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2689, "problem_number": "KP-1.30", "title": "Kirby Problem 1.30", "statement": "Does the Khovanov homology of every nontrivial knot contain\n2-torsion?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.30.\n\nLiterature notes:\nThis is a conjecture due to Shumakovitch [Shu14] originally posted\nin 2004. It has been verified for some infinite families of knots [Shu14, AP04,\n\nPS14, DM25a, DM25b]. Moreover, Gujral–Wang have shown [GW25] that if\n$K$ is a knot with no 2-torsion in its Khovanov homology, and $J$ is obtained from\n$K$ by a proper rational tangle replacement, then\nrankKh $(K) \\leq$ rankKh $(J),$\nwhere Kh refers to reduced Khovanov homology. It follows in particular that Shu-\nmakovitch’s conjecture holds for knots with unknotting number 1.\nIf proven, the conjecture would offer an alternative proof that Khovanov ho-\nmology detects the unknot.\n\nReferences cited:\n- [Shu14] Alexander N. Shumakovitch. Torsion of Khovanov homology. Fund. Math., 225(1):343–364, 2014. doi:10.4064/fm225-1-16.\n- [AP04] Marta M. Asaeda and Józef H. Przytycki. Khovanov homology: torsion and thickness. In Advances in topological quantum field theory, volume 179 of NATO Sci. Ser. II Math. Phys. Chem., pages 135–166. Kluwer Acad. Publ., Dordrecht, 2004. doi:10.1007/978-1-4020-2772-7\\\\_6.\n- [PS14] Józef H. Przytycki and Radmila Sazdanović. Torsion in Khovanov homology of semi-adequate links. Fund. Math., 225(1):277–304, 2014. doi:10.4064/fm225-1-13.\n- [DM25a] Raquel Dı́az and Pedro M. G. Manchón. A pattern for torsion in Khovanov homology. Fund. Math., 270(1):75–97, 2025. doi:10.4064/fm240810-12-3.\n- [DM25b] Raquel Dı́az and Pedro M. G. Manchón. New torsion patterns in Khovanov homology, 2025. arXiv:2508.00606.\n- [GW25] Onkar Singh Gujral and Joshua Wang. A minimality property for knots without Khovanov 2-torsion. Algebr. Geom. Topol., 25(7):4073–4075, 2025. doi:10.2140/agt.2025.25.4073.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2690, "problem_number": "KP-1.31", "title": "Kirby Problem 1.31", "statement": "(a) Compute the Khovanov homology for all torus knots $T(m, n)$.\n(b) Compute the Khovanov–Rozansky $\\mathfrak{g}\\mathfrak{l}(N)$ homology for all torus knots.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.31.\n\nLiterature notes:\n(1) The data for the Khovanov homology of $T(m, n)$ is available at [LM23b]\nfor small $m, n$. Khovanov–Rozansky homology data is available at [Lew23a,\nLew23b].\n(2) The homology is known to have a lot of torsion [MPS $^{+}18$]. Comput-\ning Khovanov homology with $\\mathbb{Z}_{2}$ coefficients and $\\mathfrak{g}\\mathfrak{l}(N)$ homology with\n$\\mathbb{Z}_{N}$ coefficients might be easier than rational coefficients [GOR13]. On\nthe other hand, an open conjecture [PS14, MPS $^{+}18$] states that the\nKhovanov homology of $T(m, n)$ for $m$ prime and $n > m$ has $m$-torsion.\n(3) The triply graded Khovanov–Rozansky homology of torus knots (with in-\nteger coefficients) is computed in [Mel22]. By [Ras15] there is a spectral\nsequence from triply graded homology to Khovanov (resp.\nKhovanov–\nRozansky) homology, but the differentials are not known explicitly.\n(4) By [Sto09], there is a well-defined limit of the Khovanov homology of\n$T(m, n)$ as $m \\to \\infty$, also known as $T(n, \\infty)$. Rozansky [Roz14] shows\nthat the Khovanov homology of $T(n, \\infty)$ provides a categorified equivalent\nof the Jones–Wenzl projector [Wen87]. The triply graded homology of\n$T(n, \\infty)$ is computed in [Hog18], and precise conjectures for the Khovanov\nhomology of $T(n, \\infty)$ are discussed in [GOR13, GL15].\n(5) In contrast, the Heegaard Floer homology of $T(m, n)$ is well known [OS05b].\n\nReferences cited:\n- [LM23b] Charles Livingston and Allison H. Moore. Knotinfo: Table of knot invariants, November 2023. https://knotinfo.org.\n- [Lew23a] Lukas Lewark. Foamho, an $\\mathfrak{sl}(3)$-homology calculator. URL: https://people.math.ethz.ch/ llewark/foamho.php, November 2023.\n- [Lew23b] Lukas Lewark. Khoca, a knot homology calculator. URL: https://people.math.ethz.ch/ llewark/khoca.php, November 2023.\n- [MPS+18] Sujoy Mukherjee, Józef H. Przytycki, Marithania Silvero, Xiao Wang, and Seung Yeop Yang. Search for torsion in Khovanov homology. Exp. Math., 27(4):488– 497, 2018. doi:10.1080/10586458.2017.1320242.\n- [GOR13] Eugene Gorsky, Alexei Oblomkov, and Jacob Rasmussen. On stable Khovanov homology of torus knots. Exp. Math., 22(3):265–281, 2013. doi:10.1080/10586458.2013.798553.\n- [PS14] Józef H. Przytycki and Radmila Sazdanović. Torsion in Khovanov homology of semi-adequate links. Fund. Math., 225(1):277–304, 2014. doi:10.4064/fm225-1-13.\n- [Mel22] Anton Mellit. Homology of torus knots. Geom. Topol., 26(1):47–70, 2022. doi: 10.2140/gt.2022.26.47.\n- [Ras15] Jacob Rasmussen. Some differentials on Khovanov-Rozansky homology. Geom. Topol., 19(6):3031–3104, 2015. doi:10.2140/gt.2015.19.3031.\n- [Sto09] Marko Stošić. Khovanov homology of torus links. Topology Appl., 156(3):533–541, 2009. doi:10.1016/j.topol.2008.08.004.\n- [Roz14] Lev Rozansky. An infinite torus braid yields a categorified Jones-Wenzl projector. Fund. Math., 225(1):305–326, 2014. doi:10.4064/fm225-1-14.\n- [Wen87] Hans Wenzl. On sequences of projections. C. R. Math. Rep. Acad. Sci. Canada, 9(1):5–9, 1987.\n- [Hog18] Matthew Hogancamp. Categorified Young symmetrizers and stable homology of torus links. Geom. Topol., 22(5):2943–3002, 2018. doi:10.2140/gt.2018.22.2943.\n- [GL15] Eugene Gorsky and Lukas Lewark. On stable sl3-homology of torus knots. Exp. Math., 24(2):162–174, 2015. doi:10.1080/10586458.2014.963746.\n- [OS05b] Peter Ozsváth and Zoltán Szabó. On knot Floer homology and lens space surgeries. Topology, 44(6):1281–1300, 2005. doi:10.1016/j.top.2005.05.001.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2691, "problem_number": "KP-1.32", "title": "Kirby Problem 1.32", "statement": "(a) Recover the Jones polynomial of links $L \\subset \\mathbb{R}^{3}$ by counting solutions to\nthe Kapustin–Witten equations on $\\mathbb{R}^{3} \\times \\mathbb{R}_{+}$ with Nahm pole boundary\nconditions;\n(b) Recover the Khovanov homology of links $L \\subset \\mathbb{R}^{3}$ by counting solutions to\nthe Haydys–Witten equations on $\\mathbb{R} \\times \\mathbb{R}^{3} \\times \\mathbb{R}_{+}$ with Nahm pole boundary\nconditions;\n\n(c) Use the Kapustin–Witten or Haydys–Witten equations to construct invari-\nants of links in other 3-manifolds.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.32.\n\nLiterature notes:\n(1) Khovanov homology [Kho00] is a link invariant defined combinatorially,\nwhose Euler characteristic is the Jones polynomial. It has many properties\nsimilar to Floer homologies (e.g., functoriality under cobordisms) and, as\nsuch, it is natural to ask if it has an interpretation in terms of gauge\ntheory. Such an interpretation was proposed by Witten in [Wit12], and\nis summarized in points (a) and (b) of the problem.\n(2) The Kapustin-Witten (KW) equations are a set of gauge invariant PDE’s\nin four dimensions introduced in [KW07].\nThe Nahm pole boundary\nconditions for these equations are described in [Wit12]; they have a more\nspecial form along the link $L \\times \\{0\\} \\subset \\mathbb{R}^{3} \\times \\{0\\}.$ Gaiotto and Witten\n[GW12] sketched an approach to showing that the count of solutions to\nthe KW equations with Nahm pole boundary conditions gives the coeffi-\ncients of the Jones polynomial. However, there is still work to be done to\nmake this mathematically rigorous.\n(3) The Haydys-Witten (HW) equations were introduced in [Wit12] and\n[Hay15]. Witten conjectured that, if one forms a complex whose gen-\nerators are the KW solutions and whose differentials count HW solutions\n(with appropriate limits and boundary conditions), then its homology is\nKhovanov homology.\n(4) Part of the appeal of these gauge-theoretic ideas is that they may pro-\nvide a definition of the Jones polynomial/Khovanov homology manifestly\nindependent of the link diagram. Moreover, we can write the same equa-\ntions with $\\mathbb{R}^{3}$ replaced by another 3-manifold, leading to part (c) of the\nproblem. Optimistically, the counts of KW solutions could give the class\nof the link $L$ in the Kauffman bracket skein module of the 3-manifold\n[Prz91, Tur88], or be related to the $\\widetilde{Z}$ invariants predicted by physi-\ncists [GPV17, GPPV20]; see Problem 3.68.\nMore ambitiously, the\ncount of HW solutions should give analogues of Khovanov homology for\nlinks in 3-manifolds.\nAt the moment, Khovanov homology is only de-\nfined in a combinatorial, concrete way for links in $S^{3}$, interval bundles\nover surfaces [APS04], $\\mathbb{R}$ P $^{3}$ [Man07, Gab13], and connected sums of\n$S^{1} \\times S^{2}$ [Roz10, Wil21]. If indeed the Haydys-Kapustin-Witten pro-\ngram can be carried through, it would also be important to compare\nthe result with the Khovanov lasagna skein modules of Morrison-Walker-\nWedrich [MWW22].\n(5) Mathematicians have made progress on understanding the analytic prop-\nerties of the KW and HW equations; see [MW14], [MW20], [Tau13],\n[LT20], [He19], [HM19a], [Tau18], [Tau19].\nSome of the compact-\nness results (e.g. [Tau18, Theorem B]) only hold when the 3-manifold\nhas positive Ricci curvature, which suggests that part (c) should first be\nattempted under this assumption.\n\nReferences cited:\n- [Kho00] Mikhail Khovanov. A categorification of the Jones polynomial. Duke Math. J., 101(3):359–426, 2000. doi:10.1215/S0012-7094-00-10131-7.\n- [Wit12] Edward Witten. Fivebranes and knots. Quantum Topol., 3(1):1–137, 2012. doi: 10.4171/QT/26.\n- [KW07] Anton Kapustin and Edward Witten. Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys., 1(1):1–236, 2007. doi:10.4310/CNTP.2007.v1.n1.a1.\n- [GW12] Davide Gaiotto and Edward Witten. Knot invariants from four-dimensional gauge theory. Adv. Theor. Math. Phys., 16(3):935–1086, 2012. doi:10.4310/atmp.2012.v16.n3.a5.\n- [Hay15] Andriy Haydys. Fukaya-Seidel category and gauge theory. J. Symplectic Geom., 13(1):151–207, 2015. doi:10.4310/JSG.2015.v13.n1.a5.\n- [Prz91] Józef H. Przytycki. Skein modules of 3-manifolds. Bull. Polish Acad. Sci. Math., 39(1-2):91–100, 1991.\n- [Tur88] V. G. Turaev. The Conway and Kauffman modules of a solid torus. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 167(Issled. Topol. 6):79–89, 190, 1988. doi:10.1007/BF01099241.\n- [GPV17] Sergei Gukov, Pavel Putrov, and Cumrun Vafa. Fivebranes and 3-manifold homology. J. High Energy Phys., 2017(7):071, front matter+80, 2017. doi:10.1007/JHEP07(2017)071.\n- [GPPV20] Sergei Gukov, Du Pei, Pavel Putrov, and Cumrun Vafa. BPS spectra and 3-manifold invariants. J. Knot Theory Ramifications, 29(2):2040003, 85, 2020. doi:10.1142/S0218216520400039.\n- [APS04] Marta M. Asaeda, Józef H. Przytycki, and Adam S. Sikora. Categorification of the Kauffman bracket skein module of I-bundles over surfaces. Algebr. Geom. Topol., 4:1177–1210, 2004. doi:10.2140/agt.2004.4.1177.\n- [Man07] Vassily Olegovich Manturov. Khovanov homology for virtual knots with arbitrary coefficients. J. Knot Theory Ramifications, 16(3):345–377, 2007. doi:10.1142/S0218216507005336.\n- [Gab13] Boštjan Gabrovšek. The categorification of the Kauffman bracket Skein module of $\\mathbb{RP}^{3}$. Bull. Aust. Math. Soc., 88(3):407–422, 2013. doi:10.1017/S0004972713000105.\n- [Roz10] Lev Rozansky. A categorification of the stable SU(2) Witten-Reshetikhin-Turaev invariant of links in $S^{2}$ $\\times$ $S^{1}$, 2010. arXiv:1011.1958.\n- [Wil21] Michael Willis. Khovanov homology for links in \\#rp$S^{2}$ $\\times$ $S^{1}$q. Michigan Math. J., 70(4):675–748, 2021. doi:10.1307/mmj/1594281620.\n- [MWW22] Scott Morrison, Kevin Walker, and Paul Wedrich. Invariants of 4-manifolds from Khovanov-Rozansky link homology. Geom. Topol., 26(8):3367–3420, 2022. doi:10.2140/gt.2022.26.3367.\n- [MW14] Rafe Mazzeo and Edward Witten. The Nahm pole boundary condition. In The influence of Solomon Lefschetz in geometry and topology, volume 621 of Contemp. Math., pages 171–226. Amer. Math. Soc., Providence, RI, 2014. doi:10.1090/conm/621/12422.\n- [MW20] Rafe Mazzeo and Edward Witten. The KW equations and the Nahm pole boundary condition with knots. Comm. Anal. Geom., 28(4):871–942, 2020. doi:10.4310/CAG.2020.v28.n4.a4.\n- [Tau13] Clifford Henry Taubes. Compactness theorems for SL(2;C) generalizations of the 4-dimensional anti-self dual equations, 2013. arXiv:1307.6447.\n- [LT20] Naichung Conan Leung and Ryosuke Takahashi. Energy bound for Kapustin-Witten solutions on $S^{3}$ $\\times$ R+. Int. Math. Res. Not. IMRN, 2020(19):6135–6148, 2020. doi: 10.1093/imrn/rny198.\n- [He19] Siqi He. A gluing theorem for the Kapustin-Witten equations with a Nahm pole. J. Topol., 12(3):855–915, 2019. doi:10.1112/topo.12102.\n- [HM19a] Siqi He and Rafe Mazzeo. The extended Bogomolny equations and generalized Nahm pole boundary condition. Geom. Topol., 23(5):2475–2517, 2019. doi:10.2140/gt.2019.23.2475.\n- [Tau18] Clifford Henry Taubes. Sequences of Nahm pole solutions to the SU(2) KapustinWitten equations, 2018. arXiv:1805.02773.\n- [Tau19] Clifford Henry Taubes. The R invariant solutions to the Kapustin Witten equations on $(0,8)$ $\\times$ $\\mathbb{R}^{2}$ $\\times$ R with generalized Nahm pole asymptotics, 2019. arXiv:1903.03539.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2692, "problem_number": "KP-1.33", "title": "Kirby Problem 1.33", "statement": "Describe topological necessary or sufficient conditions for a\nlink to have KR-parity. For example:\n(a) Are all links with KR-parity positive? Quasipositive?\n(b) Do all algebraic links have KR-parity?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.33.\n\nLiterature notes:\n(1) We say that a link has KR-parity if its triply graded Khovanov–Rozansky\nhomology is concentrated in homological degrees of the same parity. The\nmotivation for this property comes from the work of Hogancamp and\nElias [EH19], who developed a recursive procedure for computing the\nKhovanov–Rozansky homology of some links; their procedure relies on\nthe link having KR-parity.\n(2) By [Mel22, HM19b] all torus links have KR-parity.\n(3) If a link has KR-parity, then all coefficients in the HOMFLY polynomial\n$P(a, q)$ should have the same sign (depending on conventions, one might\nneed to change the variable $a$ to $-a$).\n(4) There are examples of positive links that do not have KR-parity, for ex-\nample the knot $10_{139}$ which is the closure of the braid $\\sigma_{1}\\sigma_{2}(\\sigma_{1}\\sigma_{2}\\sigma_{2}\\sigma_{1})^{2}$.\n(5) Let $\\mathrm{FT}_{i}$ denote the full twist on the first $i$ strands.\nIt is conjectured\n[GHSR20, OR23, Tur24] that the closure of the braid\n$\\mathrm{FT}_{2}^{d_{2}}\\mathrm{FT}_{3}^{d_{3}}\\cdots \\mathrm{FT}_{n}^{d_{n}}$\nhas KR-parity as long as all $d_{i} \\geq 0$.\n\nReferences cited:\n- [EH19] Ben Elias and Matthew Hogancamp. On the computation of torus link homology. Compos. Math., 155(1):164–205, 2019. doi:10.1112/s0010437x18007571.\n- [Mel22] Anton Mellit. Homology of torus knots. Geom. Topol., 26(1):47–70, 2022. doi: 10.2140/gt.2022.26.47.\n- [HM19b] Matthew Hogancamp and Anton Mellit. Torus link homology, 2019. arXiv:1909.00418.\n- [GHSR20] Eugene Gorsky, Graham Hawkes, Anne Schilling, and Julianne Rainbolt. Generalized q, t-Catalan numbers. Algebr. Comb., 3(4):855–886, 2020. doi:10.5802/alco.120.\n- [OR23] A. Oblomkov and L. Rozansky. Homfly-PT homology of Coxeter links. Transform. Groups, 28(3):1245–1275, 2023. doi:10.1007/s00031-023-09816-1.\n- [Tur24] Joshua P. Turner. Affine Springer fibers and generalized Haiman ideals (with an appendix by Eugene Gorsky and Joshua P. Turner). Int. Math. Res. Not. IMRN, 2024(16):11878–11909, 2024. doi:10.1093/imrn/rnae146.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2693, "problem_number": "KP-1.34", "title": "Kirby Problem 1.34", "statement": "(a) Khovanov and Rozansky [KR08b] used braid presentations to define a\ntriply graded link homology theory whose Euler characteristic is the HOM-\nFLYPT polynomial. Find a description of this homology in terms of arbi-\ntrary planar diagrams of the link, and determine whether some flavor of\nthis theory is functorial with respect to link cobordisms.\n(b) Do the same for Cautis’ link homologies from [Cau17], which are also\ndefined in terms of braid presentations.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.34.\n\nLiterature notes:\n(1) There are several versions of the HOMFLYPT link homology in [KR08b]:\nunreduced, middle, reduced, and totally reduced; see [Ras15]. Reduction\nrefers to choosing a basepoint and taking a mapping cone in the chain\ncomplex, which results in a smaller homology.\nOnly the totally reduced theory (where a basepoint is chosen on each\nlink component) has a chance at functoriality. For the original unreduced\ntheory, the homology of the unknot is the product of the exterior and\npolynomial algebras on one generator each, and this is infinite-dimensional\nover the ground field (i.e., over the homology of the empty link); thus, it\ncannot admit a functorial extension unless suitably modified. The totally\nreduced theory assigns finite-dimensional homology groups to all links and\nhas a chance at being functorial for link cobordisms (decorated by paths).\nA similar kind of functoriality appears in knot Floer homology [Zem19b].\n\n(2) The definition of HOMFLYPT homology in [KR08b] starts with a com-\nplex associated to a braid diagram. If one constructs the same complex\nfrom an arbitrary planar diagram, the resulting homology does not behave\nwell with respect to the Reidemeister II(b) move; see [Web07, Section 3.1]\nand [Abe17].\n(3) HOMFLYPT homology can also be interpreted as the Hochschild homol-\nogy of the Rouquier complexes formed from Soergel bimodules [Kho07].\nFunctoriality of Rouquier complexes under braid cobordisms was proved\nin [EK10].\n(4) In [Cau17], Cautis used categorical $\\mathfrak{s}\\mathfrak{l}_{n}$ actions to define triply-graded link\ninvariants that categorify the HOMFLYPT polynomial of links colored by\narbitrary partitions. He also defined a finite dimensional categorification\nof the $\\mathfrak{s}\\mathfrak{l}(n)$ link polynomial colored by symmetric powers of the standard\nrepresentation. See also [QRS18] for a combinatorial definition of Cautis’\nhomology, and [RW20] for the equivariant version.\n\nReferences cited:\n- [KR08b] Mikhail Khovanov and Lev Rozansky. Matrix factorizations and link homology. II. Geom. Topol., 12(3):1387–1425, 2008. doi:10.2140/gt.2008.12.1387.\n- [Cau17] Sabin Cautis. Remarks on coloured triply graded link invariants. Algebr. Geom. Topol., 17(6):3811–3836, 2017. doi:10.2140/agt.2017.17.3811.\n- [Ras15] Jacob Rasmussen. Some differentials on Khovanov-Rozansky homology. Geom. Topol., 19(6):3031–3104, 2015. doi:10.2140/gt.2015.19.3031.\n- [Zem19b] Ian Zemke. Link cobordisms and functoriality in link Floer homology. J. Topol., 12(1):94–220, 2019. doi:10.1112/topo.12085.\n- [Web07] Ben Webster. Khovanov-Rozansky homology via a canopolis formalism. Algebr. Geom. Topol., 7:673–699, 2007. doi:10.2140/agt.2007.7.673.\n- [Abe17] Michael Abel. HOMFLY-PT homology for general link diagrams and braidlike isotopy. Algebr. Geom. Topol., 17(5):3021–3056, 2017. doi:10.2140/agt.2017.17.3021.\n- [Kho07] Mikhail Khovanov. Triply-graded link homology and Hochschild homology of Soergel bimodules. Internat. J. Math., 18(8):869–885, 2007. doi:10.1142/S0129167X07004400.\n- [EK10] Ben Elias and Dan Krasner. Rouquier complexes are functorial over braid cobordisms. Homology Homotopy Appl., 12(2):109–146, 2010. doi:10.4310/hha.2010.v12.n2.a4.\n- [QRS18] Hoel Queffelec, David E. V. Rose, and Antonio Sartori. Annular evaluation and link homology, 2018. arXiv:1802.04131.\n- [RW20] Louis-Hadrien Robert and Emmanuel Wagner. Symmetric Khovanov-Rozansky link homologies. J. Éc. polytech. Math., 7:573–651, 2020. doi:10.5802/jep.124.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2694, "problem_number": "KP-1.35", "title": "Kirby Problem 1.35", "statement": "(a) Is symplectic Khovanov homology isomorphic to Khovanov homology, over\n$\\mathbb{Z}$?\n(b) Give a construction of odd symplectic Khovanov homology $\\mathrm{Kh}^{\\mathrm{odd}}_{\\mathrm{symp}}(K)$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.35.\n\nLiterature notes:\nSymplectic Khovanov homology, introduced in [SS06], is a can-\ndidate Floer-theoretic construction of Khovanov homology. Abouzaid and Smith\n[AS19] showed that it is isomorphic to ordinary Khovanov homology over $\\mathbb{Q}$ (or\nmore generally over fields of characteristic 0), but their proof does not work with\ninteger coefficients.\nIt is not obvious from the construction how to produce a version similarly\nrelated to odd Khovanov homology.\n\nReferences cited:\n- [SS06] Paul Seidel and Ivan Smith. A link invariant from the symplectic geometry of nilpotent slices. Duke Math. J., 134(3):453–514, 2006. doi:10.1215/S0012-7094-06-13432-4.\n- [AS19] Mohammed Abouzaid and Ivan Smith. Khovanov homology from Floer cohomology. J. Amer. Math. Soc., 32(1):1–79, 2019. doi:10.1090/jams/902.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2695, "problem_number": "KP-1.36", "title": "Kirby Problem 1.36", "statement": "Categorify the ($\\mathfrak{s}\\mathfrak{l}(2), \\mathfrak{s}\\mathfrak{l}(N)$, HOMFLYPT) skein algebras for\nsurfaces.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.36.\n\nLiterature notes:\n(1) The skein algebra of a surface, introduced by Przytycki-Sikora [PS00b]\nand Turaev [Tur91], encodes curves on the surface up to the Kauffman\nbracket skein relations.\n(2) The skein algebra of the annulus is categorified by the annular Khovanov\n(or Khovanov–Rozansky) homology [GW10, GLW18, QR18].\n(3) There are several constructions of Khovanov-type invariants of links in\nthickened surfaces [APS04, QW21]. However, it is not known if any of\nthese can support a categorical analogue of the skein product.\n\n(4) The HOMFLYPT skein algebra of the torus is closely related to the elliptic\nHall algebra [MS17b]. Categorification of the elliptic Hall algebra is an\nimportant problem in geometric representation theory [Neg22].\n\nReferences cited:\n- [PS00b] Józef H. Przytycki and Adam S. Sikora. On skein algebras and Sl2pCq-character varieties. Topology, 39(1):115–148, 2000. doi:10.1016/S0040-9383(98)00062-7.\n- [Tur91] Vladimir G. Turaev. Skein quantization of Poisson algebras of loops on surfaces. Ann. Sci. École Norm. Sup. (4), 24(6):635–704, 1991. URL: http://www.numdam.org/item?id=ASENS 1991 4 24 6 635 0.\n- [GW10] J. Elisenda Grigsby and Stephan M. Wehrli. Khovanov homology, sutured Floer homology and annular links. Algebr. Geom. Topol., 10(4):2009–2039, 2010. doi: 10.2140/agt.2010.10.2009.\n- [GLW18] J. Elisenda Grigsby, Anthony M. Licata, and Stephan M. Wehrli. Annular Khovanov homology and knotted Schur-Weyl representations. Compos. Math., 154(3):459– 502, 2018. doi:10.1112/S0010437X17007540.\n- [QR18] Hoel Queffelec and David E. V. Rose. Sutured annular Khovanov-Rozansky homology. Trans. Amer. Math. Soc., 370(2):1285–1319, 2018. doi:10.1090/tran/7117.\n- [APS04] Marta M. Asaeda, Józef H. Przytycki, and Adam S. Sikora. Categorification of the Kauffman bracket skein module of I-bundles over surfaces. Algebr. Geom. Topol., 4:1177–1210, 2004. doi:10.2140/agt.2004.4.1177.\n- [QW21] Hoel Queffelec and Paul Wedrich. Khovanov homology and categorification of skein modules. Quantum Topol., 12(1):129–209, 2021. doi:10.4171/qt/148.\n- [MS17b] Hugh Morton and Peter Samuelson. The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra. Duke Math. J., 166(5):801–854, 2017. doi:10.1215/00127094-3718881.\n- [Neg22] Andrei Neguţ. Hecke correspondences for smooth moduli spaces of sheaves. Publ. Math. Inst. Hautes Études Sci., 135:337–418, 2022. doi:10.1007/s10240-022-00131-1.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2696, "problem_number": "KP-1.37", "title": "Kirby Problem 1.37", "statement": "(a) For every link $L \\subset \\mathbb{R}^{3}$, every simple Lie algebra $\\mathfrak{g}$, and every coloring\nof the components of $L$ with irreducible representations of $\\mathfrak{g}$, construct\na bigraded link homology theory that is functorial under link cobordisms,\nand whose Euler characteristic is the Reshetikhin–Turaev link invariant\nassociated to $\\mathfrak{g}$ and $L$.\n(b) For each of the theories above, define an odd version $($ with the same mod\n2 reduction $)$ or explain the obstruction to its existence.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.37.\n\nLiterature notes:\n(1) Link homologies corresponding to $\\mathfrak{s}\\mathfrak{l}(n)$ with its standard vector represen-\ntation were constructed by Khovanov [Kho00] for $n = 2$ and Khovanov–\nRozansky [KR08a] for all $n$. Generalizations to exterior products of the\nstandard representations of $\\mathfrak{s}\\mathfrak{l}(n)$ were constructed in [Wu14], [Yon11],\n[QR16], and were proved to be functorial under link cobordisms in [ETW18].\n(2) Webster [Web17] has a categorification of the Reshetikhin–Turaev link\ninvariants for any simple Lie algebra $\\mathfrak{g}$ and any labeling of the link com-\nponents by irreducible representations of $\\mathfrak{g}$. His homology is bigraded.\nFor most $\\mathfrak{g}$ and labelings by irreducible representations, his homology of\nthe unknot is infinite-dimensional over the ground field and thus cannot\nextend to a functorial link homology theory.\nSuch a theory should at\nleast assign a commutative Frobenius algebra to the unknot, necessarily\nfinite-dimensional over the ground field. (A naive guess is that Webster\nhomology extends to a functorial theory, possibly up to overall scaling, if\nand only if all representations are minuscule.) It is also possible that a\ngood functorial theory would be more complicated algebraically, e.g. be\ndefined over a dg ring, or have more interesting gradings.\n(3) The odd version of Khovanov homology was defined by Ozsváth, Ras-\nmussen, and Szabó in [ORS13]; it is isomorphic to Khovanov homology\nover $\\mathbb{Z}/2$ but different over $\\mathbb{Z}$. The odd version appears naturally in re-\nlation to Floer homologies of the double branched cover of the link; see\n[OS05c], [Sca15]. Its functoriality (up to sign) was studied by Migdail\nand Wehrli [MW24].\n(4) Mikhaylov and Witten [MW15] studied odd Khovanov homology from\nthe perspective of physics. They argued that there should be odd versions\nfor the link homologies associated to $\\mathfrak{s}\\mathfrak{o}(2n + 1)$, but not for $\\mathfrak{s}\\mathfrak{u}(n)$; except\nfor $n = 2$, when $\\mathfrak{s}\\mathfrak{u}(2) = \\mathfrak{s}\\mathfrak{o}(3)$. The odd link homologies for $\\mathfrak{s}\\mathfrak{o}(2n+1)$ for\n$n \\geq 2$ have not yet been constructed mathematically.\n(5) Ellis and Lauda [EL16] gave an odd categorification of quantum $\\mathfrak{s}\\mathfrak{l}(2)$,\nusing a covering Kac-Moody algebra. They pointed out that the theory\nof covering Kac-Moody algebras only exists in finite type for $\\mathfrak{s}\\mathfrak{o}(2n + 1)$.\n\n(6) Khovanov, Putyra and Vaz [KPV24] constructed odd analogues of So-\nergel bimodules and Rouquier complexes, and pointed out an obstruction\nto defining an odd HOMFLYPT homology.\n\nReferences cited:\n- [Kho00] Mikhail Khovanov. A categorification of the Jones polynomial. Duke Math. J., 101(3):359–426, 2000. doi:10.1215/S0012-7094-00-10131-7.\n- [KR08a] Mikhail Khovanov and Lev Rozansky. Matrix factorizations and link homology. Fund. Math., 199(1):1–91, 2008. doi:10.4064/fm199-1-1.\n- [Wu14] Hao Wu. A colored $\\mathfrak{sl}(N)$ homology for links in $S^{3}$. Dissertationes Math., 499:217, 2014. doi:10.4064/dm499-0-1.\n- [Yon11] Yasuyoshi Yonezawa. Quantum $\\mathfrak{sl}(n), \\wedge^n$ link invariant and matrix factorizations. Nagoya Math. J., 204:69–123, 2011. doi:10.1215/00277630-1431840.\n- [QR16] Hoel Queffelec and David E. V. Rose. The sln foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality. Adv. Math., 302:1251–1339, 2016. doi:10.1016/j.aim.2016.07.027.\n- [ETW18] Michael Ehrig, Daniel Tubbenhauer, and Paul Wedrich. Functoriality of colored link homologies. Proc. Lond. Math. Soc. (3), 117(5):996–1040, 2018. doi:10.1112/plms.12154.\n- [Web17] Ben Webster. Knot invariants and higher representation theory. Mem. Amer. Math. Soc., 250(1191):v+141, 2017. doi:10.1090/memo/1191.\n- [ORS13] Peter Ozsváth, Jacob Rasmussen, and Zoltán Szabó. Odd Khovanov homology. Algebr. Geom. Topol., 13(3):1465–1488, 2013. doi:10.2140/agt.2013.13.1465.\n- [OS05c] Peter Ozsváth and Zoltán Szabó. On the Heegaard Floer homology of branched double-covers. Adv. Math., 194(1):1–33, 2005. doi:10.1016/j.aim.2004.05.008.\n- [Sca15] Christopher W. Scaduto. Instantons and odd Khovanov homology. J. Topol., 8(3):744–810, 2015. doi:10.1112/jtopol/jtv012.\n- [MW24] Jacob Migdail and Stephan Wehrli. Functoriality of odd and generalized Khovanov homology in $\\mathbb{R}^{3}$ $\\times$ I, 2024. arXiv:2206.14710.\n- [MW15] Victor Mikhaylov and Edward Witten. Branes and supergroups. Comm. Math. Phys., 340(2):699–832, 2015. doi:10.1007/s00220-015-2449-y.\n- [EL16] Alexander P. Ellis and Aaron D. Lauda. An odd categorification of $U_q(\\mathfrak{sl}_2)$. Quantum Topol., 7(2):329–433, 2016. URL: https://doi-org.stanford.idm.oclc.org/10.4171/QT/78, doi:10.4171/QT/78.\n- [KPV24] Mikhail Khovanov, Krzysztof Putyra, and Pedro Vaz. Odd two-variable Soergel bimodules and Rouquier complexes. In Algebraic and topological aspects of representation theory, volume 791 of Contemp. Math., pages 205–227. Amer. Math. Soc., [Providence], RI, [2024] ©2024. URL: https://doi-org.stanford.idm.oclc.org/10.1090/conm/791/15876, doi:10.1090/conm/791/15876.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2697, "problem_number": "KP-1.38", "title": "Kirby Problem 1.38", "statement": "What is the structure of the smooth knot concordance group?\n(a) Is there a torsion element of the smooth concordance group $\\mathcal{C}$ having order\nother than two? Is there a torsion element in $\\mathcal{C}_{TS}$, the smooth concordance\ngroup of topologically slice knots, having order other than two?\n(b) Is there a knot $K$ having concordance order two in $\\mathcal{C}$ that is not concordant\nto a strongly negatively amphichiral knot?\n(c) Do there exist infinitely divisible elements in $\\mathcal{C}$, that is, does there exist $K$\nsuch that for infinitely many $n \\in \\mathbb{N}$, there is a $J$ such that $[K] = n[J]$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.38.\n\nLiterature notes:\n(1) This is [Kir97, Problem 1.32]. Fox and Milnor’s original work on the\nconcordance group shows there exist elements of order two [FM66]; Hed-\nden, Kim, and Livingston [HKL16a] show the existence of an infinite\nsubgroup of topologically slice knots of order two.\nMany elements of\ninfinite order are known; for example, $\\mathcal{C}_{TS}$ contains subgroup isomor-\nphic to $\\mathbb{Z}_{\\infty}$ [End95], and in fact a direct summand isomorphic to $\\mathbb{Z}_{\\infty}$\n[DHST21, OSS17b].\n(2) Any infinitely divisible element of $\\mathcal{C}$ must be sent to zero under any integer-\nvalued homomorphism from the concordance group, such as the Ozsváth-\nSzabó $\\tau$-invariant or the Rasmussen $s$-invariant.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [FM66] Ralph H. Fox and John W. Milnor. Singularities of 2-spheres in 4-space and cobordism of knots. Osaka Math. J., 3:257–267, 1966. http://projecteuclid.org/euclid.ojm/1200691730.\n- [HKL16a] Matthew Hedden, Se-Goo Kim, and Charles Livingston. Topologically slice knots of smooth concordance order two. J. Differential Geom., 102(3):353–393, 2016. http://projecteuclid.org/euclid.jdg/1456754013.\n- [End95] Hisaaki Endo. Linear independence of topologically slice knots in the smooth cobordism group. Topology Appl., 63(3):257–262, 1995. doi:10.1016/0166-8641(94) 00062-8.\n- [DHST21] Irving Dai, Jennifer Hom, Matthew Stoffregen, and Linh Truong. More concordance homomorphisms from knot Floer homology. Geom. Topol., 25(1):275–338, 2021. doi:10.2140/gt.2021.25.275.\n- [OSS17b] Peter Ozsváth, András I. Stipsicz, and Zoltán Szabó. Concordance homomorphisms from knot Floer homology. Adv. Math., 315:366–426, 2017. doi:10.1016/j.aim.2017.05.017.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2698, "problem_number": "KP-1.39", "title": "Kirby Problem 1.39", "statement": "(a) Do the algebraic knots freely generate a subgroup of the smooth concor-\ndance group $\\mathcal{C}$?\n(b) Do the algebraic knots freely generate a subgroup of the topological con-\ncordance group $\\mathcal{C}^{\\mathrm{TOP}}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.39.\n\nLiterature notes:\n(1) This question is originally due to Rudolph [Rud76], although it is not\nspecified whether $\\mathcal{C}$ or $\\mathcal{C}^{\\mathrm{TOP}}$ is intended. Litherland showed that torus\nknots are independent in the smooth concordance group [Lit79]; his proof\nyields the topological version as well. Litherland’s theorem uses Levine-\nTristram signatures, but various authors have shown that there exist con-\nnected sums of algebraic knots and their mirrors that are algebraically\nslice but not smoothly slice [LM83, HKL12] implying that Litherland’s\nstrategy is not sufficient to answer the question above.\n(2) This problem is especially interesting in light of its connection to the slice-\nribbon conjecture (Problem 1.50): a result of Miyazaki shows that non-\ntrivial linear combinations of iterated torus knots are not ribbon [Miy94,\nCorollary 8.4], so the slice-ribbon conjecture implies Rudolph’s conjec-\nture.\nIndeed, Baker [Bak16] and Abe-Tagami [AT16a] observed that\nthe slice-ribbon conjecture implies a stronger form of Rudolph’s conjec-\nture, to wit, that the set of prime fibered strongly quasi-positive knots is\nlinearly independent in the smooth concordance group. Large families of\nknots satisfying both Rudolph’s conjecture and this strengthening were\nexhibited by Conway, Kim, and Politarczyk [CKP23].\n(3) In a related direction, Baker showed that the slice-ribbon conjecture im-\nplies that if two fibered knots supporting the tight contact structure on\n$S^{3}$ are concordant, then they are in fact isotopic [Bak16, Corollary 4].\nAbe and Tagami observed that it would then follow that the set of fibered\nknots supporting the tight contact structure on $S^{3}$ is linearly indepen-\ndent in the smooth concordance group and contains the algebraic knots\n[AT16a, Observation 1.3]. This can be thought of as a generalization of\nRudolph’s question.\n\nReferences cited:\n- [Rud76] Lee Rudolph. How independent are the knot-cobordism classes of links of plane curve singularities? Notices Amer. Math. Soc, 23:410, 1976.\n- [Lit79] R. A. Litherland. Signatures of iterated torus knots. In Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), volume 722 of Lecture Notes in Math., pages 71–84. Springer, Berlin, 1979.\n- [LM83] Charles Livingston and Paul Melvin. Algebraic knots are algebraically dependent. Proc. Amer. Math. Soc., 87(1):179–180, 1983. doi:10.2307/2044377.\n- [HKL12] Matthew Hedden, Paul Kirk, and Charles Livingston. Non-slice linear combinations of algebraic knots. J. Eur. Math. Soc. (JEMS), 14(4):1181–1208, 2012. doi:10.4171/JEMS/330.\n- [Miy94] Katura Miyazaki. Nonsimple, ribbon fibered knots. Trans. Amer. Math. Soc., 341(1):1–44, 1994. doi:10.2307/2154613.\n- [Bak16] Kenneth L. Baker. A note on the concordance of fibered knots. J. Topol., 9(1):1–4, 2016. doi:10.1112/jtopol/jtv024.\n- [AT16a] Tetsuya Abe and Keiji Tagami. Fibered knots with the same 0-surgery and the slice-ribbon conjecture. Math. Res. Lett., 23(2):303–323, 2016. doi:10.4310/MRL.2016.v23.n2.a1.\n- [CKP23] Anthony Conway, Min Hoon Kim, and Wojciech Politarczyk. Nonslice linear combinations of iterated torus knots. Algebr. Geom. Topol., 23(2):765–802, 2023. doi:10.2140/agt.2023.23.765.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2699, "problem_number": "KP-1.40", "title": "Kirby Problem 1.40", "statement": "A satellite operator $P \\subset S^{1} \\times D^{2}$ induces an operation $P$ on\nthe concordance group $\\mathcal{C}$ [Gor75].\n(a) Let $P$ be a winding number one satellite operator with $P(\\operatorname{U}) \\simeq \\operatorname{U}$.\nIf\n$P(K) \\simeq \\operatorname{U}$, does this mean that $K \\simeq \\operatorname{U}$?\n(b) Is the Whitehead doubling operator injective on $\\mathcal{C}$? Can we exhibit any\n(winding-number-zero) operator that is injective on $\\mathcal{C}$?\n(c) Conjecture (Hedden; see [HPC21]). The only homomorphisms on $\\mathcal{C}$ in-\nduced by satellite operators are the zero map, the identity, and the invo-\nlution induced by orientation reversal.\n(d) Conjecture (Hedden, Pinzón Caicedo) Let $P$ be a non-constant satellite\noperator with winding number zero. Does $\\\\langle \\\\operatorname{im} P \\\\rangle$ necessarily have infinite\nrank? Some special cases are:\n(i) Does there always exist a fixed knot $K$ such that $\\langle\\{P(nK)\\}_{n\\in\\mathbb{Z}}\\rangle$ has\ninfinite rank?\n(ii) If $\\{nK\\}_{n\\in\\mathbb{Z}}$ is any rank-one subgroup of $\\mathcal{C}$, does $\\langle\\{P(nK)\\}_{n\\in\\mathbb{Z}}\\rangle$ have\ninfinite rank?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.40.\n\nLiterature notes:\n(1) Satellite operators are defined in the Introduction to this section.\n(2) The overall conjecture (d) is easily verified in the case of non-zero winding\nnumber. It was shown in [HK12b] that Whitehead doubling has infinite\nrank, with many other patterns treated in [HPC21]. The authors subse-\nquently posed the general conjecture, as well as subconjecture (i).\n(3) In a positive answer to [DHMS24], subconjecture (i) was announced\nfor the Whitehead doubling operation. Indeed, $K$ was taken to be the\n(right-handed) trefoil; this was previously unknown. The authors gave\na general class of patterns (as well as a flexible Floer-theoretic condi-\ntion on $K$) for subconjecture (i) to hold; they subsequently made the\nstronger subconjecture (ii). Roughly speaking, the latter indicates that\nany (non-constant, winding number zero) pattern expands rank in the\nknot concordance group.\n\nReferences cited:\n- [Gor75] C. McA. Gordon. Knots, homology spheres, and contractible 4-manifolds. Topology, 14:151–172, 1975. doi:10.1016/0040-9383(75)90024-5.\n- [HPC21] Matthew Hedden and Juanita Pinzón-Caicedo. Satellites of infinite rank in the smooth concordance group. Invent. Math., 225(1):131–157, 2021. doi:10.1007/s00222-020-01026-w.\n- [HK12b] Matthew Hedden and Paul Kirk. Instantons, concordance, and Whitehead doubling. J. Differential Geom., 91(2):281–319, 2012. URL: http://projecteuclid.org/euclid.jdg/1344430825.\n- [DHMS24] Irving Dai, Matthew Hedden, Abhishek Mallick, and Matthew Stoffregen. Rankexpanding satellites, Whitehead doubles, and Heegaard Floer homology. J. Topol., 17(4):Paper No. e70008, 40, 2024. doi:10.1112/topo.70008.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2700, "problem_number": "KP-1.41", "title": "Kirby Problem 1.41", "statement": "This problem is concerned with the stable 4-genus $g_{s}(K)$ of a\nknot $K$, defined below.\n(a) Is there a knot $K$ such that $g_{s}(K) \\in \\mathbb{Q}\\setminus\\mathbb{Z}$?\n(b) Is there a knot $K$ such that $g_{s}(K) \\notin \\mathbb{Q}$?\n(c) Is there a knot $K$ such that $0 < g_{s}(K) < 1/2$?\n(d) [Liv10, Question 1] Does $K$ represent a torsion element in the concor-\ndance group if and only if $g_{s}(K) = 0$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.41.\n\nLiterature notes:\n(1) The stable 4-genus of a knot $K$ is defined by\n\n$$\ng_s(K)=\\lim_{n\\to\\infty}\\frac{g_4(nK)}{n}.\n$$\n\n(2) One instance under which the stable 4-genus $g_{s}(K)$ of a knot $K$ may\nbecome strictly less than $g_{4}(K)$, is when for some $m \\geq 2$, the knot $mK$\nhas $g_{4}(mK) < mg_{4}(K)$, for in this case\n\n$$\ng_s(K)=\\lim_{\\ell\\to\\infty}\\frac{g_4(\\ell mK)}{\\ell m}\n\\leq \\lim_{\\ell\\to\\infty}\\frac{\\ell g_4(mK)}{\\ell m}\n=\\frac{g_4(mK)}{m} 0$, there exists a knot $K_{\\epsilon}$ with\nstable 4-genus near $\\frac{1}{2}$, in the sense of this double inequality:\n\n$$\n\\frac{1-\\epsilon}{2}\\leq g_s(K_{\\epsilon})\\leq \\frac{1}{2}.\n$$\n\nSimilarly, in [Ilt22] it is shown that if $K_{n}$ denotes the twist knot with $n$\nfull twists, where $n$ is such that the Pell equation $x^{2} - (4n + 1)y^{2} = -1$\nhas integral solutions in $x$ and $y$, then\n\n$$\n\\frac{1}{2}-\\frac{6}{2n+7}\\leq g_s(K_n)\\leq \\frac{1}{2}.\n$$\n\nThe above bounds can be interpreted as suggesting that $\\frac{1}{2}$ may have a\nspecial significance with regards to the stable 4-genus, motivating part (c).\n(4) The lower bound of $0 < g_{s}(K)$ is required in part (c) of the question to\nexclude knots $K$ that represent torsion elements in the concordance group;\nany such knot will have stable 4-genus equal to 0. This leads to part (d),\nwhich suggests a connection with torsion in the concordance group (See\nProblem 1.38).\n\nReferences cited:\n- [Liv10] Charles Livingston. The stable 4-genus of knots. Algebr. Geom. Topol., 10(4):2191– 2202, 2010. doi:10.2140/agt.2010.10.2191.\n- [Ilt22] Damian Iltgen. A lower bound on the stable 4-genus of knots. Algebr. Geom. Topol., 22(5):2239–2265, 2022. doi:10.2140/agt.2022.22.2239.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2701, "problem_number": "KP-1.42", "title": "Kirby Problem 1.42", "statement": "Do there exist algebraically concordant Seifert forms $V_{1}$ and\n$V_{2}$ for which there do not exist concordant knots $K_{1}$ and $K_{2}$ with Seifert forms $V_{1}$\nand $V_{2}$, respectively?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.42.\n\nLiterature notes:\n(1) Seifert forms $V_{1}$ and $V_{2}$ are algebraically concordant if $V_{1} \\oplus -V_{2}$ is meta-\nbolic. This problem is open regardless of whether one works with topo-\nlogical or smooth concordance of knots.\n\n(2) As an example, the Seifert forms\n\n$$\nV_1=\\begin{pmatrix}3&2\\\\1&3\\end{pmatrix}\n\\qquad\\text{and}\\qquad\nV_2=\\begin{pmatrix}1&2\\\\1&9\\end{pmatrix}\n$$\n\nare algebraically concordant: the set of vectors $\\{(0, -3, 6, -1), (2, -1, 0, 1)\\}$\nis a basis for a metabolizer of $V_{1} \\oplus -V_{2}$. Does there exist a pair of con-\ncordant knots $K_{1}$ and $K_{2}$ having these Seifert forms? Conjecture 1.11\nof [Liv01] posits that any knots $K_{1}$ and $K_{2}$ having as Seifert forms these\nparticular $V_{1}$ and $V_{2}$, respectively, cannot be concordant.\n\nReferences cited:\n- [Liv01] Charles Livingston. Examples in concordance, 2001. arXiv:math/0101035v1.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2702, "problem_number": "KP-1.43", "title": "Kirby Problem 1.43", "statement": "Does knot Floer homology give a categorification of the Fox–\nMilnor condition?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.43.\n\nLiterature notes:\n(1) Fox and Milnor [FM66] prove that if $K$ is slice in $B^{4}$, then the Alexander\npolynomial factors as\n\n$$\n\\Delta_{K}(t) = f(t)f(t^{-1}).\n$$\n\nGiven that the Euler characteristic of knot Floer homology is the Alexan-\nder polynomial, one might expect that the knot Floer homology of a slice\nknot $K$ factors as\n\n$$\n\\widehat{\\mathrm{HFK}}(K) \\simeq V \\otimes V^{*}\n$$\n\nwhere $V$ is a bigraded vector space, and $V^{*}$ denotes its dual. This is\nfalse, since the knot Floer homology of the Kinoshita-Terasaka knot has\nrank 33. One precise conjecture would be that there is a natural spectral\nsequence from such a vector space of the form $V \\otimes V^{*}$ to $\\widehat{\\mathrm{HFK}}(K)$.\n(2) Note that Gilmer [Gil84] proved that if there is a ribbon concordance\nfrom $K_{1}$ to $K_{0}$, then\n\n$$\n\\Delta_{K_1}(t) = \\Delta_{K_0}(t) \\cdot f(t)f(t^{-1}).\n$$\n\nOne could ask if this relation, or similar relations, also admit meaningful\nlifts to knot Floer homology.\n(3) One could similarly ask for categorifications of restrictions on the Alexan-\nder polynomial coming from symmetries on the three-manifold or the knot.\nFor example, the Alexander polynomial of the preimage $\\widetilde{K}$ of a knot $K$\nin its $n$-fold branched cover $\\Sigma_{n}(K)$ is\n\n$$\n\\Delta_{\\widetilde K}(t) \\doteq \\prod_{i=1}^{n} \\Delta_K(\\xi_n^i t^{1/n}),\n$$\n\nwhere $\\xi_n$ is a primitive $n$th root of unity. Similarly, let $K$ be a $q$-periodic\nknot with quotient $\\overline K$, and let $\\lambda$ be the linking number of $\\overline K$ with the\naxis of periodicity $\\overline A$. Let $\\overline L=\\overline K\\cup\\overline A$. Murasugi [Mur71] shows that the\nAlexander polynomial of $K$ is\n\n$$\n\\Delta_K(t) \\doteq \\Delta_{\\overline K}(t)\\prod_{i=1}^{q}\\Delta_{\\overline L}(t,\\xi_q^i),\n$$\n\nimplying in particular that $\\Delta_{\\overline K}(t)$ divides $\\Delta_K(t)$ and that\n\n$$\n\\Delta_K(t) \\doteq (1+t+\\cdots+t^{\\lambda-1})^{q-1}\\Delta_{\\overline K}(t) \\pmod{q}.\n$$\n\nwhere the equivalences above are up to a factor of $\\pm t^{\\pm i}$. Work of Hendricks\nshows that knot Floer homology recovers this last condition [Hen15], but\na precise categorification of the first two remains unclear.\n\nReferences cited:\n- [FM66] Ralph H. Fox and John W. Milnor. Singularities of 2-spheres in 4-space and cobordism of knots. Osaka Math. J., 3:257–267, 1966. http://projecteuclid.org/euclid.ojm/1200691730.\n- [Gil84] Patrick M. Gilmer. Ribbon concordance and a partial order on S-equivalence classes. Topology Appl., 18(2-3):313–324, 1984. doi:10.1016/0166-8641(84)90016-6.\n- [Mur71] Kunio Murasugi. On periodic knots. Comment. Math. Helv., 46:162–174, 1971. doi:10.1007/BF02566836.\n- [Hen15] Kristen Hendricks. Localization of the link Floer homology of doubly-periodic knots. J. Symplectic Geom., 13(3):545–608, 2015. doi:10.4310/JSG.2015.v13.n3.a2.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2703, "problem_number": "KP-1.44", "title": "Kirby Problem 1.44", "statement": "(a) If $K \\in \\mathcal{F}_{n}$ for all $n$, is $K$ topologically slice?\n(b) If $K \\in \\mathcal{T}_{n}$ for all $n$, is $K$ smoothly slice?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.44.\n\nLiterature notes:\n(1) Let $\\{\\mathcal{F}_{n}\\}_{n\\in \\frac{1}{2}\\mathbb{N}}$ denote the solvable filtration of $\\mathcal{C}$ due to Cochran-Orr-\nTeichner [COT03], and let $\\{\\mathcal{T}_{n}\\}_{n\\in\\mathbb{N}}$ denote the bipolar filtration of $\\mathcal{T}$ due\nto Cochran-Harvey-Horn [CHH13].\n(2) It is known that $\\mathcal{T} \\subseteq \\bigcap_{n} \\mathcal{F}_{n}$. The solvable filtration subsumes several\ntopological concordance invariants, e.g. a knot $K$ is in $\\mathcal{F}_{0}$ if and only if\nArf $(K) = 0$; it is in $\\mathcal{F}_{0.5}$ if and only if it is algebraically slice; and if $K$ is\nin $\\mathcal{F}_{1.5}$ then all its Casson-Gordon sliceness obstructions vanish [COT03].\nLikewise, the bipolar filtration [CHH13] subsumes many smooth concor-\ndance invariants, e.g. if a knot $K \\in \\mathcal{T}_{0}$ then $\\tau(K) = \\upsilon(K) = \\Upsilon(K) =$\n$\\nu^{+}(K) = 0$, referring to invariants from Heegaard Floer homology.\nIt\nis not known whether $s(K) = 0$, referring to the $s$-invariant from Kho-\nvanov homology. However $s^{\\#}(K) = 0$ [KM13b] (see also [Gon21]) and\n$\\widetilde{s}(K) = 0$ [DIS $^{+}25$].\n\nReferences cited:\n- [COT03] Tim D. Cochran, Kent E. Orr, and Peter Teichner. Knot concordance, Whitney towers and L2-signatures. Ann. of Math. (2), 157(2):433–519, 2003. doi:10.4007/annals.2003.157.433.\n- [CHH13] Tim D. Cochran, Shelly Harvey, and Peter Horn. Filtering smooth concordance classes of topologically slice knots. Geom. Topol., 17(4):2103–2162, 2013. doi:10.2140/gt.2013.17.2103.\n- [KM13b] P. B. Kronheimer and T. S. Mrowka. Gauge theory and Rasmussen’s invariant. J. Topol., 6(3):659–674, 2013. doi:10.1112/jtopol/jtt008.\n- [Gon21] Sherry Gong. On the Kronheimer-Mrowka concordance invariant. J. Topol., 14(1):1–28, 2021. doi:10.1112/topo.12175.\n- [DIS+25] Aliakbar Daemi, Hayato Imori, Kouki Sato, Christopher Scaduto, and Masaki Taniguchi. Instantons, special cycles and knot concordance. Geom. Topol., 29(8):4189–4298, 2025. doi:10.2140/gt.2025.29.4189.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2704, "problem_number": "KP-1.45", "title": "Kirby Problem 1.45", "statement": "(a) For arbitrary $n \\geq 2.5$ and $g > 1$, does there exist a knot in $\\mathcal{F}_{n}$ with\ntopological slice genus at least $g$?\n(b) For arbitrary $n \\geq 0$ and $g > 1$, does there exist a knot in $\\mathcal{T}_{n}$ with smooth\nslice genus at least $g$?\n(c) Could the $L^{(2)}$-signature invariants used to obstruct membership in deeper\nlevels of the solvable/bipolar filtration be used to give bounds on the slice\ngenus?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.45.\n\nLiterature notes:\n(1) The problem originates from Cha [Cha08, Remark 5.6].\n(2) The filtrations $\\mathcal{F}_{n}$ and $\\mathcal{T}_{n}$ of the smooth concordance group $\\mathcal{C}$ and the\nsubgroup $\\mathcal{T}$ of topologically slice knots, respectively, are defined in Prob-\nlem 1.44.\n(3) For (a), an affirmative answer for the $n = 0, 1$ cases can be shown using\nLevine-Tristram and Casson–Gordon signatures respectively. Similar, an\naffirmative answer to the $n = 2$ case was established by Cha–Miller–Powell\nin [CMP21]. Unlike (a), question (b) is open even in the case $n = 0$.\n(4) A knot is closer to being slice if it has low slice genus. Similarly it is closer\nto being slice if it lies deeper in the solvable/bipolar filtrations. There-\nfore the question is asking whether these two methods of approximating\nsliceness are related.\n(5) We note that there are many examples of highly solvable/bipolar knots\nthat are not known to have low slice genera–that is, the issue in part (c)\nis not a lack of potential examples but rather a lack of obstructions.\n\nReferences cited:\n- [Cha08] Jae Choon Cha. Topological minimal genus and L2-signatures. Algebr. Geom. Topol., 8(2):885–909, 2008. doi:10.2140/agt.2008.8.885.\n- [CMP21] Jae Choon Cha, Allison N. Miller, and Mark Powell. Two-solvable and two-bipolar knots with large four-genera. Math. Res. Lett., 28(2):331–382, 2021. doi:10.4310/MRL.2021.v28.n2.a2.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2705, "problem_number": "KP-1.46", "title": "Kirby Problem 1.46", "statement": "(a) Determine the topological slice genera of torus knots. In particular, does\nthe topological slice genus of a torus knot equal half the absolute value of\nits maximal Levine–Tristram signature?\n(b) Is a two-bridge knot topologically slice if and only if it is ribbon, if and\nonly if it is topologically homotopy-ribbon?\n(c) Are the Casson–Gordon ribbon and topological sliceness obstructions [CG86]\ncomplete for two-bridge knots?\n(d) Can the difference between the smooth and topological slice genus of a\ntwo-bridge knot be arbitrarily large?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.46.\n\nLiterature notes:\n(1) Most known methods to bound the topological slice genus from above are\nbased on Freedman–Quinn’s theorem that knots with Alexander polyno-\nmial 1 are topologically slice [FQ90, BKK $^{+}21$]; the exception to this\nis [FT05].\n(2) The smooth slice genera of torus knots were determined by Kronheimer\nand Mrowka using gauge theory: $g_{4}(T(p, q)) = (p - 1)(q - 1)/2$ [KM94].\nThe first person to show that the topological slice genus of torus knots is\ntypically less than their 3-genus was Rudolph [Rud84]. The topological\nslice genus $g^{\\mathrm{top}}_{4}(T(p,q))$ is less than the smooth slice genus $g_{4}(T(p,q))$\nunless $T(p,q)$ has maximal signature, and in general\n$g^{\\mathrm{top}}_{4}(K) 3$ were produced by Batson, who\nused an elegant construction and the Heegaard Floer $d$ invariant to prove that\n$\\gamma_{4}(T_{2k,2k-1}) = k - 1$ [Bat14]. Lobb subsequently showed that Batson’s construc-\ntion does not always yield a minimal-genus nonorientable surface for torus knots\n[Lob19]. Allen added context by asking how the nonorientable 4-genus interacts\nwith the normal Euler number of the bounding surfaces for torus (and other) knots\n[All23]. See [BKST25, JVC21, OSS17a, Sab23], among others, for further\nexplorations of the nonorientable 4-genus for torus knots. In [All23], Allen made\nthree conjectures (Conjectures 1.6, 1.7, and 1.8); Sato [Sat24a] has given coun-\nterexamples to Conjectures 1.6 and 1.8.\nThere are now many knots for which the nonorientable four-genus is known,\nincluding all knots with at most 10 crossings [Gha22]. There are also currently\na variety of lower bounds for $\\gamma_{4}$ using modern knot invariants.\nMany of these\nare what Sato [Sat24b] terms “unoriented slice-torus invariants”, including the $\\upsilon$\ninvariant from Heegaard Floer theory [OSS17a], the $t$ invariant from Khovanov\nhomology [Bal20], the $h_{\\mathbb{Z}}$ invariant from instanton homology [DS24a]. Additional\ninformation about the nonorientable four-genus can be obtained by considering\nversions of Seiberg-Witten Floer homology of the double branched cover that take\ninto account the involution coming from the covering transformation; see [BH24b,\nKMT23b].\n\nReferences cited:\n- [All23] Samantha Allen. Nonorientable surfaces bounded by knots: a geography problem. New York J. Math., 29:1038–1059, 2023. http://nyjm.albany.edu/j/2023/29-41v.pdf.\n- [MY00] Hitoshi Murakami and Akira Yasuhara. Four-genus and four-dimensional clasp number of a knot. Proc. Amer. Math. Soc., 128(12):3693–3699, 2000. doi:10.1090/S0002-9939-00-05461-7.\n- [Vir75] Oleg Ja. Viro. Positioning in codimension 2, and the boundary. Uspehi Mat. Nauk, 30(1(181)):231–232, 1975.\n- [Yas96] Akira Yasuhara. Connecting lemmas and representing homology classes of simply connected 4-manifolds. Tokyo J. Math., 19(1):245–261, 1996. doi:10.3836/tjm/1270043232.\n- [GL11] Patrick M. Gilmer and Charles Livingston. The nonorientable 4-genus of knots. J. Lond. Math. Soc. (2), 84(3):559–577, 2011. doi:10.1112/jlms/jdr024.\n- [Bat14] Joshua Batson. Nonorientable slice genus can be arbitrarily large. Math. Res. Lett., 21(3):423–436, 2014. doi:10.4310/MRL.2014.v21.n3.a1.\n- [Lob19] Andrew Lobb. A counterexample to Batson’s conjecture. Math. Res. Lett., 26(6):1789, 2019. doi:10.4310/MRL.2019.v26.n6.a8.\n- [BKST25] Fraser Binns, Sungkyung Kang, Jonathan Simone, and Paula Truöl. On the nonorientable four-ball genus of torus knots. Algebr. Geom. Topol., 25(4):2209–2251, 2025. doi:10.2140/agt.2025.25.2209.\n- [JVC21] Stanislav Jabuka and Cornelia A. Van Cott. On a nonorientable analogue of the Milnor conjecture. Algebr. Geom. Topol., 21(5):2571–2625, 2021. doi:10.2140/agt.2021.21.2571.\n- [OSS17a] Peter Ozsváth, András Stipsicz, and Zoltán Szabó. Unoriented knot Floer homology and the unoriented four-ball genus. Int. Math. Res. Not. IMRN, 2017(17):5137– 5181, 2017. doi:10.1093/imrn/rnw143.\n- [Sab23] Joshua M. Sabloff. On a refinement of the non-orientable 4-genus of torus knots. Proc. Amer. Math. Soc. Ser. B, 10:242–251, 2023. doi:10.1090/bproc/166.\n- [Sat24a] Kouki Sato. Counterexamples to Allen’s conjectures, 2024. arXiv:2407.12049.\n- [Gha22] Nakisa Ghanbarian. The non-orientable 4-genus for knots with 10 crossings. J. Knot Theory Ramifications, 31(5):Paper No. 2250034, 46, 2022. doi:10.1142/S0218216522500341.\n- [Sat24b] Kouki Sato. An unoriented analogue of slice-torus invariant, 2024. arXiv:2404.04056.\n- [Bal20] William Ballinger. Concordance invariants from the Ep-1q spectral sequence on Khovanov homology, 2020. arXiv:2004.10807.\n- [DS24a] Aliakbar Daemi and Christopher Scaduto. Chern–Simons functional, singular instantons, and the four-dimensional clasp number. J. Eur. Math. Soc. (JEMS), 26(6):2127–2190, 2024. doi:10.4171/jems/1320.\n- [BH24b] David Baraglia and Pedram Hekmati. New invariants of involutions from SeibergWitten-Floer theory, 2024. arXiv:2403.00203.\n- [KMT23b] Hokuto Konno, Jin Miyazawa, and Masaki Taniguchi. Involutions, links, and Floer cohomologies, 2023. arXiv:2304.01115.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2707, "problem_number": "KP-1.48", "title": "Kirby Problem 1.48", "statement": "(a) Suppose $K$ and $K\\#J$ are (smoothly) doubly slice knots.\nMust $J$ be a\n(smoothly) doubly slice knot?\n(b) Does there exist a knot that is smoothly slice, topologically doubly slice, and\nnot smoothly doubly slice but such that $K\\#K$ is smoothly doubly slice?\n(c) Let $M$ and $N$ be closed 3-manifolds such that $M$ and $M\\#N$ embed (smoothly)\nin $S^{4}$. Must $N$ embed (smoothly) in $S^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.48.\n\nLiterature notes:\n(1) A knot $K \\subset S^{3}$ is called (smoothly) doubly slice if there is a (smoothly)\nunknotted 2-sphere $\\mathcal{\\operatorname{U}} \\subset S^{4}$ such that $K = S^{3} \\cap \\mathcal{\\operatorname{U}}$.\n(2) It is natural to try to equip the set of knots with the relation in which\n$K_{0} \\sim K_{1}$ if $K_{0}\\#K_{1}$ is doubly slice. However, to prove this is an equiv-\nalence relation, one must provide an affirmative answer to part (a) of\nthe problem. For more details, see [Mei15] and [Sto78]. This question\nis also interesting in the topological category: Replace “smooth” with\n“topologically locally flat” above. Question (a) is originally due to Stoltz-\nfus [Sto78].\n(3) One motivation for Question (a) is that it would allow for the formulation\nof the Grothendieck group of knots modulo the relation discussed above.\nIn the absence of a positive answer to Question (a), there is an alternative\napproach. Two knots $K_{0}$ and $K_{1}$ are called (smoothly) doubly concordant\nif there exist (smoothly) doubly slice knots $J_{0}$ and $J_{1}$ such that $K_{0}\\#J_{0} =$\n$K_{1}\\#J_{1}$. Let $\\mathcal{C}_{\\mathcal{D}}$ denote the group (under connected sum) of knots modulo\nthis equivalence relation, which is called the (smooth) double concordance\n\ngroup.\nAn affirmative answer to Question (a) would mean the double\nconcordance group is the same as the aforementioned Grothendieck group.\nA natural problem is to exhibit nontrivial elements in these groups, as we\nnow outline.\nThere is a topological version $\\mathcal{C}_{\\mathcal{D}}^{\\mathrm{top}}$\nof this group obtained by replacing\n“smooth” with “topologically locally flat” above. There are epimorphisms\nfrom $\\mathcal{C}_{\\mathcal{D}}$ to $\\mathcal{C}_{\\mathcal{D}}^{\\mathrm{top}}$\nand to the smooth concordance group $\\mathcal{C}$. Let $\\mathcal{K}$ denote the\nintersection of the kernels of these two homomorphisms. It is known that\n$\\mathcal{K}$ contains an infinitely generated subgroup $S$ with generators of order at\nleast 3 [Mei15].\nConjecture. These generators of $S$ have infinite order.\n(4) There are many techniques for producing and studying 2-torsion in the\nsmooth concordance group $\\mathcal{C}$; see [HKL16a] for a nice overview. However,\nthese techniques seem difficult to apply to $\\mathcal{C}_{\\mathcal{D}}$, which motivates Question\n(b).\n(5) If $K$ is doubly slice, then any cyclic cover of $S^{3}$ branched along $K$ embeds\n(smoothly) in $S^{4}$. This suggests Question (c), which asks about embed-\nding 3-manifolds in $S^{4}$. The connection between doubly slice knots, em-\nbedding problems and homology cobordisms was observed by Gilmer and\nLivingston [GL83]. See also Problem 4.25.\n\nReferences cited:\n- [Mei15] Jeffrey Meier. Distinguishing topologically and smoothly doubly slice knots. J. Topol., 8(2):315–351, 2015. doi:10.1112/jtopol/jtu027.\n- [Sto78] Neal W. Stoltzfus. Algebraic computations of the integral concordance and double null concordance group of knots. In Knot theory (Proc. Sem., Plans-sur-Bex, 1977), volume 685 of Lecture Notes in Math., pages 274–290. Springer, Berlin, 1978.\n- [HKL16a] Matthew Hedden, Se-Goo Kim, and Charles Livingston. Topologically slice knots of smooth concordance order two. J. Differential Geom., 102(3):353–393, 2016. http://projecteuclid.org/euclid.jdg/1456754013.\n- [GL83] Patrick M. Gilmer and Charles Livingston. On embedding 3-manifolds in 4-space. Topology, 22(3):241–252, 1983. doi:10.1016/0040-9383(83)90011-3.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2708, "problem_number": "KP-1.49", "title": "Kirby Problem 1.49", "statement": "(a) What is the structure of the equivariant concordance groups?\n(b) Is the strongly negative amphichiral concordance group abelian?\n(c) For any type of knot involution, is it possible to exhibit $K_{1}$ and $K_{2}$ that\nare equivariantly concordant but not standardly equivariantly concordant?\n(d) Does there exist a freely periodic slice knot that is not equivariantly slice?\n(e) (Boyle-Rouse) Is every periodic or freely periodic $L$-space knot either a\ntorus knot or an iterated torus knot?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.49.\n\nLiterature notes:\n(1) Knots may have various sorts of symmetries; an equivariant concordance\nis a usual concordance that admits the same sort of symmetry. In the\ntopological case, we require that the concordance be locally flat, and that\nany group action be locally linear. There are several variations discussed\nin this problem, each with its own particular set of concerns.\n(2) Equivariant concordance groups can be defined for the classes of strongly\ninvertible and strongly negative amphichiral knots. The strongly invert-\nible concordance group has been studied by several authors: for instance,\nthere exists a homomorphism to the Laurent polynomial ring $\\mathbb{Z}[t, t^{-1}]$\n[Sak86]. These groups are not a priori abelian due to (possible) depen-\ndence on the choice of the connected sum point, which must be taken to be\none of the two fixed points of the symmetry on the knot. In the strongly\ninvertible case, it is known that the group is not abelian [DP23b], but in\nthe amphichiral case this is not known.\n\nUnderstanding strongly negative amphichiral knots is especially inter-\nesting, since the branched covers of such knots may provide examples of\n2-torsion in the integer homology cobordism group. Examples of strongly\nnegative amphichiral knots with determinant one were exhibited by Boyle\nand Chen in [BC24b].\n(3) In part (c), we say that two knots $K_{1}$ and $K_{2}$ (equipped with the same\ntype of involution) are standardly equivariantly concordant if they are\nconnected by a smooth annular cobordism in $S^{3}\\times I$ that is invariant under\nan involution on $S^{3} \\times I$ that restricts to a standard involution on each\n$S^{3} \\times \\{t\\}$. One can also consider more general equivariant cobordisms, in\nwhich one requires that the annular cobordism be fixed by some involution\non $S^{3} \\times I$ which extends the actions on $S^{3} \\times \\{0\\}$ and $S^{3} \\times \\{1\\}$. It is not\nknown in any of the possible cases whether these notions coincide. The\nquestion may be asked in either the smooth or the topological category.\nIn the case of 2-periodic and strongly invertible knots, it is known\nthat there exists a nonstandard involution on $S^{3} \\times I$ that restricts to the\nstandard involution on $S^{3}\\times\\{0\\}$ and $S^{3}\\times\\{1\\}$. To construct one, start with\nan involution on $S^{4}$ with fixed-point set a knotted $S^{2}$ [Sum75]; punctur-\ning such an example gives a nonstandard involution on $S^{3} \\times I$ that has\nfixed-point set $S^{1}$ on each of the two ends. It is not known whether an\ninteresting such involution exists in the freely periodic or strongly am-\nphichiral cases.\n(4) Question (d) appears as Question 1 in [BM23]. Manolescu and Willis\n[MW25] gave examples of freely 2-periodic knots that are concordant\nbut not (standardly) equivariantly concordant.\n(5) Evidence for Question (e) comes from classic work of Murasugi [Mur71]\nand Hartley [Har81], who established factorization results for the Alexan-\nder polynomials of periodic or freely periodic knots. We also have several\nwell-known restrictions on the Alexander polynomials of L-space knots;\nsee [OS05b] or [HW18]. As discussed by Boyle and Rouse [BR23b],\nthe only known examples of Alexander polynomials that simultaneously\nsatisfy all of these conditions are products of cyclotomic polynomials. On\nthe other hand, a conjecture of Li and Ni states that the only L-space\nknots with Alexander polynomials which are products of cyclotomic poly-\nnomials are torus knots or iterated torus knots [LN15]. Based on this,\nBoyle and Rouse [BR23b] conjectured a positive answer to Question (e).\n\nReferences cited:\n- [Sak86] Makoto Sakuma. On strongly invertible knots. In Algebraic and topological theories (Kinosaki, 1984), pages 176–196. Kinokuniya, Tokyo, 1986.\n- [DP23b] Alessio Di Prisa. The equivariant concordance group is not abelian. Bull. Lond. Math. Soc., 55(1):502–507, 2023. doi:10.1112/blms.12741.\n- [BC24b] Keegan Boyle and Wenzhao Chen. Equivariant topological slice disks and negative amphichiral knots. Indiana Univ. Math. J., 73(5):1623–1637, 2024.\n- [Sum75] D. W. Sumners. Smooth $\\mathbb{Z}_p$-actions on spheres which leave knots pointwise fixed. Trans. Amer. Math. Soc., 205:193–203, 1975. doi:10.2307/1997199.\n- [BM23] Keegan Boyle and Jeffrey Musyt. Equivariant cobordisms between freely periodic knots. Canad. Math. Bull., 66(2):450–457, 2023. doi:10.4153/S000843952200042X.\n- [MW25] Ciprian Manolescu and Michael Willis. A Rasmussen invariant for links in $\\mathbb{RP}^{3}$. Trans. Amer. Math. Soc. Ser. B, 12:789–830, 2025. doi:10.1090/btran/221.\n- [Mur71] Kunio Murasugi. On periodic knots. Comment. Math. Helv., 46:162–174, 1971. doi:10.1007/BF02566836.\n- [Har81] Richard Hartley. Knots with free period. Canadian J. Math., 33(1):91–102, 1981. doi:10.4153/CJM-1981-009-7.\n- [OS05b] Peter Ozsváth and Zoltán Szabó. On knot Floer homology and lens space surgeries. Topology, 44(6):1281–1300, 2005. doi:10.1016/j.top.2005.05.001.\n- [HW18] Matthew Hedden and Liam Watson. On the geography and botany of knot Floer homology. Selecta Math. (N.S.), 24(2):997–1037, 2018. doi:10.1007/s00029-017-0351-5.\n- [BR23b] Keegan Boyle and Nicholas Rouse. Obstructions to free periodicity and symmetric L-space knots, 2023. arXiv:2310.01705.\n- [LN15] Eileen Li and Yi Ni. Half-integral finite surgeries on knots in $S^{3}$. Ann. Fac. Sci. Toulouse Math. (6), 24(5):1157–1178, 2015. doi:10.5802/afst.1479.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2709, "problem_number": "KP-1.50", "title": "Kirby Problem 1.50", "statement": "(a) Is every slice knot a ribbon knot?\n(b) Is every slice link ribbon?\n(c) Suppose $K$ is a knot with smooth four-genus $g_{4}(K) = g$. Does $K$ bound a\nsmooth, genus $g$, ribbon surface $F$ in $B^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.50.\n\nLiterature notes:\n(1) The Slice-Ribbon Conjecture (a) was first posed as a question by Fox\nin 1961 [Fox62, Problem 25], and was discussed in [Kir97, Problem\n4.22].\nVery limited progress has been made on this problem, though\nthe conjecture has been verified for 2-bridge knots by Lisca [Lis07] and\nmost 3-stranded pretzel knots by Greene and Jabuka [GJ11] and by\nLecuona [Lec15].\n(2) There are other notions of “ribbonness” that can be considered: A knot\n$K \\subset S^{3}$ is called homotopy-ribbon if there is a slice disk $D$ for $K$ with a\nsurjection $\\pi_{1}(S^{3}\\setminus K) \\twoheadrightarrow\\pi_{1}(B^{4}\\setminus D)$ and is called handle-ribbon if there is\na slice disk $D$ for $K$ such that $B^{4}\\setminus\\nu(D)$ admits a handle-decomposition\nwith only 0-, 1-, and 2-handles; see [Gor81, LM15, MZ23].\n(3) The implications\nribbon $\\Rightarrow$ handle ribbon $\\Rightarrow$ homotopy-ribbon $\\Rightarrow$ slice\nare immediate, but all three converse implications are open.\n(4) For many examples of handle ribbon knots and links that are not known\nto be ribbon, as well as connections to the Generalized Property R Con-\njecture, see [GST10, MZ22] and Problem 1.10.\n(5) Miyazaki gave many examples of non-simple knots that are not homotopi-\ncally ribbon, such as certain linear combinations of algebraic knots and,\nfamously, the $(2, 1)$-cable of the figure-8 knot [Miy94]. Some of these ex-\namples are now known to be non-slice [HKL12, DKM $^{+}24$, ACM $^{+}26$].\nBaker draws connections between the Slice-Ribbon Conjecture for fibered\nknots and contact topology in [Bak16]. See Problem 1.54 for more ques-\ntions about ribbon concordances.\n(6) In part (b), we say that an $n$-component link $L \\subset S^{3}$ is called (strongly)\nslice if there is a disjoint collection $D \\subset B^{4}$ of $n$ disks with $\\partial D = L$;\nthe notions of ribbon, handle-ribbon, and homotopy-ribbon have obvious\nanalogs for links.\n(7) Many examples of handle ribbon links that are not known to be ribbon\nare constructed in [GST10] and [MZ22, MZ25].\n(8) Eisermann [Eis09] gave an obstruction for a link (with at least two com-\nponents) to be ribbon, in terms of the Jones polynomial. It is not clear\nwhether his obstruction vanishes when the link is handle ribbon (or homotopy-\nribbon, or slice) so, in principle, it could be used to show that a handle\nribbon link is not ribbon.\n(9) The question in part (c) generalizes the Slice-Ribbon Conjecture, and so\nit may be easier to find a counterexample. If $g_{4}(K) = g(K)$, then the\npush-in of a minimal genus Seifert surface is a ribbon surface realizing\n\nthe minimum possible value of $g_{4}(K)$, so the question has an affirmative\nanswer in this case.\n(10) Finally, the question of whether slice implies homotopy-ribbon can be\nphrased in the more general setting of knots in homology 3–spheres bound-\ning disks in contractible 4–manifolds. For more details, see [Kir97, Prob-\nlem 4.22] and [CG83a].\n\nReferences cited:\n- [Fox62] Ralph Fox. Some problems in knot theory. In M. K. Fort, Jr., editor, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961). Prentice-Hall, Englewood Cliffs, N.J., 1962.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Lis07] Paolo Lisca. Lens spaces, rational balls and the ribbon conjecture. Geom. Topol., 11:429–472, 2007. doi:10.2140/gt.2007.11.429.\n- [GJ11] Joshua Greene and Stanislav Jabuka. The slice-ribbon conjecture for 3-stranded pretzel knots. Amer. J. Math., 133(3):555–580, 2011. doi:10.1353/ajm.2011.0022.\n- [Lec15] Ana G. Lecuona. On the slice-ribbon conjecture for pretzel knots. Algebr. Geom. Topol., 15(4):2133–2173, 2015. doi:10.2140/agt.2015.15.2133.\n- [Gor81] C. McA. Gordon. Ribbon concordance of knots in the 3-sphere. Math. Ann., 257(2):157–170, 1981. doi:10.1007/BF01458281.\n- [LM15] Kyle Larson and Jeffrey Meier. Fibered ribbon disks. J. Knot Theory Ramifications, 24(14):1550066, 22, 2015. doi:10.1142/S0218216515500662.\n- [MZ23] Maggie Miller and Alexander Zupan. Equivalent characterizations of handle-ribbon knots. Comm. Anal. Geom., 31(9):2157–2193, 2023. doi:10.4310/cag.2023.v31.n9.a1.\n- [GST10] Robert E. Gompf, Martin Scharlemann, and Abigail Thompson. Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures. Geom. Topol., 14(4):2305–2347, 2010. doi:10.2140/gt.2010.14.2305.\n- [MZ22] Jeffrey Meier and Alexander Zupan. Generalized square knots and homotopy 4-spheres. J. Differential Geom., 122(1):69–129, 2022. doi:10.4310/jdg/1668186788.\n- [Miy94] Katura Miyazaki. Nonsimple, ribbon fibered knots. Trans. Amer. Math. Soc., 341(1):1–44, 1994. doi:10.2307/2154613.\n- [HKL12] Matthew Hedden, Paul Kirk, and Charles Livingston. Non-slice linear combinations of algebraic knots. J. Eur. Math. Soc. (JEMS), 14(4):1181–1208, 2012. doi:10.4171/JEMS/330.\n- [DKM+24] Irving Dai, Sungkyung Kang, Abhishek Mallick, JungHwan Park, and Matthew Stoffregen. The $(2,1)$-cable of the figure-eight knot is not smoothly slice. Invent. Math., 238(2):371–390, 2024. doi:10.1007/s00222-024-01286-w.\n- [ACM+26] Paolo Aceto, Nickolas A Castro, Maggie Miller, JungHwan Park, and András Stipsicz. Slice Obstructions From Genus Bounds in Definite 4-Manifolds. Int. Math. Res. Not. IMRN, 2026(2):Paper No. rnaf377, 2026. doi:10.1093/imrn/rnaf377.\n- [Bak16] Kenneth L. Baker. A note on the concordance of fibered knots. J. Topol., 9(1):1–4, 2016. doi:10.1112/jtopol/jtv024.\n- [MZ25] Jeffrey Meier and Alexander Zupan. Knots bounding nonisotopic ribbon disks. J. Topol., 18(4):Paper No. e70047, 18, 2025. doi:10.1112/topo.70047.\n- [Eis09] Michael Eisermann. The Jones polynomial of ribbon links. Geom. Topol., 13(2):623– 660, 2009. doi:10.2140/gt.2009.13.623.\n- [CG83a] A. J. Casson and C. McA. Gordon. A loop theorem for duality spaces and fibred ribbon knots. Invent. Math., 74(1):119–137, 1983. doi:10.1007/BF01388533.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2710, "problem_number": "KP-1.51", "title": "Kirby Problem 1.51", "statement": "Does every ribbon knot arise as a symmetric union?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.51.\n\nLiterature notes:\nSymmetric unions were introduced by Kinoshita and Terasaka to\nconstruct knots with Alexander polynomial 1 (by taking a symmetric union of an\nunknot diagram) [KT57]. It is straightforward to see that every symmetric union is\na ribbon knot. A census shows that most small ribbon knots do arise as symmetric\nunions [Lam21], but there are candidates for ribbon knots that are not symmetric\nunions, such as $3_{1}\\#8_{10}$ and $3_{1}\\#8_{11}$. Other potential counterexamples obtained as\ncertain satellites of knots of the form $K\\# - K$ are given by Aceto [Ace14]. These\nknots have ribbon number two, but the symmetric ribbon number is shown to be\narbitrarily large, if not infinite.\n\nReferences cited:\n- [KT57] Shin’ichi Kinoshita and Hidetaka Terasaka. On unions of knots. Osaka Math. J., 9:131–153, 1957.\n- [Lam21] Christoph Lamm. The search for nonsymmetric ribbon knots. Exp. Math., 30(3):349–363, 2021. doi:10.1080/10586458.2018.1540313.\n- [Ace14] Paolo Aceto. Symmetric ribbon disks. J. Knot Theory Ramifications, 23(9):1450048, 9, 2014. doi:10.1142/S0218216514500485.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2711, "problem_number": "KP-1.52", "title": "Kirby Problem 1.52", "statement": "Given $K$ in $S^{3}$, is there an algorithm to detect if $K$ is slice?\nRibbon?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.52.\n\nLiterature notes:\n(1) This is Problem 1.34 from [Kir97], posed by A. Casson.\n(2) There is still no algorithm, but there have been some interesting computer-\naided searches: Owens and Swenton search for ribbon bands for alternat-\ning knots [OS23] and Gukov, Halverson, Manolescu, and Ruehle apply\nmachine learning algorithms to the problem [GHMR23]. Dunfield and\nGong [DG25] have announced the results of a search for ribbon knots,\nslice knots, and concordances among the knots with at most 19 crossings.\n(3) Casson and Long gave an algorithm to determine if a surface automor-\nphism compresses [CL85]. In conjunction with the main results of [CG83a],\nthis algorithm can be used to determine whether a given fibered knot\nbounds a fibered, homotopy-ribbon disk in a homotopy four-ball; see\nalso [Lon86].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [OS23] Brendan Owens and Frank Swenton. An algorithm to find ribbon disks for alternating knots. Experimental Mathematics, 0(0):1–19, 2023. doi:10.1080/10586458.2022.2158968.\n- [GHMR23] Sergei Gukov, James Halverson, Ciprian Manolescu, and Fabian Ruehle. Searching for ribbons with machine learning, 2023. arXiv:2304.09304.\n- [DG25] Nathan M. Dunfield and Sherry Gong. Ribbon concordances and slice obstructions: experiments and examples, 2025. arXiv:2512.21825.\n- [CL85] A. J. Casson and D. D. Long. Algorithmic compression of surface automorphisms. Invent. Math., 81(2):295–303, 1985. doi:10.1007/BF01389054.\n- [CG83a] A. J. Casson and C. McA. Gordon. A loop theorem for duality spaces and fibred ribbon knots. Invent. Math., 74(1):119–137, 1983. doi:10.1007/BF01388533.\n- [Lon86] D. D. Long. Discs in compression bodies. Pacific J. Math., 122(1):129–146, 1986. http://projecteuclid.org/euclid.pjm/1102702126.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2712, "problem_number": "KP-1.53", "title": "Kirby Problem 1.53", "statement": "(a) Which knot properties are hereditary under ribbon concordance? Is the\nproperty of being alternating hereditary under ribbon concordance?\n(b) Which knot properties imply minimality under ribbon concordance? Are\npositive knots minimal under ribbon concordance?\n(c) Which knot invariants are monotone under ribbon concordance? Is cross-\ning number, bridge number, or braid index monotone under ribbon con-\ncordance?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.53.\n\nLiterature notes:\n(1) A knot property is a subset $\\mathcal{P}$ of the set of knots. It is hereditary under a\ndirected relation $\\mathcal{R}$ on the set of knots if $K_{1} \\in \\mathcal{P}, K_{0}\\mathcal{R}K_{1} \\Longrightarrow K_{0} \\in \\mathcal{P}$.\n(2) Using techniques of Silver and the solution of Rapoport’s conjecture given\nby Kochloukova, Miyazaki showed that the property of being fibered is\nhereditary under ribbon concordance [Koc06, Miy18, Sil92] in the sense\nthat if $K_{1}$ is ribbon concordant to $K_{0}$ and $K_{1}$ is fibered, then so is $K_{0}$; see\nalso work of Zemke [Zem19a]. It is conjectured that the property of being\nalternating is hereditary under ribbon concordance [GO23, Conjecture 9].\n(3) A knot $K_{1}$ is minimal under $\\mathcal{R}$ if $K_{0}\\mathcal{R}K_{1} \\Longrightarrow K_{1} = K_{0}$. It is conjectured\nthat positive knots are minimal under ribbon concordance [BG24, Con-\njecture 1.6]. Baker and Stoimenow have asked related questions [Bak16,\nSto15]. See also Problem 1.54.\n(4) Most knot invariants support natural directed relations $\\preceq$ on their codomains:\nnumerical (real-valued) invariants with respect to the standard ordering\n$\\leq$; polynomial invariants with respect to divisibility; group-valued invari-\nants with respect to injections; etc. A knot invariant $\\iota$ is monotone under\na directed relation $\\mathcal{R}$ if $K_{0}\\mathcal{R}K_{1} \\Longrightarrow \\iota(K_{0}) \\preceq \\iota(K_{1})$.\nSeveral knot invariants are known to be monotone under ribbon con-\ncordance. For example, if $K_{1}$ is ribbon concordant to $K_{0}$, the Alexander\npolynomial of $K_{0}$ divides that of $K_{1}$ [Gil84, FP20]. Zemke showed that\na ribbon concordance from $K_{1}$ to $K_{0}$ induces a bigraded injection from the\nknot Floer homology of $K_{0}$ to $K_{1}$, with the consequence that the Seifert\ngenus of $K_{0}$ is less than or equal to that of $K_{1}$ [Zem19a]. It is known that\nif $K_{1}$ has bridge number 2, then $K_{0}$ has bridge number at most 2 [GO23,\nParagraph after Conjecture 9].\n(5) Gordon’s pioneering paper on ribbon concordance contains several fasci-\nnating problems related to the ones raised here [Gor81].\n\nReferences cited:\n- [Koc06] Dessislava H. Kochloukova. Some Novikov rings that are von Neumann finite and knot-like groups. Comment. Math. Helv., 81(4):931–943, 2006. doi:10.4171/CMH/81.\n- [Miy18] Katura Miyazaki. A note on genera of band sums that are fibered. J. Knot Theory Ramifications, 27(12):1871002, 3, 2018. doi:10.1142/S0218216518710025.\n- [Sil92] D. S. Silver. On knot-like groups and ribbon concordance. J. Pure Appl. Algebra, 82(1):99–105, 1992. doi:10.1016/0022-4049(92)90013-6.\n- [Zem19a] Ian Zemke. Knot Floer homology obstructs ribbon concordance. Ann. of Math. (2), 190(3):931–947, 2019. doi:10.4007/annals.2019.190.3.5.\n- [GO23] Joshua Evan Greene and Brendan Owens. Alternating links, rational balls, and cube tilings, 2023. J. Eur. Math. Soc., to appear. arXiv:2212.06248.\n- [BG24] Joe Boninger and Joshua Evan Greene. Special alternating knots are band prime. Int. Math. Res. Not., 2024(10):8758–8763, 2024. doi:10.1093/imrn/rnae009.\n- [Bak16] Kenneth L. Baker. A note on the concordance of fibered knots. J. Topol., 9(1):1–4, 2016. doi:10.1112/jtopol/jtv024.\n- [Sto15] A. Stoimenow. Application of braiding sequences III: Concordance of positive knots. Internat. J. Math., 26(7):1550050, 36, 2015. doi:10.1142/S0129167X15500500.\n- [Gil84] Patrick M. Gilmer. Ribbon concordance and a partial order on S-equivalence classes. Topology Appl., 18(2-3):313–324, 1984. doi:10.1016/0166-8641(84)90016-6.\n- [FP20] Stefan Friedl and Mark Powell. Homotopy ribbon concordance and Alexander polynomials. Arch. Math. (Basel), 115(6):717–725, 2020. doi:10.1007/s00013-020-01517-5.\n- [Gor81] C. McA. Gordon. Ribbon concordance of knots in the 3-sphere. Math. Ann., 257(2):157–170, 1981. doi:10.1007/BF01458281.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2713, "problem_number": "KP-1.54", "title": "Kirby Problem 1.54", "statement": "This problem is concerned with the restriction of the partial\nordering $\\geq$ coming from ribbon concordance to the concordance class $[K]$ of a knot\n$K$.\n(a) What can be said about the order type of $[K]$?\n(b) Does it depend on $[K]$? Must it contain a (unique) minimal element?\n(c) (Gordon) Does there exist a concordance class of knots that contains two\nminimal elements with respect to ribbon concordance?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.54.\n\nLiterature notes:\n(1) The ribbon concordance relation was defined by Gordon in [Gor81], who\nproves that some classes $[K]$ contain minimal elements, and asked Ques-\ntion (c).\n(2) The ribbon concordance relation was conjectured to be a partial ordering\nby Gordon. This was proved by Agol [Ago22]. Notice that for any knots\n$K$ and $L$, the map from $[K]$ to $[L]$ defined for $J \\in [K]$ by\n\n$$\nJ \\mapsto J \\# -K \\# L\n$$\n\nis an order preserving injection.\n(3) The Slice-Ribbon Conjecture can be formulated to say the unknot $\\operatorname{U}$ is\nthe unique minimal element of $[\\operatorname{U}]$; see Problem 1.50.\nSee Problem 1.56 for a Heegaard Floer theory perspective.\n\nReferences cited:\n- [Gor81] C. McA. Gordon. Ribbon concordance of knots in the 3-sphere. Math. Ann., 257(2):157–170, 1981. doi:10.1007/BF01458281.\n- [Ago22] Ian Agol. Ribbon concordance of knots is a partial ordering. Comm. Amer. Math. Soc., 2:374–379, 2022. doi:10.1090/cams/15.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2714, "problem_number": "KP-1.55", "title": "Kirby Problem 1.55", "statement": "Suppose that $C$ is a ribbon concordance from a fibered knot $K_{1}$\nto a fibered knot $K_{0}$.\n(a) Does the capped-off monodromy of $K_{1}$ (i.e. extended over a disk) extend to\na compression body whose lower boundary gives the capped-off monodromy\nof $K_{0}$?\n(b) Is $C$ fibered by compression bodies?\n(c) Suppose a fibered knot in a homology 3-sphere bounds a homotopy-ribbon\ndisk $D$ in a contractible 4-manifold $V$ . Is $V \\setminus\\nu(D)$ fibered by handlebodies?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.55.\n\nLiterature notes:\n(1) Part (a) holds when $K_{0}$ is the unknot by the main theorem of [CG83a].\n(2) Miller answers part (b) in the affirmative under the hypothesis that the\nribbon concordance is defined by a collection of ribbon bands that are\ntransverse to the fibration of the exterior of $K_{1}$ [Mil21, Theorem 1.9].\n(3) Part (c) was posed by Casson and Gordon in the Remark before Corol-\nlary 5.4 of [CG83a].\n\nReferences cited:\n- [CG83a] A. J. Casson and C. McA. Gordon. A loop theorem for duality spaces and fibred ribbon knots. Invent. Math., 74(1):119–137, 1983. doi:10.1007/BF01388533.\n- [Mil21] Maggie Miller. Extending fibrations of knot complements to ribbon disk complements. Geom. Topol., 25(3):1479–1550, 2021. doi:10.2140/gt.2021.25.1479.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2715, "problem_number": "KP-1.56", "title": "Kirby Problem 1.56", "statement": "(Hom). If $K_{0}$ and $K_{1}$ are ribbon concordant and\n\n$$\n\\widehat{\\mathrm{HFK}}(K_{0}) \\cong \\widehat{\\mathrm{HFK}}(K_{1}),\n$$\n\nare $K_{0}$ and $K_{1}$ isotopic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.56.\n\nLiterature notes:\n(1) Recall that Zemke [Zem19a] proved that a ribbon concordance induces\na split injection on knot Floer homology.\nGordon asked the following\n[Gor81, Question 6.2]: If $K_{1} \\geq K_{2} \\geq...$ , does there exist some $m$\nsuch that $K_{n} = K_{m}$ for all $n \\geq m$? An affirmative answer to Problem\n1.56 would answer Gordon’s question in the affirmative. One large class of\nconcordant knots with the same knot Floer homology is given in [Wan22,\n\nTheorem 1.2] (see also [HW18, Theorem 1]), but none of these are ribbon\nconcordant to any other, by [Wan22, Theorem 1.8] and [LZ19].\n(2) The same question could be asked with Khovanov homology\n$\\widetilde{\\mathrm{Kh}}$ replacing knot Floer homology $\\widehat{\\mathrm{HFK}}$,\nor indeed other homology theories as treated\nin [Kan22a].\n\nReferences cited:\n- [Zem19a] Ian Zemke. Knot Floer homology obstructs ribbon concordance. Ann. of Math. (2), 190(3):931–947, 2019. doi:10.4007/annals.2019.190.3.5.\n- [Gor81] C. McA. Gordon. Ribbon concordance of knots in the 3-sphere. Math. Ann., 257(2):157–170, 1981. doi:10.1007/BF01458281.\n- [Wan22] Joshua Wang. The cosmetic crossing conjecture for split links. Geom. Topol., 26(7):2941–3053, 2022. doi:10.2140/gt.2022.26.2941.\n- [HW18] Matthew Hedden and Liam Watson. On the geography and botany of knot Floer homology. Selecta Math. (N.S.), 24(2):997–1037, 2018. doi:10.1007/s00029-017-0351-5.\n- [LZ19] Adam Simon Levine and Ian Zemke. Khovanov homology and ribbon concordances. Bull. Lond. Math. Soc., 51(6):1099–1103, 2019. doi:10.1112/blms.12303.\n- [Kan22a] Sungkyung Kang. Link homology theories and ribbon concordances. Quantum Topol., 13(1):183–205, 2022. doi:10.4171/qt/162.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2716, "problem_number": "KP-1.57", "title": "Kirby Problem 1.57", "statement": "(a) In either the smooth or topological settings, are 0-shake slice knots slice?\n(b) Does there exist a knot $K$ whose topological 0-shake slice genus is strictly\nless than the topological slice genus?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.57.\n\nLiterature notes:\n(1) The smooth analogue of the second question was [Kir97, Problem 1.41],\nwhich was answered in the negative by Piccirillo in [Pic19]. Her construc-\ntion uses knots with diffeomorphic 0-traces whose smooth slice genera are\ndistinct. A negative answer to part (a) would imply the existence of a knot\nthat is slice in an integer homology ball but not in $B^{4}$; see Problem 1.60.\n(2) One may also ask analogous questions for $n$-traces of knots, where some\nnegative answers are known.\nFor more on this see [Akb77, Lic79,\nAkb93, AJOT13].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Pic19] Lisa Piccirillo. Shake genus and slice genus. Geom. Topol., 23(5):2665–2684, 2019. doi:10.2140/gt.2019.23.2665.\n- [Akb77] Selman Akbulut. On 2-dimensional homology classes of 4-manifolds. Math. Proc. Cambridge Philos. Soc., 82(1):99–106, 1977. doi:10.1017/S0305004100053718.\n- [Lic79] W. B. Raymond Lickorish. Shake-slice knots. In Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), volume 722 of Lecture Notes in Math., pages 67–70. Springer, Berlin, 1979.\n- [Akb93] S. Akbulut. Knots and exotic smooth structures on 4-manifolds. J. Knot Theory Ramifications, 2(1):1–10, 1993. doi:10.1142/S0218216593000027.\n- [AJOT13] Tetsuya Abe, In Dae Jong, Yuka Omae, and Masanori Takeuchi. Annulus twist and diffeomorphic 4-manifolds. Math. Proc. Cambridge Philos. Soc., 155(2):219–235, 2013. doi:10.1017/S0305004113000194.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2717, "problem_number": "KP-1.58", "title": "Kirby Problem 1.58", "statement": "What concordance information about a knot $K$ is contained in\nits 0-trace $X_{0}(K)$ and in its 0-surgery $S^{3}_{0}(K)$?\nSpecifically,\n(a) Suppose $K$ and $K'$ have homeomorphic 0-traces.\n(i) Are $K$ and $K'$ topologically concordant?\n(ii) What if one just assumes $K$ and $K'$ have homeomorphic 0-surgeries?\n(iii) What if one just assumes $K$ and $K'$ have 0-surgeries that are topolog-\nically homology cobordant, preserving the homology class of a merid-\nian.\n\n(b) Suppose $K$ and $K'$ have homeomorphic 0-surgeries, where the image of\nthe meridian of $K$ is freely homotopic to the meridian of $K'$. Are $K$ and\n$K'$ smoothly concordant?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.58.\n\nLiterature notes:\n(1) This problem originates in a conjecture of Akbulut and Kirby [Kir97,\nProblem 1.19] that knots with the same 0-surgery are concordant. It was\nmotivated, at least in part, by the fact [KM78] that it holds when one\nof the knots is unknotted. (The result of [KM78] predates the proof of\nProperty R, which is of course much stronger.)\nAlthough the original\nconjecture is false, there are several variations, as one can ask about the\nhomeomorphism or diffeomorphism type of $X_{0}(K)$ and about smooth or\ntopological concordance.\n(2) All of the questions in part (a) have negative answers if one is looking for\nsmooth concordances; see [CFHH13], [Yas17], [MP18]. With reference\nto Question (a)(iii) and part (b), note that a concordance would preserve\nthe free homotopy class (and hence homology class) of a meridian.\n(3) Part (b) has a negative answer without the assumption on the meridian,\nsee [Yas17]. There are no examples in the literature of homeomorphisms,\nsatisfying the assumption on the meridian, between 0-surgeries on distinct\nknots.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [KM78] Robion Kirby and Paul Melvin. Slice knots and property R. Invent. Math., 45(1):57– 59, 1978. doi:10.1007/BF01406223.\n- [CFHH13] Tim D. Cochran, Bridget D. Franklin, Matthew Hedden, and Peter D. Horn. Knot concordance and homology cobordism. Proc. Amer. Math. Soc., 141(6):2193–2208, 2013. doi:10.1090/S0002-9939-2013-11471-1.\n- [Yas17] Kouichi Yasui. Corks, exotic 4-manifolds and knot concordance, 2017. arXiv:1505.02551.\n- [MP18] Allison N. Miller and Lisa Piccirillo. Knot traces and concordance. J. Topol., 11(1):201–220, 2018. doi:10.1112/topo.12054.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2718, "problem_number": "KP-1.59", "title": "Kirby Problem 1.59", "statement": "Let $K \\subset S^{3}$ be a slice knot.\n(a) Determine the set $\\mathcal{R}(K)$ of ribbon disks bounded by $K$ modulo isotopy.\n(b) Determine the set $\\mathcal{D}(K)$ of slice disks bounded by $K$ modulo isotopy and\nlocal knotting.\n(c) Determine the set $\\mathcal{H}\\mathcal{R}(K)$ of homotopy-ribbon disks bounded by $K$ modulo\nisotopy.\n(d) If $K$ is fibered, determine the set $\\mathcal{F}\\mathcal{H}\\mathcal{R}(K)$ of fibered, homotopy-ribbon\ndisks bounded by $K$ in some homotopy 4-ball, modulo isotopy.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.59.\n\nLiterature notes:\n(1) There is no knot for which part (a) has been solved. A knot with $|\\mathcal{R}(K)| =$\n0 would be a counter-example to the Slice-Ribbon Conjecture. It is un-\nknown whether or not $|\\mathcal{R}(\\operatorname{U})| = 1$ for the unknot $\\operatorname{U}$, though Scharlemann\nhas shown that any non-standard ribbon disk for the unknot must have at\nleast three minima [Sch85a]. It is known that $\\mathcal{R}(Q_{p,2})$ is infinite [MZ25],\nwhere $Q_{p,q} = T_{p,q}\\#\\overline{T}_{p,q}$ is the connected sum of a torus knot with its mir-\nror.\n(2) There are many knots that bound isotopic ribbon disks that are dis-\ntinct when considered up to isotopy rel. boundary: Any knot of the form\n$J\\# - J$ bounds infinitely many isotopic ribbon disks that are pairwise\nnon-isotopic rel. boundary, as follows from Zeeman’s twist-spinning con-\nstruction [Zee65]; see, for example, [MZ25, Proposition 2.2]. Addition-\nally, many simple knots such as the stevedore knot $6_{1}$ and pretzel knots\n\n$P(p, -p, p)$ bound pairs of such disks. The problems of studying disks\nmodulo isotopy and modulo isotopy rel. boundary are closely related via\nan understanding of the symmetries of the boundary knot; see the proofs\nof [JZ20b, Lemma 2.2], [MZ25, Theorem 1.5], and [MM25b, Theo-\nrem 4.2].\n(3) The set $\\mathcal{D}(K)$ has recently been studied using modern invariants [JZ20b],\nclassical invariants [MP19], and geometric techniques [MM25b]. It is\nimmediate that $|\\mathcal{D}(\\operatorname{U})| = 1$ for $\\operatorname{U}$ the unknot, but $\\mathcal{D}(K)$ is not known for\nany nontrivial knot $K$.\n(4) Consider the map $\\iota_{K}: \\mathcal{H}\\mathcal{R}(K) \\to \\mathcal{D}(K)$. Is there a knot $K$ such that\nthis map is injective, surjective, or has finite image? See [MM25b, Sec-\ntion 1.1]. There is no knot for which part (c) has been solved.\n(5) Given the apparent difficulty in solving the above problems, it is rea-\nsonable to look for more tractable alternatives in which there is more\nstructure, such as part (d).\nCasson and Gordon found a beautiful connection between fibered,\nhomotopy-ribbon disks bounded by a fibered knot $K$ and handlebody\nextensions of the closed monodromy of $K$ [CG83a].\nHere, the extra\nstructure yields more positive results. First, $|\\mathcal{F}\\mathcal{H}\\mathcal{R}(\\operatorname{U})| = 1$. Second, for\nall coprime $p > q > 1$, the set $\\mathcal{F}\\mathcal{H}\\mathcal{R}(Q_{p,q})$ has been explicitly determined;\nthe set $\\mathcal{F}\\mathcal{H}\\mathcal{R}_{\\partial}(Q_{p,q})$ of fibered, homotopy-ribbon disks modulo isotopy\nrel. boundary is in explicit bijection with $\\{c/d \\in \\mathbb{Q}: c$ even $\\}$ [MZ25].\n\nReferences cited:\n- [Sch85a] Martin Scharlemann. Smooth spheres in $\\mathbb{R}^{4}$ with four critical points are standard. Invent. Math., 79(1):125–141, 1985. doi:10.1007/BF01388659.\n- [MZ25] Jeffrey Meier and Alexander Zupan. Knots bounding nonisotopic ribbon disks. J. Topol., 18(4):Paper No. e70047, 18, 2025. doi:10.1112/topo.70047.\n- [Zee65] E. C. Zeeman. Twisting spun knots. Trans. Amer. Math. Soc., 115:471–495, 1965. doi:10.2307/1994281.\n- [JZ20b] András Juhász and Ian Zemke. Distinguishing slice disks using knot Floer homology. Selecta Math. (N.S.), 26(1):Paper No. 5, 18, 2020. doi:10.1007/s00029-019-0531-6.\n- [MM25b] Jeffrey Meier and Allison N. Miller. Slice disks modulo local knotting, 2025. arXiv: 2503.09870.\n- [MP19] Allison N. Miller and Mark Powell. Stabilization distance between surfaces. Enseign. Math., 65(3-4):397–440, 2019. doi:10.4171/lem/65-3/4-4.\n- [CG83a] A. J. Casson and C. McA. Gordon. A loop theorem for duality spaces and fibred ribbon knots. Invent. Math., 74(1):119–137, 1983. doi:10.1007/BF01388533.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2719, "problem_number": "KP-1.60", "title": "Kirby Problem 1.60", "statement": "Is there a knot in $S^{3}$ that is not smoothly slice in $B^{4}$ but is\nsmoothly slice in an integer homology ball? What about a $\\mathbb{Z}$/2-homology ball?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.60.\n\nLiterature notes:\n(1) As discussed in [FGMW10], one strategy to disprove the smooth Poincaré\nconjecture is to find a homotopy four-sphere $X$ and a knot $K$ in $S^{3}$ that is\nsmoothly slice in $X - B^{4}$ but not in $B^{4}$; see also Problem 4.1. It is worth\nremarking that just about all known slice obstructions necessarily vanish\nfor a knot that is slice in an integer homology ball. One invariant that\ncan potentially make this distinction is the $s$-invariant from Khovanov\nhomology [Ras10]. As a warm-up, it would be interesting to find a knot\n$K$ with $s(K) \\neq 0$ that is smoothly slice in a rational homology ball.\n(2) Fintushel-Stern showed the figure-eight knot is smoothly slice in a rational\nhomology ball with $\\pi_{1} = \\mathbb{Z}/2$ [FS84], but is not even topologically slice\nin $B^{4}$ by the Fox-Milnor condition. For all known examples of knots that\nare slice in a rational ball but not $B^{4}$, the rational ball has 2-torsion\nin $H_{1}$. Kawauchi shows that all strongly negative-amphichiral knots are\nrationally slice [Kaw09] and Levine showed they are slice in the same\nrational homology ball $Z_{0}$ [Lev23]; in fact, he asks if it is possible that\nall rationally slice knots are slice in boundary sums of $Z_{0}$.\n\nReferences cited:\n- [FGMW10] Michael Freedman, Robert Gompf, Scott Morrison, and Kevin Walker. Man and machine thinking about the smooth 4-dimensional Poincaré conjecture. Quantum Topol., 1(2):171–208, 2010. doi:10.4171/QT/5.\n- [Ras10] Jacob Rasmussen. Khovanov homology and the slice genus. Invent. Math., 182(2):419–447, 2010. doi:10.1007/s00222-010-0275-6.\n- [FS84] Ronald Fintushel and Ronald J. Stern. A µ-invariant one homology 3-sphere that bounds an orientable rational ball. In Four-manifold theory (Durham, N.H., 1982), volume 35 of Contemp. Math., pages 265–268. Amer. Math. Soc., Providence, RI, 1984. doi:10.1090/conm/035/780582.\n- [Kaw09] Akio Kawauchi. Rational-slice knots via strongly negative-amphicheiral knots. Commun. Math. Res., 25(2):177–192, 2009.\n- [Lev23] Adam Simon Levine. A note on rationally slice knots. New York J. Math., 29:1363– 1372, 2023. https://nyjm.albany.edu/j/2023/29-52p.pdf.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2720, "problem_number": "KP-1.61", "title": "Kirby Problem 1.61", "statement": "A knot in $S^{3}$ bounds a topological disk in $B^{4}$ by coning (not\nnecessarily locally flat); this problem asks about topological disks that a knot in a\nhomology sphere $Y$ might bound in a homology or homotopy ball bounded by $Y$ .\n(a) Given a locally flat knot in an integer homology 3-sphere, is it concordant\nto a knot in $S^{3}$, via a locally flat annulus in a topological integer homology\ncobordism? Variation: Can the homology cobordism be taken to be simply\nconnected?\n(b) Given an arbitrary knot in an integer homology 3-sphere $Y$ , does it bound\nan embedded disc (not necessarily locally flat) in $\\Delta$, the Freedman filling\nof $Y$ ? Variation: Is there such a disk in a homology ball bounded by $Y$ ?\n(c) Is there a link in an integer homology 3-sphere, each of whose components\nis independently concordant to a knot in $S^{3}$, that is not itself concordant\nto a link in $S^{3}$.\n(d) Suppose that $\\mathbb{Z}_{n}$ acts semi-freely on the homology sphere $Y$ , with fixed\npoint set a knot $K$. Does the action of $\\mathbb{Z}_{n}$ extend semi-freely over $\\Delta$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.61.\n\nLiterature notes:\n(1) By the proof of [Fre82, Theorem 1.4'] every integer homology 3-sphere\nbounds a compact, contractible 4-manifold $\\Delta$, which we call the Freed-\nman filling of $Y$ . It is known to be unique up to homeomorphism, by\nFreedman’s $h$-cobordism theorem [Fre82, Theorems 1.3 and 1.6].\n(2) The smooth analogue of (a) was disproved by A. Levine [Lev16], answer-\ning Problem 1.31 in [Kir97]; see also [HLL22b, Zho21]. An affirmative\nanswer to (a) gives an affirmative answer to (b) for locally flat knots,\nvia an embedding with an isolated, piecewise linear singularity obtained\nby coning off the knot in $S^{3}$.\nNote that topological embeddings need\nnot be approximable by piecewise linear or locally flat embeddings. Such\na wild topological embedding of a disk was constructed by Giffen using\na shift-spinning construction in unpublished work, explained in [DV09,\nSection 6.6]. Recent work of Davis [Dav20, Dav25] gives evidence in\nsupport of an affirmative answer to (a), in terms of the solvable filtration\nof the knot concordance group [COT03].\n(3) Part (d) is related to questions about extending topological group actions,\nwithout any requirement of local linearity.\nRecall [FQ90] that a free\naction of a cyclic group on a homology sphere $Y$ extends to an action\non the Freedman filling $\\Delta$ of $Y$ with one fixed point; the action is not\nnecessarily locally linear at that point.\nSuppose instead that $\\mathbb{Z}_{n}$ acts\nsemi-freely on $Y$ , with fixed point set a knot $K$. If $n$ is a prime, say $p$,\nthen the fixed point set of the extension would be a $\\mathbb{Z}_{p}$-homology disk\nwith boundary equal to $K$. So in some weak sense, $K$ would be slice in\n$\\Delta$. This problem is rather different in character from the smooth version,\ndiscussed in problem 3.75.\n\nReferences cited:\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [Lev16] Adam Simon Levine. Nonsurjective satellite operators and piecewise-linear concordance. Forum Math. Sigma, 4:Paper No. e34, 47, 2016. doi:10.1017/fms.2016.31.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [HLL22b] Jennifer Hom, Adam Simon Levine, and Tye Lidman. Knot concordance in homology cobordisms. Duke Math. J., 171(15):3089–3131, 2022. doi:10.1215/00127094-2021-0110.\n- [Zho21] Hugo Zhou. Homology concordance and an infinite rank free subgroup. J. Topol., 14(4):1369–1395, 2021. doi:10.1112/topo.12211.\n- [DV09] Robert J. Daverman and Gerard A. Venema. Embeddings in manifolds, volume 106 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2009. doi:10.1090/gsm/106.\n- [Dav20] Christopher W. Davis. Topological concordance of knots in homology spheres and the solvable filtration. J. Topol., 13(1):343–355, 2020. doi:10.1112/topo.12126.\n- [Dav25] Christopher W. Davis. Whitney tower concordance and knots in homology spheres. Algebr. Geom. Topol., 25(6):3503–3521, 2025. doi:10.2140/agt.2025.25.3503.\n- [COT03] Tim D. Cochran, Kent E. Orr, and Peter Teichner. Knot concordance, Whitney towers and L2-signatures. Ann. of Math. (2), 157(2):433–519, 2003. doi:10.4007/annals.2003.157.433.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2721, "problem_number": "KP-1.62", "title": "Kirby Problem 1.62", "statement": "(a) Are all good boundary links topologically slice? Freely topologically slice?\n(b) A special case of interest: Is the Whitehead double of the Borromean rings\n(with any choice of clasps) topologically slice?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.62.\n\nLiterature notes:\n(1) A boundary link $(L, \\varphi)$ is an $n$-component link $L \\subseteq S^{3}$ together with a\nsurjection $\\varphi: \\pi_{1}(S^{3}\\setminus L) \\twoheadrightarrow F_{n}$, sending a set of meridians of $L$ to a free\ngenerating set of the free group on $n$ generators. A link $L$ admits such\na homomorphism $\\varphi$ if and only if the components of $L$ bound pairwise\ndisjoint Seifert surfaces in $S^{3}$ [Smy66] (see also [CS80, Proposition 1.1]).\nFollowing [FQ90, Section 11.7C], a boundary link $(L, \\varphi)$ is said to be a\ngood boundary link if the kernel of $\\varphi$ is a perfect group. Note that a 1-\ncomponent good boundary link is a knot with Alexander polynomial one.\nWhitehead doubles of links with vanishing pairwise linking numbers are\ngood boundary links.\n(2) Part (a) is posed in [Kir97, Problem 1.36], with a slightly different def-\ninition of good boundary links than appears here. The first half of that\nproblem asked whether Alexander polynomial one knots are topologically\nslice. The answer to that question is now known to be yes [GT04], [FQ90,\nTheorem 11.7B], [Fre84, Theorem 7] (see also [BPR21, Theorem 1.14]\nand [PR21, Section 21.6.3]).\n(3) If the surgery sequence is exact (see Problem 4.46), then a link in $S^{3}$\nis freely slice if and only if it is a good boundary link [FQ90, Corol-\nlary 11.7C]. More surprisingly, the topological surgery sequence in dimen-\nsion four is defined and exact if and only if all good boundary links are\nfreely slice [FQ90, Corollary 12.3C] (see also [KOPR21a, Section 23.2.1]).\n(4) [CKP20, Corollary 2.2] gives a characterization of good boundary links\nin terms of Seifert matrices. This is very useful in confirming whether\na given boundary link is good. For example, this gives an easy method\nto verify that Whitehead doubles of links with vanishing pairwise linking\nnumbers are good boundary links.\n(5) Several families of good boundary links have been shown to be freely\ntopologically slice. The most general result is due to Cha, Kim, and Pow-\nell [CKP20, Theorem A], generalizing previous work focusing on White-\nhead doubles, e.g. in [FT95b].\n(6) Not all good boundary links are smoothly slice, even with more than one\ncomponent.\nFor example, Levine [Lev12] showed that the Whitehead\ndouble of the Borromean rings, with all positive clasps, is not smoothly\nslice. The question of whether the Whitehead double of the Borromean\nrings is topologically slice appeared as Problem 4.46 in [Kir97], and we\nrestate it here in part (b) as a special case of the problem.\n\nQuestion. Is the Whitehead double of the Borromean rings (with any\nchoice of clasps) topologically slice?\n\nReferences cited:\n- [Smy66] N Smythe. Boundary links. In Topology Seminar, Wisconsin, 1965, volume 60 of Annals of Mathematics Studies, pages 69–72. Princeton University Press, Princeton, N.J., 1966. Edited by R. H. Bing and R. J. Bean.\n- [CS80] Sylvain E. Cappell and Julius L. Shaneson. Link cobordism. Comment. Math. Helv., 55(1):20–49, 1980. doi:10.1007/BF02566673.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [GT04] Stavros Garoufalidis and Peter Teichner. On knots with trivial Alexander polynomial. J. Differential Geom., 67(1):167–193, 2004. http://projecteuclid.org/euclid.jdg/1099587731.\n- [Fre84] Michael H. Freedman. The disk theorem for four-dimensional manifolds. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 647–663. PWN, Warsaw, 1984.\n- [BPR21] Stefan Behrens, Mark Powell, and Arunima Ray. Context for the disc embedding theorem. In The disc embedding theorem, pages 1–26. Oxford Univ. Press, Oxford, 2021.\n- [PR21] Mark Powell and Arunima Ray. The development of topological 4-manifold theory. In The disc embedding theorem, pages 295–330. Oxford Univ. Press, Oxford, 2021.\n- [KOPR21a] Min Hoon Kim, Patrick Orson, JungHwan Park, and Arunima Ray. Good groups. In The disc embedding theorem, pages 273–282. Oxford Univ. Press, Oxford, 2021.\n- [CKP20] Jae Choon Cha, Min Hoon Kim, and Mark Powell. A family of freely slice good boundary links. Math. Ann., 376(3-4):1009–1030, 2020. doi:10.1007/s00208-019-01907-3.\n- [FT95b] Michael H. Freedman and Peter Teichner. 4-manifold topology. II. Dwyer’s filtration and surgery kernels. Invent. Math., 122(3):531–557, 1995. doi:10.1007/BF01231455.\n- [Lev12] Adam Simon Levine. Slicing mixed Bing-Whitehead doubles. J. Topol., 5(3):713– 726, 2012. doi:10.1112/jtopol/jts019.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2722, "problem_number": "KP-1.63", "title": "Kirby Problem 1.63", "statement": "Is there a knot type with Legendrian representatives that do\nnot destabilize but have arbitrarily negative Thurston–Bennequin number?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.63.\n\nLiterature notes:\n(1) This is Etnyre-Ng [EN03, Question 44]; see also [CGH09, Question 61].\n\n(2) It is known by work of Etnyre and Honda that for any $N$ there is a\nknot type having peaks of height $N$ in its Legendrian mountain range\n[EH03]. For any choice of knot type and Thurston-Bennequin number,\nthere are finitely many distinct Legendrian representatives, so this ques-\ntion is equivalent to asking if there is a knot type with infinitely many\nnon-destabilizable representatives. This has been proved for many knot\ntypes, including the unknot [EF09], torus knots [EH01a], cables of torus\nknots [ELT12, LaF10], and twist knots [ENV13].\n(3) The finiteness of non-destabilizable Legendrian representatives problem\nis equivalent to whether there are finitely many non-destabilizable grid\ndiagrams for a given knot [DP13].\nIt is proved in [Dyn06] that any\nnon-minimal rectangular diagram of the unknot can be destabilized. This\ngives an algorithm to detect the unknot using monotonic simplification.\nA first step towards generalizing this result to arbitrary topological knots\nis to answer the question above.\n\nReferences cited:\n- [EN03] John B. Etnyre and Lenhard L. Ng. Problems in low dimensional contact topology. In Topology and geometry of manifolds (Athens, GA, 2001), volume 71 of Proc. Sympos. Pure Math., pages 337–357. Amer. Math. Soc., Providence, RI, 2003. doi: 10.1090/pspum/071/2024641.\n- [CGH09] Vincent Colin, Emmanuel Giroux, and Ko Honda. Finitude homotopique et isotopique des structures de contact tendues. Publ. Math. Inst. Hautes Études Sci., 109:245–293, 2009. doi:10.1007/s10240-009-0022-y.\n- [EH03] John B. Etnyre and Ko Honda. On connected sums and Legendrian knots. Adv. Math., 179(1):59–74, 2003. doi:10.1016/S0001-8708(02)00027-0.\n- [EF09] Yakov Eliashberg and Maia Fraser. Topologically trivial Legendrian knots. J. Symplectic Geom., 7(2):77–127, 2009. http://projecteuclid.org/euclid.jsg/1239974381.\n- [EH01a] John B. Etnyre and Ko Honda. Knots and contact geometry. I. Torus knots and the figure eight knot. J. Symplectic Geom., 1(1):63–120, 2001. http://projecteuclid.org/euclid.jsg/1092316299.\n- [ELT12] John B. Etnyre, Douglas J. LaFountain, and Bülent Tosun. Legendrian and transverse cables of positive torus knots. Geom. Topol., 16(3):1639–1689, 2012. doi:10.2140/gt.2012.16.1639.\n- [LaF10] Douglas J. LaFountain. Studying uniform thickness. I. Legendrian simple iterated torus knots. Algebr. Geom. Topol., 10(2):891–916, 2010. doi:10.2140/agt.2010.10.891.\n- [ENV13] John B. Etnyre, Lenhard L. Ng, and Vera Vértesi. Legendrian and transverse twist knots. J. Eur. Math. Soc. (JEMS), 15(3):969–995, 2013. doi:10.4171/JEMS/383.\n- [DP13] I. A. Dynnikov and M. V. Prasolov. Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions. Trans. Moscow Math. Soc., pages 97–144, 2013. doi:10.1090/s0077-1554-2014-00210-7.\n- [Dyn06] I. A. Dynnikov. Arc-presentations of links: monotonic simplification. Fund. Math., 190:29–76, 2006. doi:10.4064/fm190-0-3.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2723, "problem_number": "KP-1.64", "title": "Kirby Problem 1.64", "statement": "(a) Let $L \\subset (S^{3}, \\xi_{std})$ be a transverse link such that the branched double cover\n$(\\Sigma_{2}(L), \\xi_{L})$ is Stein fillable. Is $L$ transversely isotopic to the closure of a\nquasipositive braid?\n(b) Is the same true if $(\\Sigma_{n}(L), \\xi_{L})$ is Stein fillable for some $n > 2$?\n(c) Does Stein fillability of the branched cover imply that the slice-Bennequin\ninequality must be sharp for the given link?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.64.\n\nLiterature notes:\n(1) Here, $(S^{3}, \\xi_{std})$ denotes the standard contact structure on $S^{3}$.\nIf $L \\subset$\n$(S^{3}, \\xi_{std})$ is a transverse link, then the cyclic $n$-fold branched cover $\\Sigma_{n}(L)$\nhas an induced contact structure $\\xi_{L}, n \\geq 2$.\n(2) See Problems 1.69 and 1.70 for definitions, background on quasipositive\nbraids, and related questions.\n(3) A closed braid is quasipositive if and only if it can be represented as\na transverse intersection $S^{3} \\cap \\mathcal{V}$ for some smooth, complex curve $\\mathcal{V} \\subset$\n$\\mathbb{C}^{2}$ [Rud83, BO01].Thus if $L$ is isotopic to the closure of a quasiposi-\ntive braid, then $(\\Sigma_{n}(L), \\xi_{L})$ is Stein fillable, because it bounds the Stein\nmanifold obtained as the cover of $B^{4}$ branched over the complex curve\n$\\mathcal{V} \\cap B^{4}$. The question asks whether fillability of branched covers charac-\nterizes quasipositive braids. For more results on the branched covers of\ntransverse links, see [Pla06]. More discussion of relations between dif-\nferent monoids related to quasipositivity and contact geometry is given\nin [EVHM15].\n\nReferences cited:\n- [Rud83] Lee Rudolph. Algebraic functions and closed braids. Topology, 22(2):191–202, 1983. doi:10.1016/0040-9383(83)90031-9.\n- [BO01] Michel Boileau and Stepan Orevkov. Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe. C. R. Acad. Sci. Paris Sér. I Math., 332(9):825–830, 2001. doi:10.1016/S0764-4442(01)01945-0.\n- [Pla06] Olga Plamenevskaya. Transverse knots, branched double covers and Heegaard Floer contact invariants. J. Symplectic Geom., 4(2):149–170, 2006. doi:10.4310/jsg.2006.v4.n2.a2.\n- [EVHM15] John B. Etnyre and Jeremy Van Horn-Morris. Monoids in the mapping class group. In Interactions between low-dimensional topology and mapping class groups, volume 19 of Geom. Topol. Monogr., pages 319–365. Geom. Topol. Publ., Coventry, 2015. doi:10.2140/gtm.2015.19.319.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2724, "problem_number": "KP-1.65", "title": "Kirby Problem 1.65", "statement": "Decomposable Lagrangian cobordisms between Legendrian knots\nor links in $\\mathbb{R}^{3}$ are compositions of certain simple pieces admitting diagrammatic de-\nscriptions [EHK16]. Such cobordisms necessarily have no index 2 critical points.\nIs every exact Lagrangian cobordism without index 2 critical points Lagrangian iso-\ntopic to a decomposable Lagrangian cobordism?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.65.\n\nLiterature notes:\n(1) At the time of writing, a candidate for a non-decomposable cobordism is\ngiven in [CNS16, Conjecture 3.3]. This can be built using another ele-\nmentary construction, Guadagni’s tangle move, see [BLL $^{+}21$, GSY22],\nwhich could in principle produce additional non-decomposable cobordisms.\n(2) The hypotheses on index 2 critical points is necessary for a variety of\nreasons. For example, Lagrangian caps (see, e.g., [Lin16]) are, by defini-\ntion, not decomposable; for decomposability, the positive boundary of the\ncobordism must be nonempty or else index 2 critical points are necessary.\nIn the recent preprint [DRG24], an example of a non-decomposable La-\ngrangian concordance between knots is given. However, the authors there\nshow that the concordance must contain index 2 critical points.\n(3) Conway, Etnyre, and Tosun prove that decomposable Lagrangians are reg-\nular [CET21], which means there is a Weinstein structure on the cobor-\ndism where the Liouville vector field is tangent to the cobordism. Using\nthis, [EGL20] implies the exterior of a decomposable Lagrangian disk\nnecessarily has a Stein structure, and hence the exterior can be built out\nof particularly constrained elementary pieces, akin to the decomposable\nLagrangian itself. (It is worth pointing out that the non-decomposable\nexamples of [DRG24] above are not regular either.)\n(4) Here is a variant of the problem: Is every filling (exact Lagrangian cobor-\ndism with empty negative boundary) Lagrangian isotopic to a decom-\nposable Lagrangian cobordism? This question is sometimes stated more\ngenerally for exact Lagrangian cobordisms with negative boundary given\nby any (possibly empty) non-destabilizable Lagrangian link; the non-\ndestabilizability condition rules out the presence of a Lagrangian cap as\nin [Lin16].\nThe possibly non-decomposable candidate from [CNS16],\nwhich is a concordance with a Legendrian trefoil as its negative boundary,\ncan be glued to a filling of the trefoil to produce a potential counterexam-\nple to this version of the problem.\n\nReferences cited:\n- [EHK16] Tobias Ekholm, Ko Honda, and Tamás Kálmán. Legendrian knots and exact Lagrangian cobordisms. J. Eur. Math. Soc. (JEMS), 18(11):2627–2689, 2016. doi: 10.4171/JEMS/650.\n- [CNS16] Christopher Cornwell, Lenhard Ng, and Steven Sivek. Obstructions to Lagrangian concordance. Algebr. Geom. Topol., 16(2):797–824, 2016. doi:10.2140/agt.2016.16.797.\n- [BLL+21] Sarah Blackwell, Noémie Legout, Caitlin Leverson, Maÿlis Limouzineau, Ziva Myer, Yu Pan, Samantha Pezzimenti, Lara Simone Suárez, and Lisa Traynor. Constructions of Lagrangian cobordisms. In Research directions in symplectic and contact geometry and topology, volume 27 of Assoc. Women Math. Ser., pages 245– 272. Springer, Cham, [2021] ©2021. doi:10.1007/978-3-030-80979-9\\\\_5.\n- [GSY22] Roberta Guadagni, Joshua M. Sabloff, and Matthew Yacavone. Legendrian satellites and decomposable cobordisms. J. Knot Theory Ramifications, 31(13):Paper No. 2250071, 33, 2022. doi:10.1142/s0218216522500717.\n- [Lin16] Francesco Lin. Exact Lagrangian caps of Legendrian knots. J. Symplectic Geom., 14(1):269–295, 2016. doi:10.4310/JSG.2016.v14.n1.a10.\n- [DRG24] Georgios Dimitroglou Rizell and Roman Golovko. Instability of Legendrian knottedness, and non-regular Lagrangian concordances of knots, 2024. arXiv:2409.00290.\n- [CET21] James Conway, John B. Etnyre, and Bülent Tosun. Symplectic fillings, contact surgeries, and Lagrangian disks. Int. Math. Res. Not. IMRN, 2021(8):6020–6050, 2021. doi:10.1093/imrn/rny291.\n- [EGL20] Yakov Eliashberg, Sheel Ganatra, and Oleg Lazarev. Flexible Lagrangians. Int. Math. Res. Not. IMRN, 2020(8):2408–2435, 2020. doi:10.1093/imrn/rny078.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2725, "problem_number": "KP-1.66", "title": "Kirby Problem 1.66", "statement": "For Legendrian links $\\Lambda_{1}, \\Lambda_{2} \\subset (\\mathbb{R}^{3}, \\xi_{std})$, write $\\Lambda_{1} \\preceq \\Lambda_{2}$ if\nthere is an exact Lagrangian cobordism in the symplectization $\\mathbb{R} \\times \\mathbb{R}^{3}$ from $\\Lambda_{1}$ to\n$\\Lambda_{2}$. Is $\\preceq$ a partial order?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.66.\n\nLiterature notes:\n(1) One can ask an analogous question in any contact 3-manifold, as well as\nin higher dimensions.\n(2) The fact that $\\preceq$ is not symmetric was first shown by Chantraine [Cha15].\n\n(3) As for ribbon concordance in the smooth setting, many invariants of Leg-\nendrian knots are monotone under $\\preceq$. For example, the behavior of rul-\nings [CNS16], contact homology [Pan17a] and the contact invariant in\nmonopole Floer homology [BS18a] support this conjecture.\n(4) One can ask an analogous question about symplectic concordances be-\ntween transverse knots.\n\nReferences cited:\n- [Cha15] Baptiste Chantraine. Lagrangian concordance is not a symmetric relation. Quantum Topol., 6(3):451–474, 2015. doi:10.4171/QT/68.\n- [CNS16] Christopher Cornwell, Lenhard Ng, and Steven Sivek. Obstructions to Lagrangian concordance. Algebr. Geom. Topol., 16(2):797–824, 2016. doi:10.2140/agt.2016.16.797.\n- [Pan17a] Yu Pan. The augmentation category map induced by exact Lagrangian cobordisms. Algebr. Geom. Topol., 17(3):1813–1870, 2017. doi:10.2140/agt.2017.17.1813.\n- [BS18a] John A. Baldwin and Steven Sivek. Invariants of Legendrian and transverse knots in monopole knot homology. J. Symplectic Geom., 16(4):959–1000, 2018. doi:10.4310/JSG.2018.v16.n4.a3.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2726, "problem_number": "KP-1.67", "title": "Kirby Problem 1.67", "statement": "Given a Legendrian link in the standard contact $\\mathbb{R}^{3}$ besides the\nstandard unknot or Hopf link, classify its exact Lagrangian fillings up to compactly\nsupported Hamiltonian isotopy.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.67.\n\nLiterature notes:\nEliashberg-Polterovich [EP96] showed that the unknot has a single\nfilling. A recent preprint of Thomson claims that the Hopf link has exactly 2 fillings\n[Tho25]. These are the only links with known classification. The right-handed tre-\nfoil famously has 5 fillings [EHK16], distinguished through augmentations of the\nLegendrian contact homology. Casals and Gao showed there are examples of links\nwith infinitely many non-isotopic fillings [CG22]. Casals has a more general con-\njectured classification for some large families of links in [Cas22, Conjecture 5.1]. A\nspecial case would be that the $T(2, n)$ torus link has exactly $C_{n}$ fillings up to isotopy\nthrough exact Lagrangians, where $C_{n}$ is the $n^{th}$ Catalan number, generalizing the\ncount for the unknot and Hopf link. (See the work of Ekholm-Honda-Kalman for\nearlier questions on classifying Lagrangian fillings by augmentations of Legendrian\ncontact homology [EHK16], including for the $T(2, n)$ torus links.) A lower bound\nof $C_{n}$ in this case is due to Pan [Pan17b] and independently Shende-Treumann-\nWilliams-Zaslow [STWZ19].\n\nReferences cited:\n- [EP96] Y. Eliashberg and L. Polterovich. Local Lagrangian 2-knots are trivial. Ann. of Math. (2), 144(1):61–76, 1996. doi:10.2307/2118583.\n- [Tho25] Bryce Thomson. The Legendrian Hopf Link has exactly two Lagrangian fillings, 2025. arXiv:2506.15111.\n- [EHK16] Tobias Ekholm, Ko Honda, and Tamás Kálmán. Legendrian knots and exact Lagrangian cobordisms. J. Eur. Math. Soc. (JEMS), 18(11):2627–2689, 2016. doi: 10.4171/JEMS/650.\n- [CG22] Roger Casals and Honghao Gao. Infinitely many Lagrangian fillings. Ann. of Math. (2), 195(1):207–249, 2022. doi:10.4007/annals.2022.195.1.3.\n- [Cas22] Roger Casals. Lagrangian skeleta and plane curve singularities. J. Fixed Point Theory Appl., 24(2):Paper No. 34, 43, 2022. doi:10.1007/s11784-022-00939-8.\n- [Pan17b] Yu Pan. Exact Lagrangian fillings of Legendrian $(2,n)$ torus links. Pacific J. Math., 289(2):417–441, 2017. doi:10.2140/pjm.2017.289.417.\n- [STWZ19] Vivek Shende, David Treumann, Harold Williams, and Eric Zaslow. Cluster varieties from Legendrian knots. Duke Math. J., 168(15):2801–2871, 2019. doi: 10.1215/00127094-2019-0027.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2727, "problem_number": "KP-1.68", "title": "Kirby Problem 1.68", "statement": "Determine the smooth knot types that have Legendrian repre-\nsentatives with orientable exact Lagrangian fillings.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.68.\n\nLiterature notes:\n(1) This problem appears in [HS15] together with a conjecture that such fill-\nings exist if and only if the knot type $K$ is quasi-positive and tb $(K) =$\n$-$ deg $_{a}P_{K} - 1$, where tb $(K)$ denotes the maximal Thurston-Bennequin\nnumber of a Legendrian representative of $K$ and $P_{K}(a, z)$ the HOM-\nFLYPT polynomial.\nThese conditions are necessary for existence of a\nfilling (see [HS15] and references therein) as is the sharpness of the slice-\nBennequin inequality, tb $(K) = 2g_{4}(K)-1$, as shown in [Cha10, Theorem\n1.4].\n(2) For alternating knots [CNS16], an answer is that a filling exists if and\nonly if the knot type is positive. In general, constructions of fillings exist\nfor all positive knots [HS15] and all almost positive knots with diagrams\nof Type II [Tag19]. A variant of the question allows for nonorientable\nLagrangian fillings; see [CCPR $^{+}24$].\n\n(3) For comparison, the analogous question involving transverse knots has a\nsuccinct answer: Transverse knots with symplectic fillings exist within\na smooth knot type if and only if the knot type is quasi-positive; see\n[Rud83, BO01].\n\nReferences cited:\n- [HS15] Kyle Hayden and Joshua M. Sabloff. Positive knots and Lagrangian fillability. Proc. Amer. Math. Soc., 143(4):1813–1821, 2015. doi:10.1090/S0002-9939-2014-12365-3.\n- [Cha10] Baptiste Chantraine. Lagrangian concordance of Legendrian knots. Algebr. Geom. Topol., 10(1):63–85, 2010. doi:10.2140/agt.2010.10.63.\n- [CNS16] Christopher Cornwell, Lenhard Ng, and Steven Sivek. Obstructions to Lagrangian concordance. Algebr. Geom. Topol., 16(2):797–824, 2016. doi:10.2140/agt.2016.16.797.\n- [Tag19] Keiji Tagami. On the Lagrangian fillability of almost positive links. J. Korean Math. Soc., 56(3):789–804, 2019. doi:10.4134/JKMS.j180399.\n- [CCPR+24] Linyi Chen, Grant Crider-Phillips, Braeden Reinoso, Joshua Sabloff, and Leyu Yao. Non-orientable Lagrangian fillings of Legendrian knots. Math. Proc. Cambridge Philos. Soc., 176(1):123–153, 2024. doi:10.1017/s0305004123000440.\n- [Rud83] Lee Rudolph. Algebraic functions and closed braids. Topology, 22(2):191–202, 1983. doi:10.1016/0040-9383(83)90031-9.\n- [BO01] Michel Boileau and Stepan Orevkov. Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe. C. R. Acad. Sci. Paris Sér. I Math., 332(9):825–830, 2001. doi:10.1016/S0764-4442(01)01945-0.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2728, "problem_number": "KP-1.69", "title": "Kirby Problem 1.69", "statement": "Let $L \\subset (S^{3}, \\xi_{std})$ be a transverse link with\n\n$$\nsl_{\\Sigma}(L) = -\\chi(\\Sigma),\n$$\n\nfor some Seifert surface $\\Sigma$. Must $L$ be strongly quasipositive?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.69.\n\nLiterature notes:\n(1) The self-linking number of an oriented link transverse to the standard\ncontact structure $\\xi_{std}$ on $S^{3}$ satisfies the well-known Bennequin bound\n[Ben83, Etn05]:\n\n$$\nsl_{\\Sigma}(L) \\leq -\\chi(\\Sigma),\n$$\n\nwhere $\\Sigma$ is any Seifert surface for $L$. This problem therefore attempts to\ncharacterize when this bound is sharp.\n(2) Strongly quasipositive links are those which possess a quasipositive Seifert\nsurface, constructed from parallel disks by attaching bands with positive\nhalf-twists [Rud92]. Such a surface is properly isotopic to the intersection\nof the 4-ball with a non-singular complex curve [Rud83]. It is known\nthat if a link is strongly quasipositive, or bounds a complex curve of\nEuler characteristic equal to that of some Seifert surface for the link, then\nthe Bennequin bound is sharp (see [Shu07, Proposition 1.G] or [Hed10,\nProof of Theorem 1.5] for a proof).\n(3) In light of the above remark, one can ask if sharpness of the Bennequin\nbound implies $\\Sigma$ is quasipositive.\nIf so, then it follows that that any\nminimal genus Seifert surface for $L$ is quasipositive. This is because the\nself-linking number depends only on the relative homology class of $\\Sigma$,\nhence the Bennequin bound will be sharp for any other minimal genus\nSeifert surface $\\Sigma'$.\n(4) The question in this problem has an affirmative answer for fibered links\n[Hed10], links represented by minimal-index braids with fractional Dehn\ntwist coefficient $> 1$ [IK19], 3-braids [IK19], positive knots [Rud99],\nalmost positive knots [FLL23], and knots whose canonical and Seifert\ngenus agree [FLL23]. Hedden and Tovstopyat-Nelip have announced an\naffirmative answer for links with two dimensional top group of Heegaard\nknot Floer homology.\n(5) These notions generalize to arbitrary contact 3-manifolds, with the Ben-\nnequin bound generalized by Eliashberg to tight contact 3-manifolds [Eli92]\nand strong quasipositivity generalized via braids in open books [BEH $^{+}15$,\nIK19, Hay21b]. Geometrically, the class of quasipositive Seifert surfaces\nis the same as the class of ribbons of Legendrian graphs [BI09, Hay22].\nOne can therefore ask the following more general question: Suppose there\nexists a transverse link $L \\subset (Y, \\xi)$ for which\n\n$$\nsl_{\\xi,\\Sigma}(Y, L) = -\\chi(\\Sigma),\n$$\n\nfor some Seifert surface $\\Sigma$. Is $\\Sigma$ isotopic to the Legendrian ribbon of a\nLegendrian graph $\\Gamma \\subset (Y, \\xi)$? This generalized question has an affirmative\nanswer for overtwisted contact structures [BCV09].\n\nReferences cited:\n- [Ben83] Daniel Bennequin. Entrelacements et équations de Pfaff. In Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982), volume 107-108 of Astérisque, pages 87–161. Soc. Math. France, Paris, 1983.\n- [Etn05] John B. Etnyre. Legendrian and transversal knots. In Handbook of knot theory, pages 105–185. Elsevier B. V., Amsterdam, 2005. doi:10.1016/B978-044451452-3/50004-6.\n- [Rud92] Lee Rudolph. Constructions of quasipositive knots and links. III. A characterization of quasipositive Seifert surfaces. Topology, 31(2):231–237, 1992. doi:10.1016/0040-9383(92)90017-C.\n- [Rud83] Lee Rudolph. Algebraic functions and closed braids. Topology, 22(2):191–202, 1983. doi:10.1016/0040-9383(83)90031-9.\n- [Shu07] Alexander N. Shumakovitch. Rasmussen invariant, slice-Bennequin inequality, and sliceness of knots. J. Knot Theory Ramifications, 16(10):1403–1412, 2007. doi: 10.1142/S0218216507005889.\n- [Hed10] Matthew Hedden. Notions of positivity and the Ozsváth-Szabó concordance invariant. J. Knot Theory Ramifications, 19(5):617–629, 2010. doi:10.1142/S0218216510008017.\n- [IK19] Tetsuya Ito and Keiko Kawamuro. The defect of the Bennequin-Eliashberg inequality and Bennequin surfaces. Indiana Univ. Math. J., 68(3):799–833, 2019. doi:10.1512/iumj.2019.68.7662.\n- [Rud99] Lee Rudolph. Positive links are strongly quasipositive. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 555–562. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.555.\n- [FLL23] Peter Feller, Lukas Lewark, and Andrew Lobb. Almost positive links are strongly quasipositive. Math. Ann., 385(1-2):481–510, 2023. doi:10.1007/s00208-021-02328-x.\n- [Eli92] Yakov Eliashberg. Contact 3-manifolds twenty years since J. Martinet’s work. Ann. Inst. Fourier (Grenoble), 42(1-2):165–192, 1992. URL: http://www.numdam.org/item?id=AIF 1992 42 1-2 165 0.\n- [BEH+15] R. İnanç Baykur, John Etnyre, Matthew Hedden, Keiko Kawamuro, and Jeremy Van Horn-Morris. Contact and symplectic geometry and the mapping class groups. American Institute of Mathematics, Official report of the 2nd SQuaRE meeting, 2015.\n- [Hay21b] Kyle Hayden. Quasipositive links and Stein surfaces. Geom. Topol., 25(3):1441– 1477, 2021. doi:10.2140/gt.2021.25.1441.\n- [BI09] Sebastian Baader and Masaharu Ishikawa. Legendrian graphs and quasipositive diagrams. Ann. Fac. Sci. Toulouse Math. (6), 18(2):285–305, 2009. URL: http: //afst.cedram.org/item?id=AFST 2009 6 18 2 285 0.\n- [Hay22] Kyle Hayden. Legendrian ribbons and strongly quasipositive links in an open book. J. Math. Pures Appl. (9), 165:42–57, 2022. doi:10.1016/j.matpur.2022.07.002.\n- [BCV09] Sebastian Baader, Kai Cieliebak, and Thomas Vogel. Legendrian ribbons in overtwisted contact structures. J. Knot Theory Ramifications, 18(4):523–529, 2009. doi:10.1142/S0218216509006999.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2729, "problem_number": "KP-1.70", "title": "Kirby Problem 1.70", "statement": "(a) Let $L \\subset (S^{3}, \\xi_{std})$ be a transverse link with\n\n$$\nsl_{\\Sigma}(L) = -\\chi(\\Sigma),\n$$\n\nfor some smooth surface $\\Sigma \\subset B^{4}$ bounded by $L$. Is $L$ quasipositive?\n(b) Is the class of strongly quasipositive links equal to the class of quasipositive\nlinks whose Seifert and smooth slice genera are equal?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.70.\n\nLiterature notes:\n(1) Quasipositivity implies that $L$ is isotopic to the transverse intersection\nof a complex curve in $\\mathbb{C}^{2}$ with the unit 3-sphere [Rud83], and any link\narising from such an intersection is quasipositive [BO01]. It is not true,\nhowever, that any surface in $B^{4}$ bounding $L$ realizing equality in the slice-\nBennequin bound is isotopic to a subsurface of a complex curve. Indeed,\na complex curve is ribbon (in the sense that the radius function on the 4-\nball, restricted to the curve, can be assumed to be Morse without critical\npoints of index 2).\nOn the other hand, tubing a complex curve to a\nsmoothly knotted 2-sphere in its complement will produce a surface that\nis not ribbon, yet will still realize equality in the slice-Bennequin bound.\nOne can ask, however, whether a ribbon surface realizing equality in the\nslice-Bennequin bound is properly isotopic to a piece of complex curve.\n(2) Since quasipositive links have transverse representatives for which the\nslice-Bennequin bound is sharp (again, see [Shu07, Proposition 1.G] or\n[Hed10, Proof of Theorem 1.5]) we observe that Problem 1.69 implies an\naffirmative answer to part (b).\n(3) Part (b) is asking whether there exists a link $L$ bounding a complex curve\nwhose Euler characteristic is the same as that of some Seifert surface\nfor $L$, but for which no complex curve bounded by $L$ is isotopic to a\nSeifert surface. The existence of such a link would answer part (b) in\nthe negative, from which it would follow that Problem 1.69 is false. Note\nthat part (b) has an affirmative answer for fibered links, by [Hed10].\nSince complex curves are ribbon and Seifert surfaces are ribbon-immersed\nwithout singularities, this problem can be thought of as asking about the\nminimum number of ribbon singularities required across complex curves\nbounded by $L$.\n\nReferences cited:\n- [Rud83] Lee Rudolph. Algebraic functions and closed braids. Topology, 22(2):191–202, 1983. doi:10.1016/0040-9383(83)90031-9.\n- [BO01] Michel Boileau and Stepan Orevkov. Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe. C. R. Acad. Sci. Paris Sér. I Math., 332(9):825–830, 2001. doi:10.1016/S0764-4442(01)01945-0.\n- [Shu07] Alexander N. Shumakovitch. Rasmussen invariant, slice-Bennequin inequality, and sliceness of knots. J. Knot Theory Ramifications, 16(10):1403–1412, 2007. doi: 10.1142/S0218216507005889.\n- [Hed10] Matthew Hedden. Notions of positivity and the Ozsváth-Szabó concordance invariant. J. Knot Theory Ramifications, 19(5):617–629, 2010. doi:10.1142/S0218216510008017.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2730, "problem_number": "KP-1.71", "title": "Kirby Problem 1.71", "statement": "Does a Gordian unknot exist?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.71.\n\nLiterature notes:\nA physical link is a collection of disjoint embedded loops of given\nfinite lengths in a configuration with thickness at least 1.\nThickness has many\nequivalent definitions [GM99, CKS02], but an appealing one is the same as Fed-\nerer’s definition [Fed59] of reach: the infimal $r$ so that every point in $\\mathbb{R}^{3}$ within $r$\nof the curve has a unique nearest neighbor on the curve. Alternatively, for smooth\nknots, thickness 1 means that a radius-1 tubular neighborhood is also embedded\n[Sim02]. A physical isotopy preserves the length of each loop while respecting this\nthickness constraint. The thickness lower bound yields an upper bound 1 for the\ncurvature of a physical link, implying a least-length configuration exists in each\nphysical isotopy class [CKS02]. By a Gordian pair we mean physical link config-\nurations that are isotopic but not physically isotopic; see [Nab95, LN21] which\nprovide Gordian pairs of higher dimensional knots, the latter giving an explicit con-\nstruction of a Gordian unknot in $S^{4}$. Gordian pairs with two or more components\n[CH15, KK23] are known — even for unlinks [AH25, KK25] — but it remains\nunknown for knots, and especially for the unknot.\n\nReferences cited:\n- [GM99] Oscar Gonzalez and John H. Maddocks. Global curvature, thickness, and the ideal shapes of knots. Proc. Natl. Acad. Sci. USA, 96(9):4769–4773, 1999. doi:10.1073/pnas.96.9.4769.\n- [CKS02] Jason Cantarella, Robert B. Kusner, and John M. Sullivan. On the minimum ropelength of knots and links. Inventiones Mathematicae, 150:257–286, 2002. URL: http://www.springerlink.com/index/10.1007/s00222-002-0234-y, doi: 10.1007/s00222-002-0234-y.\n- [Fed59] Herbert Federer. Curvature measures. Trans. Amer. Math. Soc., 93:418–491, 1959. doi:10.2307/1993504.\n- [Sim02] Jonathan Simon. Physical knots. In Physical knots: knotting, linking, and folding geometric objects in $\\mathbb{R}^{3}$ (Las Vegas, NV, 2001), volume 304 of Contemp. Math., pages 1–30. Amer. Math. Soc., Providence, RI, 2002. doi:10.1090/conm/304/05181.\n- [Nab95] Alexander Nabutovsky. Non-recursive functions, knots “with thick ropes”, and self-clenching “thick” hyperspheres. Comm. Pure Appl. Math., 48(4):381–428, 1995. doi:10.1002/cpa.3160480402.\n- [LN21] Boris Lishak and Alexander Nabutovsky. Complexity of unknotting of trivial 2-knots. J. Topol. Anal., 13(3):623–657, 2021. doi:10.1142/$S^{1}$793525320500272.\n- [CH15] Alexander Coward and Joel Hass. Topological and physical link theory are distinct. Pacific J. Math., 276(2):387–400, 2015. doi:10.2140/pjm.2015.276.387.\n- [KK23] Rob Kusner and Wöden Kusner. A Gordian pair of links. Geom. Dedicata, 217(1):Paper No. 47, 2023. doi:10.1007/s10711-023-00783-1.\n- [AH25] José Ayala and Joel Hass. Gordian unlinks, 2025. arXiv:2502.08499.\n- [KK25] Rob Kusner and Wöden Kusner. The Xarax unlinks are physically Gordian, 2025. arXiv:CirclePackings,MinimalSurfaces, andDiscreteDifferentialGeometryPosterSessionAbstracts,ICERM2/11/25, https://app.icerm.brown.edu/assets/521/8574/8574\\_4906\\_021120251500\\_ Slides.pdf.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2731, "problem_number": "KP-1.72", "title": "Kirby Problem 1.72", "statement": "(The equilateral stuck unknots conjecture.). Are there equilat-\neral embedded polygons that are unknotted yet cannot be unknotted through polygons\npreserving edge lengths? Such an unknot is said to be stuck.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.72.\n\nLiterature notes:\n(1) The celebrated Carpenter’s Rule problem asks whether an embedded poly-\ngon in the plane may be deformed to a convex configuration through em-\nbedded polygons with the same edge lengths. Solutions in the affirmative\nwere given more or less simultaneously by Streinu [Str00] and Connelly,\nDemaine, and Rote [CDR03].\n(2) The natural extension of this theorem to PL knots in codimension two\nasks whether polygons of knot type $K$ may be deformed into one another\nthrough embedded polygons of the same edge lengths. For arbitrary edge\nlengths, the answer is known to be ‘no’: there are stuck unknots with as\nfew as six edges [CKS98, Tou01, AET04] (see Figure 2) which cannot be\nconvexified through embedded configurations with the same edge lengths.\n(3) Interestingly, all of the known examples of stuck unknots have very dif-\nferent edge lengths, which they seem to require in an essential way. This\nleads to the question: are there equilateral stuck unknots?\n\nFigure 2. Stuck unknots\n\nThere is some reason to believe that there are not. Khoi [Kho05]\nproved that the symplectic volume of the space of equilateral $n$-gons in\n$\\mathbb{R}^{3}$ is the largest symplectic volume of any space of $n$-gons with the same\ntotal length.\nThat is, in this sense, equilateral polygons are the most\nflexible polygons. On the other hand, it is quite difficult to imagine an\nalgorithm for unfolding an equilateral unknot of $n$ edges.\n\nReferences cited:\n- [Str00] Ileana Streinu. A combinatorial approach to planar non-colliding robot arm motion planning. In Proceedings 41st Annual Symposium on Foundations of Computer Science, pages 443–453, 2000. doi:10.1109/SFCS.2000.892132.\n- [CDR03] Robert Connelly, Erik D. Demaine, and Günter Rote. Straightening polygonal arcs and convexifying polygonal cycles. Discrete \\& Computational Geometry, 30:205– 239, 2003. doi:10.1007/s00454-003-0006-7.\n- [CKS98] Jason Cantarella, Robert B. Kusner, and John M. Sullivan. Tight knot values deviate from linear relations. Nature, 392:237, 1998. URL: http://torus.math.uiuc.edu/jms/Papers/scicor/scicor.ps.gzpapers3://publication/uuid/A1C$D^{2}$FF8-BAEE-4CCC-A98C-BB5ECC1DC83A.\n- [Tou01] Godfried Toussaint. A new class of stuck unknots in Pol6. Beiträge Algebra Geom., 42(2):301–306, 2001.\n- [AET04] Greg Aloupis, Günter Ewald, and Godfried Toussaint. More classes of stuck unknotted hexagons. Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry, 45:429–434, 2004.\n- [Kho05] Vu The Khoi. On the symplectic volume of the moduli space of spherical and Euclidean polygons. Kodai Mathematical Journal, 28:199–208, 2005. doi:10.2996/kmj/1111588046.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2732, "problem_number": "KP-1.73", "title": "Kirby Problem 1.73", "statement": "(The 15 pearls conjecture). Is the pearl number of the trefoil\nequal to 15?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.73.\n\nLiterature notes:\n(1) A pearl necklace with $n$ pearls is an embedding of an equilateral $n$-gon\nin $\\mathbb{R}^{3}$ so that each vertex is surrounded by a sphere with diameter equal\nto the edgelength and the interiors of all the spheres are disjoint. This\nis a common off-lattice model of a self-avoiding polygon, usually called\nthe hard sphere polymer (see [SJ23]). The pearl number of a knot $K$ is\nthe minimum number of pearls in any necklace whose polygon has knot\ntype $K$.\n(2) A self-avoiding polygon on a cubic lattice is a common model for random\nknots with self-interactions. The “minimum step numbers” are known for\nself-avoiding polygons on the cubic lattice for a number of knot types by a\ncombination of geometric arguments and computer enumeration [SIA $^{+}09$].\nFor instance, the minimal step number of the trefoil on the simple cubic\nlattice is 24 [Dia93].\n(3) Very little is known about the pearl number. Oshiro and Maehara [MO99]\ngive a construction for a 15-pearl trefoil and conjecture that this configu-\nration is minimal (so the pearl number of the trefoil is 15). Interestingly,\ntheir configuration is similar to the 15 step embedding of the trefoil on\nthe face-centered cubic lattice (shown in Figure 3) found by Rechnitzer\nand Janse van Rensburg [RR11], which those authors conjecture to be\nminimal. By putting balls of radius $1/2$ on the vertices of a realization\non the cubic lattice one obtains a hard sphere polymer. Thus the pearl\nnumber is bounded above by the length of any lattice representation.\nConversely, the pearl number is an upper bound on the equilateral stick\nnumber of any knot. The difference between the three integer invariants\n\nFigure 3. Face-centered cubic lattice embedding of trefoil\n\nshould be quite large for most knot types as can be seen for the trefoil\nwith an equilateral stick number of six [Jin97].\n(4) In [Mae07], Maehara proves that the pearl number of the trefoil is at\nleast 11.\n\nReferences cited:\n- [SJ23] Stefan Schnabel and Wolfhard Janke. Monte Carlo simulation of long hard-sphere polymer chains in two to five dimensions. Macromolecular Theory and Simulations, 32:2200080, 2023. doi:10.1002/mats.202200080.\n- [SIA+09] Rob Scharein, Kai Ishihara, Javier Arsuaga, Yuanan Diao, Koya Shimokawa, and Mariel Vazquez. Bounds for the minimum step number of knots in the simple cubic lattice. Journal of Physics A: Mathematical and Theoretical, 42, 2009. doi:10.1088/1751-8113/42/47/475006.\n- [Dia93] Yuanan Diao. Minimal knotted polygons on the cubic lattice. Journal of Knot Theory and Its Ramifications, 02:413–425, 1993. doi:10.1142/S0218216593000234.\n- [MO99] Hiroshi Maehara and Ai Oshiro. On knotted necklaces of pearls. European Journal of Combinatorics, 20:411–420, 1999. doi:10.1006/eujc.1998.0279.\n- [RR11] E. J. Janse Van Rensburg and Andrew Rechnitzer. Generalized atmospheric sampling of knotted polygons. Journal of Knot Theory and its Ramifications, 20:1145– 1171, 2011. doi:10.1142/S0218216511009170.\n- [Jin97] Gyo Taek Jin. Polygon indices and superbridge indices of torus knots and links. J. Knot Theory Ramifications, 6(2):281–289, 1997. doi:10.1142/S0218216597000170.\n- [Mae07] Hiroshi Maehara. On configurations of solid balls in 3-space: Chromatic numbers and knotted cycles. Graphs and Combinatorics, 23:307–320, 2007. doi:10.1007/s00373-007-0702-7.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2733, "problem_number": "KP-1.74", "title": "Kirby Problem 1.74", "statement": "How does ropelength behave under connected sum of knots?\nHere are two conjectures, the second a weakening of the first.\n(a) For any knot or link types $K_{1}$ and $K_{2}$,\nRop $(K_{1}\\#K_{2}) \\leq$ Rop $(K_{1}) +$ Rop $(K_{2}) - (4\\pi - 4).$\n(b) For any knot or link types $K_{1}$ and $K_{2}$,\n\n$$\nRop(K_{1}\\#K_{2}) \\leq Rop(K_{1}) + Rop(K_{2}) - c,\n$$\n\nwhere $c > 0$ is some constant independent of knot type.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.74.\n\nLiterature notes:\n(1) The (minimum) ropelength Rop $(K)$ of a knot is defined to be the small-\nest quotient of length and thickness (defined in Problem 1.71) among\nall rectifiable curves in $\\mathbb{R}^{3}$ realizing the knot type. It is known that ro-\npelength minimizers exist in every knot and link type [GMSvdM02,\nGL03, CKS02], but their shapes (and rope-lengths) are known only in\na few special cases of links with planar components. In all other case we\nonly have numerical results approximating the rope-length.\n(2) The ropelength minimizing Hopf link $2^{2}_{1}$ is a pair of circles of equal radius\nin orthogonal planes, passing through each other’s centers. Ropelength\nis scale-invariant, so we can assume that the circles have radius 2. Their\nthickness is then 1 and $\\operatorname{Rop}(2^{2}_{1})=8\\pi$. The ropelength-minimizing com-\nposite link $2^{2}_{1}\\#2^{2}_{1}$ (a chain of three linked circles) is composed of two round\n\ncircles in the same plane, each passing through a center of the semicircular\narc of a stadium curve in the orthogonal plane. This has ropelength $8\\pi$\n(for the circles) plus $4\\pi + 4$ (for the stadium curve), so\n\n$$\n\\operatorname{Rop}(2^{2}_{1}\\#2^{2}_{1}) = 12\\pi+4\n= \\operatorname{Rop}(2^{2}_{1})+\\operatorname{Rop}(2^{2}_{1})-(4\\pi-4),\n$$\n\nand we have saved some rope by splicing these links together. Katritch\net al. [KOP $^{+}97$] conjectured in 1997 that we can save at least as much\nrope on any connect sum; leading to Conjecture (a).\n(3) The work of Diao [Dia24] on alternating knots uses the braid index as\na lower bound on ropelength.\nThe braid index of a connected sum is\n$b(K_{1}\\#K_{2}) = b(K_{1}) + b(K_{2}) - 1$ for two alternating knots $K_{1}$ and $K_{2}$.\nThus the lower bound on ropelength of a connected sum of the two knots\n$b(K_{1}\\#K_{2})$ will be slightly less than the sum of the lower bounds of the\ntwo knots $K_{1}$ and $K_{2}$.\nHence it is worth considering the weaker version, Conjecture (b), with\nan undetermined constant.\n(4) In general, it would be important to have any theoretical lower bound on\nropelength for knot types that approximates the numerical results. This\nhas been done for the trefoil, but the techniques used do not generalize\nto other knot types, for which there remains a wide gap between the\nnumerical results and the theoretical lower bounds.\n\nReferences cited:\n- [GMSvdM02] O. Gonzalez, J. H. Maddocks, F. Schuricht, and H. von der Mosel. Global curvature and self-contact of nonlinearly elastic curves and rods. Calc. Var. Partial Differential Equations, 14(1):29–68, 2002. doi:10.1007/s005260100089.\n- [GL03] Oscar Gonzalez and R. De La Llave. Existence of ideal knots. Journal of Knot Theory and its Ramifications, 12:123–133, 2003. doi:10.1142/S0218216503002354.\n- [CKS02] Jason Cantarella, Robert B. Kusner, and John M. Sullivan. On the minimum ropelength of knots and links. Inventiones Mathematicae, 150:257–286, 2002. URL: http://www.springerlink.com/index/10.1007/s00222-002-0234-y, doi: 10.1007/s00222-002-0234-y.\n- [KOP+97] Vsevolod Katritch, Wilma K. Olson, Piotr Pieranski, Jacques Dubochet, and Andrzej Stasiak. Properties of ideal composite knots. Nature, 388:148–151, 1997. doi:10.1038/40582.\n- [Dia24] Yuanan Diao. The ropelength conjecture of alternating knots. Math. Proc. Cambridge Philos. Soc., 177(2):367–369, 2024. doi:10.1017/S0305004124000288.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2734, "problem_number": "KP-1.75", "title": "Kirby Problem 1.75", "statement": "(a) Find some knot energy on the space of smoothly embedded unknotted circles\nin $\\mathbb{S}^{3}$ for which all unknotted critical points are great circles.\n(b) Define a gradient flow for this knot energy which yields a deformation\nretract of the space of unknots to the subspace of great circles.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.75.\n\nLiterature notes:\n(1) These questions were posed by Freedman, He, and Wang [FHW94]. The\nmotivation is one of the many equivalent formulations of the Smale conjec-\nture mentioned in the Appendix to Hatcher’s 1983 paper (which actually\nproves this conjecture). Variation (7) states [Hat83, p. 606] that “the\nspace of smoothly embedded unknotted circles in $\\mathbb{S}^{3}$ deformation retracts\nonto the space of great circles in $\\mathbb{S}^{3}$, (i.e., O $(4)/$ O $(2) \\times$ O $(2))$.”Of course,\nit would be of some interest to construct a more or less explicit example\nof such a retract.\n(2) Beginning in the 1980s, several repulsive functionals, so-called knot ener-\ngies, have been defined in the pursuit of disentangling complicated knot-\nted curves and deforming them into ‘simpler’ curves within the same knot\nclass; see, e.g., the survey by Strzelecki et al. [SSvdM13] and references\ntherein. The most prominent example is the Möbius energy introduced\nby O’Hara [O’H91] in 1991.\n(3) As pointed out by Freedman, He, and Wang [FHW94], it is tempting to\nconjecture that a suitable gradient flow for a knot energy actually defines\n\na retraction as stated by Hatcher, i.e., it will deform any curve from the\nunknot class to a round circle. Of course, this can only work if, except\nfor the circles, there are no critical points within the unknot class; cf.\nBudney’s post [Bud16]. Currently, we do not know whether any of the\nmany smooth knot energies that have been proposed so far enjoys this\nproperty.\n\nReferences cited:\n- [FHW94] Michael H. Freedman, Zheng-Xu He, and Zhenghan Wang. Möbius energy of knots and unknots. Ann. of Math. (2), 139(1):1–50, 1994. doi:10.2307/2946626.\n- [Hat83] Allen E. Hatcher. A proof of the Smale conjecture, Diffp$S^{3}$q » $O(4)$. Ann. of Math. (2), 117(3):553–607, 1983. doi:10.2307/2007035.\n- [SSvdM13] Pawel Strzelecki, Marta Szumańska, and Heiko von der Mosel. On some knot energies involving Menger curvature. Topology Appl., 160(13):1507–1529, 2013. doi:10.1016/j.topol.2013.05.022.\n- [O’H91] Jun O’Hara. Energy of a knot. Topology, 30(2):241–247, 1991. doi:10.1016/0040-9383(91)90010-2.\n- [Bud16] Ryan Budney. A gorgeous but incomplete proof of “The Smale Conjecture”, 2016. https://ldtopology.wordpress.com/2016/10/02/a-gorgeous-but-incomplete-proof-of-the-smale-conjecture/. Accessed: 2024-01-25.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2735, "problem_number": "KP-1.76", "title": "Kirby Problem 1.76", "statement": "(a) Is there an algorithm to detect the unknot that runs in polynomial time\n(as a function of the number of crossings in an input diagram)?\n(b) What is the structure of the unknot Reidemeister graph?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.76.\n\nLiterature notes:\n(1) There are now many algorithms to detect the unknot. The first was due to\nHaken [Hak61] and used normal surfaces. Hass, Lagarias and Pippenger\n[HLP99] refined his methods to prove that unknot recognition lies in NP.\nDynnikov [Dyn06] used grid diagrams to find a very different algorithm.\nHe showed that any grid diagram for the unknot can be reduced to the\ntrivial diagram through a sequence of moves, none of which increases the\ngrid number.\n(2) There are now many invariants that detect the unknot, including Kho-\nvanov homology [KM11] and Heegaard Floer homology [OS04b]. How-\never, none of these invariants seem to be computable in polynomial time.\nA further significant result was proved by Kuperberg [Kup14] who showed\nthat unknot recognition lies in co-NP, assuming the Generalized Riemann\n\nHypothesis. In other words, if a knot is nontrivial, then there is an efficient\nway of certifying this. This result was proved unconditionally, removing\nthe GRH assumption, by Lackenby [Lac21a] using sutured manifold hi-\nerarchies.\n(3) The unknot Reidemeister graph in part (b) is formed by assigning a vertex\nto each diagram of the unknot and connecting two vertices by an edge if\nthere is a Reidemeister move connecting them; it is locally finite. There\nare pairs of vertices representing two diagrams with $n$ crossings such that\nthe shortest path connecting them has length at least $n^{2}/25$ [HN10]. No\ntwo vertices representing diagrams with $n$ crossings have distance greater\nthan $2(236n)^{11}$ [Lac15].\nThis problem is complicated by the existence of “hard” unknot dia-\ngrams [BCL $^{+}24$, PZ16], which have the property that any sequence of\nReidemeister moves taking them to the trivial diagram must go via dia-\ngrams with higher crossing number. The question may be more tractable\nif different sets of basic moves are allowed in addition to or in place of\nthe standard Reidemeister moves; in the related setting of grid diagrams,\nmonotonic simplification has been established [Dyn06].\n\nReferences cited:\n- [Hak61] Wolfgang Haken. Theorie der Normalflächen. Acta Math., 105:245–375, 1961. doi: 10.1007/BF02559591.\n- [HLP99] Joel Hass, Jeffrey C. Lagarias, and Nicholas Pippenger. The computational complexity of knot and link problems. J. ACM, 46(2):185–211, 1999. doi:10.1145/301970.301971.\n- [Dyn06] I. A. Dynnikov. Arc-presentations of links: monotonic simplification. Fund. Math., 190:29–76, 2006. doi:10.4064/fm190-0-3.\n- [KM11] P. B. Kronheimer and T. S. Mrowka. Khovanov homology is an unknotdetector. Publ. Math. Inst. Hautes Études Sci., 113:97–208, 2011. doi:10.1007/s10240-010-0030-y.\n- [OS04b] Peter Ozsváth and Zoltán Szabó. Holomorphic disks and genus bounds. Geom. Topol., 8:311–334, 2004. doi:10.2140/gt.2004.8.311.\n- [Kup14] Greg Kuperberg. Knottedness is in NP, modulo GRH. Adv. Math., 256:493–506, 2014. doi:10.1016/j.aim.2014.01.007.\n- [Lac21a] Marc Lackenby. The efficient certification of knottedness and Thurston norm. Adv. Math., 387:Paper No. 107796, 142, 2021. doi:10.1016/j.aim.2021.107796.\n- [HN10] Joel Hass and Tahl Nowik. Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle. Discrete Comput. Geom., 44(1):91–95, 2010. doi:10.1007/s00454-009-9156-4.\n- [Lac15] Marc Lackenby. A polynomial upper bound on Reidemeister moves. Ann. of Math. (2), 182(2):491–564, 2015. doi:10.4007/annals.2015.182.2.3.\n- [BCL+24] Benjamin A. Burton, Hsien-Chih Chang, Maarten Löffler, Clément Maria, Arnaud de Mesmay, Saul Schleimer, Eric Sedgwick, and Jonathan Spreer. Hard diagrams of the unknot. Exp. Math., 33(3):482–500, 2024. doi:10.1080/10586458.2022.2161676.\n- [PZ16] Carlo Petronio and Adolfo Zanellati. Algorithmic simplification of knot diagrams: new moves and experiments. J. Knot Theory Ramifications, 25(10):1650059, 2016. doi:10.1142/S0218216516500590.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2736, "problem_number": "KP-1.77", "title": "Kirby Problem 1.77", "statement": "How many Reidemeister moves are required to relate two dia-\ngrams of a knot (as a function of their numbers of crossings)?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.77.\n\nLiterature notes:\n(1) One reason why this question is interesting is that a computable upper\nbound leads to an algorithm to decide whether two knots are equivalent.\n(2) Lackenby [Lac15] proved that, for a diagram of the unknot with $c$ cross-\nings, there is a sequence of Reidemeister moves taking it to the trivial\ndiagram with length at most $(236c)^{11}$. It is not inconceivable that there is\na polynomial upper bound that applies to every knot type. However, the\nbest known upper bound on the number of Reidemeister moves required\nto relate two diagrams of a knot with $c_{1}$ and $c_{2}$ crossings is due to Coward\nand Lackenby [CL14]:\n\n$$\n2^{2^{\\cdot^{\\cdot^{\\cdot^{2}}}}},\n$$\n\nwhere the height of the tower is $k^{c_1+c_2}$ and where $k = 10^{1000000}$.\n(3) A nontrivial lower bound was established by Hass and Nowik [HN10],\nwho proved that for each natural number $n$ there exists a diagram of the\nunknot with $7n-1$ crossings and where the number of Reidemeister moves\nrequired to take it to the trivial diagram is at least $2n^{2} + 3n - 2$.\n\nReferences cited:\n- [Lac15] Marc Lackenby. A polynomial upper bound on Reidemeister moves. Ann. of Math. (2), 182(2):491–564, 2015. doi:10.4007/annals.2015.182.2.3.\n- [CL14] Alexander Coward and Marc Lackenby. An upper bound on Reidemeister moves. Amer. J. Math., 136(4):1023–1066, 2014. doi:10.1353/ajm.2014.0027.\n- [HN10] Joel Hass and Tahl Nowik. Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle. Discrete Comput. Geom., 44(1):91–95, 2010. doi:10.1007/s00454-009-9156-4.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2737, "problem_number": "KP-1.78", "title": "Kirby Problem 1.78", "statement": "Let $D$ be any diagram of the unknot with $n$ crossings. Let $h(D)$\nbe the smallest number such that some series of Reidemeister moves that transforms\n\n$D$ to a 0-crossing diagram has the property that all intermediate diagrams have at\nmost $h(D)$ crossings. Define $h(n)$ to be the maximum of $h(D)$ over all $n$-crossing\ndiagrams of the unknot. A diagram represents a hard unknot if $h(D) > n$. What\nis $h(n)$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.78.\n\nLiterature notes:\nIt is known [BCL $^{+}24$] that $h(n) \\geq n + 3$ for certain values of $n$.\n\nReferences cited:\n- [BCL+24] Benjamin A. Burton, Hsien-Chih Chang, Maarten Löffler, Clément Maria, Arnaud de Mesmay, Saul Schleimer, Eric Sedgwick, and Jonathan Spreer. Hard diagrams of the unknot. Exp. Math., 33(3):482–500, 2024. doi:10.1080/10586458.2022.2161676.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2738, "problem_number": "KP-1.79", "title": "Kirby Problem 1.79", "statement": "Are there additional moves that, when added to the three Rei-\ndemeister moves, allow for strict monotonic descent in the crossing number of an\nunknot diagram?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.79.\n\nLiterature notes:\n(1) Additional moves allow increased efficiency in diagram simplification. Sev-\neral attempts at finding such moves have been given, but none are known\nto suffice [PZ16]. Work of Dynnikov gives a monotonic, but not strictly\nmonotonic, process [Dyn03].\n(2) A strictly monotonic process would lead to a polynomial time algorithm\nfor unknot detection; see Problem 1.76.\n(3) The collection of additional moves can grow with the number of crossings,\ne.g. moving an arc across a twist region of a diagram. However the set of\nadditional moves cannot be arbitrary; moves consisting of any sequence\nof Reidemeister moves would allow any knot equivalence to be carried out\nin one step. A reasonable restriction is to allow moves that reduce the\ncrossing number and can be found in polynomial time (polynomial in the\ncrossing number).\n\nReferences cited:\n- [PZ16] Carlo Petronio and Adolfo Zanellati. Algorithmic simplification of knot diagrams: new moves and experiments. J. Knot Theory Ramifications, 25(10):1650059, 2016. doi:10.1142/S0218216516500590.\n- [Dyn03] I. A. Dynnikov. Recognition algorithms in knot theory. Uspekhi Mat. Nauk, 58(6(354)):45–92, 2003. doi:10.1070/RM2003v058n06ABEH000675.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2739, "problem_number": "KP-1.80", "title": "Kirby Problem 1.80", "statement": "Is unknotting number computable? Is there even an algorithm\nto decide whether a knot has unknotting number one?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.80.\n\nLiterature notes:\n(1) Many elementary knot invariants, such as the crossing number, are known\nto be computable. However, it seems extremely challenging to determine\nwhether the unknotting number is computable.\n(2) Recall that the unknotting number of a knot is the minimum number of\ncrossing changes required to unknot the knot, where the minimum is taken\nover all diagrams of the knot. One can also consider the unlinking number\nof a link, which is the minimal number of crossing changes required to\nturn it into an unlink. Closely related is the splitting number, which is the\nminimal number of crossing changes required to turn into a split link.\n(3) Some lower bounds on the computational complexity of determining un-\nlinking number are known: it was shown by de Mesmay, Rieck, Sedgwick\nand Tancer [dMRST21] and Koenig and Tsvietkova [KT21] that deter-\nmining the unlinking number of a link is NP-hard.\n\n(4) The second question in the problem seems more tractable, since many\nstructural results are known about knots with unknotting number one\n[ST89], [Sch85b]. The analogous problem of determining whether a link\nhas unlinking number one was solved for a large class of links by Lackenby\n[Lac21b].\n(5) This problem was also stated in [Lac17a].\n\nReferences cited:\n- [dMRST21] Arnaud de Mesmay, Yo’av Rieck, Eric Sedgwick, and Martin Tancer. The unbearable hardness of unknotting. Adv. Math., 381:Paper No. 107648, 36, 2021. doi:10.1016/j.aim.2021.107648.\n- [KT21] Dale Koenig and Anastasiia Tsvietkova. NP-hard problems naturally arising in knot theory. Trans. Amer. Math. Soc. Ser. B, 8:420–441, 2021. doi:10.1090/btran/71.\n- [ST89] Martin Scharlemann and Abigail Thompson. Link genus and the Conway moves. Comment. Math. Helv., 64(4):527–535, 1989. doi:10.1007/BF02564693.\n- [Sch85b] Martin G. Scharlemann. Unknotting number one knots are prime. Invent. Math., 82(1):37–55, 1985. doi:10.1007/BF01394778.\n- [Lac21b] Marc Lackenby. Links with splitting number one. Geom. Dedicata, 214:319–351, 2021. doi:10.1007/s10711-021-00618-x.\n- [Lac17a] Marc Lackenby. Elementary knot theory. In Lectures on geometry, Clay Lect. Notes, pages 29–64. Oxford Univ. Press, Oxford, 2017.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2740, "problem_number": "KP-1.81", "title": "Kirby Problem 1.81", "statement": "(a) Are all knots trivial?\n(b) Conjecture: The Bing sling is knotted.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.81.\n\nLiterature notes:\n(1) For this problem, a knot is a topological embedding (a homeomorphism\nonto its image) of a circle in $S^{3}$. A knot is trivial if it is isotopic, i.e. con-\nnected by an arc of embeddings, to the unknot. A smooth knot in a string\nis trivial in this sense by simply pulling the string tight so that the knot\nbecomes a point. A similar argument works for locally flat embeddings:\nthe problem is only interesting for wild embeddings.\n(2) The Bing Sling [Bin56, DV09] (see also [Nan18, amd22] and [Shi73])\nis drawn in Figure 1.12.\n(3) Brin [Bri83] shows that there are knots that at each point are locally\nequivalent to the Bing sling but are non-ambiently isotopic to the unknot.\n\nFigure 4. The Bing Sling\n\nReferences cited:\n- [Bin56] R. H. Bing. A simple closed curve that pierces no disk. J. Math. Pures Appl. (9), 35:337–343, 1956.\n- [DV09] Robert J. Daverman and Gerard A. Venema. Embeddings in manifolds, volume 106 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2009. doi:10.1090/gsm/106.\n- [Nan18] Ollie Nanyes. Wild and non-compact knot theory. WordPress, 2018. https://wildandnoncompactknots.wordpress.com.\n- [amd22] amd1234. Bing sling isotopy to unknot, 2022. by user https://mathoverflow.net/users/170240/amd1234. See https://mathoverflow.net/q/437106 (version: 2022-12-28).\n- [Shi73] Arnold C. Shilepsky. Homogeneity by isotopy for simple closed curves. Duke Math. J., 40:463–472, 1973. URL: http://projecteuclid.org/euclid.dmj/1077309868.\n- [Bri83] M. Brin. Curves isotopic to tame curves. In Continua, decompositions, manifolds (Austin, Tex., 1980), pages 163–166. Univ. Texas Press, Austin, TX, 1983.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2741, "problem_number": "KP-1.82", "title": "Kirby Problem 1.82", "statement": "(a) What is a positive knot?\n(b) Describe a simple set of moves to convert between two positive diagrams\nof the same knot or link.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.82.\n\nLiterature notes:\n(1) Positive knots and alternating knots are both defined in diagrammatic\nterms, but somewhat orthogonally to one another.\nThe corresponding\nproblems for alternating knots have been settled.\n(2) The first problem is meant to evoke Ralph Fox’s question, ‘What is an\nalternating knot?’ [Lic97, p.32]. It seeks a characterization of the class\nof positive knots in terms intrinsic to the knot exterior, e.g.\na condi-\ntion on the knot group or the existence of a special kind of geometric\nstructure or spanning surface. It echoes a related question of Rudolph,\nwho proved that positive links are strongly quasipositive. He asked, as\na kind of converse to his result: “Can positive links be characterized as\nstrongly quasipositive links that satisfy some extra geometric conditions?”\n[Rud99, Question, p.556]. Baader made progress on Rudolph’s question\nby showing that a knot is positive if and only if it is strongly quasiposi-\ntive and homogeneous [Baa05]. In light of this, an answer to Problem 1\ncould be obtained by giving a geometric characterization of homogeneity.\nNote that Fox’s question was addressed in related papers by Greene and\nby Howie [Gre17, How17], which led to new applications for alternating\nknots.\n(3) The second problem is meant to evoke the Tait flyping conjecture for alter-\nnating links, which was proven by Menasco and Thistlethwaite [MT93].\nThe quest for a set of moves satisfying Problem (b) is complicated by the\nfact that a knot can have reduced positive diagrams with different crossing\nnumbers.\n\nReferences cited:\n- [Lic97] W. B. Raymond Lickorish. An introduction to knot theory, volume 175 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. doi:10.1007/978-1-4612-0691-0.\n- [Rud99] Lee Rudolph. Positive links are strongly quasipositive. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 555–562. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.555.\n- [Baa05] S. Baader. Quasipositivity and homogeneity. Math. Proc. Cambridge Philos. Soc., 139(2):287–290, 2005. doi:10.1017/S0305004105008698.\n- [Gre17] Joshua Evan Greene. Alternating links and definite surfaces. Duke Math. J., 166(11):2133–2151, 2017. With an appendix by András Juhász and Marc Lackenby. doi:10.1215/00127094-2017-0004.\n- [How17] Joshua A. Howie. A characterisation of alternating knot exteriors. Geom. Topol., 21(4):2353–2371, 2017. doi:10.2140/gt.2017.21.2353.\n- [MT93] William Menasco and Morwen Thistlethwaite. The classification of alternating links. Ann. of Math. (2), 138(1):113–171, 1993. doi:10.2307/2946636.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2742, "problem_number": "KP-1.83", "title": "Kirby Problem 1.83", "statement": "Determine the algebraic structure of the concordance group $\\mathcal{O}$\nof open strings.\n(a) Is it abelian?\n(b) Does it contain torsion?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.83.\n\nLiterature notes:\n(1) Virtual strings, also known as flat virtual knots [Kau99], were introduced\nby Turaev [Tur04] as homotopy classes of immersions of $S^{1}$ in compact\noriented surfaces, up to stabilization of the surface.\n(2) Two virtual strings in $\\Sigma_{1}$ and $\\Sigma_{2}$ are said to be concordant if they cobound\nan immersed annulus in an oriented 3-manifold $M$ with boundary $\\partial M =$\n$\\Sigma_{1} \\cup -\\Sigma_{2}$. A virtual string $\\alpha$ in $\\Sigma$ is said to be slice if there exists an\noriented 3-manifold $M$ and immersed disk $D$ in $M$ such that $\\partial M = \\Sigma$ and\n\n$\\partial D = \\alpha$. The first example of an immersed curve in a surface that does\nnot bound an immersed disk in any 3-manifold is due to Carter [Car91].\n(3) Concordance classes of open strings form a group $\\mathcal{O}$, which is known to be\ninfinitely generated [Tur04]. Turaev also introduced the related notion of\nalgebraic concordance for virtual strings, and Jie Chen [Che23, Example\n3.19] has found examples of flat knots that are algebraically slice but not\nslice. There is a surjection to $\\mathcal{O}$ from the concordance group of virtual\nknots (see the following Problem 1.84), which was shown to be non-abelian\nby Chrisman [Chr22].\n\nReferences cited:\n- [Kau99] Louis H. Kauffman. Virtual knot theory. European J. Combin., 20(7):663–690, 1999. doi:10.1006/eujc.1999.0314.\n- [Tur04] Vladimir Turaev. Virtual strings. Ann. Inst. Fourier (Grenoble), 54(7):2455–2525, 2004. URL: http://aif.cedram.org/item?id=AIF 2004 54 7 2455 0.\n- [Car91] J. Scott Carter. Extending immersions of curves to properly immersed surfaces. Topology Appl., 40(3):287–306, 1991. doi:10.1016/0166-8641(91)90111-X.\n- [Che23] Jie Chen. Flat knots and invariants. PhD thesis, McMaster University, 2023. URL: https://www.math.mcmaster.ca/\\%7Eboden/students/Chen-PhD.pdf.\n- [Chr22] Micah Chrisman. Milnor’s concordance invariants for knots on surfaces. Algebr. Geom. Topol., 22(5):2293–2353, 2022. doi:10.2140/agt.2022.22.2293.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2743, "problem_number": "KP-1.84", "title": "Kirby Problem 1.84", "statement": "For a classical knot, does its slice genus as a virtual knot agree\nwith its slice genus as a classical knot?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.84.\n\nLiterature notes:\n(1) Virtual knots were originally defined combinatorially by Kauffman [Kau99].\nFor the purposes of this problem a virtual knot $K$ may be regarded as an\nembedding of $S^{1}$ into $\\Sigma \\times [0, 1],$ where $\\Sigma$ is a compact oriented surface.\nThe virtual slice genus of $K$ is the minimal genus, taken over all sur-\nfaces $F$ and all 3-manifolds $M$, such that $F$ embeds into $M \\times [0, 1]$ with\n$\\partial F = K$. Here $M$ is a compact oriented 3-manifold with $\\partial M = \\Sigma$. A\nvirtual knot is said to be virtually slice if it bounds a disk in $M \\times [0, 1]$\nfor some 3-manifold $M$.\n(2) This question is due to Dye-Kaestner-Kauffman [DKK17, §6], who gave\nan affirmative answer any knot for which the slice genus is determined by\nthe Rasmussen invariant, including torus knots. Their results support the\nconjecture that for classical knots, their slice genus as virtual knots agrees\nwith their slice genus as classical knots. It is known that a classical knot\nis virtually slice if and only if it is slice [BN17], thus the conjecture is\ntrue for knots with slice genus one.\n(3) This problem has two versions, one for the smooth slice genus and the\nother for the topological (locally flat) slice genus.\n\nReferences cited:\n- [Kau99] Louis H. Kauffman. Virtual knot theory. European J. Combin., 20(7):663–690, 1999. doi:10.1006/eujc.1999.0314.\n- [DKK17] Heather A. Dye, Aaron Kaestner, and Louis H. Kauffman. Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots. J. Knot Theory Ramifications, 26(3):1741001, 57, 2017. doi:10.1142/S0218216517410012.\n- [BN17] Hans U. Boden and Matthias Nagel. Concordance group of virtual knots. Proc. Amer. Math. Soc., 145(12):5451–5461, 2017. doi:10.1090/proc/13667.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2744, "problem_number": "KP-1.85", "title": "Kirby Problem 1.85", "statement": "Let $K$ be a hyperbolic knot in $S^{3}$ and $\\chi(K)$ the space of con-\njugacy classes of $\\operatorname{PSL}_{2}(\\mathbb{C})$ representations of $\\pi_{1}(S^{3} - K)$. There is a distinguished\ncomponent of $\\chi(K)$ that contains the discrete faithful representation coming from\nthe finite volume hyperbolic structure on the complement of $K$. Does this component\nalways contain an arc of representations to $\\operatorname{SO}(3)$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.85.\n\nLiterature notes:\n(1) Chinburg-Reid-Stover conjectured [CRS22, Conjecture 1.9] that there\nalways is such an arc of representations. A positive answer would in fact\nprove that the knot groups of hyperbolic knots always have faithful $\\operatorname{SO}(3)$\nrepresentations.\n(2) It is explained in [Ago22] that an affirmative answer to this question\nwould prove there are no infinite descending chains of hyperbolic knots in\nthe ribbon concordance partial order [Gor81, Question 6.2]. See Prob-\nlem 1.56 for a knot Floer homology perspective on Gordon’s question.\n\nReferences cited:\n- [CRS22] Ted Chinburg, Alan W. Reid, and Matthew Stover. Azumaya algebras and canonical components. Int. Math. Res. Not. IMRN, 2022(7):4969–5036, 2022. doi:10.1093/imrn/rnaa209.\n- [Ago22] Ian Agol. Ribbon concordance of knots is a partial ordering. Comm. Amer. Math. Soc., 2:374–379, 2022. doi:10.1090/cams/15.\n- [Gor81] C. McA. Gordon. Ribbon concordance of knots in the 3-sphere. Math. Ann., 257(2):157–170, 1981. doi:10.1007/BF01458281.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2745, "problem_number": "KP-1.86", "title": "Kirby Problem 1.86", "statement": "(a) Every connected cubic $($ i.e. trivalent $)$ graph has freeness index at least 2.\n(b) Every graph has freeness index at least two.\n(c) There is a graph with freeness index two.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.86.\n\nLiterature notes:\n(1) A knot in the 3-sphere is the trivial knot if and only if its complement is\na solid torus. One can extend the notion of triviality from knots to em-\nbedded (finite) graphs by considering a graph embedded in the 3-sphere\nto be “unknotted” if its complement is a connected sum of handlebod-\nies. Every connected graph has an embedding into $S^{3}$ with handlebody\ncomplement; in fact, using induction and 2-handle addition, one can show\nthat every graph has an unknotted embedding with the additional feature\nthat every subgraph obtained by erasing the interior of a single edge is\nalso unknotted [Tho22].\n(2) Conway and Gordon showed that some graphs (for example, the complete\ngraph with 7 vertices, $K_{7}$) have some knotted subgraphs no matter how\nthey are embedded into $S^{3}$ [CG83b]. Planar graphs, by contrast, can\nbe embedded in $S^{3}$ such that every subgraph is unknotted; so can some\nnon-planar graphs, such as $K_{5}$.\n(3) One can define an integer invariant of a finite (connected) graph $\\Gamma$, the\nfreeness index, which measures the extent to which $\\Gamma$ can be embedded in\nthe 3-sphere such that it and its subgraphs are unknotted.\nDefinition. An embedding $f$ of $\\Gamma$ into $S^{3}$ such that the complement\nof $\\Gamma'$ is unknotted for every subgraph of $\\Gamma'$ of $\\Gamma$ obtained by deleting up to\n$k$ edges from $\\Gamma$ is called $k$-free. The freeness index of $\\Gamma$ is the maximum\nof this integer over all embeddings $f$.\n\n(4) In [Tho22] it is shown that every graph has freeness index at least one,\nwhile the graph $K_{6}$ has freeness index 8 and the Petersen graph has free-\nness index 4. [Tho22].\n(5) We can relate the freeness index to the long-standing orientable cycle\ndouble cover conjecture (OCDCC) from combinatorics ([Sze73, Sey80]).\nThe OCDCC for (bridgeless) cubic graphs implies the OCDCC for all\ngraphs. For bridgeless cubic graphs $\\Gamma$, the OCDCC is equivalent to the\n\nstatement that $\\Gamma$ has an embedding into a closed orientable surface $F$ such\nthat every complementary region is a disk and the closure of every com-\nplementary region is also a disk (this is a strong embedding of $\\Gamma$ into $F$).\nIn order to make a connection with the freeness index, one can exploit the\nfact that once a graph is embedded into a closed orientable surface $F, F$\ncan then be embedded into $S^{3}$ as a Heegaard surface. If both the embed-\nding of the graph into the surface, and of the surface into the 3-sphere, are\nsufficiently “nice”, one can conclude that the induced embedding of the\ngraph into the 3-sphere satisfies some strong freeness conditions. This can\nbe used to show that a cubic graph satisfying the OCDCC has freeness\nindex at least 2 [Tho22].\n(6) Note that a connected bridgeless cubic graph with freeness index 1 would\nprovide a counter-example to the OCDCC.\n\nReferences cited:\n- [Tho22] Abigail Thompson. The freeness index of a graph, 2022. arXiv:2206.12939.\n- [CG83b] J. H. Conway and C. McA. Gordon. Knots and links in spatial graphs. J. Graph Theory, 7(4):445–453, 1983. doi:10.1002/jgt.3190070410.\n- [Sze73] G. Szekeres. Polyhedral decompositions of cubic graphs. Bulletin of the Australian Mathematical Society, 8(3):367–387, 1973. doi:10.1017/S0004972700042660.\n- [Sey80] P. D. Seymour. Disjoint paths in graphs. Discrete Math., 29(3):293–309, 1980. doi: 10.1016/0012-365X(80)90158-2.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2746, "problem_number": "KP-1.87", "title": "Kirby Problem 1.87", "statement": "Is every fibered link in $S^{3}$ realized as the link of an isolated\nsingular point of a polynomial map $\\mathbb{R}^{4} \\to \\mathbb{R}^{2}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.87.\n\nLiterature notes:\n(1) This problem is due to Benedetti–Shiota [BS98], and originates in the\ncelebrated Milnor book [Mil68b].\n(2) Let $f: \\mathbb{R}^{4} \\to \\mathbb{R}^{2}$ be a polynomial map with $f(0) = 0$ and set $V = f^{-1}(0)$.\nAssume that $0 \\in \\mathbb{R}^{4}$ is an isolated point of the intersection of $V$ and the\nset of singular points of $f$. Such a point is often called a weakly isolated\nsingularity.\nAs instances of Milnor’s cone structure and fibration theorems [Mil68b],\nwe have:\n(i) There exists a small positive real number $\\epsilon_{0}$ such that for every $0 <$\n$\\epsilon < \\epsilon_{0}$, the sphere $S^{3}_{\\epsilon}$ in $\\mathbb{R}^{4}$ with radius $\\epsilon$ centered at 0 is transverse\nto the surface $V$, and the link type $L$ of\n$L_{\\epsilon}=(1/\\epsilon)(S^{3}_{\\epsilon}\\cap V)\\subset S^{3}$\ndoes not depend on the choice of $\\epsilon$.\n(ii) Under the stronger assumption that $0 \\in \\mathbb{R}^{4}$ is an isolated singular\npoint of the map $f$, the link $L \\subset S^{3}$ is fibered.\n(3) In [Mil68b], one is mainly concerned with the links of (the realifications\nof) complex polynomial maps $\\mathbb{C}^{2} \\to \\mathbb{C}$ at an isolated singular point; in\nsuch a case, a weakly isolated singularity in (1) above is equivalent to\na usual isolated singularity as in (2), and we obtain a distinguished and\nwell understood family of fibered links (see [Neu03, Web08]). In the\ngeneral case, there are no evident obstructions against a positive answer\nto the problem. As remarked in [Mil68b, Page 84], the first example of\na fibered knot to be considered not belonging to that family for complex\npolynomials, was the figure-eight knot. Later, an explicit realization of the\nfigure-eight knot as the link of a real isolated singularity was obtained in\n[Per82]. Further partial positive results have been worked out in [Loo71,\nRud87, Pic05, Bod19, Bod20, Bod23, AdSSQ24].\n\n(4) It is known [AK81] that every link $L$ in $S^{3}$ can be realized as the link\nof a polynomial map as in (i) above, only requiring that 0 is an isolated\npoint of the intersection of $V$ and the set of singular points of $f$.\n\nReferences cited:\n- [BS98] R. Benedetti and M. Shiota. On real algebraic links in $S^{3}$. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1(3):585–609, 1998.\n- [Mil68b] John Milnor. Singular points of complex hypersurfaces, volume No. 61 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1968. https://www.jstor.org/stable/j.ctt1bd6kvv.\n- [Neu03] W. Neumann. Topology of hypersurface singularities. In Erich Kähler Mathematische Werken, pages 727–736. de Gruyter, 2003. See also https://arxiv.org/abs/1706.04386.\n- [Web08] Claude Weber. On the topology of singularities. In Singularities II, volume 475 of Contemp. Math., pages 217–251. Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/conm/475/09285.\n- [Per82] B. Perron. Le nœud “huit” est algébrique réel. Invent. Math., 65(3):441–451, 1981/82. doi:10.1007/BF01396628.\n- [Loo71] Eduard Looijenga. A note on polynomial isolated singularities. Indag. Math., 33:418–421, 1971. Nederl. Akad. Wetensch. Proc. Ser. A 74.\n- [Rud87] Lee Rudolph. Isolated critical points of mappings from $\\mathbb{R}^{4}$ to $\\mathbb{R}^{2}$ and a natural splitting of the Milnor number of a classical fibered link. I. Basic theory; examples. Comment. Math. Helv., 62(4):630–645, 1987. doi:10.1007/BF02564467.\n- [Pic05] Anne Pichon. Real analytic germs fg and open-book decompositions of the 3-sphere. Internat. J. Math., 16(1):1–12, 2005. doi:10.1142/S0129167X05002710.\n- [Bod19] Benjamin Bode. Constructing links of isolated singularities of polynomials $\\mathbb{R}^{4}$ $\\to$ $\\mathbb{R}^{2}$. J. Knot Theory Ramifications, 28(1):1950009, 21, 2019. doi:10.1142/S0218216519500093.\n- [Bod20] Benjamin Bode. Real algebraic links in $S^{3}$ and braid group actions on the set of n-adic integers. J. Knot Theory Ramifications, 29(6):2050039, 44, 2020. doi: 10.1142/S021821652050039X.\n- [Bod23] Benjamin Bode. Twisting and satellite operations on P-fibered braids. Comm. Anal. Geom., 31(8):2013–2038, 2023. doi:10.4310/cag.2023.v31.n8.a5.\n- [AdSSQ24] Raimundo N. Araújo dos Santos and Eder L. Sanchez Quiceno. On real algebraic links in the 3-sphere associated with mixed polynomials. Res. Math. Sci., 11(2):Paper No. 22, 22, 2024. doi:10.1007/s40687-024-00424-3.\n- [AK81] S. Akbulut and H. King. All knots are algebraic. Comment. Math. Helv., 56(3):339– 351, 1981. doi:10.1007/BF02566217.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2747, "problem_number": "KP-1.88", "title": "Kirby Problem 1.88", "statement": "Are there infinitely many congruence arithmetic links in the\n3-sphere?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.88.\n\nLiterature notes:\n(1) This problem relates to the more general question posed by Thurston\nto find the topological and geometric properties of quotients of $\\mathbb{H}^{3}$ by\narithmetic subgroups of $\\operatorname{PSL}(2, \\mathbb{C})$ [Thu82].\n(2) A link complement is arithmetic if it is of the form $\\mathbb{H}^{3}/\\Gamma$ for some discrete\nsubgroup $\\Gamma \\leq \\operatorname{PSL}(2, \\mathbb{C})$ with the property that $\\Gamma$ is commensurable with\n$\\operatorname{PSL}(2, O_{d})$, where $O_{d}$ the ring of integers in $\\mathbb{Q}($\n$?$\n$-d)$. Famously, Reid\n[Rei91] proved that the figure-eight knot is the only arithmetic knot in\nthe 3-sphere. However, there are infinitely many arithmetic links (see for\nexample the discussion in [BR23a]).\n(3) The question is concerned with a particular type of arithmetic link called\ncongruence links, which are defined as follows. A principal congruence\nsubgroup $\\Gamma$ is equal to the kernel of\n\n$$\n\\operatorname{PSL}(2, O_{d}) \\to \\operatorname{PSL}(2, O_{d}/I)\n$$\n\nfor some ideal $I$ in $O_{d}$. More generally, $\\Gamma$ is called congruence if it contains\na principal congruence subgroup. When $\\mathbb{H}^{3}/\\Gamma$ is a link complement, the\nlink is said to be principal congruence and congruence respectively.\n(4) In [BGR19], Baker, Goerner and Reid gave a complete classification of\nprincipal congruence arithmetic links in the 3-sphere. In particular, there\nare finitely many such links. (See [BGR22] for all known link diagrams\nof principal congruence links.) The question asks whether this remains\ntrue for congruence arithmetic links.\n(5) Much information is known about arithmetic links in the 3-sphere and in\nparticular about congruence links. For example, any arithmetic link in\nthe 3-sphere commensurable with $\\operatorname{PSL}(2, O_{d})$ must have\n\n$$\nd \\in \\{1, 2, 3, 5, 6, 7, 11, 15, 19, 23, 31, 39, 47, 71\\}.\n$$\n\nReferences cited:\n- [Thu82] William P. Thurston. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.), 6(3):357–381, 1982. doi:10.1090/S0273-0979-1982-15003-0.\n- [Rei91] Alan W. Reid. Arithmeticity of knot complements. J. London Math. Soc. (2), 43(1):171–184, 1991. doi:10.1112/jlms/s2-43.1.171.\n- [BR23a] Mark D. Baker and Alan W. Reid. Infinitely many arithmetic alternating links. Algebr. Geom. Topol., 23(6):2857–2866, 2023. doi:10.2140/agt.2023.23.2857.\n- [BGR19] M. D. Baker, M. Goerner, and A. W. Reid. All principal congruence link groups. J. Algebra, 528:497–504, 2019. doi:10.1016/j.jalgebra.2019.02.023.\n- [BGR22] Mark D. Baker, Matthias Goerner, and Alan W. Reid. All known principal congruence links, 2022. arXiv:1902.04426.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2748, "problem_number": "KP-1.89", "title": "Kirby Problem 1.89", "statement": "(a) Fix a long link L. What is the homotopy type of the embedding space of\nlinks isotopic to L?\n(b) Fix a link L in a 3-manifold M. What is the homotopy type of the embed-\nding space of links in M isotopic to L?\n\n(c) Given a link $L$ obtained by infecting a link $L_{1}$ by another link $L_{2}$, describe\nthe homotopy type of component of $L$ in the space of links in terms of the\ncomponents of $L_{1}$ and $L_{2}$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.89.\n\nLiterature notes:\n(1) A long link is an embedding of a disjoint union of lines that agree with\na fixed collection of affine lines outside of a box. The answer to the first\nquestion, due to Hatcher and Budney, is known for long knots in the sense\nthat the knot spaces are shown to be homotopy equivalent to standard\ncombinations of standard spaces; see Remark (3) below.\n(2) Spaces of long links have applications in 4-manifold theory as well as high-\ndimensional manifold theory. For example, the recent work of Budney and\nGabai [BG19] on the homotopy-type of $\\operatorname{Diff}(S^{1} \\times D^{n-1})$ for $n \\geq 4$ uses\nthe low-dimensional homotopy groups of the space of long embeddings\n$\\operatorname{Emb}(I, S^{1} \\times D^{n-1})$. This space can be thought of as the homotopy fiber\nof the forgetful map $\\operatorname{Emb}(I \\sqcup D^{n-2}, D^{n}) \\to \\operatorname{Emb}(D^{n-2}, D^{n})$, i.e. it is the\nspace of 2-component ‘long links’ where one component is trivial.\n(3) A major step in the description of long knot spaces was Hatcher’s proof\nof the Smale conjecture, which is equivalent [Hat83, Appendix] to the\nstatement that $\\operatorname{Diff}(D^{3}; \\partial D^{3})$ is contractible, which in turn implies that\nthe component of the long unknot is contractible. Combined with work of\nHatcher [Hat76] and Ivanov [Iva76] on diffeomorphisms of 3-manifolds,\nit also implies that each component in the space of long knots is a $K(\\pi, 1)$\nspace [Bud07].\nHatcher [Hat02] determined the homotopy type of a\nlong torus knot and, building on his work with McCullough [HM97a], of\na long hyperbolic knot.\nBudney [Bud10] determined the homotopy type of the component of\nany long knot $K$ in $\\operatorname{Emb}(\\mathbb{R}, \\mathbb{R}^{3})$ in terms of the satellite decomposition\nof $K$, i.e. in terms of splicing operations, thus answering part (a) above\nwhen $L$ has one component.\nThe homotopy type is roughly a twisted\nproduct of factors of $S^{1}, S^{1} \\times S^{1}$, and configuration spaces of points in\nthe plane, with one factor for each knot or link appearing in the satellite\ndecomposition.\n(4) Havens and Koytcheff [HK21] generalized Hatcher and Budney’s results\nto the space of closed links in $S^{3}$ modulo rotations, whose components are\n$K(\\pi, 1)$ spaces for irreducible links. They did not completely determine\nthe homotopy types of spaces of split links because $\\operatorname{Diff}(M)$ for a reducible\n3-manifold $M$ is not completely understood. However, Boyd and Bregman\n[BB25] obtained more detailed information on spaces of split links, espe-\ncially their fundamental groups, using semi-simplicial spaces that model\nthem. Kosanović [Kos24] related Vassiliev invariants to the Goodwillie–\nWeiss embedding calculus in the setting of a long knot $K$ in any compact\noriented 3-manifold $M$ with boundary.\n(5) In part (c), infection generalizes splicing from long knots to string (a.k.a. long)\nlinks. Budney [Bud07, Bud12] developed the operations of connect-sum\nand splicing at the space level, parameterizing them by operads.\nFor\nboth operations, he obtained space-level decomposition results, which for\nconnect-sum he further developed at the homological level with F. Cohen\n[BC09].\n\nBurke and Koytcheff [BK15b] generalized Budney’s splicing operad\nto a (colored) operad for string link infection. They obtained a partial\ndecomposition result for 2-component string links, building upon a re-\nsult on isotopy classes in their joint work with Blair [BBK15]. This de-\ncomposition was completed for 2-component string links by Batelier and\nDucoulombier [BD23]. It is essentially in terms of the stacking of string\nlinks rather than the more general infection operation. An answer to part\n(c) would be related to a decomposition of a 3-manifold analogous to the\nJSJ decomposition but with tori replaced by surfaces of higher genus.\n\nReferences cited:\n- [BG19] Ryan Budney and David Gabai. Knotted 3-balls in $S^{4}$, 2019. arXiv:1912.09029.\n- [Hat83] Allen E. Hatcher. A proof of the Smale conjecture, Diffp$S^{3}$q » $O(4)$. Ann. of Math. (2), 117(3):553–607, 1983. doi:10.2307/2007035.\n- [Hat76] Allen Hatcher. Homeomorphisms of sufficiently large P2-irreducible 3-manifolds. Topology, 15(4):343–347, 1976. doi:10.1016/0040-9383(76)90027-6.\n- [Iva76] N. V. Ivanov. Diffeomorphism groups of Waldhausen manifolds. J Math Sci., 12:115–118, 1976. doi:10.1007/BF01098421.\n- [Bud07] Ryan Budney. Little cubes and long knots. Topology, 46(1):1–27, 2007. doi:10.1016/j.top.2006.09.001.\n- [Hat02] Allen Hatcher. Topological moduli spaces of knots. https://pi.math.cornell.edu/„hatcher/Papers/knotspaces.pdf, 2002.\n- [HM97a] Allen Hatcher and Darryl McCullough. Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds. Geom. Topol., 1:91–109, 1997. doi:10.2140/gt.1997.1.91.\n- [Bud10] Ryan Budney. Topology of knot spaces in dimension 3. Proc. Lond. Math. Soc. (3), 101(2):477–496, 2010. doi:10.1112/plms/pdp058.\n- [HK21] Andrew Havens and Robin Koytcheff. Spaces of knots in the solid torus, knots in the thickened torus, and links in the 3-sphere. Geom. Dedicata, 214:671–737, 2021. doi:10.1007/s10711-021-00633-y.\n- [BB25] Rachael Boyd and Corey Bregman. Embedding spaces of split links. Adv. Math., 470:Paper No. 110235, 41, 2025. doi:10.1016/j.aim.2025.110235.\n- [Kos24] Danica Kosanović. Embedding calculus and grope cobordism of knots. Adv. Math., 451:Paper No. 109779, 118, 2024. doi:10.1016/j.aim.2024.109779.\n- [Bud12] Ryan Budney. An operad for splicing. J. Topol., 5(4):945–976, 2012. doi:10.1112/jtopol/jts024.\n- [BC09] Ryan Budney and Fred Cohen. On the homology of the space of knots. Geom. Topol., 13(1):99–139, 2009. doi:10.2140/gt.2009.13.99.\n- [BK15b] John Burke and Robin Koytcheff. A colored operad for string link infection. Algebr. Geom. Topol., 15(6):3371–3408, 2015. doi:10.2140/agt.2015.15.3371.\n- [BBK15] Ryan Blair, John Burke, and Robin Koytcheff. A prime decomposition theorem for the 2-string link monoid. J. Knot Theory Ramifications, 24(2):1550005, 24, 2015. doi:10.1142/S0218216515500054.\n- [BD23] Etienne Batelier and Julien Ducoulombier. Operadic actions on long knots and 2-string links. Algebr. Geom. Topol., 23(2):833–882, 2023. doi:10.2140/agt.2023.23.833.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2749, "problem_number": "KP-2.1", "title": "Kirby Problem 2.1", "statement": "(Ivanov conjecture). Let $S$ be an orientable surface of finite type\nwith genus at least three. If $G \\leq \\operatorname{Mod}(S)$ is a subgroup of finite index, does $G$ have\nfinite abelianization?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.1.\n\nLiterature notes:\n(1) The Ivanov conjecture is a weaker version of the question of whether or\nnot the mapping class group has property (T). (The unpublished preprint\n[And07] announced a negative answer to the latter, which does not resolve\nthe Ivanov conjecture in any case). It was recently shown that automor-\nphism groups of most free groups do have property (T) and hence every\nfinite index subgroup has finite abelianization: this was resolved for rank\n5 by Kaluba–Nowak–Ozawa [KNO19] and for $\\operatorname{rank} \\geq 6$ by [KKN21a].\nA resolution for rank 4 was announced by Nitsche [Nit22].\n(2) The Ivanov conjecture has an equivalent reformulation in terms of map-\nping class group actions on the homology of finite covers of surfaces [PW13].\nThe corresponding statement about these actions has become known as\nthe Putman–Wieland conjecture. Some progress on the Putman–Wieland\nconjecture has been made [LL23b, LL23a, MT24].\n(3) For braid groups and mapping class groups of surfaces of genus two or\nlower, virtual surjections to the integers exist by virtue of the existence of\nsuch maps for braid groups.\n\nReferences cited:\n- [And07] Jorgen Ellegaard Andersen. Mapping class groups do not have Kazhdan’s property (t), 2007. arXiv:0706.2184.\n- [KNO19] Marek Kaluba, Piotr W. Nowak, and Narutaka Ozawa. $\\mathrm{Aut}(F_5)$ has property $(T)$. Math. Ann., 375(3-4):1169–1191, 2019. doi:10.1007/s00208-019-01874-9.\n- [KKN21a] Marek Kaluba, Dawid Kielak, and Piotr W. Nowak. On property (T) for $\\mathrm{Aut}(F_n)$ and $\\mathrm{SL}_n(\\mathbb{Z})$. Ann. of Math. (2), 193(2):539–562, 2021. doi:10.4007/annals.2021.193.2.3.\n- [Nit22] Martin Nitsche. Computer proofs for Property (T), and SDP duality, 2022. arXiv: 2009.05134.\n- [PW13] Andrew Putman and Ben Wieland. Abelian quotients of subgroups of the mappings class group and higher Prym representations. J. Lond. Math. Soc. (2), 88(1):79–96, 2013. doi:10.1112/jlms/jdt001.\n- [LL23b] Aaron Landesman and Daniel Litt. An introduction to the algebraic geometry of the Putman-Wieland conjecture. Eur. J. Math., 9(2):Paper No. 40, 25, 2023. doi: 10.1007/s40879-023-00637-w.\n- [LL23a] Aaron Landesman and Daniel Litt. Applications of the algebraic geometry of the Putman-Wieland conjecture. Proc. Lond. Math. Soc. (3), 127(1):116–133, 2023. doi:10.1112/plms.12539.\n- [MT24] Vladimir Marković and Ognjen Tošić. The second variation of the Hodge norm and higher Prym representations. J. Topol., 17(1):Paper No. e12322, 23, 2024. doi:10.1112/topo.12322.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2750, "problem_number": "KP-2.2", "title": "Kirby Problem 2.2", "statement": "(Congruence subgroup problem). Does every finite-index sub-\ngroup of the mapping class group of $S$ contain a congruence subgroup?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.2.\n\nLiterature notes:\n(1) Let $S$ be a finite-type surface, and let $K \\leq \\pi_{1}(S)$ be a finite-index charac-\nteristic (i.e., $\\operatorname{Aut}(\\pi_{1}(S))$-invariant) subgroup. A congruence subgroup of\nthe mapping class group of $S$ is the kernel of the natural map from the\nmapping class group of $S$ to $\\operatorname{Out}(\\pi_{1}(S)/K)$. This definition is analogous\nto congruence subgroups of integral special linear groups.\n(2) The congruence subgroup problem is known to have a positive answer\nin certain low-genus cases. For punctured spheres, this was established\nby Diaz–Donagi–Harbater [DDH89], with subsequent alternative proofs\ngiven by Asada [Asa01], Thurston, and McReynolds [McR12]. The case\nof genus one was originally addressed by Asada [Asa01], with further\nwork done by Kent [Ken16] and Bux–Ershov–Rapinchuk [BER11].\n(3) A positive resolution to the congruence subgroup problem has applications\nin anabelian geometry [Mar19].\n\nReferences cited:\n- [DDH89] Steven Diaz, Ron Donagi, and David Harbater. Every curve is a Hurwitz space. Duke Math. J., 59(3):737–746, 1989. doi:10.1215/S0012-7094-89-05933-4.\n- [Asa01] Mamoru Asada. The faithfulness of the monodromy representations associated with certain families of algebraic curves. J. Pure Appl. Algebra, 159(2-3):123–147, 2001. doi:10.1016/S0022-4049(00)00056-6.\n- [McR12] D. B. McReynolds. The congruence subgroup problem for pure braid groups: Thurston’s proof. New York J. Math., 18:925–942, 2012. http://nyjm.albany.edu: 8000/j/2012/18 925.html.\n- [Ken16] Autumn Kent. Congruence kernels around affine curves. J. Reine Angew. Math., 713:1–20, 2016. doi:10.1515/crelle-2014-0023.\n- [BER11] Kai-Uwe Bux, Mikhail V. Ershov, and Andrei S. Rapinchuk. The congruence subgroup property for Aut F2: a group-theoretic proof of Asada’s theorem. Groups Geom. Dyn., 5(2):327–353, 2011. doi:10.4171/GGD/130.\n- [Mar19] Dan Margalit. Problems, questions, and conjectures about mapping class groups. In Breadth in contemporary topology, volume 102 of Proc. Sympos. Pure Math., pages 157–186. Amer. Math. Soc., Providence, RI, 2019. doi:10.1090/pspum/102/12.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2751, "problem_number": "KP-2.3", "title": "Kirby Problem 2.3", "statement": "Is the mapping class group of a surface of finite type linear?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.3.\n\nLiterature notes:\n(1) Recall that a group is linear if it can be embedded as a subgroup of $\\operatorname{GL}_{n}(\\mathbb{C})$\nfor some $n$.\n(2) Following the linearity of the braid group (see Remark (4) below), there\nare some low-complexity cases for which linearity is known to hold. This\nis true, in particular, for mapping class groups of punctured spheres and\nhyperelliptic mapping class groups, including the full mapping class group\nin genus 2 [Kor00, BB01]. In addition, if one can prove that a non-\nclosed mapping class group is nonlinear (say of genus $g$ with one puncture\nor boundary component) then all mapping class groups of genus $h > g$\nhave nonlinear mapping class groups. Unsurprisingly, there are examples\nof infinite-type surfaces with nonlinear mapping class groups [APV21].\n(3) The analogous question of linearity for $\\operatorname{Aut}(F_{n})$, the automorphism group\nof a free group $F_{n}$ of $\\operatorname{rank} n$, has been resolved in the negative by Formanek–\nProcesi [FP92]. They construct certain “poison subgroups” of $\\operatorname{Aut}(F_{n})$\nthat are known to be nonlinear. Subsequently, such poison subgroups were\nshown not to exist in mapping class groups by Brendle–Hamidi-Tehrani\n[BHT01].\n(4) On the other hand, braid groups are known to be linear, due to the work\nof Bigelow and Krammer [Big01, Kra02].\nThe Lawrence–Krammer–\nBigelow representations used to certify this are constructed as the action\non the homology of a certain 4-manifold given as an infinite abelian cover\nof the configuration space of two points in the $n$-punctured disk. It is\ntempting to speculate about carrying out an analogous construction in\nhigher genus, but various complications arise; ultimately, the natural ho-\nmology representations of braid groups are simply symmetric groups, and\nthe Lawrence–Krammer–Bigelow representation arises as a deformation of\na symmetric group representation [Jon87]. The corresponding homology\n\nactions of mapping class groups of surfaces in positive genus have infinite\nimage.\n(5) One possible approach to establishing nonlinearity would be to make effec-\ntive the residual finiteness of mapping class groups. In particular, super-\npolynomial residual finiteness growth would imply nonlinearity by work\nof Bou-Rabee–McReynolds [BRM15].\n(6) There is a categorified faithful, finite-dimensional, linear representation\nwith mapping classes acting by functors; see [LOT13].\n(7) Certain constraints on the properties of a faithful representation have\nbeen established. In [Kor23], Korkmaz shows that for $g \\geq 3$, any faith-\nful representation must have dimension $n > 3g - 3$.\nFor any finite-\ndimensional linear representation of a mapping class group in genus $g \\geq 3$,\nBridson [Bri10] proved that Dehn twists are necessarily sent to quasi-\nunipotent matrices (i.e., ones whose eigenvalues are roots of unity). From\nthis one can easily show that mapping class groups have no faithful linear\nrepresentations in positive characteristic. For more recent further devel-\nopments in this direction, see [AS16, But19, KLS19].\n(8) For a much more extensive discussion and bibliography, see the article of\nMargalit [Mar19].\n\nReferences cited:\n- [Kor00] Mustafa Korkmaz. On the linearity of certain mapping class groups. Turkish J. Math., 24(4):367–371, 2000.\n- [BB01] Stephen J. Bigelow and Ryan D. Budney. The mapping class group of a genus two surface is linear. Algebr. Geom. Topol., 1:699–708, 2001. doi:10.2140/agt.2001.1.699.\n- [APV21] Tarik Aougab, Priyam Patel, and Nicholas G. Vlamis. Isometry groups of infinite genus hyperbolic surfaces. Math. Ann., 381:459–498, 2021.\n- [FP92] Edward Formanek and Claudio Procesi. The automorphism group of a free group is not linear. J. Algebra, 149(2):494–499, 1992. doi:10.1016/0021-8693(92)90029-L.\n- [BHT01] Tara E. Brendle and Hessam Hamidi-Tehrani. On the linearity problem for mapping class groups. Algebr. Geom. Topol., 1:445–468, 2001. doi:10.2140/agt.2001.1.445.\n- [Big01] Stephen J. Bigelow. Braid groups are linear. J. Amer. Math. Soc., 14(2):471–486, 2001. doi:10.1090/S0894-0347-00-00361-1.\n- [Kra02] Daan Krammer. Braid groups are linear. Ann. of Math. (2), 155(1):131–156, 2002. doi:10.2307/3062152.\n- [Jon87] V. F. R. Jones. Hecke algebra representations of braid groups and link polynomials. Ann. of Math. (2), 126(2):335–388, 1987. doi:10.2307/1971403.\n- [BRM15] Khalid Bou-Rabee and D. B. McReynolds. Extremal behavior of divisibility functions. Geom. Dedicata, 175:407–415, 2015. doi:10.1007/s10711-014-9955-5.\n- [LOT13] Robert Lipshitz, Peter Ozsváth, and Dylan Thurston. A faithful linear-categorical action of the mapping class group of a surface with boundary. J. Eur. Math. Soc. (JEMS), 15(4):1279–1307, 2013. doi:10.4171/JEMS/392.\n- [Kor23] Mustafa Korkmaz. Low-dimensional linear representations of mapping class groups. J. Topol., 16(3):899–935, 2023. doi:10.1112/topo.12305.\n- [Bri10] Martin R. Bridson. Semisimple actions of mapping class groups on $\\mathrm{CAT}(0)$ spaces. In Geometry of Riemann surfaces, volume 368 of London Math. Soc. Lecture Note Ser., pages 1–14. Cambridge Univ. Press, Cambridge, 2010.\n- [AS16] Javier Aramayona and Juan Souto. Rigidity phenomena in the mapping class group. In Handbook of Teichmüller theory. Vol. VI, volume 27 of IRMA Lect. Math. Theor. Phys., pages 131–165. Eur. Math. Soc., Zürich, 2016.\n- [But19] Jack Oliver Button. Aspects of non positive curvature for linear groups with no infinite order unipotents. Groups Geom. Dyn., 13(1):277–292, 2019. doi:10.4171/GGD/484.\n- [KLS19] Thomas Koberda, Feng Luo, and Hongbin Sun. An effective Lie-Kolchin theorem for quasi-unipotent matrices. Linear Algebra Appl., 581:304–323, 2019. doi:10.1016/j.laa.2019.07.023.\n- [Mar19] Dan Margalit. Problems, questions, and conjectures about mapping class groups. In Breadth in contemporary topology, volume 102 of Proc. Sympos. Pure Math., pages 157–186. Amer. Math. Soc., Providence, RI, 2019. doi:10.1090/pspum/102/12.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2752, "problem_number": "KP-2.4", "title": "Kirby Problem 2.4", "statement": "Let $S_{1}$ and $S_{2}$ be orientable surfaces of finite type. Under what\nconditions do injective maps from (finite-index subgroups of) the mapping class\ngroup of $S_{1}$ to the mapping class group of $S_{2}$ necessarily arise from “manipulations”\n(e.g., inclusions, (branched) coverings, etc.) taking $S_{1}$ to $S_{2}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.4.\n\nLiterature notes:\n(1) It is known that all such injections arise from inclusions of surfaces pro-\nvided that the genus of $S_{2}$ is not much larger than the genus of $S_{1}$ [AS12].\nFor braid groups, a detailed study of injective maps in a certain range\nwas announced by Chen–Kordek–Margalit [CKM19]; see also Problem\n\\section*{2.7. Both of these results pertain only to injections at the level of the full}\nmapping class group; the case of arbitrary finite-index subgroups seems\nmuch harder.\n(2) There are many natural variations on this problem, replacing mapping\nclass groups by curve graphs and related graphs associated to surfaces.\nThe literature on this subject is vast, with two central results being [Iva97]\nand [BM19]. It has been announced that a (model-theoretic) interpreta-\ntion between curve graphs of surfaces induces a virtual injection between\nmapping class group [DKdlNG20].\n\nReferences cited:\n- [AS12] Javier Aramayona and Juan Souto. Homomorphisms between mapping class groups. Geom. Topol., 16(4):2285–2341, 2012. doi:10.2140/gt.2012.16.2285.\n- [CKM19] Lei Chen, Kevin Kordek, and Dan Margalit. Homomorphisms between braid groups, 2019. arXiv:1910.00712.\n- [Iva97] Nikolai V. Ivanov. Automorphism of complexes of curves and of Teichmüller spaces. Internat. Math. Res. Notices, 1997(14):651–666, 1997. doi:10.1155/$S^{1}$073792897000433.\n- [BM19] Tara E. Brendle and Dan Margalit. Normal subgroups of mapping class groups and the metaconjecture of Ivanov. J. Amer. Math. Soc., 32(4):1009–1070, 2019. doi:10.1090/jams/927.\n- [DKdlNG20] Valentina Disarlo, Thomas Koberda, and J. de la Nuez González. The model theory of the curve graph, 2020. arXiv:2008.10490.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2753, "problem_number": "KP-2.5", "title": "Kirby Problem 2.5", "statement": "For $g \\geq 3$, determine a finite presentation for the Torelli\ngroup $\\mathcal{I}_{g}$, or show that no finite presentation exists.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.5.\n\nLiterature notes:\n(1) The Torelli group $\\mathcal{I}_{g} \\leq \\operatorname{Mod}(S_{g})$ is the subgroup of $\\operatorname{Mod}(S_{g})$ consisting of\nmapping classes acting trivially on $H_{1}(S_{g}; \\mathbb{Z})$. Work of Johnson [Joh83]\nshows that $\\mathcal{I}_{g}$ is finitely generated for $g \\geq 3$.\n(2) This is [Kir78, Problem 2.9]. The question was raised by Magnus, and\ndescribed by Birman in [Bir71].\n(3) For $g = 1$, the Torelli group is trivial, and for $g = 2$, Mess showed it is a\nfree group of infinite rank [Mes92].\n(4) For any finitely presented group $G$, the second homology $H_{2}(G; \\mathbb{Q})$ is finite\ndimensional, though the converse does not hold. Recent work of Minahan–\nPutman [MP25] announced that $H_{2}(\\mathcal{I}_{g}; \\mathbb{Q})$ is finite dimensional for $g \\geq 6$.\nIt is still open whether or not $H_{2}(\\mathcal{I}_{g}; \\mathbb{Z})$ is finitely generated for large $g$.\n\nReferences cited:\n- [Joh83] Dennis Johnson. The structure of the Torelli group. I. A finite set of generators for I. Ann. of Math. (2), 118(3):423–442, 1983. doi:10.2307/2006977.\n- [Kir78] Rob Kirby. Problems in low dimensional manifold theory. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 273–312. Amer. Math. Soc., Providence, R.I., 1978.\n- [Bir71] Joan S. Birman. On Siegel’s modular group. Math. Ann., 191:59–68, 1971. doi: 10.1007/BF01433472.\n- [Mes92] Geoffrey Mess. The Torelli groups for genus 2 and 3 surfaces. Topology, 31(4):775– 790, 1992. doi:10.1016/0040-9383(92)90008-6.\n- [MP25] Daniel Minahan and Andrew Putman. The second rational homology of the torelli group, 2025. arXiv:2504.00211.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2754, "problem_number": "KP-2.6", "title": "Kirby Problem 2.6", "statement": "Give a classification or enumeration of the finite-index sub-\ngroups of $\\operatorname{Mod}(S_{g})$ that are generated by Dehn twists, Dehn multitwists, or powers\nthereof. Which of these subgroups are “geometrically meaningful”, in the sense that\nthey are given as the stabilizers of some kind of geometric or topological struc-\nture on the surface? Conversely, when is a geometrically meaningful subgroup (not\nnecessarily of finite index) generated by powers of Dehn multitwists?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.6.\n\nLiterature notes:\n(1) The classical Dehn–Lickorish theorem asserts that the mapping class group\nof a closed orientable surface is generated by a finite collection of Dehn\ntwists.\nAny finite-index subgroup contains some power of every Dehn\ntwist, but a general finite-index subgroup need not contain any single twist\n$T_{c}$ at all, nor is a general finite-index subgroup generated by the powers\nof Dehn twists that it does contain. Indeed, work of Funar [Fun99] and\nMasbaum [Mas99] shows that for general values of $r$, the subgroup of\n$\\operatorname{Mod}(\\Sigma_{g})$ generated by $r^{th}$ powers of Dehn twists is of infinite index.\n(2) Salter and Calderon–Salter [Sal19, CS21] show that one such example\nis the class of $r$-spin mapping class groups, i.e., the stabilizers of a chosen\n$r$-spin structure on $S_{g}$, for $g \\geq 5$. In [CS23], they proved that the framed\nmapping class group for a framing (i.e., the stabilizer of the isotopy class\nof that framing) is likewise generated by Dehn twists; these subgroups are\nof infinite index. Salter–Sane [SS23] announced a generating set for the\nstabilizer of a mod-2 homology class $[x]$ consisting of Dehn twists together\nwith a single square-twist $T_{c}^{2}$.\n(3) The terms “geometrically meaningful” and “geometric or topological struc-\nture” in the problem statement are intentionally vague, but the examples\ndiscussed in the remarks above are meant to be representative. To dis-\nambiguate, we do not (necessarily) mean “geometric structure” in the\nThurstonian sense of a $(G, X)$-structure.\n\n(4) One motivation for this question comes from the theory of (achiral) Lef-\nschetz pencils, which are encoded in the mapping class group as sequences\nof Dehn twists whose product is the boundary multi-twist. When these\nDehn twists lie in some “geometrically meaningful” subgroup, one can\nask if this endows the ambient 4–manifold with an additional structure.\nOne instance of this is the case of $r = 2$ (i.e., classical) spin mapping class\ngroups, where the question of whether a symplectic 4-manifold is also spin\nis related to whether it admits a symplectic Lefschetz pencil whose twists\nlie in a spin mapping class group [Sti01, BHM23, BH24c, AB23].\n\nReferences cited:\n- [Fun99] Louis Funar. On the TQFT representations of the mapping class groups. Pacific J. Math., 188(2):251–274, 1999. doi:10.2140/pjm.1999.188.251.\n- [Mas99] Gregor Masbaum. An element of infinite order in TQFT-representations of mapping class groups. In Low-dimensional topology (Funchal, 1998), volume 233 of Contemp. Math., pages 137–139. Amer. Math. Soc., Providence, RI, 1999. doi: 10.1090/conm/233/03423.\n- [Sal19] Nick Salter. Monodromy and vanishing cycles in toric surfaces. Invent. Math., 216(1):153–213, 2019. doi:10.1007/s00222-018-0845-6.\n- [CS21] Aaron Calderon and Nick Salter. Higher spin mapping class groups and strata of abelian differentials over Teichmüller space. Adv. Math., 389:Paper No. 107926, 56, 2021. doi:10.1016/j.aim.2021.107926.\n- [CS23] Aaron Calderon and Nick Salter. Framed mapping class groups and the monodromy of strata of abelian differentials. J. Eur. Math. Soc. (JEMS), 25(12):4719–4790, 2023. doi:10.4171/jems/1290.\n- [SS23] Nick Salter and Abdoul Karim Sane. Connected components of the topological surgery graph of a unicellular collection, 2023. arXiv:2308.09165.\n- [Sti01] András I. Stipsicz. Spin structures on Lefschetz fibrations. Bull. London Math. Soc., 33(4):466–472, 2001. doi:10.1017/S0024609301008232.\n- [BHM23] R. İnanç Baykur, Kenta Hayano, and Naoyuki Monden. Unchaining surgery and topology of symplectic 4-manifolds. Math. Z., 303(3):Paper No. 77, 32, 2023. doi: 10.1007/s00209-023-03204-x.\n- [BH24c] R. İnanç Baykur and Noriyuki Hamada. Lefschetz fibrations with arbitrary signature. J. Eur. Math. Soc. (JEMS), 26(8):2837–2895, 2024. doi:10.4171/jems/1326.\n- [AB23] Mihail Arabadji and R. İnanç Baykur. Spin Lefschetz fibrations are abundant. Pacific J. Math., 326(1):1–16, 2023. doi:10.2140/pjm.2023.326.1.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2755, "problem_number": "KP-2.7", "title": "Kirby Problem 2.7", "statement": "Classify the homomorphisms from the braid group $B_{n}$ on $n$\nstrands to the braid group $B_{m}$ on $m$ strands, where $n, m \\in \\mathbb{N}$ are arbitrary.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.7.\n\nLiterature notes:\n(1) There are several works in this direction including those of Artin [Art47],\nLin [Lin04], Dyer–Grossman [DG81], Bell–Margalit [BM06], Castel [Cas16]\nand, finally, Chen–Kordek–Margalit [CKM19], where the most general\nresult known to date was announced (the complete classification for $m \\leq$\n$2n$ and $n \\geq 5$).\n(2) There is a more precise formulation of this question. We view the group\n$B_{m}$ as the mapping class group $\\operatorname{Mod}(S_{0,m}^{1})$ of the $m$-punctured disk and\ndenote by $\\mathcal{C}(S_{0,m}^{1})$ the curve complex of $S_{0,m}^{1}$. We say that a homomor-\nphism $\\phi: B_{n} \\to B_{m} = \\operatorname{Mod}(S_{0,m}^{1})$ is reducible if there exists a nonempty\nsimplex $\\mathcal{A}$ of $\\mathcal{C}(S_{0,m}^{1})$ such that $\\phi(\\beta)(\\mathcal{A}) = \\mathcal{A}$ for all $\\beta \\in B_{n}$. That is,\nevery element of $\\phi(B_{n})$ is reducible, and moreover preserves the same set\nof curves. Then we have the following conjecture, which can be compared\nwith the Nielsen–Thurston classification.\nConjecture. Let $n, m \\in \\mathbb{N}$ with $5 \\leq n \\leq m$, and let $\\phi: B_{n} \\to B_{m}$ be\na homomorphism. Then we have one of the following three possibilities.\n(a) The image of $\\phi$ is a cyclic group.\n(b) $n = m$, and $\\phi$ is an injective homomorphism as described in [BM06].\n(c) $\\phi$ is reducible.\n\n(3) In light of the identification of $B_{m}$ with the mapping class group $\\operatorname{Mod}(S_{0,m}^{1})$,\nthis question admits a natural generalization to the problem of classifying\nall homomorphisms between mapping class groups of surfaces; see Prob-\nlem 2.4. For further discussion, see the survey article [AS16].\n\nReferences cited:\n- [Art47] E. Artin. Braids and permutations. Ann. of Math. (2), 48:643–649, 1947. doi: 10.2307/1969131.\n- [Lin04] Vladimir Lin. Braids and permutations, 2004. arXiv:math/0404528.\n- [DG81] Joan L. Dyer and Edna K. Grossman. The automorphism groups of the braid groups. Amer. J. Math., 103(6):1151–1169, 1981. doi:10.2307/2374228.\n- [BM06] Robert W. Bell and Dan Margalit. Braid groups and the co-Hopfian property. J. Algebra, 303(1):275–294, 2006. doi:10.1016/j.jalgebra.2005.10.038.\n- [Cas16] Fabrice Castel. Geometric representations of the braid groups. Astérisque, 378:vi+175, 2016.\n- [CKM19] Lei Chen, Kevin Kordek, and Dan Margalit. Homomorphisms between braid groups, 2019. arXiv:1910.00712.\n- [AS16] Javier Aramayona and Juan Souto. Rigidity phenomena in the mapping class group. In Handbook of Teichmüller theory. Vol. VI, volume 27 of IRMA Lect. Math. Theor. Phys., pages 131–165. Eur. Math. Soc., Zürich, 2016.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2756, "problem_number": "KP-2.8", "title": "Kirby Problem 2.8", "statement": "Fix distinct trivial tangles $\\tau_{1}, \\tau_{2}$ for which $\\tau_{1} \\cup\\tau_{2}$ is the unknot.\nDescribe the intersection of the associated wicket groups $W_{n}(\\tau_{1}) \\cap W_{n}(\\tau_{2}) \\leq B_{2n}$.\nIs this group finitely generated? Finitely presented?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.8.\n\nLiterature notes:\n(1) A tangle on $n$ strands is a proper embedding of $n$ disjoint arcs into the\nupper half-space in $\\mathbb{R}^{3}$, considered up to isotopy rel boundary. Tangles\n$\\tau_{1}, \\tau_{2}$ based at the same set of $2n$ points in $\\mathbb{R}^{2} \\subset \\mathbb{R}^{3}$ can be combined\ninto the link $\\tau_{1} \\cup \\tau_{2}$, where $\\overline{\\tau}_{2}$ denotes the reflection of $\\tau_{2}$ through the\n$xy$-plane. A trivial tangle is any tangle that can be isotoped so that each\nstrand has a single local maximum with respect to the standard height\nfunction on $\\mathbb{R}^{3}$. There is a natural transitive action of the braid group\n$B_{2n}$ on the set of trivial tangles based at a fixed set of $2n$ points, and the\nstabilizer of a fixed trivial tangle $\\tau$ is known as a wicket group, written\n$W_{n}$ or $W_{n}(\\tau)$ when we wish to emphasize the particular trivial tangle.\n(2) The questions in the problem originate from the works of S. Hirose,\nD. Iguchi, E. Kin, and Y. Koda that are discussed below.\n(3) Note that $W_{n}(\\tau_{1})$ and $W_{n}(\\tau_{2})$ are conjugate in $B_{2n}$.\nBrendle–Hatcher\n[BH13] determined a finite presentation for $W_{n}$. This problem has a nat-\nural generalization in which one allows $\\tau_{1} \\cup \\overline{\\tau}_{2}$ to be an unlink of $c$ com-\nponents for $c \\geq 1$ [HIKK22, Question 2.10]. The group $W_{n}(\\tau_{1})\\cap W_{n}(\\tau_{2})$\ncould be called the Goeritz group of the bridge splitting $\\tau_{1} \\cup \\overline{\\tau}_{2}$ of the un-\nlink. Hirose–Iguchi–Kin–Koda defined and studied this group as the hy-\nperelliptic Goeritz group, proving that this group is finitely presented when\nthe Heegaard splitting is a 3-bridge splitting of a 2-bridge link [HIKK22].\nIf the distance of an $n$-bridge decomposition of a link in $S^{3}$ with $n \\geq 3$\nis at least 5, then its Goeritz group is a finite group [IK20, Theorem 0.1].\n(4) There is a connection with the study of knotted surfaces through the the-\nory of bridge trisections [MZ17a]: Any smooth surface-link in $\\mathbb{R}^{4}$ can be\nrepresented by a tri-plane diagram in which the first two tangle diagrams\nare arranged to have no crossings. Given this, the Goeritz group of the\nbridge splitting of the unlink given by the first two tangles is precisely the\nset of braids that act on the tri-plane diagram without changing the first\ntwo tangles. Thus, an understanding of this group could be applied to the\nproblem of finding equivalences between bridge trisected surface-links.\n\nReferences cited:\n- [BH13] Tara E. Brendle and Allen Hatcher. Configuration spaces of rings and wickets. Comment. Math. Helv., 88(1):131–162, 2013. doi:10.4171/CMH/280.\n- [HIKK22] Susumu Hirose, Daiki Iguchi, Eiko Kin, and Yuya Koda. Goeritz groups of bridge decompositions. Int. Math. Res. Not. IMRN, 2022(12):9308–9356, 2022. doi:10.1093/imrn/rnab001.\n- [IK20] Daiki Iguchi and Yuya Koda. Twisted book decompositions and the Goeritz groups. Topology Appl., 272:107064, 15, 2020. doi:10.1016/j.topol.2020.107064.\n- [MZ17a] Jeffrey Meier and Alexander Zupan. Bridge trisections of knotted surfaces in $S^{4}$. Trans. Amer. Math. Soc., 369(10):7343–7386, 2017. doi:10.1090/tran/6934.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2757, "problem_number": "KP-2.9", "title": "Kirby Problem 2.9", "statement": "Is there a nice presentation of the $n$-stranded braid group whose\ngenerating set is the set of all positive elementary braid half-twists?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.9.\n\nLiterature notes:\n(1) The adjective “nice” here should at least mean that all relations consist\nof equating two words in the positive generators of the same length, and\nperhaps also that there be a finite number of local pictures describing the\nrequired relations.\n(2) The positive monoid in the standard Artin presentation consists of all\npositive braids. In the Birman–Ko–Lee presentation [BKL98], the pos-\nitive monoid is the strongly quasipositive braids. The positive monoid\nfor the desired presentation would correspond to the quasipositive braids\nand would hopefully provide interesting algorithmic tools similar to Gar-\nside [Gar69] for solving questions regarding quasipositivity. Similar infi-\nnite presentations hold for the mapping class groups of surfaces (Gervais\n[Ger96]) as well as the Torelli group (Putman [Put07]).\n\nReferences cited:\n- [BKL98] Joan Birman, Ki Hyoung Ko, and Sang Jin Lee. A new approach to the word and conjugacy problems in the braid groups. Adv. Math., 139(2):322–353, 1998. doi:10.1006/aima.1998.1761.\n- [Gar69] F. A. Garside. The braid group and other groups. Quart. J. Math. Oxford Ser. (2), 20:235–254, 1969. doi:10.1093/qmath/20.1.235.\n- [Ger96] Sylvain Gervais. Presentation and central extensions of mapping class groups. Trans. Amer. Math. Soc., 348(8):3097–3132, 1996. doi:10.1090/S0002-9947-96-01509-7.\n- [Put07] Thomas Andrew Putman. An infinite presentation of the Torelli group. ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)–The University of Chicago. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt=info: ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri:pqdiss:3262286.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2758, "problem_number": "KP-2.10", "title": "Kirby Problem 2.10", "statement": "(a) Is there an efficient algorithm to compute distances in the curve complex\nof a surface? The input to the algorithm should be the surface and the\ncurves. One may ask similar questions for related complexes, such as the\narc complex of a surface with boundary.\n(b) Is there an algorithm (efficient or otherwise) to compute the distance be-\ntween quasi-convex subsets of the curve complex? This question is partic-\nularly interesting in the case where the quasi-convex subsets are the disk\nsets of handlebodies.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.10.\n\nLiterature notes:\n(1) The curve complex has proved to be particularly important in Kleinian\ngroup theory and in the study of the mapping class groups of surfaces\n[Min06].\nOne notable feature of $\\mathcal{C}(S)$, proved by Masur and Minsky\n[MM99], is that it is Gromov hyperbolic.\n(2) The complex is not locally finite, and so there is no naive way of comput-\ning distances. However, distances in $\\mathcal{C}(S)$ are known to be computable,\nby work of Leasure [Lea02], Shackleton [Sha12], Webb [Web15], Bir-\nman, Margalit, and Menasco [BMM16] and Watanabe [Wat16]. Bell\nand Webb [BW16a] announced an algorithm with running time that is\nbounded above by a polynomial function of the logarithm of the weight of\n\nthe two curves, given as normal curves in some triangulation of $S$. How-\never, the running time of their algorithm grows very rapidly as the surface\n$S$ is allowed to vary. The aim of Problem (a) is to find such an algorithm\nthat runs in polynomial time even as the surface varies.\nBaroni [Bar24] has announced an efficient algorithm to compute dis-\ntances up to a bounded multiplicative error, the bound depending only\non $S$. He used this to find an efficient algorithm to determine whether a\ngiven surface automorphism is pseudo-Anosov, reducible, or periodic.\n(3) A related problem is as follows. Given a reducible element of the mapping\nclass group, can an invariant multi-curve be found efficiently (in polyno-\nmial time as function of the ‘size’ of the mapping class and also in the\nEuler characteristic of $S$)?\n(4) When $S$ is the boundary of a handlebody $H$, the associated disk set con-\nsists of the subcomplex of $\\mathcal{C}(S)$ induced by the curves bounding disks\nin $H$. This was proved to be quasi-convex in $\\mathcal{C}(S)$ by Masur and Min-\nsky [MM04]. When $S$ is a Heegaard surface for a closed 3-manifold, it\nbounds a handlebody on each side and so there are two associated disk\nsets. The distance between them in $\\mathcal{C}(S)$ is the Hempel distance of the\nsplitting [Hem01]. This is not known to be computable. However, it\nwas proved by Masur and Schleimer [MS13] that it is computable up to\na bounded additive error. The Hempel distance of a Heegaard splitting\nencodes important information. For example, the Hempel distance is zero\nif and only if the splitting is reducible, and the Hempel distance is one if\nand only if the splitting is weakly reducible but irreducible.\n\nReferences cited:\n- [Min06] Yair N. Minsky. Curve complexes, surfaces and 3-manifolds. In International Congress of Mathematicians. Vol. II, pages 1001–1033. Eur. Math. Soc., Zürich, 2006.\n- [MM99] Howard A. Masur and Yair N. Minsky. Geometry of the complex of curves. I. Hyperbolicity. Invent. Math., 138(1):103–149, 1999. doi:10.1007/s002220050343.\n- [Lea02] Jason Paige Leasure. Geodesics in the complex of curves of a surface. ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)–The University of Texas at Austin. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt= info:ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri:pqdiss:3114766.\n- [Sha12] Kenneth J. Shackleton. Tightness and computing distances in the curve complex. Geom. Dedicata, 160:243–259, 2012. doi:10.1007/s10711-011-9680-2.\n- [Web15] Richard C. H. Webb. Combinatorics of tight geodesics and stable lengths. Trans. Amer. Math. Soc., 367(10):7323–7342, 2015. doi:10.1090/tran/6301.\n- [BMM16] Joan Birman, Dan Margalit, and William Menasco. Efficient geodesics and an effective algorithm for distance in the complex of curves. Math. Ann., 366(3-4):1253– 1279, 2016. doi:10.1007/s00208-015-1357-y.\n- [Wat16] Yohsuke Watanabe. Intersection numbers in the curve graph with a uniform constant. Topology Appl., 204:157–167, 2016. doi:10.1016/j.topol.2016.03.009.\n- [BW16a] Mark C. Bell and Richard C. H. Webb. Polynomial-time algorithms for the curve graph, 2016. arXiv:1609.09392.\n- [Bar24] Filippo Baroni. Uniformly polynomial-time classification of surface homeomorphisms, 2024. arXiv:2402.00231.\n- [MM04] Howard A. Masur and Yair N. Minsky. Quasiconvexity in the curve complex. In In the tradition of Ahlfors and Bers, III, volume 355 of Contemp. Math., pages 309–320. Amer. Math. Soc., Providence, RI, 2004. doi:10.1090/conm/355/06460.\n- [Hem01] John Hempel. 3-manifolds as viewed from the curve complex. Topology, 40(3):631– 657, 2001. doi:10.1016/S0040-9383(00)00033-1.\n- [MS13] Howard Masur and Saul Schleimer. The geometry of the disk complex. J. Amer. Math. Soc., 26(1):1–62, 2013. doi:10.1090/S0894-0347-2012-00742-5.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2759, "problem_number": "KP-2.11", "title": "Kirby Problem 2.11", "statement": "Find precise estimates for both the extremal and average behav-\nior of the simple lifting degree of curves, in terms of combinatorial (e.g., intersection\nnumber) and/or geometric (e.g., geodesic length, word length) data.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.11.\n\nLiterature notes:\n(1) A result of Peter Scott [Sco78] tells us that for any closed geodesic $\\gamma$ on\na hyperbolic surface $S$, there exists a finite degree cover $S' \\to S$ such that\n$\\gamma$ lifts to $S'$ (i.e., choosing basepoints, $\\gamma \\in \\pi_{1}(S') \\leq \\pi_{1}(S)$) and is a simple\nclosed curve in $S'$. The simple lifting degree of $\\gamma$, which we will write as\n$\\deg(\\gamma)$, is defined to be the minimal degree of such a covering $S'$ to which\n$\\gamma$ lifts simply.\n(2) To precisely formulate the relationship between simple lifting degree and\ncombinatorial/geometric complexity, define the functions\n\n$$\nf_{S}(n)=\\max\\{\\deg(\\gamma): i(\\gamma,\\gamma)\\leq n\\}.\n$$\n\nand, for a choice of hyperbolic metric $\\rho$ on $S$,\n\n$$\nf_{\\rho}(L)=\\max\\{\\deg(\\gamma): \\ell_{\\rho}(\\gamma)\\leq L\\}.\n$$\n\nThe extremal behavior (i.e., upper bounds) of these functions has been\nstudied in work of Patel [Pat14], Gaster [Gas16], and Aougab–Gaster–\nPatel–Sapir [AGPS17]. A lower bound for average behavior of $\\deg(\\gamma)$ was\n\nobtained by Aougab–Gaster [AG22], but work of Sisto–Taylor [ST19]\nimplies that some mechanism other than what Gaster finds in the extremal\ncase is responsible for the average behavior. What is this mechanism?\nProgress in the study of $f_{S}(n)$ has been obtained by Arenas and\nNeumann-Coto [ANC20]. They establish the inequality $f_{S}(n) \\leq 5(n+1)$;\nnote that this is independent of the surface $S$. They conjecture that the\ntighter bound $f_{S}(n) \\leq 2(n + 1)$ should hold. At the time of this writing,\nno examples exist with $f_{S}(n) - n$ more than 2.\n(3) Variations of these questions can be posed by, e.g., restricting to the class\nof regular covers or to families of curves intersecting on a surface $S$. These\nhave been largely unexplored.\n\nReferences cited:\n- [Sco78] Peter Scott. Subgroups of surface groups are almost geometric. J. London Math. Soc. (2), 17(3):555–565, 1978. doi:10.1112/jlms/s2-17.3.555.\n- [Pat14] Priyam Patel. On a theorem of Peter Scott. Proc. Amer. Math. Soc., 142(8):2891– 2906, 2014. doi:10.1090/S0002-9939-2014-12031-4.\n- [Gas16] Jonah Gaster. Lifting curves simply. Int. Math. Res. Not. IMRN, 2016(18):5559– 5568, 2016. doi:10.1093/imrn/rnv316.\n- [AGPS17] Tarik Aougab, Jonah Gaster, Priyam Patel, and Jenya Sapir. Building hyperbolic metrics suited to closed curves and applications to lifting simply. Math. Res. Lett., 24(3):593–617, 2017. doi:10.4310/MRL.2017.v24.n3.a1.\n- [AG22] Tarik Aougab and Jonah Gaster. Combinatorially random curves on surfaces, 2022. arXiv:2209.11309.\n- [ST19] Alessandro Sisto and Samuel J. Taylor. Largest projections for random walks and shortest curves in random mapping tori. Math. Res. Lett., 26(1):293–321, 2019. doi:10.4310/MRL.2019.v26.n1.a14.\n- [ANC20] Macarena Arenas and Max Neumann-Coto. Measuring complexity of curves on surfaces. Geom. Dedicata, 204:25–41, 2020. doi:10.1007/s10711-019-00443-3.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2760, "problem_number": "KP-2.12", "title": "Kirby Problem 2.12", "statement": "(a) What is the maximum number of systoles on a closed, hyperbolic surface\nof genus $g$?\n(b) What is the maximum cardinality of a set of pairwise non-isotopic, essen-\ntial, simple closed curves on a closed surface of genus $g$ with the property\nthat no two curves in the set intersect in more than one point?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.12.\n\nLiterature notes:\n(1) A systole on a closed hyperbolic surface is a closed geodesic of shortest\nlength. A systole is necessarily an essential, simple closed curve. On a\nclosed surface, any two systoles intersect in at most one point (in the\npresence of punctures, systoles may intersect twice; see [FP15]). Hence\nthe first problem arises as a special case of the second. This quantity is\nsometimes called the “kissing number”; kissing numbers of lattices have\nbeen studied in a variety of contexts.\n(2) The first problem was first seriously studied by Schmutz Schaller [Sch96].\nThe answer is known to lie between $C_{1}(\\epsilon)\\cdot g^{4/3-\\epsilon}$ and $C_{2} \\cdot g^{2}/ \\log(g)$. Here\n$C_{1}(\\epsilon)$ is a function of an arbitrary value $\\epsilon > 0$ and $C_{2}$ is an absolute\nconstant. The lower bound is a construction of Schmutz Schaller using\narithmetic surfaces [SS97]. The upper bound is due to Parlier using com-\nbinatorial and hyperbolic techniques [Par13b]. Schmutz Schaller conjec-\ntured that the lower bound is optimal, in that the maximum cardinality\ncannot exceed $C_{3} \\cdot g^{4/3}$ for some absolute constant $C_{3}$ [SS97, Conjecture].\n(3) The second problem was raised independently by Juvan, Malnic, and Mo-\nhar [JMM96] and by Farb and Leininger, as recorded in [MRT14, Ques-\ntion 1].\nThe answer is known to lie between $C_{1} \\cdot g^{2}$ and $C_{2} \\cdot g^{2} \\log g$.\nThe lower bound is a construction due to Malestein, Rivin, and Theran\n[MRT14]. The upper bound is due to Greene, using probabilistic com-\nbinatorics and geometric techniques [Gre19]. The lower bound is conjec-\ntured to be optimal, i.e., there exists an upper bound of the form $C_{3} \\cdot g^{2}$\nto match it.\n\nPrzytycki solved the corresponding problem for arcs: if $S$ is a con-\nnected, orientable, punctured surface of finite type, then the maximum\nnumber of pairwise non-isotopic, essential, simple arcs that limit to the\npunctures is $(|\\chi(S)| + 1)(|\\chi(S)| + 2)$ [Prz15]. His methods involve hyper-\nbolic geometry.\nReplacing “one” by $k$ in the problem, the answer is known to lie\nbetween $C_{1}(k) \\cdot g^{k+1}$ and $C_{2}(k) \\cdot g^{k+1} \\log g$, for some functions $C_{1}(k)$ and\n$C_{2}(k)$ independent of $g$. Once more, the lower bound is conjectured to\nbe optimal, i.e., there exists an upper bound of the form $C_{3}(k) \\cdot g^{k+1}$ to\nmatch it [Gre19, Conjecture 1].\n\nReferences cited:\n- [FP15] Federica Fanoni and Hugo Parlier. Systoles and kissing numbers of finite area hyperbolic surfaces. Algebr. Geom. Topol., 15(6):3409–3433, 2015. doi:10.2140/agt.2015.15.3409.\n- [Sch96] Paul Schmutz. Compact Riemann surfaces with many systoles. Duke Math. J., 84(1):191–198, 1996. doi:10.1215/S0012-7094-96-08406-9.\n- [SS97] Paul Schmutz Schaller. Extremal Riemann surfaces with a large number of systoles. In Extremal Riemann surfaces (San Francisco, CA, 1995), volume 201 of Contemp. Math., pages 9–19. Amer. Math. Soc., Providence, RI, 1997. doi:10.1090/conm/201/02617.\n- [Par13b] Hugo Parlier. Kissing numbers for surfaces. J. Topol., 6(3):777–791, 2013. doi: 10.1112/jtopol/jtt012.\n- [JMM96] M. Juvan, A. Malnič, and B. Mohar. Systems of curves on surfaces. J. Combin. Theory Ser. B, 68(1):7–22, 1996.\n- [MRT14] Justin Malestein, Igor Rivin, and Louis Theran. Topological designs. Geom. Dedicata, 168:221–233, 2014. doi:10.1007/s10711-012-9827-9.\n- [Gre19] Joshua Evan Greene. On loops intersecting at most once. Geom. Funct. Anal., 29(6):1828–1843, 2019. doi:10.1007/s00039-019-00517-0.\n- [Prz15] Piotr Przytycki. Arcs intersecting at most once. Geom. Funct. Anal., 25(2):658–670, 2015. doi:10.1007/s00039-015-0320-0.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2761, "problem_number": "KP-2.13", "title": "Kirby Problem 2.13", "statement": "Let $S$ be a surface, and let $\\Gamma_{1}, \\Gamma_{2}$ be isotopy classes of embedded\ngraphs in $S$. Determine when $\\Gamma_{1}$ and $\\Gamma_{2}$ are related by a sequence of surgeries.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.13.\n\nLiterature notes:\n(1) A surgery of an embedded graph $\\Gamma \\subset S$ is defined as follows. Suppose\nsome component of $S\\setminus\\Gamma$ is a disk $D$. Let $x$ and $y$ be oriented edges in $\\Gamma$\nbounding $D$. There is a unique isotopy class $\\lambda_{x,y}$ of arc connecting $x$ to\n$y$ inside $D$. A regular neighborhood $N_{x,y}$ of $\\lambda_{x,y}$ is a rectangle with two\nopposite sides comprising subarcs of $x$ and $y$, and remaining sides running\nparallel to $\\lambda_{x,y}$. The surgery of $\\Gamma$ along $\\lambda_{x,y}$ is the graph $\\Gamma'$ obtained from\n$\\Gamma$ by deleting the edges of $N_{x,y}$ lying on $x, y$ and replacing them with the\ntwo edges parallel to $\\lambda_{x,y}$. Note that surgery can change the topology of\nthe complement; there is a simple combinatorial criterion to determine\nwhen this does or does not happen [San21].\n(2) The number of vertices and edges are manifestly preserved under surgery,\nbut the isomorphism type of the graph can change. Sane has investigated\nthe “surgery graph” that tracks when two embedded graphs are related\nby a surgery and has studied connectivity properties of this graph in a\nnumber of cases, discussed below.\nWhen every vertex of $\\Gamma$ has even valence, $\\Gamma$ determines a class in\n$H_{1}(S; \\mathbb{Z}/2\\mathbb{Z})$. This homology class is readily seen to be invariant under\nsurgery. The case where every vertex has valence 4 and the complement\nis a single disk was studied by Salter–Sane [SS23], who announced that\nin this case, graphs $\\Gamma$ and $\\Gamma'$ are related via a sequence of surgeries if and\nonly if their mod-2 homology classes are equal. In the case where every\nvertex has valence 3 and the complement is a single disk, Sane announced\nin [San20] that any two such graphs on the same surface can be connected\nvia a sequence of surgeries.\n(3) This problem is more topological (and less purely combinatorial) than it\nmight appear to be at first glance. It is closely connected to the action of\nthe mapping class group on embedded graphs, and to the general prob-\nlem of understanding subgroups of the mapping class group. A surgery\nequivalence class determines a “stabilizer subgroup” of the mapping class\ngroup of $S$, consisting of all mapping classes $f$ for which $f(\\Gamma)$ can be\n\ntaken to $\\Gamma$ via surgery. The result of Salter–Sane mentioned above can\nbe rephrased as the assertion that the stabilizer subgroup for the class of\ngraphs they consider is the stabilizer of a mod-2 homology class. Other\nclasses of graphs appear to have smaller stabilizer subgroups that are not\nso easily characterized. Given the enduring mystery of classifying sub-\ngroups of the mapping class group (especially those of finite index), a\nprimary motivation for understanding graph surgery is that it may lead\nto novel examples and phenomena in this arena.\n(4) The literature mentioned above restricts attention to the unicellular case,\nwhen $S\\setminus\\Gamma$ is a single disk (note that these authors further restrict to surg-\neries that preserve this property). One reason to be particularly interested\nin this setting is that the enumeration of such “unicellular graphs” is a key\ningredient in Harer and Zagier’s computation of the Euler characteristic\nof the moduli space of Riemann surfaces [HZ86].\n\nReferences cited:\n- [San21] Abdoul Karim Sane. Curves on surfaces and surgeries. European J. Combin., 93:Paper No. 103281, 20, 2021. doi:10.1016/j.ejc.2020.103281.\n- [SS23] Nick Salter and Abdoul Karim Sane. Connected components of the topological surgery graph of a unicellular collection, 2023. arXiv:2308.09165.\n- [San20] Abdoul Karim Sane. Unicellular maps and filtrations of the mapping class group, 2020. arXiv:2006.15880.\n- [HZ86] J. Harer and D. Zagier. The Euler characteristic of the moduli space of curves. Invent. Math., 85(3):457–485, 1986. doi:10.1007/BF01390325.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2762, "problem_number": "KP-2.14", "title": "Kirby Problem 2.14", "statement": "(a) Does every Jordan curve in the Euclidean plane contain the vertices of a\nsquare?\n(b) Does every Jordan curve in the Euclidean plane contain the vertices of the\naffine image of a regular hexagon?\n(c) Let $\\gamma \\subset \\mathbb{C}$ be a smooth Jordan curve, and let $S \\subset \\mathbb{C}$ consist of $2n \\geq 4$\nconcyclic points. Does there exist a non-constant polynomial $p(z) \\in \\mathbb{C}[z]$\nof degree $< n$ such that $p(S) \\subset \\gamma$?\n(d) Suppose that $h$ is a continuous, real-valued function defined on the unit\nsphere $S^{2} \\subset \\mathbb{R}^{3}$ and that points $x_{1},..., x_{4} \\in S^{2}$ form the vertices of\na square.\nIs there an isometry $\\rho \\in \\operatorname{SO}(3)$ such that $h$ is constant on\n$\\rho(x_{1}),..., \\rho(x_{4})$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.14.\n\nLiterature notes:\n(1) The first problem is known as the Square Peg Problem. It was posed by\nOtto Toeplitz in 1911 [Toe11]. It is solved affirmatively for several classes\nof curves, including $C^{1}$ curves (see [Mat14]).\nMore generally, a curve inscribes an $n$-gon (and the $n$-gon inscribes\nin the curve) if there exists an orientation-preserving similar copy of the\n$n$-gon whose vertices lie on the curve. Every Jordan curve inscribes ev-\nery triangle (a noncolinear 3-gon) [Nie92], but no two dissimilar ellipses\ninscribe the same pentagon. Vaughn proved that every Jordan curve in-\nscribes a rectangle (of some uncontrolled aspect ratio) [Mey81]. Greene\nand Lobb proved the optimal result for smooth curves: every smooth Jor-\ndan curve inscribes every cyclic quadrilateral (i.e., a quadrilateral whose\nvertices lie on a circle) [GL23]. By contrast, Pak observed that the quadri-\nlaterals that inscribe in all triangles are the isosceles trapezoids [Pak08].\nIt is possible that the widest class of quadrilaterals that inscribe in all\nJordan curves are the isosceles trapezoids.\n\n(2) The second problem was raised as a conjecture by Grünbaum [Gru72,\nConjecture 4.3]. It is known to hold if the curve is convex [Gru72, p.84].\nIt is known that every $C^{1}$ Jordan curve either contains the vertices of the\naffine image of a regular hexagon or six colinear points that are the limit\nof the vertices of a sequence of affine images of regular hexagons [Vv11,\nTheorem 7]. However, the problem is open in general, even for smooth\ncurves.\n(3) The third problem was raised as a conjecture by Greene and Lobb [GL24,\nConjecture 1.1]. The case $n = 2$ is true and is equivalent to the result\nabout cyclic quadrilaterals noted above in the discussion of the first prob-\nlem.\nThe case $n = 3$, and some special instances when $n \\geq 4$, was\nannounced in [GL24]. These results all involve symplectic geometry.\n(4) The fourth problem is known as the Table Problem, and it is related to the\nKnaster problem (which has been solved in the negative). It is posed as\na conjecture in [Mat14, Conjecture 13]. The case in which $x_{1},..., x_{4}$ are\nequally spaced around a great circle was solved affirmatively by Dyson\n[Dys51].\nThe case in which $h$ is an even function was announced by\nNaseri Sadr, who also announced a proof of a variation for Riemannian\nsurfaces [NS24]. Call a set of four points $x_{1},..., x_{4} \\in S^{2}$ balancing if for\nevery $h$ there exists a $\\rho$ as in the problem statement. The Table Problem\ntherefore asserts that the vertices of a square are a balancing set. The\ncase of the standard height function shows that a balancing set of points\nmust be concyclic. On the other hand, there exist concyclic $x_{1},..., x_{4}$\nthat are not a balancing set [Kar13]. Livesay proved that the vertices\nof a rectangle contained in a great circle form a balancing set [Liv54].\nIt is possible that the vertices of any rectangle on $S^{2}$ form a balancing\nset. A proof was claimed in [Gri91], but that paper was later invalidated\n[Mat14].\nRemarks on these and related problems appear in Klee and Wagon’s book [KW91]\nand Matschke’s survey article [Mat14].\n\nReferences cited:\n- [Toe11] Otto Toeplitz. Ueber einige Aufgaben der Analysis situs. Verhandlungen der Schweizerischen Naturforschenden Gesellschaft, 4:197, 1911.\n- [Mat14] Benjamin Matschke. A survey on the square peg problem. Notices Amer. Math. Soc., 61(4):346–352, 2014. doi:10.1090/noti1100.\n- [Nie92] Mark J. Nielsen. Triangles inscribed in simple closed curves. Geom. Dedicata, 43(3):291–297, 1992. doi:10.1007/BF00151519.\n- [Mey81] Mark D. Meyerson. Balancing acts. Topology Proc., 6(1):59–75 (1982), 1981.\n- [GL23] Joshua Evan Greene and Andrew Lobb. Cyclic quadrilaterals and smooth Jordan curves. Invent. Math., 234(3):931–935, 2023. doi:10.1007/s00222-023-01212-6.\n- [Pak08] Igor Pak. The discrete square peg problem, 2008. arXiv:0804.0657.\n- [Gru72] Branko Grunbaum. Arrangements and spreads, volume No. 10 of Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI, 1972.\n- [Vv11] Siniša T. Vrećica and Rade T. Živaljević. Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem. Israel J. Math., 184:221–249, 2011. doi:10.1007/s11856-011-0066-9.\n- [GL24] Joshua Evan Greene and Andrew Lobb. Polynomial inscriptions, 2024. arXiv:2206.14710.\n- [Dys51] F. J. Dyson. Continuous functions defined on spheres. Ann. of Math. (2), 54:534– 536, 1951. doi:10.2307/1969487.\n- [NS24] Ali Naseri Sadr. A Table theorem for surfaces with odd Euler characteristic, 2024. arXiv:2206.14710.\n- [Kar13] R. N. Karasëv. On two conjectures of Makeev. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 415:5–14, 2013. doi:10.1007/s10958-016-2679-3.\n- [Liv54] George R. Livesay. On a theorem of F. J. Dyson. Ann. of Math. (2), 59:227–229, 1954. doi:10.2307/1969689.\n- [Gri91] H. B. Griffiths. The topology of square pegs in round holes. Proc. London Math. Soc. (3), 62(3):647–672, 1991. doi:10.1112/plms/s3-62.3.647.\n- [KW91] Victor Klee and Stan Wagon. Old and new unsolved problems in plane geometry and number theory, volume 11 of The Dolciani Mathematical Expositions. Mathematical Association of America, Washington, DC, 1991.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2763, "problem_number": "KP-2.15", "title": "Kirby Problem 2.15", "statement": "Is the genus $g$ Goeritz group $\\mathcal{G}_{g}$ finitely generated when $g \\geq 4$?\nIf so, find a set of generators.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.15.\n\nLiterature notes:\n(1) The Goeritz group $\\mathcal{G}_{g}$ is the group of isotopy classes of orientation-preserving\nhomeomorphisms of $S^{3}$ that preserve an unknotted genus $g$ handlebody,\nthat is, one side of a genus $g$ Heegaard splitting. It is a rough analogue\nfor $S^{3}$ of the braid group for $S^{2}$. The notion can be extended to Heegaard\nsplittings of other 3-manifolds, as well [JM13].\n(2) Goeritz [Goe33] exhibited a complete set of three generators for $\\mathcal{G}_{2}$ and\nAkbas [Akb08] found associated relators, providing a complete presen-\ntation of $\\mathcal{G}_{2}$. Powell [Pow80] believed he had found a complete set of\n5 generators for $\\mathcal{G}_{g}$, for any $g$, but there was a serious gap in the proof\n(see footnote of [Sch04]). This is now known as the Powell Conjecture.\nIn [Sch20] it is shown that one of Powell’s proposed five generators is\nredundant.\nFreedman and Scharlemann [FS18] announced a proof of the Powell\nConjecture for $g = 3$. Their methods break down for higher genus, and\nit is suspected that for $g \\geq 4$ the group $\\mathcal{G}_{g}$ is not even finitely generated.\nFreedman [Fre22] suggests a method by which this might be proven.\n(3) In [Sch24], Scharlemann generalizes Powell’s proposed generators and\nshows that this generalized set suffices.\nThe set itself is infinite, but\nScharlemann announced that it can be used to demonstrate that the Pow-\nell Conjecture is stably true [Sch25]. That is, any element of $\\mathcal{G}_{g}$, when\nviewed as acting on the genus $g + 1$ unknotted handlebody obtained by\nadding a single genus 1 summand to the original genus $g$ handlebody, is\na consequence of the 5 Powell elements of $\\mathcal{G}_{g+1}$.\n(4) In another approach, Zupan [Zup20] identifies a certain graph $\\mathcal{R}_{g}$ in the\ncurve complex of a genus $g$ Heegaard surface and shows that, in general,\nthe genus $k$ Powell conjecture is true for all $k \\leq g$ if and only if $\\mathcal{R}_{k}$ is\nconnected for all $k \\leq g$.\n\nReferences cited:\n- [JM13] Jesse Johnson and Darryl McCullough. The space of Heegaard splittings. J. Reine Angew. Math., 679:155–179, 2013. doi:10.1515/crelle.2012.016.\n- [Goe33] L. Goeritz. Die Abbildungen der Brezelfläche und der Volbrezel vom Gesschlect 2. Abh. Math. Sem. Univ. Hamburg, 9:244–259, 1933.\n- [Akb08] Erol Akbas. A presentation for the automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. Pacific J. Math., 236(2):201–222, 2008. doi:10.2140/pjm.2008.236.201.\n- [Pow80] Jerome Powell. Homeomorphisms of $S^{3}$ leaving a Heegaard surface invariant. Trans. Amer. Math. Soc., 257(1):193–216, 1980. doi:10.2307/1998131.\n- [Sch04] Martin Scharlemann. Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. Bol. Soc. Mat. Mexicana (3), 10:503–514, 2004.\n- [Sch20] Martin Scharlemann. One Powell generator is redundant. Proc. Amer. Math. Soc. Ser. B, 7:138–141, 2020. doi:10.1090/bproc/58.\n- [FS18] Michael Freedman and Martin Scharlemann. Powell moves and the Goeritz group, 2018. arXiv:1804.05909.\n- [Fre22] Michael Freedman. The 2-width of embedded 3-manifolds. Peking Math. J., 5(1):21–35, 2022. doi:10.1007/s42543-021-00035-9.\n- [Sch24] Martin Scharlemann. Generating the Goeritz group of $S^{3}$. J. Assoc. Math. Res., 2(2):209–335, 2024.\n- [Sch25] Martin Scharlemann. Powell’s conjecture on the Goeritz group of $S^{3}$ is stably true. Algebr. Geom. Topol., 25(6):3775–3787, 2025. doi:10.2140/agt.2025.25.3775.\n- [Zup20] Alexander Zupan. The Powell conjecture and reducing sphere complexes. J. Lond. Math. Soc. (2), 101(1):328–348, 2020. doi:10.1112/jlms.12272.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2764, "problem_number": "KP-2.16", "title": "Kirby Problem 2.16", "statement": "If two hyperbolic surfaces have the same unmarked simple\nlength spectra (i.e., the same multiset of lengths that correspond to simple closed\ncurves), are they isometric?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.16.\n\nLiterature notes:\n(1) Let $M$ be a closed Riemannian manifold with negative sectional curva-\nture, and let $\\mathcal{C}$ denote the set of free homotopy classes of closed curves in\n$M$. In each homotopy class, there is a unique closed geodesic. This de-\nfines a marked length spectrum function $\\ell: \\mathcal{C} \\to \\mathbb{R}_{>0}$ that assigns to each\n\nclass $g$ the length $\\ell(g)$ of this closed geodesic. Burns and Katok asked\nwhether the function $\\ell$ determines $M$ up to isometry [BK85]. While this\nquestion is open in general, the answer is yes in the case of surfaces; this\nis called marked length spectrum rigidity. Fricke and Klein showed this\nholds for closed hyperbolic surfaces, in which case there are $6g - 5$ geo-\ndesic curves whose lengths determine the surface us to isometry [FK65].\nOtal and, independently, Croke showed that surfaces with variable neg-\native curvature are also marked length spectrum rigid [Ota90, Cro90];\nsee [CFF92, HP97] for other classes of metrics with respect to which\nsurfaces are marked length spectrum rigid.\nIn this problem, the length spectrum is unmarked, in the sense that\nwe no longer consider the function $\\ell$ but only the image of $\\ell$ as a multiset\nin $\\mathbb{R}_{>0}$; that is, we simply have a list of the lengths of closed geodesics on\nthe surface, counted with multiplicity. Moreover, we consider only lengths\ncorresponding to simple closed curves.\nIf we drop the assumption that curves we consider are simple, then the\nanswer is no; that is, hyperbolic surfaces do not enjoy unmarked length\nspectrum rigidity. Vignéras constructed the first examples of isospectral\nbut non-isometric closed surfaces [Vig80], and Sunada gave a general\nconstruction of such surfaces [Sun85].\nHowever, the length spectrum\ndoes determine the isometry class of the surface in some low-complexity\ncases, such as a one-holed torus with one boundary component [Haa85,\nBS88b].\nOn the other hand, the question of simple length spectrum rigidity is\nstill open. It is known that the examples of isospectral but non-isometric\nsurfaces constructed by Sunada can be distinguished by their simple length\nspectra [Mau13].\n(2) One can ask a family of analogous questions for each natural number:\nsuppose two hyperbolic surfaces have the same unmarked “$n$-spectra”,\nwhich is the super-set of lengths corresponding to all curves with at most\n$n$ self-intersections.\nAre they isometric?\nAbove, “superset of lengths”\nmeans the sets of lengths counting multiplicity.\n(3) Baik–Choi–Kim [BCK21] recently announced that there is an open dense\nset in Teichmüller space where the unmarked simple length spectrum is in-\ndeed rigid. The introduction to this paper contains a thorough discussion\nof the history and context of this problem.\n(4) See also Problem 3.17 for a discussion of length spectrum regidity for\nhigher-dimensional hyperbolic manifolds.\n\nReferences cited:\n- [BK85] K. Burns and A. Katok. Manifolds with nonpositive curvature. Ergodic Theory Dynam. Systems, 5(2):307–317, 1985. doi:10.1017/S0143385700002935.\n- [FK65] Robert Fricke and Felix Klein. Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausführungen und die Andwendungen, volume Bande 3, 4 of Bibliotheca Mathematica Teubneriana. Johnson Reprint Corp., New York; B. G. Teubner Verlagsgesellschaft, Stuttgart, 1965.\n- [Ota90] Jean-Pierre Otal. Le spectre marqué des longueurs des surfaces à courbure négative. Ann. of Math. (2), 131(1):151–162, 1990. doi:10.2307/1971511.\n- [Cro90] Christopher B. Croke. Rigidity for surfaces of nonpositive curvature. Comment. Math. Helv., 65(1):150–169, 1990. doi:10.1007/BF02566599.\n- [CFF92] C. Croke, A. Fathi, and J. Feldman. The marked length-spectrum of a surface of nonpositive curvature. Topology, 31(4):847–855, 1992. doi:10.1016/0040-9383(92) 90013-8.\n- [HP97] Sa’ar Hersonsky and Frédéric Paulin. On the rigidity of discrete isometry groups of negatively curved spaces. Comment. Math. Helv., 72(3):349–388, 1997. doi:10.1007/s000140050022.\n- [Vig80] Marie-France Vignéras. Variétés riemanniennes isospectrales et non isométriques. Ann. of Math. (2), 112(1):21–32, 1980. doi:10.2307/1971319.\n- [Sun85] Toshikazu Sunada. Riemannian coverings and isospectral manifolds. Ann. of Math. (2), 121(1):169–186, 1985. doi:10.2307/1971195.\n- [Haa85] Andrew Haas. Length spectra as moduli for hyperbolic surfaces. Duke Math. J., 52(4):923–934, 1985. doi:10.1215/S0012-7094-85-05249-4.\n- [BS88b] P. Buser and K.-D. Semmler. The geometry and spectrum of the one-holed torus. Comment. Math. Helv., 63(2):259–274, 1988. doi:10.1007/BF02566766.\n- [Mau13] Rasimate Maungchang. The Sunada construction and the simple length spectrum. Geom. Dedicata, 163:349–360, 2013. doi:10.1007/s10711-012-9753-x.\n- [BCK21] Hyungryul Baik, Inhyeok Choi, and Dongryul M. Kim. Simple length spectra as moduli for hyperbolic surfaces and rigidity of length identities, 2021. arXiv:2012.05652.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2765, "problem_number": "KP-2.17", "title": "Kirby Problem 2.17", "statement": "Suppose that $c$ is a geodesic current on a hyperbolic surface,\nand suppose that, on the space of hyperbolic metrics on the surface, $c$ has the same\nlength function as a closed geodesic.\nMust $c$ be a convex combination of closed\ncurves?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.17.\n\nLiterature notes:\n(1) For an introduction to the theory of geodesic currents in general, see the\nfoundational work of Bonahon [Bon86, Bon88], as well as the contem-\nporary book of Erlandsson–Souto [ES22].\n(2) As an example, suppose that $c_{1}$ and $c_{2}$ are so-called “length-twins,” i.e.,\nnon-homotopic closed curves that have the same length in any hyperbolic\nmetric. Then the geodesic current $\\frac{1}{2}c_{1}+\\frac{1}{2}c_{2}$ is not a closed curve, but it\nwill also have the same length as $c_{1}$ and $c_{2}$. It is a convex combination of\n$c_{1}$ and $c_{2}$; thus, the question asks whether there exists some more exotic\ntype of current (for example one whose support in $\\mathbb{H}^{2}$ is not a discrete set\nof geodesics) that can also have the same length as either $c_{1}$ or $c_{2}$.\n\nReferences cited:\n- [Bon86] Francis Bonahon. Bouts des variétés hyperboliques de dimension 3. Annals of Mathematics, 124(1):71–158, 1986. URL: http://www.jstor.org/stable/1971388.\n- [Bon88] Francis Bonahon. The geometry of Teichmüller space via geodesic currents. Inventiones mathematicae, 92(1):139–162, 1988. doi:10.1007/BF01393996.\n- [ES22] Viveka Erlandsson and Juan Souto. Mirzakhani’s curve counting and geodesic currents, volume 345 of Progress in Mathematics. Birkhäuser/Springer, Cham, [2022] ©2022. doi:10.1007/978-3-031-08705-9.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2766, "problem_number": "KP-2.18", "title": "Kirby Problem 2.18", "statement": "What is the best lower bound on the volume of a fibered hyper-\nbolic 3-manifold one can give in terms of the translation length of the monodromy\nwith respect to various metric spaces, such as Teichmüller space with the Weil–\nPeterson metric or the pants complex?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.18.\n\nLiterature notes:\n(1) The monodromy of a hyperbolic 3–manifold that fibers over the circle is\na pseudo-Anosov homeomorphism, which acts with positive translation\nlength on several well-studied spaces, including Teichmüller space with\neither the Teichmüller or Weil-Peterson metric and the pants complex.\nBrock [Bro03] shows that for a finite-type surface $S$ and a pseudo-\nAnosov homeomorphism $f$ of $S$, the hyperbolic volume of the mapping\ntorus $M_{f}$ is comparable to the translation length $\\tau_{WP}(f)$ of $f$ acting on\nthe Teichmüller space of $S$ with the Weil-Peterson metric. In particular,\nhe shows that there exist constants $\\kappa_{1}$ and $\\kappa_{2}$, depending on the topology\nof $S$, such that\n\n$$\n\\kappa_{2}\\tau_{WP}(f) \\leq \\operatorname{Vol}(M_{f}) \\leq \\kappa_{1}\\tau_{WP}(f).\n$$\n\n(3)\nHe also proves the analogous inequality with the translation length of $f$\nacting on the pants complex.\nThe heart of the question, therefore, lies in making the constant $\\kappa_{2}$\n(or the analogous constant for other metric spaces) effective by describing\nprecisely how it depends on the genus $g$ of $S$.\n(2) Kojima–McShane [KM18] and Brock–Bromberg [BB16] give effective\nupper bounds on the volume in terms of the translation length of the\nmonodromy with respect to the Teichmüller and Weil-Peterson metrics on\nTeichmüller space, respectively. A related question is to find an effective\nupper bound on $\\operatorname{Vol}(M_{f})$ in terms of the translation length with respect\nto the pants complex.\nA lower bound was obtained by Bridgeman–Brock–Bromberg for the\nWeil-Peterson translation length in the case the 3–manifold is relatively\nacylindrical [BBB23], but the bound is not effective. In the unpublished\n\nwork of Aougab–Taylor–Webb, they show that when considering the trans-\nlation length of $f$ on the pants complex, the constant $\\kappa_{2}$ decays like\n$1/g!$.\nNo effective lower bounds are known.\n(3) There is no such lower bound in terms of the translation length of the mon-\nodromy with respect to the Teichmüller metric on Teichmüller space. Mc-\nMullen gives an explicit counterexample in which the Teichmüller transla-\ntion lengths of the monodromies $f_{n}$ tend to infinity but the mapping tori\n$M_{fn}$ have bounded volume [McM14].\n(4) The analogue of the upper bound of the inequality (3) was proved for irre-\nducible end-periodic homeomorphisms of infinite-type surfaces by Field–\nKim–Leininger–Loving [FKLL23], and the analogue of the lower bound\nwas recently announced by Field–Kent–Leininger–Loving [FKLL25]. They\ngive a universal constant for $\\kappa_{1}$ and show that the constant $\\kappa_{2}$ depends\non the topology of the finite-type subsurface on which the irreducible\nend-periodic is acting “interestingly” (not just by translation). It would\ninteresting to make their lower bound effective and compare it to the\ndescription of $\\kappa_{2}$ one might achieve in the finite-type case.\n(5) This problem falls into the larger category of questions asking what hyperbolic-\ngeometric data about a fibered 3-manifold one can obtain from the surface\nand homeomorphism data used to construct it. For instance, in any of\nthe cases mentioned above, is it possible to gather data about the short\ngeodesics in the 3-manifold $M_{f}$ from information about $S$ and $f$?\n\nReferences cited:\n- [Bro03] Jeffrey F. Brock. Weil-Petersson translation distance and volumes of mapping tori. Comm. Anal. Geom., 11(5):987–999, 2003. doi:10.4310/CAG.2003.v11.n5.a6.\n- [KM18] Sadayoshi Kojima and Greg McShane. Normalized entropy versus volume for pseudo-Anosovs. Geom. Topol., 22(4):2403–2426, 2018. doi:10.2140/gt.2018.22.2403.\n- [BB16] Jeffrey F. Brock and Kenneth W. Bromberg. Inflexibility, Weil-Peterson distance, and volumes of fibered 3-manifolds. Math. Res. Lett., 23(3):649–674, 2016. doi: 10.4310/MRL.2016.v23.n3.a4.\n- [BBB23] Martin Bridgeman, Jeffrey Brock, and Kenneth Bromberg. The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores. Geom. Topol., 27(8):3183–3228, 2023. doi:10.2140/gt.2023.27.3183.\n- [McM14] Curt McMullen. Dynamics, geometry, and moduli spaces seminar, 2014: Renormalized volume. Seminar notes, 2014.\n- [FKLL23] Elizabeth Field, Heejoung Kim, Christopher Leininger, and Marissa Loving. Endperiodic homeomorphisms and volumes of mapping tori. J. Topol., 16(1):57–105, 2023. doi:10.1112/topo.12277.\n- [FKLL25] Elizabeth Field, Autumn Kent, Christopher Leininger, and Marissa Loving. A lower bound on volumes of end-periodic mapping tori. J. Topol., 18(3):Paper No. e70037, 36, 2025. doi:10.1112/topo.70037.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2767, "problem_number": "KP-2.19", "title": "Kirby Problem 2.19", "statement": "Which mapping classes give rise to arithmetic hyperbolic 3-\nmanifolds as their mapping tori?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.19.\n\nLiterature notes:\n(1) Roughly speaking, arithmetic hyperbolic manifolds are a special class of\nhyperbolic manifolds constructed by “number-theoretic” methods. They\nenjoy many special properties, e.g., possessing a very rich group of com-\nmensurations (isometries of finite covers). For a general introduction, see\n[MR03]; also see Problem 3.10.\n(2) This is likely to be a deep and difficult question, since Thurston’s proof\nof the existence of the hyperbolic structure on $M_{f}$ is non-constructive.\nIt would be useful to amass some systematic data on the monodromies\nof known fibered arithmetic 3-manifolds. Are the elements that appear\ndistinctive or exceptional from the point of view of the mapping class\ngroup?\nAs alluded to above, arithmetic hyperbolic manifolds are distinguished\nin the class of all hyperbolic manifolds by the property that their funda-\nmental groups have infinite index in their commensurators (indeed, the\ncommensurator is dense in $\\operatorname{PSL}_{2}(\\mathbb{C})$). One place to start would be to bet-\nter understand how the commensurator of the mapping torus is encoded\nin the mapping class itself.\n\n(3) In the world of knots, there is an analogous question which, remarkably,\nhas been resolved.\nAlan Reid [Rei91] has shown that the only knot\n$K \\subset S^{3}$ for which the complement $S^{3}\\setminus K$ is arithmetic hyperbolic is the\nfigure-8 knot. The figure-8 knot is fibered with fiber a once-punctured\ntorus. More generally, work of Bowditch-Maclachlan-Reid [BMR95] clas-\nsifies the monodromies of all arithmetic hyperbolic once-punctured torus\nbundles over $S^{1}$. In spite of this, it is not clear what special properties\nthese elements possess that single them out from other elements of $\\operatorname{SL}_{2}(\\mathbb{Z})$.\n(4) The paper [BMR95] also proves some other results of relevance.\nFor\nexample, it shows that there are at most finitely many cyclic commen-\nsurability classes of arithmetic hyperbolic surface bundles with surface of\nfixed non-compact topological type.\n(5) Associated to any hyperbolic manifold is an invariant called the invari-\nant trace field, the field generated by the traces of squares of elements of\nthe fundamental group, viewed via the hyperbolic structure as isometries\nin $\\operatorname{PSL}_{2}(\\mathbb{C})$. Arithmetic hyperbolic 3-manifolds are known to be char-\nacterizable in terms of properties of their traces (see [MR03]). A more\ngeneral problem is to determine the invariant trace field associated to any\npseudo-Anosov element.\n\nReferences cited:\n- [MR03] Colin Maclachlan and Alan W. Reid. The arithmetic of hyperbolic 3-manifolds, volume 219 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2003. doi:10.1007/978-1-4757-6720-9.\n- [Rei91] Alan W. Reid. Arithmeticity of knot complements. J. London Math. Soc. (2), 43(1):171–184, 1991. doi:10.1112/jlms/s2-43.1.171.\n- [BMR95] B. H. Bowditch, C. Maclachlan, and A. W. Reid. Arithmetic hyperbolic surface bundles. Math. Ann., 302(1):31–60, 1995. doi:10.1007/BF01444486.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2768, "problem_number": "KP-2.20", "title": "Kirby Problem 2.20", "statement": "Can one detect holomorphicity from a monodromy factoriza-\ntion of a Lefschetz pencil, fibration, or a surface bundle over a surface? What are\nthe special properties of monodromy factorizations of holomorphic fibrations?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.20.\n\nLiterature notes:\n(1) Unless $\\ell= g -1 = 0$, any fibration corresponding to a monodromy factor-\nization as above can be made symplectic and $J$-holomorphic for an almost\ncomplex structure $J$ compatible with the symplectic form. However, $J$ is\nnot necessarily integrable.\n(2) Let $f: X \\to S_{h}$ be a relatively minimal genus–$g$ Lefschetz fibration.\nClearly, for it to be holomorphic, the total space $X$ should have the ho-\nmotopy type of a compact complex surface, e.g., the fundamental group\none reads from the monodromy factorization should be a Kähler group,\nthe odd Betti numbers should be even, and such.\nSet $\\chi_{f}:=\\frac{1}{4}(\\chi(X)+\\sigma(X))-(g-1)(h-1)$ and\n$K_{f}^{2}:=c_{1}^{2}(X)-8(g-1)(h-1)$. A few other known constraints are:\n$\\bullet$ Beauville’s inequality [Bea79]: $\\chi_{f} \\geq 0$;\n$\\bullet$ Arakelov’s inequality [Ara71]: $K_{f}^{2} \\geq 0$; and\n$\\bullet$ Xiao’s slope inequality [Xia87]: $(4 - 4/g)\\chi_{f} \\leq K_{f}^{2} \\leq 12\\chi_{f}$.\nFurther constraints on $\\chi_{f}$ and $K_{f}^{2}$ are available for holomorphic fibrations\nof special types (hyperelliptic, non-hyperelliptic, trigonal, non-trigonal,\nbielliptic, etc.); see e.g., [AK02].\n(3) There is an additional constraint formulated in terms of the Nielsen-\nThurston classification. A subgroup $\\Gamma < \\operatorname{Mod}(S_{g})$ is reducible if there\nis a multi-curve $C \\subset S_{g}$ with only essential components fixed up to ho-\nmotopy by each element of $\\Gamma$. The monodromy group $\\Gamma$ of a nontrivial\nholomorphic surface bundle over $B\\setminus f(\\operatorname{Crit}(f))$ must be infinite and irre-\nducible [Shi97].\n\nReferences cited:\n- [Bea79] Arnaud Beauville. L’application canonique pour les surfaces de type général. Invent. Math., 55(2):121–140, 1979. doi:10.1007/BF01390086.\n- [Ara71] S. Ju. Arakelov. Families of algebraic curves with fixed degeneracies. Izv. Akad. Nauk SSSR Ser. Mat., 35:1269–1293, 1971.\n- [Xia87] Gang Xiao. Fibered algebraic surfaces with low slope. Math. Ann., 276(3):449–466, 1987. doi:10.1007/BF01450841.\n- [AK02] Tadashi Ashikaga and Kazuhiro Konno. Global and local properties of pencils of algebraic curves. In Algebraic geometry 2000, Azumino (Hotaka), volume 36 of Adv. Stud. Pure Math., pages 1–49. Math. Soc. Japan, Tokyo, 2002. doi:10.2969/aspm/03610001.\n- [Shi97] Hiroshige Shiga. On monodromies of holomorphic families of Riemann surfaces and modular transformations. Math. Proc. Cambridge Philos. Soc., 122(3):541– 549, 1997. doi:10.1017/S0305004197001825.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2769, "problem_number": "KP-2.21", "title": "Kirby Problem 2.21", "statement": "What is the minimum number, $m_{g,b}$, of right-handed Dehn\ntwists along essential curves into which the boundary multi-twist, $\\Delta:= T_{\\delta_{1}} \\cdots T_{\\delta_{b}}$,\ncan be factorized in $\\operatorname{Mod}(S_{g}^{b})$? Find factorizations realizing the minimum.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.21.\n\nLiterature notes:\n(1) A factorization $T_{c_1} \\cdots T_{c_\\ell}= \\Delta$, where no $c_{i}$ is homotopic to a boundary\ncomponent or point, is often called a positive factorization of $\\Delta$. A positive\nfactorization of the boundary multi-twist $\\Delta= T_{\\delta_{1}} \\cdots T_{\\delta_{b}}$ in $\\operatorname{Mod}(S_{g}^{b})$,\nwhere each $\\delta_{j}$ is parallel to a distinct boundary component of $S_g^b$ and\n$b > 0$, corresponds to a genus-$g$ Lefschetz pencil with $b$ base points and $\\ell$\nnodes.\nWith these in mind, the problem amounts to determining the small-\nest second Betti number $b_{2}$ of genus-$g$ Lefschetz pencils/fibrations with\n$b$ base points.\nIt is motivated by a central problem regarding the ex-\nistence of small symplectic and exotic 4-manifolds.\nSee, e.g., [BK17,\n\nBay22, BH24c] for some successful implementations of this approach to\n4-dimensional exotica.\n(2) The pairs $g, b \\in \\mathbb{N}$ for which there is no positive factorization of the bound-\nary multi-twist are determined in [BMVHM17]; namely, when $g = 1$ and\n$b > 9$, or $g \\geq 2$ and $b > 4g + 4$. In all other cases, the minimum is known\nto exist. The only cases when the problem has been completely settled\nare $g = 1$ [Kas77, Moi77b] and $g = 2$ [BK17], for any $b \\geq 0$.\nFor $g \\geq 3$, there are reasonable upper bounds for every genus; the\npositive factorizations discovered in [Mat96, Cad98, Kor01, Ham17]\nimply that $m_{g,b} \\leq 2g+4$ when $g$ is even and $b \\leq 4$, and $m_{g,b} \\leq 2g+10$ when\n$g$ is odd and $b \\leq 8$. These bounds are known to not be sharp, at least in\nlow genera. For example, $m_{g,b} = 7$ when $g = 2$ and $b \\leq 3$ [BK17, Bay22],\nwhereas $m_{g,b} \\leq 12$ when $g = 3$ and $b \\leq 4$ [Smi01b, Bay22, HH18a].\nFor large $b \\leq 4g + 4$, another upper bound is $m_{g,b} \\leq 8g + 4$, which is\nrealized by genus–$g$ pencils on $S^{2} \\times S^{2}$ [SS94, Tan12].\nThere are also lower bounds available in every genus, derived from\nSeiberg-Witten theory of symplectic 4-manifolds, roughly implying that\n$m_{g,b} \\geq g$ [Li00, Sti99, BK03]—which is unlikely to be sharp in any\ngenus.\n(3) There are several variations of this problem studied in the literature.\nThe minimum number of Dehn twists in positive factorizations of any\n$h \\geq 1$ commutators in $\\operatorname{Mod}(S_{g})$ (which correspond to genus-$g$ Lefschetz\nfibrations over $S_{h}$) is almost completely determined in [KO01, Mon12,\nHam14, SY17, BH23]; the only open cases are when $g \\geq 3$ and $h = 1$.\nWhen $b > 0$, one can also inquire about the maximum number of\nDehn twists, related to the uniform topology of Stein fillings of con-\ntact 3-manifolds. This problem was completely answered in [BVHM15,\nBVHM16, BMVHM17], establishing in particular that in many cases\none may have arbitrarily long positive factorizations of $\\Delta$ in $\\operatorname{Mod}(S_{g}^{b})$.\nIn some cases, one can obtain more leverage on the stated problem by\nimposing additional conditions, say by assuming that the factorization lies\nin a smaller mapping class group (e.g., [Alt20]) or by fixing some topo-\nlogical invariants of the corresponding Lefschetz pencils (e.g., [Sti02]).\n\nReferences cited:\n- [BK17] R. İnanç Baykur and Mustafa Korkmaz. Small Lefschetz fibrations and exotic 4-manifolds. Math. Ann., 367(3-4):1333–1361, 2017. doi:10.1007/s00208-016-1466-2.\n- [Bay22] R. İnanç Baykur. Small exotic 4-manifolds and symplectic Calabi-Yau surfaces via genus-3 pencils. In Gauge theory and low-dimensional topology—progress and interaction, volume 5 of Open Book Ser., pages 185–221. Math. Sci. Publ., Berkeley, CA, 2022. https://msp.org/obs/2022/5-1/p09.xhtml.\n- [BH24c] R. İnanç Baykur and Noriyuki Hamada. Lefschetz fibrations with arbitrary signature. J. Eur. Math. Soc. (JEMS), 26(8):2837–2895, 2024. doi:10.4171/jems/1326.\n- [BMVHM17] R. İnanç Baykur, Naoyuki Monden, and Jeremy Van Horn-Morris. Positive factorizations of mapping classes. Algebr. Geom. Topol., 17(3):1527–1555, 2017. doi: 10.2140/agt.2017.17.1527.\n- [Kas77] A. Kas. On the deformation types of regular elliptic surfaces. In Complex analysis and algebraic geometry, pages 107–111. Iwanami Shoten Publishers, Tokyo, 1977.\n- [Moi77b] Boris Moishezon. Complex surfaces and connected sums of complex projective planes. Lecture Notes in Mathematics, Vol. 603. Springer-Verlag, Berlin-New York, 1977. With an appendix by R. Livne.\n- [Mat96] Yukio Matsumoto. Lefschetz fibrations of genus two—a topological approach. In Topology and Teichmüller spaces (Katinkulta, 1995), pages 123–148. World Sci. Publ., River Edge, NJ, 1996.\n- [Cad98] Carlos Alberto Cadavid. On a remarkable set of words in the mapping class group. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–The University of Texas at Austin. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\& rft val fmt=info:ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri: pqdiss:9936983.\n- [Kor01] Mustafa Korkmaz. Noncomplex smooth 4-manifolds with Lefschetz fibrations. Internat. Math. Res. Notices, 2001(3):115–128, 2001. doi:10.1155/$S^{1}$07379280100006X.\n- [Ham17] Noriyuki Hamada. Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration, 2017. arXiv:1610.08458.\n- [Smi01b] Ivan Smith. Torus fibrations on symplectic four-manifolds. Turkish J. Math., 25(1):69–95, 2001.\n- [HH18a] Noriyuki Hamada and Kenta Hayano. Topology of holomorphic Lefschetz pencils on the four-torus. Algebr. Geom. Topol., 18(3):1515–1572, 2018. doi:10.2140/agt.2018.18.1515.\n- [SS94] Masa-Hiko Saitō and Ken-Ichi Sakakibara. On Mordell-Weil lattices of higher genus fibrations on rational surfaces. J. Math. Kyoto Univ., 34(4):859–871, 1994. doi: 10.1215/kjm/1250518890.\n- [Tan12] Shunsuke Tanaka. On sections of hyperelliptic Lefschetz fibrations. Algebr. Geom. Topol., 12(4):2259–2286, 2012. doi:10.2140/agt.2012.12.2259.\n- [Li00] Tian-Jun Li. Symplectic Parshin-Arakelov inequality. Internat. Math. Res. Notices, 2000(18):941–954, 2000. doi:10.1155/$S^{1}$073792800000490.\n- [Sti99] András I. Stipsicz. On the number of vanishing cycles in Lefschetz fibrations. Math. Res. Lett., 6(3-4):449–456, 1999. doi:10.4310/MRL.1999.v6.n4.a7.\n- [BK03] V. Braungardt and D. Kotschick. Clustering of critical points in Lefschetz fibrations and the symplectic Szpiro inequality. Trans. Amer. Math. Soc., 355(8):3217–3226, 2003. doi:10.1090/S0002-9947-03-03290-2.\n- [KO01] Mustafa Korkmaz and Burak Ozbagci. Minimal number of singular fibers in a Lefschetz fibration. Proc. Amer. Math. Soc., 129(5):1545–1549, 2001. doi:10.1090/S0002-9939-00-05676-8.\n- [Mon12] Naoyuki Monden. On minimal number of singular fibers in a genus-2 Lefschetz fibration. Tokyo J. Math., 35(2):483–490, 2012. doi:10.3836/tjm/1358951332.\n- [Ham14] Noriyuki Hamada. Upper bounds for the minimal number of singular fibers in a Lefschetz fibration over the torus. Michigan Math. J., 63(2):275–291, 2014. doi: 10.1307/mmj/1401973051.\n- [SY17] András I. Stipsicz and Ki-Heon Yun. On the minimal number of singular fibers in Lefschetz fibrations over the torus. Proc. Amer. Math. Soc., 145(8):3607–3616, 2017. doi:10.1090/proc/13480.\n- [BH23] R. Inanc Baykur and Noriyuki Hamada. Exotic 4-manifolds with signature zero, 2023. Selecta Math., to appear. arXiv:2305.10908.\n- [BVHM15] R. İnanç Baykur and Jeremy Van Horn-Morris. Families of contact 3-manifolds with arbitrarily large Stein fillings. J. Differential Geom., 101(3):423–465, 2015. With an appendix by Samuel Lisi and Chris Wendl, http://projecteuclid.org/euclid.jdg/1445518920.\n- [BVHM16] R. İnanç Baykur and Jeremy Van Horn-Morris. Topological complexity of symplectic 4-manifolds and Stein fillings. J. Symplectic Geom., 14(1):171–202, 2016. doi:10.4310/JSG.2016.v14.n1.a7.\n- [Alt20] Tülin Altunöz. The number of singular fibers in hyperelliptic Lefschetz fibrations. J. Math. Soc. Japan, 72(4):1309–1325, 2020. doi:10.2969/jmsj/82988298.\n- [Sti02] András I. Stipsicz. Singular fibers in Lefschetz fibrations on manifolds with $b_2^+=1$. Topology Appl., 117(1):9–21, 2002. doi:10.1016/S0166-8641(00)00105-X.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2770, "problem_number": "KP-2.22", "title": "Kirby Problem 2.22", "statement": "Does every Lefschetz fibration over the 2–sphere admit a sec-\ntion?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.22.\n\nLiterature notes:\n(1) Here we assume the fibration has critical points; otherwise, there exist\n$T^{2}$–bundles over $S^{2}$ without sections.\nLefschetz fibrations that arise from blowing up the base locus of Lef-\nschetz pencils naturally admit sections—namely, the exceptional divisors.\nHowever, if say the fibration is constructed through branched covers, or ab-\nstractly via a group-theoretic factorization, it is not a priori clear whether\na section exists.\n\nEquivalently, one can ask if every positive Dehn twist factorization of\nthe identity in $\\operatorname{Mod}(S_{g})$ lifts to $\\operatorname{Mod}(S_{g}, p)$, the mapping class group of\nthe (fiber) surface fixing a point $p \\in \\Sigma_{g}$.\n(2) This question has been raised repeatedly in the literature; e.g. [Smi01a,\nAur05, Ona10, Gom25], and Problems 2.8, 4.1, 7.1 and 8.1 in [Kor06],\n[Sti15], [KS09], and [End21] respectively.\n(3) More generally we can ask about the existence of multisections (in the\nsense of [DS03, BH16b]):\nQuestion. Does every Lefschetz fibration over the 2–sphere admit a\nmultisection? How about a 2-section?\nSimilarly, one can reformulate the existence of an $n$–section in terms of\nthe existence of a lift of a positive Dehn twist factorization of the identity\nin $\\operatorname{Mod}(S_{g})$ to a positive factorization also involving positive arc twists in\n$\\operatorname{Mod}(S_{g}, \\{p_{1},..., p_{n}\\})$, the mapping class group of $S_{g}$ fixing the set of $n$\ndistinct points $p_{1},..., p_{n} \\in S_{g}$ [BH16b, BH16a].\n(4) It is worth noting that for achiral Lefschetz fibrations, where the Dehn\ntwists in the factorization can be a mix of right-handed and left-handed\ntwists, there are counterexamples to the question. For instance, there is a\ngenus–1 achiral Lefschetz fibration on $S^{4}$ [Mat82], which, for homological\nreasons, cannot admit a section. (In contrast, every genus–1 Lefschetz\nfibration admits a section.) More examples are given in a recent preprint\nof Gompf [Gom25].\n(5) An ad hoc method for finding sections of a Lefschetz fibration with an\nexplicit monodromy factorization is to reverse-engineer the monodromy\nfactorization using elementary relations (among Dehn twists) in the map-\nping class group (e.g., [KO08, Ham17, Tan12]).\nA potential counterexample to the existence of a section, where this ad\nhoc method essentially fails, is the (spin, signature-zero) genus–9 Lefschetz\nfibration of Baykur–Hamada in [BH24c].\n\nReferences cited:\n- [Smi01a] Ivan Smith. Geometric monodromy and the hyperbolic disc. Q. J. Math., 52(2):217– 228, 2001. doi:10.1093/qjmath/52.2.217.\n- [Aur05] Denis Auroux. A stable classification of Lefschetz fibrations. Geom. Topol., 9:203– 217, 2005. doi:10.2140/gt.2005.9.203.\n- [Ona10] Sinem Çelik Onaran. On sections of genus two Lefschetz fibrations. Pacific J. Math., 248(1):203–216, 2010. doi:10.2140/pjm.2010.248.203.\n- [Gom25] Robert E. Gompf. On sections of maps from 4-manifolds to the 2-sphere, 2025. URL: https://arxiv.org/abs/2506.18066, arXiv:2506.18066.\n- [Kor06] Mustafa Korkmaz. Problems on homomorphisms of mapping class groups. In Problems on mapping class groups and related topics, volume 74 of Proc. Sympos. Pure Math., pages 81–89. Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/pspum/074/2264533.\n- [Sti15] András I. Stipsicz. Symplectic 4-manifolds, Stein domains, Seiberg-Witten theory and mapping class groups. In Interactions between low-dimensional topology and mapping class groups, volume 19 of Geom. Topol. Monogr., pages 173–200. Geom. Topol. Publ., Coventry, 2015. doi:10.2140/gtm.2015.19.173.\n- [KS09] Mustafa Korkmaz and András I. Stipsicz. Lefschetz fibrations on 4-manifolds. In Handbook of Teichmüller theory. Vol. II, volume 13 of IRMA Lect. Math. Theor. Phys., pages 271–296. Eur. Math. Soc., Zürich, 2009. doi:10.4171/055-1/9.\n- [End21] Hisaaki Endo. Lefschetz fibrations. Sugaku Expositions, 34(2):175–204, 2021. Translation of [ 3675915]. doi:10.1090/suga/462.\n- [DS03] Simon Donaldson and Ivan Smith. Lefschetz pencils and the canonical class for symplectic four-manifolds. Topology, 42(4):743–785, 2003. doi:10.1016/S0040-9383(02)00024-1.\n- [BH16b] R. İnanç Baykur and Kenta Hayano. Multisections of Lefschetz fibrations and topology of symplectic 4-manifolds. Geom. Topol., 20(4):2335–2395, 2016. doi: 10.2140/gt.2016.20.2335.\n- [BH16a] R. Inanç Baykur and Kenta Hayano. Hurwitz equivalence for Lefschetz fibrations and their multisections. In Real and complex singularities, volume 675 of Contemp. Math., pages 1–24. Amer. Math. Soc., Providence, RI, 2016. doi:10.1090/conm/675.\n- [Mat82] Yukio Matsumoto. On 4-manifolds fibered by tori. Proc. Japan Acad. Ser. A Math. Sci., 58(7):298–301, 1982. http://projecteuclid.org/euclid.pja/1195515921.\n- [KO08] Mustafa Korkmaz and Burak Ozbagci. On sections of elliptic fibrations. Michigan Math. J., 56(1):77–87, 2008. doi:10.1307/mmj/1213972398.\n- [Ham17] Noriyuki Hamada. Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration, 2017. arXiv:1610.08458.\n- [Tan12] Shunsuke Tanaka. On sections of hyperelliptic Lefschetz fibrations. Algebr. Geom. Topol., 12(4):2259–2286, 2012. doi:10.2140/agt.2012.12.2259.\n- [BH24c] R. İnanç Baykur and Noriyuki Hamada. Lefschetz fibrations with arbitrary signature. J. Eur. Math. Soc. (JEMS), 26(8):2837–2895, 2024. doi:10.4171/jems/1326.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2771, "problem_number": "KP-2.23", "title": "Kirby Problem 2.23", "statement": "Does there exist a surface bundle over a surface that admits\na complete hyperbolic metric, or, more generally, a complete metric of variable\nnegative curvature?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.23.\n\nLiterature notes:\n(1) Thurston’s hyperbolization theorem provides for an enormous wealth of 3-\nmanifolds that are surface bundles over $S^{1}$, and that moreover admit com-\nplete hyperbolic metrics. Moving up a dimension, the same phenomenon\nbecomes quite mysterious. As in the 3-manifold case, it is necessary that\nthe monodromy of every nontrivial loop in the base be pseudo-Anosov. A\nconstruction of such purely pseudo-Anosov surface subgroups of mapping\nclass groups has been announced in very recent work of Kent–Leininger\n[KL24], resolving a major open question in its own right. It is not yet clear\nwhether every purely pseudo-Anosov subgroup induces a word-hyperbolic\n\nsurface-by-surface group extension, which is a necessary condition for the\ntotal space to admit a metric of negative curvature, and indeed, it is\npresently not known if the Kent-Leininger examples are word-hyperbolic.\nA weak version of Problem 2.23 is then simply to furnish examples of\nsurface bundles over surfaces with word-hyperbolic fundamental group.\n(2) Bowditch [Bow09] shows that for fixed base and fiber genus, there are\nonly finitely many possible examples of hyperbolic surface bundles over a\nsurface.\n(3) As detailed in [Rei06], the existence of a hyperbolic surface bundle over a\nsurface would contradict the long-standing conjecture of LeBrun [LeB02]\nthat the Seiberg-Witten invariants of a closed hyperbolic 4-manifold must\nvanish (see Problem 4.95).\n(4) Work of Kapovich [Kap98] shows that surface bundles over surfaces can\nnever admit complex hyperbolic metrics.\n(5) For an extensive account of the questions raised here, see [KL24, Section\n1].\n\nReferences cited:\n- [KL24] Autumn E. Kent and Christopher J. Leininger. Atoroidal surface bundles, 2024. arXiv:2405.12067.\n- [Bow09] Brian H. Bowditch. Atoroidal surface bundles over surfaces. Geom. Funct. Anal., 19(4):943–988, 2009. doi:10.1007/s00039-009-0033-3.\n- [Rei06] Alan W. Reid. Surface subgroups of mapping class groups. In Problems on mapping class groups and related topics, volume 74 of Proc. Sympos. Pure Math., pages 257– 268. Amer. Math. Soc., Providence, RI, 2006. doi:10.1090/pspum/074/2264545.\n- [LeB02] Claude LeBrun. Hyperbolic manifolds, harmonic forms, and Seiberg-Witten invariants. In Proceedings of the Euroconference on Partial Differential Equations and their Applications to Geometry and Physics (Castelvecchio Pascoli, 2000), volume 91, pages 137–154, 2002. doi:10.1023/A:1016222709901.\n- [Kap98] Michael Kapovich. On normal subgroups in the fundamental groups of complex surfaces, 1998. arXiv:math/9808085.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2772, "problem_number": "KP-2.24", "title": "Kirby Problem 2.24", "statement": "Does there exist a complex surface $X$ that admits three or more\nnon-isomorphic structures as a surface bundle over a surface?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.24.\n\nLiterature notes:\n(1) The analogous situation in the case of surface bundles over the circle is\norganized by the Thurston norm [Thu86]: if a surface bundle over the\ncircle admits two different fibrations, then it admits infinitely many. By\ncontrast, work of F.E.A. Johnson [Joh99] shows that a given 4-manifold\nadmits only finitely many structures as a surface bundle over a surface\nwith base and fiber both of genus $\\geq 2$.\n(2) Atiyah [Ati69] and Kodaira [Kod67] constructed complex surfaces $X$\nthat are surface bundles over surfaces in two distinct ways. These are\nconstructed as branched covers of products of algebraic curves; the two\nbundle structures arise from the projections onto either factor. Certain\nfamilies of these examples admit exactly two fiberings by work of Chen\n[Che18], Salter–Tshishiku [ST20], and Landesman–Litt–Sawin [LLS25].\nThe problem formulated here was also raised by Catanese [Cat17].\n(3) In the smooth category, Salter [Sal15] gave a method of constructing\nsurface bundles over surfaces (with base and fiber each of genus at least\n2) with an arbitrary finite number of fiberings. However, these are known\nnot to admit complex structures.\n\nReferences cited:\n- [Thu86] William P. Thurston. A norm for the homology of 3-manifolds. Mem. Amer. Math. Soc., 59(339):i–vi and 99–130, 1986.\n- [Joh99] F. E. A. Johnson. A rigidity theorem for group extensions. Arch. Math. (Basel), 73(2):81–89, 1999. doi:10.1007/s000130050371.\n- [Ati69] M. F. Atiyah. The signature of fibre-bundles. In Global Analysis (Papers in Honor of K. Kodaira), pages 73–84. Univ. Tokyo Press, Tokyo, 1969.\n- [Kod67] K. Kodaira. A certain type of irregular algebraic surfaces. J. Analyse Math., 19:207– 215, 1967. doi:10.1007/BF02788717.\n- [Che18] Lei Chen. The number of fiberings of a surface bundle over a surface. Algebr. Geom. Topol., 18(4):2245–2263, 2018. doi:10.2140/agt.2018.18.2245.\n- [ST20] Nick Salter and Bena Tshishiku. Arithmeticity of the monodromy of some Kodaira fibrations. Compos. Math., 156(1):114–157, 2020. doi:10.1112/s0010437x19007668.\n- [LLS25] Aaron Landesman, Daniel Litt, and Will Sawin. Big monodromy for higher Prym representations. Geom. Topol., 29(5):2733–2782, 2025. doi:10.2140/gt.2025.29.2733.\n- [Cat17] Fabrizio Catanese. Kodaira fibrations and beyond: methods for moduli theory. Jpn. J. Math., 12(2):91–174, 2017. doi:10.1007/s11537-017-1569-x.\n- [Sal15] Nick Salter. Surface bundles over surfaces with arbitrarily many fiberings. Geom. Topol., 19(5):2901–2923, 2015. doi:10.2140/gt.2015.19.2901.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2773, "problem_number": "KP-2.25", "title": "Kirby Problem 2.25", "statement": "Consider surface bundles over surfaces where both fiber $F$ and\nbase $B$ have genus $\\geq 2$ and where $\\pi_{1}(B)$ injects in the mapping class group of the\n\nfiber. Does such a bundle have a $k$–multisection for some $k > 0$ (i.e., a continuous\nchoice of $k$ everywhere distinct points)?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.25.\n\nLiterature notes:\n(1) This is Problem 2.17 in [Kir97] by G. Mess. In [Mor89], Morita gives\na cohomological obstruction for a surface bundle to admit a section (i.e.\nthe case $k = 1$). Using this obstruction, Hillman gave examples of surface\nbundles over surfaces (with both base and fiber genus $\\geq 2$) that do not\nadmit sections [Hil15]. Other examples of $S_{g}$-bundles over the 2-torus\nwith no section were constructed by Li–Litt–Salter–Srinivasan [LLSS23];\nthese were constructed with applications to arithmetic geometry in mind.\nBoth classes of examples virtually admit sections, that is, they admit\nsections after pulling back to some finite cover of the base.\n(2) In spite of the similarity between this question and Problem 2.22, it is\nlikely that substantially different techniques will be necessary.\nSection\nproblems for surface bundles tend to be of a more group-theoretic flavor,\nboiling down to obstructing a lifting of the monodromy representation\nfrom an unpointed to a pointed mapping class group.\nAvailable tech-\nniques include cohomological considerations as well as approaches based\non the Nielsen–Thurston classification. The multisection question for Lef-\nschetz fibrations is more delicate, since here it is necessary to lift the\nDehn twists in the monodromy factorization on $\\operatorname{Mod}(S_{g})$ to Dehn twists\nin $\\operatorname{Mod}(S_{g}^{b})$, not simply to arbitrary elements in $\\operatorname{Mod}(S_{g}^{b})$ projecting back\nto the original twists.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Mor89] Shigeyuki Morita. Families of Jacobian manifolds and characteristic classes of surface bundles. II. Math. Proc. Cambridge Philos. Soc., 105(1):79–101, 1989. doi:10.1017/S0305004100001389.\n- [Hil15] Jonathan A. Hillman. Sections of surface bundles. In Interactions between lowdimensional topology and mapping class groups, volume 19 of Geom. Topol. Monogr., pages 1–19. Geom. Topol. Publ., Coventry, 2015. doi:10.2140/gtm.2015.19.1.\n- [LLSS23] Wanlin Li, Daniel Litt, Nick Salter, and Padmavathi Srinivasan. Surface bundles and the section conjecture. Math. Ann., 386(1-2):877–942, 2023. doi:10.1007/s00208-022-02421-9.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2774, "problem_number": "KP-2.26", "title": "Kirby Problem 2.26", "statement": "(Kontsevich–Zorich conjecture). Understand the homotopy types\nof strata of abelian differentials. Which stratum-components are $K(\\pi, 1)$ spaces?\nWhat are the fundamental groups? For which stratum-components $\\mathcal{H}$ is some ver-\nsion of the monodromy map $\\rho: \\pi_{1}(\\mathcal{H}) \\to \\operatorname{Mod}(S_{g})$ injective? Do the answers to\nthese questions change in a predictable way as one moves from the principal stra-\ntum down to the minimal?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.26.\n\nLiterature notes:\n(1) A translation surface is a surface $T$ that admits an atlas of charts to $\\mathbb{C}$\nfor which transition maps are translations ($z \\mapsto z+c$). An abelian differ-\nential is a pair $(X, \\omega)$ of a Riemann surface together with a holomorphic\ndifferential. It is a basic but profound fact that every translation surface\ncorresponds to an abelian differential and vice-versa.\nFixing a genus, the space of all abelian differentials is identified with\nthe Hodge bundle over the moduli space of Riemann surfaces of genus $g$, the\nvector bundle whose fiber over a Riemann surface $X$ is the space $H^{1,0}(X)$\nof all holomorphic 1-forms on $X$. This space is stratified according to the\nmultiplicities of the zeroes. Every differential on a Riemann surface of\ngenus $g$ has $2g -2$ zeroes when counted with multiplicity; the generic case\n\nwhere all zeroes are simple (the partition $1^{2g-2}$) is called the principal\nstratum, and the case where there is a single zero of multiplicity $2g - 2$ is\ncalled the minimal stratum.\nIn general, the stratum $\\mathcal{H}_{\\kappa}$ associated to the partition $\\kappa = k_{1}+\\cdots +k_{n}$\nis a quasiprojective complex orbifold of complex dimension $2g + n - 1$,\nand therefore (after passing to a finite cover to resolve the orbifold issue)\nhas the homotopy type of a finite CW complex. Strata are fundamental\nobjects in the study of Teichmüller dynamics, where they host a dynamical\nsystem induced from a natural action of $\\operatorname{GL}_{2}(\\mathbb{R})$, but their topology is\nquite mysterious, despite their close relationship to the moduli space of\nRiemann surfaces and the mapping class group.\n(2) In [KZ03], Kontsevich–Zorich compute $\\pi_{0}$ for all strata. They find that\neach stratum has between between one and three connected components.\nFor the partitions $2g - 2$ and $g - 1, g - 1$, there are special “hyperelliptic”\ncomponents that can be identified with configuration spaces of points in\n$\\mathbb{C}$, resolving Problem 2.26 in this case.\nIn [KZ97], they pose a version of Problem 2.26, conjecturing that\neach stratum component should be a $K(\\pi, 1)$ and that the fundamental\ngroups should be some flavor of mapping class group.\n(3) Work of Calderon–Salter [CS23] identifies the image of $\\pi_{1}(\\mathcal{H})$ in the map-\nping class group under the natural monodromy homomorphism for all non-\nhyperelliptic stratum components in genus $g \\geq 5$. They find that these\nare “framed mapping class groups”—the stabilizer of the distinguished\nframing on the translation surface inherited from the standard framing\non $\\mathbb{C}$. This shows that whenever the monodromy homomorphism is in-\njective, the fundamental group of the stratum component is indeed some\nflavor of mapping class group, as predicted by the conjecture.\nThe question of monodromy injectivity appears to be subtle, how-\never. In the hyperelliptic setting, injectivity was established in the work\nof Kontsevich–Zorich. On the other hand, in genus 3, there is a stratum-\ncomponent known as $\\mathcal{H}^{\\mathrm{odd}}(4)$\nfor which injectivity is known not to hold.\nWork of Looijenga–Mondello [LM14] computes the fundamental group\nto be $A(E_{6})$ (the Artin group of type $E_{6}$) modulo its center. Under the\nmonodromy map, the Artin generators are sent to Dehn twists about sim-\nple closed curves whose intersection pattern is given by the $E_{6}$ Dynkin\ndiagram.\nWork of Wajnryb [Waj99] shows that this homomorphism\n$A(E_{6}) \\to \\operatorname{Mod}(\\Sigma_{3})$ has a non-central element in the kernel.\n\nReferences cited:\n- [KZ03] Maxim Kontsevich and Anton Zorich. Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math., 153(3):631–678, 2003. doi:10.1007/s00222-003-0303-x.\n- [KZ97] M. Kontsevich and A. Zorich. Lyapunov exponents and Hodge theory, 1997. arXiv: hep-th/9701164.\n- [CS23] Aaron Calderon and Nick Salter. Framed mapping class groups and the monodromy of strata of abelian differentials. J. Eur. Math. Soc. (JEMS), 25(12):4719–4790, 2023. doi:10.4171/jems/1290.\n- [LM14] Eduard Looijenga and Gabriele Mondello. The fine structure of the moduli space of abelian differentials in genus 3. Geom. Dedicata, 169:109–128, 2014. doi:10.1007/s10711-013-9845-2.\n- [Waj99] Bronislaw Wajnryb. Artin groups and geometric monodromy. Invent. Math., 138(3):563–571, 1999. doi:10.1007/s002220050353.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2775, "problem_number": "KP-2.27", "title": "Kirby Problem 2.27", "statement": "For $n \\geq 4$, does the braid group $B_{n}$ admit a finite-index sub-\ngroup that embeds in a right-angled Artin group?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.27.\n\nLiterature notes:\n(1) In CAT(0) geometry, a cube complex is called special if it satisfies certain\nconditions on its hyperplanes [HW08]. Groups acting by combinatorial\nisometries on special cube complexes are of particular interest in geometric\ngroup theory, and are closely related to subgroups of right-angled Artin\ngroups. A group is said to be virtually special if it contains a finite-index\nsubgroup isomorphic to the fundamental group of a special cube complex.\n(2) By [BKK16], any group that embeds in a right-angled Artin group acts\nby $C^{\\infty}$-diffeomorphisms on $\\mathbb{R}$. As braid groups do not admit such ac-\ntions [FF20], they do not embed in right-angled Artin groups, so it is\nnecessary to pass to proper subgroups.\n(3) One could go further and ask whether braid groups are virtually special.\nThis is strictly stronger than Question 2.27.\n(4) In light of the resolution of the congruence subgroup problem for braid\ngroups (see [Mar19] and the references therein), to resolve Problem 2.27\nin the negative, it suffices to obtain a cofinal sequence of subgroups of $B_{n}$,\nnone of which embed in a right-angled Artin group.\n\nReferences cited:\n- [HW08] Frédéric Haglund and Daniel T. Wise. Special cube complexes. Geom. Funct. Anal., 17(5):1551–1620, 2008. doi:10.1007/s00039-007-0629-4.\n- [BKK16] Hyungryul Baik, Sang-hyun Kim, and Thomas Koberda. Right-angled Artin groups in the C8 diffeomorphism group of the real line. Israel J. Math., 213(1):175–182, 2016. doi:10.1007/s11856-016-1307-8.\n- [FF20] Benson Farb and John Franks. Groups of homeomorphisms of one-manifolds, I: Actions of nonlinear groups. In What’s next?—the mathematical legacy of William P. Thurston, volume 205 of Ann. of Math. Stud., pages 116–140. Princeton Univ. Press, Princeton, NJ, 2020. doi:10.2307/j.ctvthhdvv.9.\n- [Mar19] Dan Margalit. Problems, questions, and conjectures about mapping class groups. In Breadth in contemporary topology, volume 102 of Proc. Sympos. Pure Math., pages 157–186. Amer. Math. Soc., Providence, RI, 2019. doi:10.1090/pspum/102/12.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2776, "problem_number": "KP-2.28", "title": "Kirby Problem 2.28", "statement": "Let $\\Gamma$ be a graph that is not a nontrivial join, and let $A(\\Gamma)$\nbe the associated right-angled Artin group. Does there exist an injective map from\n$A(\\Gamma)$ to the mapping class group of a surface such that every loxodromic element\nof $A(\\Gamma)$ is sent to a pseudo-Anosov mapping class?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.28.\n\nLiterature notes:\n(1) There are many injective homomorphisms from right-angled Artin groups\ninto mapping class groups [CP01, CLM12, Kob12], and general criteria\nare fairly well-understood [KK14b]. Right-angled Artin groups have a\nNielsen–Thurston classification, not unlike mapping class groups [BC12,\nKK14a], wherein each element is either elliptic or loxodromic.\n(2) The set of loxodromic elements of $A(\\Gamma)$ is nonempty if and only if $A(\\Gamma)$\ndoes not split as a nontrivial direct product, which is true if and only if $\\Gamma$\ndoes not split as a nontrivial join.\n\n(3) By [MT16, KMT17], the restriction of any such injective map to a sub-\ngroup where every nontrivial element is loxodromic is necessarily a convex\ncocompact subgroup of the mapping class group of $S$. However, any such\nsubgroup obtained in this way is free (see [KK14a]). In particular, this\ndoes not provide a strategy to construct convex cocompact surface sub-\ngroups as in Problem 2.23.\n(4) In light of [KMT17], the problem can be reformulated entirely in terms\nof the combinatorial topology of the underlying surface. Let $\\Gamma$ be fixed,\nand let $\\Lambda$ be the opposite graph, i.e., the one obtained by reversing the\nadjacency relation in $\\Gamma$. It suffices to find a surface $S$ and $\\pi_{1}$-injective,\npairwise non-nested subsurfaces $\\\\{S_{v}\\\\}_{v\\\\in V(\\\\Gamma)}$ indexed by the vertices of $\\Gamma$\nsuch that the following hold:\n(i) Each $S_{v}$ supports a pseudo-Anosov mapping class (in the mapping\nclass group of $S_{v}$). In particular, each $S_{v}$ is non-annular.\n(ii) Subsurfaces $S_{v}$ and $S_{w}$ intersect essentially if and only if $v$ and $w$ are\nnot adjacent in $\\Gamma$ (or equivalently, adjacent in $\\Lambda$).\n(iii) Let $Y \\subset \\Lambda$ be a connected subgraph such that every vertex of $\\Lambda$ is\nadjacent to a vertex of $Y$ . Then every essential, simple, nonperipheral\ncurve on $S$ essentially intersects $S_{v}$ for some vertex $v$ of $Y$ .\nThe difficulty of this reformulation is the non-canonical nature of $Y$ for a\ngeneral graph $\\Lambda$.\n\nReferences cited:\n- [CP01] John Crisp and Luis Paris. The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group. Invent. Math., 145(1):19– 36, 2001. doi:10.1007/s002220100138.\n- [CLM12] Matt T. Clay, Christopher J. Leininger, and Johanna Mangahas. The geometry of right-angled Artin subgroups of mapping class groups. Groups Geom. Dyn., 6(2):249–278, 2012. doi:10.4171/GGD/157.\n- [Kob12] Thomas Koberda. Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups. Geom. Funct. Anal., 22(6):1541–1590, 2012. doi:10.1007/s00039-012-0198-z.\n- [KK14b] Sang-Hyun Kim and Thomas Koberda. An obstruction to embedding right-angled Artin groups in mapping class groups. Int. Math. Res. Not. IMRN, 2014(14):3912– 3918, 2014. doi:10.1093/imrn/rnt064.\n- [BC12] Jason Behrstock and Ruth Charney. Divergence and quasimorphisms of right-angled Artin groups. Math. Ann., 352(2):339–356, 2012. doi:10.1007/s00208-011-0641-8.\n- [KK14a] Sang-Hyun Kim and Thomas Koberda. The geometry of the curve graph of a rightangled Artin group. Internat. J. Algebra Comput., 24(2):121–169, 2014. doi:10.1142/S021819671450009X.\n- [MT16] Johanna Mangahas and Samuel J. Taylor. Convex cocompactness in mapping class groups via quasiconvexity in right-angled Artin groups. Proc. Lond. Math. Soc. (3), 112(5):855–881, 2016. doi:10.1112/plms/pdw009.\n- [KMT17] Thomas Koberda, Johanna Mangahas, and Samuel J. Taylor. The geometry of purely loxodromic subgroups of right-angled Artin groups. Trans. Amer. Math. Soc., 369(11):8179–8208, 2017. doi:10.1090/tran/6933.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2777, "problem_number": "KP-2.29", "title": "Kirby Problem 2.29", "statement": "Determine the Artin groups that can be embedded into a map-\nping class group.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.29.\n\nLiterature notes:\n(1) If the Artin group is simply laced and the homomorphism sends each\nstandard generator to a Dehn twist, then there is an almost complete\nclassification. It is known that the Artin groups of type $A_{n}$ (see [BH73]),\n$D_{n}$ (see [PV96]) and $\\widetilde{A}_{n}$ (see [Ryf23]) can be embedded into a mapping\nclass group in this way. It is also known that an Artin group does not\nembed in this way if its Coxeter graph is connected and contains a full\nsubgraph isomorphic to either $\\widetilde{D}_{n}$ with $n \\geq 4$ (see [Lab97]), or $E_{6}$ (see\n[Waj99]). These leave a finite number of possible open cases.\n(2) Say a homomorphism from an Artin group to a mapping class group is\ngeometric if each standard generator is sent to a Dehn multi-twist. The\nquestion is very open in the case of non-geometric embeddings. There are\nby now various examples of non-geometric embeddings of Artin groups\ninto mapping class groups; see e.g., [CMM21] for the case of right-angled\nArtin groups and [Sze10, BT12] for the case of braid groups.\n(3) The question is also closely related to another folklore question on Artin\ngroups: which Artin groups are virtually special (possessing a finite-index\nsubgroup isomorphic to a subgroup of a right-angled Artin group). It is\nknown that every right-angled Artin group embeds into a mapping class\ngroup, and so obstructions to embedding an Artin group into mapping\n\nclass groups also obstruct embeddability into right-angled Artin groups.\nFrom this point of view it is also natural to consider a “virtual” version of\nthe above question, namely to determine which Artin groups have some\nfinite-index subgroup that embeds into a mapping class group.\n(4) This problem is historically attributed to J. Crisp and L. Paris [CP01].\n\nReferences cited:\n- [BH73] Joan S. Birman and Hugh M. Hilden. On isotopies of homeomorphisms of Riemann surfaces. Ann. of Math. (2), 97:424–439, 1973. doi:10.2307/1970830.\n- [PV96] B. Perron and J. P. Vannier. Groupe de monodromie géométrique des singularités simples. Math. Ann., 306(2):231–245, 1996. doi:10.1007/BF01445249.\n- [Ryf23] Levi Ryffel. Curves intersecting in a circuit pattern. Topology Appl., 332:Paper No. 108522, 15, 2023. doi:10.1016/j.topol.2023.108522.\n- [Lab97] C. Labruere. Generalized braid groups and mapping class groups. J. Knot Theory Ramifications, 6(5):715–726, 1997. doi:10.1142/S021821659700039X.\n- [Waj99] Bronislaw Wajnryb. Artin groups and geometric monodromy. Invent. Math., 138(3):563–571, 1999. doi:10.1007/s002220050353.\n- [CMM21] Matt Clay, Johanna Mangahas, and Dan Margalit. Right-angled Artin groups as normal subgroups of mapping class groups. Compos. Math., 157(8):1807–1852, 2021. doi:10.1112/S0010437X21007417.\n- [Sze10] Bl ażej Szepietowski. Embedding the braid group in mapping class groups. Publ. Mat., 54(2):359–368, 2010. URL: https://doi.org/10.5565/PUBLMAT 54210 04, doi:10.5565/PUBLMAT\\\\_54210\\\\_04.\n- [BT12] Carl-Friedrich Bödigheimer and Ulrike Tillmann. Embeddings of braid groups into mapping class groups and their homology. In Configuration spaces, volume 14 of CRM Series, pages 173–191. Ed. Norm., Pisa, 2012. URL: https://doi.org/10.1007/978-88-7642-431-1 7, doi:10.1007/978-88-7642-431-1\\\\_7.\n- [CP01] John Crisp and Luis Paris. The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group. Invent. Math., 145(1):19– 36, 2001. doi:10.1007/s002220100138.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2778, "problem_number": "KP-2.30", "title": "Kirby Problem 2.30", "statement": "Which right-angled Artin groups contain closed hyperbolic sur-\nface groups? Is there an algorithmic or graph-theoretic criterion to decide this?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.30.\n\nLiterature notes:\nAll but finitely many surface groups embed in right-angled Artin\ngroups [CW04], and right-angled Artin groups on cycles of length at least five\ncontain hyperbolic surface groups [SDS89]. Right-angled Artin groups on graphs\nup to eight vertices containing surface groups were classified by Crisp–Sageev–\nSapir [CSS08]; they also posed the question of whether a right-angled Artin group\ncontains a surface group if and only if it contains a one-ended word-hyperbolic\ngroup. See also [Kim10].\n\nReferences cited:\n- [CW04] John Crisp and Bert Wiest. Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups. Algebr. Geom. Topol., 4:439–472, 2004. doi:10.2140/agt.2004.4.439.\n- [SDS89] Herman Servatius, Carl Droms, and Brigitte Servatius. Surface subgroups of graph groups. Proc. Amer. Math. Soc., 106(3):573–578, 1989. doi:10.2307/2047406.\n- [CSS08] John Crisp, Michah Sageev, and Mark Sapir. Surface subgroups of right-angled Artin groups. Internat. J. Algebra Comput., 18(3):443–491, 2008. doi:10.1142/S0218196708004536.\n- [Kim10] Sang-hyun Kim. On right-angled Artin groups without surface subgroups. Groups Geom. Dyn., 4(2):275–307, 2010. doi:10.4171/GGD/84.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2779, "problem_number": "KP-2.31", "title": "Kirby Problem 2.31", "statement": "Let $S$ be a closed surface of genus at least 2. Show that the\nstable commutator length is rational on the commutator subgroup of $\\pi_{1}(S)$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.31.\n\nLiterature notes:\n(1) If $G$ is a group and $g \\in [G, G]$, the commutator length of $g$, denoted cl $(g)$,\nis the least number of commutators in $G$ whose product is $g$, and the\nstable commutator length of $g$, denoted scl $(g)$, is the limit of cl $(g^{n})/n$ as\n$n \\to \\infty$.\n(2) Suppose $X$ is a space with $\\pi_{1}(X) = G$, suppose $L$ is an oriented circle,\nand suppose $f: L \\to G$ takes $L$ to the conjugacy class of $g \\in [G, G]$. After\nreplacing $f$ by inclusion, consider the the affine subspace $V$ of $H_{2}(X, L; \\mathbb{R})$\nthat is mapped by the boundary map in homology to the fundamental\nclass $[L]$ of $L$ in $H_{1}(L; \\mathbb{R})$. Then the stable commutator length of $g$ is $1/4$\ntimes the infimum of the (relative) Gromov norm on the subspace $V$ .\nTaking $L$ to be a 1-manifold and $f: L \\to X$ to be a map for which\n$f_{*}[L] = 0$ in $H_{1}(X; \\mathbb{R})$ extends the definition of stable commutator length\nto all homologically trivial formal chains (a purely group-theoretic defini-\ntion is given in [Cal09b]).\nThe conjecture above may be broadened as follows.\nLet $G$ be an\narbitrary word-hyperbolic group and $f: L \\to X$ an arbitrary map from\nan oriented 1-manifold $L$ to $X$ with $f_{*}[L] = 0$ in $H_{1}(X; \\mathbb{R})$. Here are a\nfew related conjectures:\n(a) the stable commutator length of $L$ is rational;\n(b) the Gromov norm of any rational class in $H_{2}(X, L; \\mathbb{Q})$ is rational;\nand\n\n(c) the Gromov norm of any rational class in $H_{2}(X; \\mathbb{R})$ is rational.\nFor $X$ a closed surface of genus at least 2, the answer to (c) is positive,\nand for every $L$ the answer to (b) is positive for all rational $\\alpha \\in V$ mapping\nto the fundamental class of $[L]$ outside a compact interval.\n(3) This is known to be true for elements represented by curves supported in\nan essential proper subsurface $S' \\subset S$ (see [Cal09b]).\n(4) If $S$ is a compact oriented surface, possibly with boundary and every\ncomponent having negative Euler characteristic, the Gromov norm of the\nfundamental class of $S$ in $H_{2}(S, \\partial S; \\mathbb{R})$ is $-2\\chi(S)$.\nIf $f: L \\to X$ is as above, a map $F: S \\to X$ virtually bounds $f$ if there\nis a degree $n$ (oriented) covering map $\\pi: \\partial S \\to L$ for which $f\\pi = F$ on $\\partial S$\n(we allow the possibility that $L$ is empty). Such a map is extremal if it is\nan isometry for the Gromov norm. Extremal maps are $\\pi_{1}$-injective.\nA further conjecture: In each case of the conjecture in Remark (1),\nthere is an extremal map of a surface virtually bounding $L$ (or virtually\nrepresenting a given class in $H_{2}(X; \\mathbb{Q})$ if $L$ is empty).\nThis conjecture could be thought of as bearing on Gromov’s question\nof whether every one-ended hyperbolic group contains a closed surface\nsubgroup.\n(5) The questions above have a positive answer for free groups.\nSee e.g.,\n[Cal09b].\n\nReferences cited:\n- [Cal09b] Danny Calegari. scl, volume 20 of MSJ Memoirs. Mathematical Society of Japan, Tokyo, 2009. doi:10.1142/e018.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2780, "problem_number": "KP-2.32", "title": "Kirby Problem 2.32", "statement": "Does every surface bundle over a surface admit a flat connec-\ntion? What about surface bundles over 3-manifolds?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.32.\n\nLiterature notes:\n(1) An $S_{g}$-bundle $p: E \\to B$ admits a flat connection if it admits a codimen-\nsion 2 foliation everywhere transverse to the fibers. Equivalently, the bun-\ndle admits a flat connection if the monodromy map $\\rho: \\pi_{1}(B) \\to \\operatorname{Mod}(S_{g})$\nadmits a lifting $\\widetilde{\\rho}: \\pi_{1}(B) \\to \\operatorname{Diff}(S_{g})$.\n(2) Work of Morita [Mor87] shows that the MMM classes $e_{i} \\in H^{2i}(\\operatorname{Mod}(S_{g}))$\nobstruct the existence of a flat connection for $i \\geq 3$. In practical terms,\nthis means that there are surface bundles over manifolds of dimension at\nleast 6 that admit no flat connection. Conversely, the first MMM class is\nknown not to obstruct flatness; see [KM05]. New techniques, presumably\nof a more dynamical flavor, will be required to obstruct the existence of\nflat connections on surface bundles when the base is of lower dimension.\n\n(3) This question can be asked for varying degrees of regularity, all the way\nfrom lifting to the group of homeomorphisms, up to real-analytic or volume-\npreserving diffeomorphisms.\nIn the volume-preserving case, techniques from symplectic geome-\ntry may be relevant. This discussion follows ideas outlined in [KM05].\nLet $\\omega$ be a volume form on $S_{g}$, and let $\\operatorname{Diff}(S_{g}, \\omega)$ denote the group of\nvolume-preserving diffeomorphisms; the identity component is denoted\n$\\operatorname{Diff}_{0}(S_{g}, \\omega)$. An $S_{h}$-bundle over $S_{g}$ is determined by a set $a_{1}, b_{1},..., a_{h}, b_{h}$\nof mapping classes for which the surface relation $[a_{1}, b_{1}]... [a_{h}, b_{h}]$ holds.\nFrom this point of view, a flat volume-preserving connection consists of a\nchoice of lifts $\\alpha_{i}, \\beta_{i} \\in \\operatorname{Diff}(S_{g}, \\omega)$ such that $[\\alpha_{1}, \\beta_{1}]... [\\alpha_{h}, \\beta_{h}] = 1$.\nOne possible method for obstructing such a lift proceeds as follows.\nBeginning with a surface bundle specified by the mapping classes $a_{1}, b_{1},..., a_{h}, b_{h}$\nas above, the set of all lifts $\\alpha_{i}, \\beta_{i}$ is a torsor for the group $\\operatorname{Diff}_{0}(S_{g}, \\omega)^{2h}$.\nThere is a map (not a group homomorphism)\n\n$$\n\\mu: \\operatorname{Diff}_{0}(S_{g}, \\omega)^{2h} \\to \\operatorname{Diff}_{0}(S_{g}, \\omega)\n$$\n\nthat sends a set of lifts $\\alpha_{1}, \\beta_{1},..., \\alpha_{h}, \\beta_{h}$ to the surface word $\\prod_{i=1}^{h}[\\alpha_{i},\\beta_{i}]$;\nthe fiber $\\mu^{-1}(\\operatorname{id})$ then describes the set of flat volume-preserving connec-\ntions on the bundle.\n$\\operatorname{Diff}_{0}(S_{g}, \\omega)$ admits a flux homomorphism Flux: $\\operatorname{Diff}_{0}(S_{g}, \\omega) \\to H^{1}(S_{g}; \\mathbb{R})$\nwhose kernel is the group of Hamiltonian diffeomorphisms Ham $(S_{g}, \\omega)$. A\nfirst question is whether the flux homomorphism can serve as an obstruc-\ntion to admitting a flat connection, although this seems unlikely to be the\ncase.\nConsidering next the set of lifts for which $[\\alpha_{1}, \\beta_{1}]... [\\alpha_{h}, \\beta_{h}]$ is Hamil-\ntonian, one is led to wonder if the Hofer metric on Ham $(S_{g}, \\omega)$ could be\nof use in obstructing the existence of a flat connection. Perhaps it is pos-\nsible to show that the image of $\\mu$ must stay far away from 0 in the Hofer\nmetric? (See [MS17a] for an introduction to the Hofer metric.)\n\nReferences cited:\n- [Mor87] Shigeyuki Morita. Characteristic classes of surface bundles. Invent. Math., 90(3):551–577, 1987. doi:10.1007/BF01389178.\n- [KM05] D. Kotschick and S. Morita. Signatures of foliated surface bundles and the symplectomorphism groups of surfaces. Topology, 44(1):131–149, 2005. doi:10.1016/j.top.2004.05.002.\n- [MS17a] Dusa McDuff and Dietmar Salamon. Introduction to symplectic topology, volume 27. Oxford University Press, 2017.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2781, "problem_number": "KP-2.33", "title": "Kirby Problem 2.33", "statement": "Let $S$ be a closed compact surface (without boundary).\nIs\nthere a finitely generated, torsion-free group $G$ such that $G$ cannot act faithfully by\nhomeomorphisms on $S$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.33.\n\nLiterature notes:\n(1) There are non-compact surfaces for which every countable group acts by\nhomeomorphisms [APV21].\n(2) We pose the question for closed surfaces only, since the presence of bound-\nary gives access to special techniques that may be able to resolve the ques-\ntion as stated without developing new methods for the closed case. On the\nother hand, the question is open even for the disk, relative to the bound-\nary. Potential candidates include lattices in high rank Lie groups. For\nmore regular actions (i.e., smooth actions) more is known; see [BFH22],\nfor instance.\n\n(3) The torsion-free hypothesis is necessary, since there are often a priori\nbounds on the size of torsion in surface homeomorphism groups.\n(4) In the case of the plane, Le Roux [LR11] has studied actions of the\nfundamental group of the Klein bottle under the assumption that the\nstandard generators act freely and preserving orientation. Among other\nresults, he finds obstructions for the action of certain torsion-free groups\nby fixed-point free orientation-preserving homeomorphisms.\n(5) A more general question would be to find an algebraic characterization\nof groups acting on $S$ by homeomorphisms. Such characterizations are\nknown for groups acting on the interval and on the circle, via orderability.\nIt may be difficult to find concise characterizations; see e.g., [dlNG22,\nRos13, Hyd19].\n\nReferences cited:\n- [APV21] Tarik Aougab, Priyam Patel, and Nicholas G. Vlamis. Isometry groups of infinite genus hyperbolic surfaces. Math. Ann., 381:459–498, 2021.\n- [BFH22] Aaron Brown, David Fisher, and Sebastian Hurtado. Zimmer’s conjecture: subexponential growth, measure rigidity, and strong property (T). Ann. of Math. (2), 196(3):891–940, 2022. doi:10.4007/annals.2022.196.3.1.\n- [LR11] Frédéric Le Roux. Free planar actions of the Klein bottle group. Geom. Topol., 15(3):1545–1567, 2011. doi:10.2140/gt.2011.15.1545.\n- [dlNG22] Javier de la Nuez Gonzalez. Non-Roelcke precompactness of groups of surface homeomorphisms, 2022. arXiv:2202.06527.\n- [Ros13] Christian Rosendal. Global and local boundedness of Polish groups. Indiana Univ. Math. J., 62(5):1621–1678, 2013. doi:10.1512/iumj.2013.62.5133.\n- [Hyd19] James Hyde. The group of boundary fixing homeomorphisms of the disc is not leftorderable. Ann. of Math. (2), 190(2):657–661, 2019. doi:10.4007/annals.2019.190.2.5.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2782, "problem_number": "KP-2.34", "title": "Kirby Problem 2.34", "statement": "Let $S$ be a compact surface. For $0 \\leq r < s$, does there exist a\nnontrivial finitely generated subgroup $G_{r} \\leq \\operatorname{Diff}^{r}_{0}(S)$ such that every action of $G_{r}$\non $S$ by $C^{s}$ diffeomorphisms is trivial? Can these examples be torsion-free?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.34.\n\nLiterature notes:\n(1) For $0 \\leq r \\leq \\infty$, let $\\operatorname{Diff}^{r}_{0}(M)$ denote the identity component of the group\nof $C^{r}$ diffeomorphisms of a manifold $M$.\nFor nonintegral $r$, we write\n$r = k + \\epsilon$ for $k \\in \\mathbb{N}$ and $0 \\leq \\epsilon < 1$, and we consider $C^{k}$ diffeomorphisms\nwhose $k^{th}$ order derivatives are $\\epsilon$–Hölder continuous.\n(2) The condition that every higher-regularity action of $G_{r}$ be trivial can be\nrelaxed, for instance, to ask that every such action of $G_{r}$ be non-faithful,\nor abelian.\n(3) Problems of this nature have been extensively investigated in dimen-\nsion one, where Kim and Koberda gave a positive answer to this ques-\ntion [KK20, KK21]; one obtains finitely generated groups with infinite\nsimple commutator subgroups such that each smoother action has abelian\nimage. No instances are known in dimension greater than one. Prior to\nthis, Plante–Thurston showed that nilpotent groups of $C^{2}$-diffeomorphisms\nof the interval are abelian [PT76], but Farb–Franks showed that there can\nbe non-abelian nilpotent actions of regularity $C^{1}$ [FF03]. More generally,\nCastro–Jorquera–Navas build nilpotent groups that can act at a certain\nregularity but whose higher-regularity actions are all abelian [CJN14].\nFurther examples of finitely generated groups acting with critical regular-\nities on 1-manifolds are constructed in [MW23].\n(4) In the case $r = 0$ and $s \\geq 1$, it is possible to use Thurston’s Stability\nTheorem [Thu74b] to exhibit particular groups of homeomorphisms that\ncannot act faithfully by diffeomorphisms. See, for instance, [Cal06a].\n\nReferences cited:\n- [KK20] Sang-hyun Kim and Thomas Koberda. Diffeomorphism groups of critical regularity. Invent. Math., 221(2):421–501, 2020. doi:10.1007/s00222-020-00953-y.\n- [KK21] Sang-hyun Kim and Thomas Koberda. Structure and regularity of group actions on one-manifolds. Springer Monographs in Mathematics. Springer, Cham, [2021] ©2021. doi:10.1007/978-3-030-89006-3.\n- [PT76] J. F. Plante and W. P. Thurston. Polynomial growth in holonomy groups of foliations. Comment. Math. Helv., 51(4):567–584, 1976. doi:10.1007/BF02568174.\n- [FF03] Benson Farb and John Franks. Groups of homeomorphisms of one-manifolds. III. Nilpotent subgroups. Ergodic Theory Dynam. Systems, 23(5):1467–1484, 2003. doi: 10.1017/S0143385702001712.\n- [CJN14] Gonzalo Castro, Eduardo Jorquera, and Andrés Navas. Sharp regularity for certain nilpotent group actions on the interval. Math. Ann., 359(1-2):101–152, 2014. doi: 10.1007/s00208-013-0995-1.\n- [MW23] Kathryn Mann and Maxime Wolff. Reconstructing maps out of groups. Ann. Sci. Éc. Norm. Supér. (4), 56(4):1135–1154, 2023. doi:10.24033/asens.2551.\n- [Thu74b] William P. Thurston. A generalization of the Reeb stability theorem. Topology, 13:347–352, 1974. doi:10.1016/0040-9383(74)90025-1.\n- [Cal06a] Danny Calegari. Dynamical forcing of circular groups. Trans. Amer. Math. Soc., 358(8):3473–3491, 2006. doi:10.1090/S0002-9947-05-03754-2.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2783, "problem_number": "KP-2.35", "title": "Kirby Problem 2.35", "statement": "Is the first-order theory of the mapping class group of a surface\ndecidable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.35.\n\nLiterature notes:\n(1) The first order theory of a group $G$ in the language of group theory is the\nformal first-order language equipped with group multiplication and the\nidentity element. A first-order theory is decidable if there is an algorithm\n(i.e., Turing machine) that decides which sentences are true in that theory\nand which are false.\n(2) For homeomorphism groups of surfaces (and indeed all compact manifolds\nof positive dimension), the corresponding theory is not decidable because\nit interprets arithmetic [KKdlNG25].\n(3) The theory of free groups is decidable; this is part of the resolution of the\nTarski problem [Sel02, KM06].\n\nReferences cited:\n- [KKdlNG25] Sang-hyun Kim, Thomas Koberda, and J. de la Nuez González. First order rigidity of homeomorphism groups of manifolds. Commun. Am. Math. Soc., 5:144–194, 2025. doi:10.1090/cams/47.\n- [Sel02] Z. Sela. Diophantine geometry over groups and the elementary theory of free and hyperbolic groups. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 87–92. Higher Ed. Press, Beijing, 2002.\n- [KM06] Olga Kharlampovich and Alexei Myasnikov. Elementary theory of free non-abelian groups. J. Algebra, 302(2):451–552, 2006. doi:10.1016/j.jalgebra.2006.03.033.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2784, "problem_number": "KP-2.36", "title": "Kirby Problem 2.36", "statement": "Are systems of equations over mapping class groups and braid\ngroups decidable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.36.\n\nLiterature notes:\n(1) A system of equations over a group $G$ is a finite conjunction of formulae\nof the form\n\n$$\nr(x_{1},..., x_{n}, a_{1},..., a_{m}) = 1\n$$\n\nand\n\n$$\nr(x_{1},..., x_{n}, a_{1},..., a_{m}) \\neq 1,\n$$\n\nwhere $r$ is a word in a free group of $\\operatorname{rank} n + m$, where $(x_{1},..., x_{n})$ is\na tuple of variables, and where $(a_{1},..., a_{m})$ (with $m \\geq 0$) is a tuple of\nparameters in $G$. A solution to a system of equations is a tuple of elements\nof $G$ such that when the elements are substituted for variables, one obtains\na valid expression in $G$.\n(2) Decidability of equations would consist of an algorithm to decide whether\nor not a particular system has a solution or not. Precisely, one might fix\na finite generating set for $G$ (if it exists) and use a Gödel numbering to\nencode equations with parameters as natural numbers. Then, one would\nrequire a Turing machine to take a natural number corresponding to an\nequation and decide if it admits a solution.\n(3) One could allow variations on the problem; for instance, one could fix the\nunderlying surface or allow it to vary, pass to finite index subgroups, or\nrestrict the topological type of the underlying surface.\n(4) Decidability of the theory of a group or class of groups has a long his-\ntory, starting at least with the Tarski problem. The Tarski problem asked\nwhether the nonabelian free groups are elementarily equivalent to each\nother and whether their first order theories are decidable; see [Sel02,\nKM06]. For general hyperbolic groups (possibly with torsion), Dahmani\n\nand Guirardel proved that equations are solvable; see [DG10]. Conjec-\nturally, the full first order theory of a vast generalization of hyperbolic\ngroups, namely hierarchically hyperbolic groups, should be decidable. A\nparticularly interesting special case of such a general result would be the\ndecidability of the first order theory of mapping class groups and braid\ngroups, which in turn would give a positive answer to the proposed ques-\ntion.\n(5) There is a vast literature on equations over groups, including finite groups\nand free groups, though for mapping class groups and braid groups the\nproblem appears to be open. See [Rom12] for a survey.\n(6) It seems likely that equations over homeomorphism groups of surfaces\n(with or without parameters) are undecidable.\n\nReferences cited:\n- [Sel02] Z. Sela. Diophantine geometry over groups and the elementary theory of free and hyperbolic groups. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 87–92. Higher Ed. Press, Beijing, 2002.\n- [KM06] Olga Kharlampovich and Alexei Myasnikov. Elementary theory of free non-abelian groups. J. Algebra, 302(2):451–552, 2006. doi:10.1016/j.jalgebra.2006.03.033.\n- [DG10] François Dahmani and Vincent Guirardel. Foliations for solving equations in groups: free, virtually free, and hyperbolic groups. J. Topol., 3(2):343–404, 2010. doi:10.1112/jtopol/jtq010.\n- [Rom12] Vitaliı̆ Roman’kov. Equations over groups. Groups Complex. Cryptol., 4(2):191– 239, 2012. doi:10.1515/gcc-2012-0015.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2785, "problem_number": "KP-2.37", "title": "Kirby Problem 2.37", "statement": "Give a Nielsen–Thurston-type classification for the mapping\nclass groups of infinite-type surfaces. In particular, which homeomorphisms are the\nappropriate generalization of pseudo-Anosov homeomorphisms in this setting?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.37.\n\nLiterature notes:\n(1) For finite-type surfaces $S$, there is a powerful classification due to Nielsen\nand Thurston of the mapping classes of $S$: every mapping class is periodic,\nreducible, or pseudo-Anosov. The most important aspect of this classifi-\ncation is that if a mapping class $f$ is not periodic or reducible, it must be\npseudo-Anosov, meaning that $f$ is highly chaotic and that there exist two\ntransverse measured foliations $\\lambda^{+}$ and $\\lambda^{-}$ on $S$, where $f$ stretches along\n$\\lambda^{+}$ and contracts along $\\lambda^{-}$. The exact analog of this classification for\n\ninfinite-type surfaces is false. For instance, the homeomorphisms called\nhandleshifts (introduced by Patel-Vlamis [PV18]) are neither periodic,\nreducible, nor pseudo-Anosov. The classification, therefore, needs to be\nmodified in the infinite-type setting. First, the definition of a reducible\nmapping class should be generalized, but more importantly, the third cate-\ngory (the analog of pseudo-Anosov mapping classes) needs to be expanded\nand likely broken into subcategories.\n(2) Some work has been announced on this problem by Bestvina, Fanoni and\nTao [BFT23], who classify so-called “tame” homeomorphisms that satisfy\nan additional finiteness condition. Such homeomorphisms do not exhibit\nany pseudo-Anosov-like behavior.\n(3) Irreducible end-periodic mapping classes should form an important class\nwithin the third category. For end-periodic homeomorphisms of infinite-\ntype surfaces, Handel and Mosher outlined a theory paralleling that of\npseudo-Anosov homeomorphisms in unpublished work from the 1980s,\ndescribing a pair of transverse geodesic laminations on the surface pre-\nserved by the end-periodic mapping class. This theory was further de-\nveloped by Cantwell, Conlon, and Fenley [CCF21]. In addition to the\nabove, irreducible end-periodic mapping classes share many properties\nwith finite-type pseudo-Anosov mapping classes; for example, they give\nrise to hyperbolic mapping tori [FKLL23] and many strongly irreducible\nones are loxodromic isometries of complexes of curves and arcs [PT25].\n\nReferences cited:\n- [PV18] Priyam Patel and Nicholas G. Vlamis. Algebraic and topological properties of big mapping class groups. Algebr. Geom. Topol., 18(7):4109–4142, 2018. doi:10.2140/agt.2018.18.4109.\n- [BFT23] Mladen Bestvina, Federica Fanoni, and Jing Tao. Towards Nielsen-Thurston classification for surfaces of infinite type, 2023. arXiv:2303.12413.\n- [CCF21] John Cantwell, Lawrence Conlon, and Sergio R. Fenley. Endperiodic automorphisms of surfaces and foliations. Ergodic Theory Dynam. Systems, 41(1):66–212, 2021. doi:10.1017/etds.2019.56.\n- [FKLL23] Elizabeth Field, Heejoung Kim, Christopher Leininger, and Marissa Loving. Endperiodic homeomorphisms and volumes of mapping tori. J. Topol., 16(1):57–105, 2023. doi:10.1112/topo.12277.\n- [PT25] Priyam Patel and Samuel J. Taylor. Constructing endperiodic loxodromics of infinite-type arc graphs. Math. Z., 310(4):Paper No. 82, 2025. doi:10.1007/s00209-025-03784-w.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2786, "problem_number": "KP-2.38", "title": "Kirby Problem 2.38", "statement": "Give an appropriate analogue of the curve graph for infinite-\ntype surfaces, and characterize the surfaces for which no such graph exists.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.38.\n\nLiterature notes:\n(1) A motivating philosophy in geometric group theory is that one can study\nthe algebra of a group $G$ via the geometry of the spaces on which $G$ acts\nand the dynamics of such actions. When $G$ is the mapping class group of\na finite-type surface, one of the most fruitful actions to study is that of\nthe mapping class group of $S$ on the curve graph, $\\mathcal{C}(S)$, associated to $S$.\nPart of the reason that this action is so dynamically rich is due to a\nfamous result of Masur and Minsky [MM00], which shows that the curve\ngraph is infinite-diameter and Gromov-hyperbolic: there exists $\\delta > 0$\nso that, for all geodesic triangles in $\\mathcal{C}(S)$, each side of the triangle is\ncontained in the $\\delta$-neighborhood of the other two. This means that $\\mathcal{C}(S)$\nshares properties with hyperbolic space so that some of the key tools in\nhyperbolic geometry can be used to study $\\mathcal{C}(S)$ and groups acting on it.\nThe curve graph has played a pivotal role not only in the study of mapping\nclass groups, but also Teichmüller theory, Kleinian groups, and the moduli\nspace of Riemann surfaces.\n\nBy contrast, for infinite-type surfaces the curve graph has finite di-\nameter. This makes it much harder to study the above topics for infinite-\ntype surfaces via the action of groups on this graph. An analogue of the\ncurve graph would be an infinite-diameter hyperbolic graph associated to\nan infinite-type surface $S$ on which the mapping class group of $S$ acts\ncoboundedly and by isometries.\n(2) Vlamis showed that there are infinite-type surfaces whose mapping class\ngroups do not act on any metric space with unbounded orbits [Vla24b];\nadditional examples were announced in [Vla24a]. Thus, it is not possible\nto find an analogue of the curve graph that can be associated to any\ninfinite-type surface.\n(3) Among infinite-type surfaces whose mapping class groups do admit un-\nbounded actions on metric spaces, there are several associated graphs that\nhave been fruitful to study thus far, for instance the ray graph (Calegari\n[Cal09a]), the relative arc graph (Aramayona–Fossas–Parlier [AFP17]),\nthe omnipresent arc graph (Fanoni–Ghaswala–McLeay [FGM21]), and\nthe grand arc graph (Bar-Natan–Verberne [BNV23]).\nUnfortunately,\nthis is a piecemeal approach, since these graphs are not defined for all\ninfinite-type surfaces in this class, and may fail to be infinite-diameter\nand hyperbolic in general.\nIn light of this, one wishes to first classify\nwhich infinite-type surfaces admit an analogue of the curve graph, and to\nthen find a universal graph that can be associated to any such infinite-type\nsurface that is both infinite-diameter and hyperbolic.\n(4) This problem appeared on an AIM problem list for a workshop on surfaces\nof infinite type [LPRT] and was proposed for inclusion there by P. Patel\nand K. Mann.\n\nReferences cited:\n- [MM00] Howard A. Masur and Yair N. Minsky. Geometry of the complex of curves. II. Hierarchical structure. Geom. Funct. Anal., 10(4):902–974, 2000.\n- [Vla24b] Nicholas G. Vlamis. Homeomorphism groups of self-similar 2-manifolds. In In the tradition of Thurston III. Geometry and dynamics, pages 105–167. Springer, Cham,\n- [Vla24a] Nicholas G. Vlamis. Homeomorphism groups of telescoping 2–manifolds are strongly distorted, 2024. arXiv:2403.03887.\n- [Cal09a] Danny Calegari. Big mapping class groups and dynamics. Geometry and the imagination, https://lamington.wordpress.com/2009/06/22/big-mapping-class-groups-and-dynamics/, 2009.\n- [AFP17] Javier Aramayona, Ariadna Fossas, and Hugo Parlier. Arc and curve graphs for infinite-type surfaces. Proceedings of the American Mathematical Society, 145(11):4995–5006, 2017.\n- [FGM21] Federica Fanoni, Tyrone Ghaswala, and Alan McLeay. Homeomorphic subsurfaces and the omnipresent arcs. Ann. H. Lebesgue, 4:1565–1593, 2021. doi:10.5802/ahl.110.\n- [BNV23] Assaf Bar-Natan and Yvon Verberne. The grand arc graph. Math. Z., 305(2):Paper No. 20, 21, 2023. doi:10.1007/s00209-023-03337-z.\n- [LPRT] Justin Lanier, Priyam Patel, Anja Randecker, and Jing Tao. AIM Problem List: Surfaces of infinite type. available at http://aimpl.org/genusinfinity.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2787, "problem_number": "KP-2.39", "title": "Kirby Problem 2.39", "statement": "(a) Does the mapping class group of an infinite-genus surface with no planar\nends contain every countable group?\n(b) Does the mapping class group of the one-ended infinite-genus surface have\nany proper finite-index subgroups?\n(c) For an infinite-type surface, describe and characterize the subgroup of\nmapping classes with quasi-conformal representatives. In particular, when\nis it a normal subgroup of the mapping class group?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.39.\n\nLiterature notes:\n(1) The work of Aougab, Patel, and Vlamis [APV21] shows every mapping\nclass group of an infinite-genus surface with no planar ends contains every\nfinite group as a subgroup. Thus the problem is really about countably\ninfinite groups. The same work of Aougab, Patel, and Vlamis also shows\nthat the above question is true for infinite-genus surfaces with no planar\nends and self-similar end space, e.g., the Loch Ness monster surface. To do\n\nthis, they realize every countable group as the isometry group of a hyper-\nbolic structure, and moreover, show that the above surfaces are the only\nones in which this is possible. Therefore, to answer the above question,\nit is necessary to construct countably infinite subgroups of the mapping\nclass group that are not subgroups of the isometry group of the surface.\n(2) It was shown by G. Domat [Dom22] that the mapping class group of\nthe one-ended infinite genus surface has nontrivial abelianization, and\nthe abelianization has many copies of $\\mathbb{Q}$ as direct summands. It follows\nthat the mapping class group of the Loch Ness monster surface has many\nproper subgroups of countably infinite index. In the same work, Domat\nalso shows that the abelianization does contain torsion subgroups, though\nthey are not known to be direct summands.\n(3) When $S$ is a Riemann of finite type, it is a basic fact that every mapping\nclass admits a representative as a quasi-conformal homeomorphism, but\nthis is no longer true if $S$ is of infinite type. In this setting, the group\nof mapping classes with quasi-conformal representatives is sometimes re-\nferred to as the Teichmüller modular group; the reader should be aware\nthat this term is sometimes used to refer to the entire mapping class\ngroup in the finite-type setting. While this usage is consistent, there is\nthe possibility for confusion.\n(4) This problem appeared on a problem list compiled at the 2021 Nearly\nCarbon Neutral Geometric Topology conference [CPV21].\n\nReferences cited:\n- [APV21] Tarik Aougab, Priyam Patel, and Nicholas G. Vlamis. Isometry groups of infinite genus hyperbolic surfaces. Math. Ann., 381:459–498, 2021.\n- [Dom22] George Domat. Big pure mapping class groups are never perfect. Math. Res. Lett., 29(3):691–726, 2022. Appendix with Ryan Dickmann.\n- [CPV21] Yassin Chandran, Priyam Patel, and Nicholas G. Vlamis. Infinite-type surfaces and mapping class groups: Open problems. Available at https://https://www.patelp.com/uploads/2/5/7/9/25792573/inftypeproblems.pdf, 2021.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2788, "problem_number": "KP-2.40", "title": "Kirby Problem 2.40", "statement": "(a) Let $S$ be an infinite-type surface and $\\varphi$ a mapping class for which there is\na (marked) conformal structure $\\Sigma$ on $S$ with respect to which $\\varphi$ is realized\nas a quasi-conformal homeomorphism.\nThen for each quasi-conformal\nstructure in the connected component of Teichmüller space containing $\\Sigma$,\nthere is a quasiconformal homeomorphism representing $\\varphi$ which minimizes\nthe dilatation. If the infimum of the minimal dilatations for each structure\nin this component is achieved and is greater than 1, is the minimizing\nquasiconformal homeomorphism unique? What is the structure of such a\nminimizer?\n(b) If $\\varphi$ is parabolic (see Remark (1) below) for $T$, must a sequence of con-\nformal structures with dilatation converging to the infimum degenerate by\npinching along some sequence of essential simple closed curves? Is this\nsystem unique? Can it depend on the component $T$?\n(c) If $\\varphi$ fixes more than one component $T, T'$ of $\\operatorname{Teich}(S)$, can the type be dif-\nferent for $T$ and $T'$? In particular, can $\\varphi$ be parabolic for $T$ and hyperbolic\nfor $T'$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.40.\n\nLiterature notes:\n(1) Let $S$ be a surface of infinite type, and let $\\varphi$ be a mapping class for\nwhich there is some (marked) conformal structure $\\Sigma$ on $S$ for which $\\varphi$\n\nis realized by a quasiconformal homeomorphism $f: \\Sigma \\to \\Sigma$. Then there\nis some representative homeomorphism which realizes the infimum of the\ndilatation. This follows from compactness (for any finite $K$) of the space\nof $K$-quasiconformal homeomorphisms of the unit disk, normalized to fix\n3 points on the boundary (see, e.g., [Ahl06]).\nWe may then try to vary the conformal structure on $S$ in the com-\nponent $T \\subset \\operatorname{Teich}(S)$ containing $\\Sigma$, and for each point in $T$ we get a\nminimum value for the dilatation of a homeomorphism representing $\\varphi$,\nand thereby a function $K_{T}: T \\to [1, \\infty)$. Note $K_{T}$ takes the value 1 if\nand only if $\\varphi$ is represented by an isometry for some conformal structure\nin the component $T$.\nSay $\\varphi$ is elliptic for $T$ if $K_{T}$ achieves the value 1, parabolic for $T$ if the\ninfimum is not achieved, and hyperbolic for $T$ if the infimum is achieved\nand is bigger than 1. Part (a) can be restated as follows: if $\\varphi$ is hyperbolic\nfor $T$, is the minimizer unique? What is its structure?\n(2) When $\\varphi$ is a mapping class on a surface $S$ of finite type, there is a (marked)\nconformal structure on $S$ with respect to which $\\varphi$ can be realized as a qua-\nsiconformal homeomorphism. Moreover, there is a unique quasiconformal\nhomeomorphism representing $\\varphi$ that minimizes the dilatation; this is the\nTeichmüller map. Note that in the finite-type setting, Teichmüller space\nhas a single connected component.\n(3) There are elementary examples of topological surfaces of infinite type $S$\nand mapping classes $\\varphi$ that are not realized by quasiconformal maps for\nany choice of (marked) conformal structure on $S$. Perhaps the simplest\nexample is: let $S$ be the infinite ladder surface, which we think of as the\ninfinite union of twice-punctured tori $T_{n}$ indexed by integers $n \\in \\mathbb{Z}$ joined\nend to end, let $\\alpha$ be any pseudo-Anosov diffeomorphism of a single twice-\npunctured torus, and let $\\varphi$ be the mapping class that does $\\alpha^{n}$ on $T_{n}$. No\nmatter what conformal structure we choose on $S$, the restriction of $\\alpha^{n}$ to\n$T_{n}$ has dilatation at least as big as $|n|$ times the minimum dilatation of $\\alpha$\non a twice-punctured torus. This example is (highly) reducible; it would\nbe nice to have a straightforward irreducible example.\n(4) If $S$ is a surface of infinite type, it is possible to define the Teichmüller\nspace of $S$ to be the space of marked conformal surfaces $f: S \\to \\Sigma$ up\nto equivalence; see the recently announced constructions of Basmajian–\nChandran [BC24a] and Tappu [Tap23] for details. The connected com-\nponents of this space will be contractible, but (with the hypothesis that\n$S$ has infinite type) there will always be infinitely many components\n[Bas97, BK08]. Some mapping classes will fix no component (as in the\nprevious remark), some will fix some components and not others, some\nwill fix every component (e.g., those supported in compact subsurfaces).\nWhat are the possible orbit types? It was recently announced by Basma-\njian and Chandran that the subgroup that fixes every component is the\ncompactly supported subgroup [BC24a].\n(5) Recent work of Basmajian and Chandran [BC24a] announced that there\nexist mapping classes $\\varphi$ that are represented by quasiconformal home-\nomorphisms on some components of Teichmüller space but not others.\n\nIn particular, the function $K_{T}$ defined in Remark (1) above may not be\nwell-defined for every component $T \\subseteq \\operatorname{Teich}(S)$.\n(6) A version of this problem appeared on a problem list compiled at the 2021\nNearly Carbon Neutral Geometric Topology conference [CPV21].\n\nReferences cited:\n- [Ahl06] Lars V. Ahlfors. Lectures on quasiconformal mappings, volume 38 of University Lecture Series. American Mathematical Society, Providence, RI, second edition, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. doi:10.1090/ulect/038.\n- [BC24a] A. Basmajian and Y. Chandran. A Bers type classification of big mapping class groups, 2024. arXiv:2410.05606.\n- [Tap23] Chaitanya Tappu. A moduli space of marked hyperbolic structures for big surfaces, 2023. arXiv:2311.01551.\n- [Bas97] Ara Basmajian. Large parameter spaces of quasiconformally distinct hyperbolic structures. J. Anal. Math., 71:75–85, 1997. doi:10.1007/BF02788023.\n- [BK08] Ara Basmajian and Youngju Kim. Geometrically infinite surfaces with discrete length spectra. Geom. Dedicata, 137:219–240, 2008. doi:10.1007/s10711-008-9294-5.\n- [CPV21] Yassin Chandran, Priyam Patel, and Nicholas G. Vlamis. Infinite-type surfaces and mapping class groups: Open problems. Available at https://https://www.patelp.com/uploads/2/5/7/9/25792573/inftypeproblems.pdf, 2021.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2789, "problem_number": "KP-2.41", "title": "Kirby Problem 2.41", "statement": "Give a finite list of practically computable invariants of the\nmapping class group or pure mapping class group of an infinite-type surface $S$ that\ndetermine the topology of $S$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.41.\n\nLiterature notes:\n(1) Let $S$ be a surface, possibly of infinite type. When $S = S_{g,b}^{p}$ is of finite\ntype, the homeomorphism type of the surface is encoded in the isomor-\nphism type of its mapping class group, with two pairs of exceptions for\nsmall values of $g, b$, and $p$. Most cases can be distinguished by consid-\nering well-known group invariants: work of Birman–Lubotsky–McCarthy\n[BLM $^{+}83$] computes the algebraic rank, and Harer [Har86] computes\nthe virtual cohomological dimension, both in terms of $g$ and $b$. Combin-\ning these formulas, it is possible to determine the topology of the surface\n$S$, although note that this argument breaks down when either $g = 0$ or\n$b = 0$ (see [RS11, Remark A.2]).\n(2) The classification of infinite-type surfaces tells us that an infinite-type sur-\nface is determined by its end space $E$, the closed subgroup of non-planar\nends $E^{g}$ of $S$, the genus, and the number of compact boundary compo-\nnents of $S$.\nHowever, the end space of an infinite-type surface can be\nany closed subset of a Cantor set, and in practice, it is hard to tell when\ntwo such subsets are homeomorphic. In addition, Bavard–Dowdall–Rafi\n[BDR20] prove algebraic rigidity for mapping class groups of infinite-type\nsurfaces: if Map $(S)$ is isomorphic to Map $(S')$, then $S$ and $S'$ are home-\nomorphic. But again, big mapping class groups are uncountable groups\nand determining when two of these groups are isomorphic in general is\ndifficult. Thus, we pose the question above.\n(3) There has been one result by Aougab–Patel–Vlamis [APV21, Theorem\n8.1] in this direction for $n$-ended orientable infinite-genus surfaces with no\nplanar ends.\n\nReferences cited:\n- [BLM+83] Joan S Birman, Alex Lubotzky, John McCarthy, et al. Abelian and solvable subgroups of the mapping class groups. Duke Mathematical Journal, 50(4):1107–1120, 1983.\n- [Har86] John L. Harer. The virtual cohomological dimension of the mapping class group of an orientable surface. Invent. Math., 84(1):157–176, 1986. doi:10.1007/BF01388737.\n- [RS11] Kasra Rafi and Saul Schleimer. Curve complexes are rigid. Duke Math. J., 158(2):225–246, 2011. doi:10.1215/00127094-1334004.\n- [BDR20] Juliette Bavard, Spencer Dowdall, and Kasra Rafi. Isomorphisms between big mapping class groups. International Mathematics Research Notices, 2020(10):3084– 3099, 2020.\n- [APV21] Tarik Aougab, Priyam Patel, and Nicholas G. Vlamis. Isometry groups of infinite genus hyperbolic surfaces. Math. Ann., 381:459–498, 2021.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2790, "problem_number": "KP-2.42", "title": "Kirby Problem 2.42", "statement": "Is the geodesic flow in almost every direction on the Chamanara\nsurface ergodic? What about on the translation surface considered by Bruin and\nLukina, which is similar to the Chamanara surface?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.42.\n\nLiterature notes:\n(1) Translation surfaces have geodesic flows in every direction that preserve\nLebesgue measure. Kerckhoff, Masur, and Smillie proved that for every\nfinite-type translation surface, this is the only probability measure pre-\nserved by the flow (that is, the flow is uniquely ergodic) in almost every\ndirection [KMS86]. As a corollary, they obtain that the flow in almost\nevery direction is ergodic with respect to the Lebesgue measure.\nIt is\nnatural to ask to what extent this generalizes to infinite-type surfaces.\nFraczek and Ulcigrai produced many examples of infinite-area, infinite-\ntype translation surfaces where the straight line flows are not ergodic in\nalmost every direction [FU14]. For finite-area translation surfaces, how-\never, this is widely open.\n(2) The Chamanara surface, first described by Reza Chamanara in [Cha04],\nis one of the best known translation surfaces. Chamanara defines a family\nof translation surfaces with a parameter $\\alpha \\in [0, 1]$. For the case $\\alpha = 1/2$,\ncalled the standard Chamanara surface, consider a square with edge length\n1, and divide the top and bottom sides into two halves. Identify the top\nright half with the bottom left half via a translation, and then divide the\nunidentified top and bottom edges in half, identify the top right half with\nthe bottom left half via a translation, and repeat. Do the same for the\nleft and right edges, always identifying the upper part of the right edge\nwith the lower part of the left edge. Excluding the corners of the square\nand the points where we divided the edges, we obtain the Chamanara\nsurface, a one-ended infinite-genus translation surface of finite area. See\nalso [DHV24, Example 2.4.22] for another description of the surface.\nThe Chamanara surface has a close relationship to baker’s map and is\nsometimes called the baker’s map surface; see [CGL06] for further details.\nBruin and Lukina consider a family of translation surfaces similar to\nthe Chamanara surfaces [BL23], but their surfaces lack certain metric\nsymmetries enjoyed by the Chamana surfaces, and so different techniques\nmay be necessary to approach this problem.\n(3) Similarly to proving that there exists a billiard in a polygon where the\nbilliard flow is ergodic (in the 3-dimensional unit tangent bundle), one\ncan use a Baire Category argument to show that there exists a finite-\narea, infinite-type surface where the flow in almost every direction is er-\ngodic with respect to Lebesgue measure. Moreover, by work of Vorobets\n[Vor97], one can even find an explicit example, but these arguments do\nnot seem applicable in the two situations above.\n\nReferences cited:\n- [KMS86] Steven Kerckhoff, Howard Masur, and John Smillie. Ergodicity of billiard flows and quadratic differentials. Ann. of Math. (2), 124(2):293–311, 1986. doi:10.2307/1971280.\n- [FU14] Krzysztof Fraczek and Corinna Ulcigrai. Non-ergodic Z-periodic billiards and infinite translation surfaces. Invent. Math., 197(2):241–298, 2014. doi:10.1007/s00222-013-0482-z.\n- [Cha04] R. Chamanara. Affine automorphism groups of surfaces of infinite type. In In the tradition of Ahlfors and Bers, III, volume 355 of Contemp. Math., pages 123–145. Amer. Math. Soc., Providence, RI, 2004. doi:10.1090/conm/355/06449.\n- [DHV24] V. Delecroix, P. Hubert, and F. Valdez. Infinite translation surfaces in the wild, 2024. arXiv:2403.05424.\n- [CGL06] R. Chamanara, F. P. Gardiner, and N. Lakic. A hyperelliptic realization of the horseshoe and baker maps. Ergodic Theory Dynam. Systems, 26(6):1749–1768, 2006. doi:10.1017/S0143385706000484.\n- [BL23] Henk Bruin and Olga Lukina. Rotated odometers. J. Lond. Math. Soc. (2), 107(6):1983–2024, 2023. doi:10.1112/jlms.12731.\n- [Vor97] Ya.B̃. Vorobets. Ergodicity of billiards in polygons. Mat. Sb., 188(3):65–112, 1997. doi:10.1070/SM1997v188n03ABEH000211.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2791, "problem_number": "KP-2.43", "title": "Kirby Problem 2.43", "statement": "Let $X$ be a compact, totally disconnected subset of $\\mathbb{R}^{2}$ with\n$|X| \\geq 2$, and let $\\Gamma_{X}$ denote the mapping class group of $\\mathbb{R}^{2} - X$.\n(a) For $Y \\subset X$, let $\\Gamma_{X,Y}$ be the subgroup of $\\Gamma_{X}$ permuting $Y$ . Classify in-\nvariants $q$ that are functorial.\n(b) If $X \\subset \\mathbb{R}^{2}$ is totally disconnected and $\\alpha$ is a homeomorphism of $\\mathbb{R}^{2}$ fixing\n$X$ as a set, let $C_{\\alpha}(X)$ denote the space of $\\alpha$-invariant closed subsets of\n\n$X$, in the Hausdorff topology. Let $q$ be functorial as in part (a). Classify\nthose $q$ that are continuous or semi-continuous.\n(c) What is the relationship between the translation length $\\tau_{X}$ and topological\nentropy? More precisely, is there a positive constant $C$ so that if $\\alpha$ is a\nhomeomorphism of $\\mathbb{R}^{2}$ permuting $X$ and $g$ is the class of $\\alpha$ in $\\Gamma_{X}$, then\nthe (topological) entropy of $\\alpha$ is at least as big as $C \\cdot \\tau_{X}(g)$?\n(d) Suppose $X$ is a Cantor set in the plane, and let $P\\Gamma_{X}$ denote the ‘pure’\nsubgroup of $\\Gamma_{X}$ fixing $X$ pointwise. Let $N$ be a subgroup of $P\\Gamma_{X}$ that is\nnormal in $\\Gamma_{X}$, and for each integer $n$, let $N_{n}$ be the subgroup of $P\\Gamma_{n}$ that\nis the image of $N$ restricted to any $n$-element subset of $X$. What possible\nsequences of subgroups $N_{n}$ can arise in this way?\n(e) For which $g \\in \\Gamma_{X}$ is there an embedding $X \\to \\mathbb{R}^{2}$ for which $g$ is represented\nby a diffeomorphism $\\alpha$ which is $C^{\\infty}$? Or $C^{k}$ for fixed $k$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.43.\n\nLiterature notes:\n(1) In Part (a), an invariant is functorial if for each $X$, there is a map\n$q_{X}: \\Gamma_{X} \\to \\mathbb{R}$ so that the following holds.\nFor each inclusion $Y \\subset X$\nand each $g \\in \\Gamma_{X,Y}$ with image $g_{Y} \\in \\Gamma_{Y}$ under the natural restriction map\n$\\Gamma_{X,Y} \\to \\Gamma_{Y}$ , there is an inequality $q_{X}(g) \\geq q_{Y} (g_{Y})$. In part (b), an in-\nvariant is (semi-)continuous if the map $C_{\\alpha}(X) \\to \\mathbb{R}$ taking $Y$ to $q_{Y} ([\\alpha])$\nis (semi-)continuous for all $\\alpha$, where $[\\alpha]$ is the class of $\\alpha$ in $\\Gamma_{Y}$ .\n(2) The ray graph $\\mathcal{R}_{X}$ is the graph whose vertices are isotopy classes of rays in\n$\\mathbb{R}^{2} - X$ from infinity to a point in $X$, and whose edges are pairs that may\nbe realized disjointly. This graph is hyperbolic, connected, and infinite\ndiameter, and its flag complex is contractible. The group $\\Gamma_{X}$ acts on it\nby isometries.\nThere is a cyclic order on the set of vertices of $\\mathcal{R}_{X}$ according to\nthe cyclic order on the geodesic representatives at infinity (with respect\nto any complete hyperbolic structure on $\\mathbb{R}^{2} - X$) which may be order\ncompleted to a circle $S^{1}_{X}$ on which $\\Gamma_{X}$ acts by (orientation-preserving)\nhomeomorphisms.\nNumerical invariants of elements of $\\Gamma_{X}$ may be defined in terms of\nthese actions, including\n(a) rotation number rot $_{X}(g) \\in \\mathbb{R}/\\mathbb{Z}$ for $g \\in \\Gamma_{X}$ acting on $S^{1}_{X}$;\n(b) translation length $\\tau_{X}(g)$ for $g \\in \\Gamma_{X}$ acting on $\\mathcal{R}_{X}$; and\n(c) counting quasimorphisms $H_{X,\\sigma}(g)$ for $g \\in \\Gamma_{X}$ and $\\sigma$ a $\\Gamma_{X}$-orbit of\npath in $\\mathcal{R}_{X}$.\nSome of these invariants have a well-defined ‘name’ (rotation number,\ntranslation length) independent of $X$. Others depend on choices that are\nspecial to $X$.\nAnother way to express the fact that some invariants have a well-\ndefined name is to say that they are functorial in the sense of part (a).\nFor example, an inclusion $Y \\to X$ induces a 1-Lipschitz map $\\mathcal{R}_{X} \\to \\mathcal{R}_{Y}$\nwell defined up to bounded distance, and translation length $\\tau_{X}$ in the\ngraph $\\mathcal{R}_{X}$ is functorial.\n(3) There is a partial order $<$ due to Boyland [Boy88] on the set of all\nconjugacy classes in all (finite type) braid groups where $g < h$ if for every\nhomeomorphism $\\alpha$ of the disk fixing the boundary with a finite $\\alpha$-invariant\nset $X$ such that $\\alpha$ represents $g$ relative to $X$, there is another $\\alpha$-invariant\n\nset $Y$ such that $\\alpha$ relative to $Y$ represents $h$. We say that $g$ forces $h$, and\nthis partial order is called braid forcing. A numerical invariant $q$ from\n(finite type) braid groups to $\\mathbb{R}$ is monotone if it is monotone with respect\nto braid forcing partial order.\nAll monotone invariants arise as follows: let $G$ denote the group of all\nhomeomorphisms of the disk fixed on the boundary, and for each braid $g$,\nlet $G_{g}$ be the subset of $G$ of homeomorphisms representing $g$ relative to\nsome finite invariant set. Then $g < h$ if and only if $G_{g} \\subset G_{h}$. For any func-\ntion $f: G \\to \\mathbb{R}$, we obtain a monotone invariant by $f(g) = \\sup_{\\alpha\\in Gg} f(\\alpha)$.\nBraid forcing makes perfect sense for mapping class groups instead of\nbraid groups, and one may extend this notion in the obvious way to a par-\ntial order on all conjugacy classes in all $\\Gamma_{X}$ simultaneously. Thus, another\nnatural question is: What is the relationship between (semi-)continuity\nand Boyland’s braid forcing?\n(4) In [CC22] the individual subgroups $N_{n}$ that can arise as in part (d) are\nidentified; they satisfy the so-called ‘inertia’ condition.\n(5) Let $\\mathcal{R}_{X}$ be the ray graph and $S^{1}_{X}$ the circle on which $\\Gamma_{X}$ acts by home-\nomorphisms as in Remark 2. An additional question is: Can the (non)-\nrealization property of part (e) be related to numerical properties of $g$ that\ncan be computed from the action on $\\mathcal{R}_{X}$ or $S^{1}_{X}$?\n\nReferences cited:\n- [Boy88] Philip Boyland. An analog of Sharkovski’s theorem for twist maps. In Hamiltonian dynamical systems (Boulder, CO, 1987), volume 81 of Contemp. Math., pages 119– 133. Amer. Math. Soc., Providence, RI, 1988. doi:10.1090/conm/081/986261.\n- [CC22] Danny Calegari and Lvzhou Chen. Normal subgroups of big mapping class groups. Trans. Amer. Math. Soc. Ser. B, 9:957–976, 2022. doi:10.1090/btran/108.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2792, "problem_number": "KP-2.44", "title": "Kirby Problem 2.44", "statement": "Given an infinite-type surface $S$, which homeomorphisms $f: S \\to$\n$S$ give rise to mapping tori $M_{f}$ that admit a hyperbolic structure? For those which\ndo admit a hyperbolic metric, is $M_{f}$ homeomorphic to the interior of a compact\nhyperbolic manifold with totally geodesic boundary?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.44.\n\nLiterature notes:\n(1) Thurston’s theorem on the hyperbolization of mapping tori states that\nfor a finite-type surface $S$ and a homeomorphism $f$ of the surface $S$, the\nmapping torus\n\n$$\nM_{f} = S \\times [0, 1]/(x, 0) \\sim (f(x), 1)\n$$\n\nis hyperbolic if and only if $f$ is pseudo-Anosov.\nThis question works\ntowards an analogue of the theorem for infinite-type surfaces.\nA first approach to this problem could be to give a list of criteria\non $f$ that ensures certain properties necessary for the mapping torus to\nbe hyperbolic. For instance, to give criteria on $f$ that ensure that $M_{f}$ is\natoroidal.\n(2) It is known from work of Field–Kim–Leininger–Loving [FKLL23] that for\na strongly irreducible end-periodic homeomorphism $f$ of an infinite-type\nsurface $S$, the mapping torus $M_{f}$ is hyperbolic and homeomorphic to the\ninterior of a compact hyperbolic manifold with totally geodesic boundary.\nThis motivates the second part of the question above.\n\nReferences cited:\n- [FKLL23] Elizabeth Field, Heejoung Kim, Christopher Leininger, and Marissa Loving. Endperiodic homeomorphisms and volumes of mapping tori. J. Topol., 16(1):57–105, 2023. doi:10.1112/topo.12277.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2793, "problem_number": "KP-2.45", "title": "Kirby Problem 2.45", "statement": "Compute the end-periodic cobordism group $\\Delta^{e}_{2}$ of end-periodic\nautomorphisms (diffeomorphisms or homeomorphisms) of surfaces.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.45.\n\nLiterature notes:\n(1) The (oriented) cobordism group of $n$-dimensional manifolds, denoted $\\Delta_{n}$,\nconsists of equivalence classes of pairs $(M^{n}, f)$ where $M$ is a compact\noriented manifold and $f$ is a homeomorphism. The equivalence relation\nis given by $(M_{0}, f_{0}) \\sim (M_{1}, f_{1})$ if there is a compact cobordism between\n$M_{0}$ and $M_{1}$ with a homeomorphism extending $f_{0}$ and $f_{1}$. This group was\ncomputed for $n \\geq 4$ by Kreck [Kre84b] and for $n = 3$ by Melvin [Mel79].\nBonahon [Bon83] (see also [EE82]) showed that $\\Delta_{2} \\cong \\mathbb{Z}^{\\infty} \\oplus (\\mathbb{Z}/2)^{\\infty}$. As in\nusual cobordism theories, compactness is used to ensure that one gets a\nnontrivial group.\n(2) An end-periodic manifold $M$ is a noncompact manifold with finitely many\nends, such that each end $\\epsilon$ has a neighborhood that is half of an infinite\ncyclic cover of a compact manifold $X_{\\epsilon}$. End-periodic manifolds behave\nin many respects (topological, geometric, and analytic) as if they were\nactually compact. An end periodic automorphism $f: M \\to M$ is a home-\nomorphism or homeomorphism that is a covering translation over $X_{\\epsilon}$ on\nsome neighborhood of each end $\\epsilon$. End-periodic homeomorphisms arise\nnaturally in the study of depth-one foliations [CCF21], with the behav-\nior in the end corresponding to the limiting behavior of a non-compact\nleaf as it approaches a compact leaf. Cobordisms of end-periodic auto-\nmorphisms are defined as in the compact case, with the extension required\nto be end-periodic, and one defines the end-periodic cobordism group $\\Delta^{e}_{n}$\nas above.\n(3) This problem is meaningful in all dimensions; the calculation of the usual\n$\\Delta_{2}$ makes use of special techniques related to the Nielsen-Thurston classifi-\ncation of surface automorphisms. A similar end-periodic Nielsen-Thurston\ntheory, originating in unpublished work of Handel-Miller, is developed in\ndepth in [CCF21], and it seems reasonable to approach the computation\nof $\\Delta^{e}_{2}$ using similar tools.\n\nReferences cited:\n- [Kre84b] Matthias Kreck. Bordism of diffeomorphisms and related topics. Springer-Verlag, Berlin, 1984. With an appendix by Neal W. Stoltzfus.\n- [Mel79] Paul Melvin. Bordism of diffeomorphisms. Topology, 18(2):173–175, 1979. doi:10.1016/0040-9383(79)90034-X.\n- [Bon83] Francis Bonahon. Cobordism of automorphisms of surfaces. Ann. Sci. École Norm. Sup. (4), 16(2):237–270, 1983. URL: http://www.numdam.org/item?id=ASENS 1983 4 16 2 237 0.\n- [EE82] A. L. Edmonds and J. H. Ewing. Remarks on the cobordism group of surface diffeomorphisms. Math. Ann., 259(4):497–504, 1982. doi:10.1007/BF01466055.\n- [CCF21] John Cantwell, Lawrence Conlon, and Sergio R. Fenley. Endperiodic automorphisms of surfaces and foliations. Ergodic Theory Dynam. Systems, 41(1):66–212, 2021. doi:10.1017/etds.2019.56.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2794, "problem_number": "KP-2.46", "title": "Kirby Problem 2.46", "statement": "(a) Which coarsely boundedly generated mapping class groups of infinite-type\nsurfaces are hyperbolic?\n(b) Consider the class of surfaces with $n \\geq 2$ ends, all accumulated by genus.\nAre the mapping class groups of these surfaces quasi-isometric for different\nvalues of $n$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.46.\n\nLiterature notes:\n(1) Big mapping class groups are not finitely, countably, or compactly gener-\nated. There is a generalization of compactness called coarse boundedness,\n\ndefined and studied extensively by C. Rosendal [Ros13, Ros22] for Pol-\nish groups. It turns out that when the (pure) mapping class group of an\ninfinite-type surface admits a coarsely bounded generating set, the quasi-\nisometry type of the group is well-defined. Mann–Rafi [MR23] and T.\nHill [Hil25] give a classification of those surfaces whose mapping class\ngroups and pure mapping class groups have this property, respectively.\nThis problem asks which coarsely boundedly generated mapping class\ngroups of infinite-type surfaces are quasi-isometric to a hyperbolic metric\nspace. In other words, for which infinite-type surfaces is the mapping class\ngroup with the word metric coming from a coarsely bounded generating\nset a hyperbolic metric space?\n(2) The mapping class groups of some surfaces are not only coarsely bound-\nedly generated, they are themselves coarsely bounded.\nIn particular,\nthey are finite diameter (when equipped with a word metric coming from\na coarsely bounded generating set), and so are elementary hyperbolic.\nSchaeffer-Cohen [SC24] showed that the mapping class group of a plane\nminus a Cantor set is quasi-isometric to the loop graph, which is an infi-\nnite diameter hyperbolic graph associated to the surface. This is the only\nknown example of a mapping class group of an infinite-type surface that\nis non-elementary hyperbolic.\n(3) Problem 2.46.(b) is a special case of the broad (and possibly unanswerable)\nproblem of determining the quasi-isometry type of coarsely boundedly\ngenerated mapping class groups of infinite-type surfaces.\n\nReferences cited:\n- [Ros13] Christian Rosendal. Global and local boundedness of Polish groups. Indiana Univ. Math. J., 62(5):1621–1678, 2013. doi:10.1512/iumj.2013.62.5133.\n- [Ros22] Christian Rosendal. Coarse geometry of topological groups, volume 223 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2022.\n- [MR23] Kathryn Mann and Kasra Rafi. Large-scale geometry of big mapping class groups. Geom. Topol., 27(6):2237–2296, 2023. doi:10.2140/gt.2023.27.2237.\n- [Hil25] Thomas Hill. Large-scale geometry of pure mapping class groups of infinite-type surfaces. Proc. Amer. Math. Soc., 153(6):2667–2680, 2025. doi:10.1090/proc/17181.\n- [SC24] Anschel Schaffer-Cohen. Graphs of curves and arcs quasi-isometric to big mapping class groups. Groups Geom. Dyn., 18(2):705–735, 2024. doi:10.4171/ggd/751.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2795, "problem_number": "KP-2.47", "title": "Kirby Problem 2.47", "statement": "(a) Given a mapping class $\\psi$ of a based surface $S$, there is an induced endo-\nmorphism of the symmetric product $\\operatorname{Sym}^{i}(S)$ and hence an endofunctor\n$\\psi_{*}$ of the partially wrapped Fukaya category $\\mathcal{F}(S)$ as constructed, say, by\nAuroux [Aur10]. Compute the categorical entropy of $\\psi_{*}$, in the sense\nof [DHKK14], for $\\psi$ pseudo-Anosov.\n(b) Given a fibered knot $K$ with pseudo-Anosov monodromy, what is\n\n$$\n\\limsup_{n\\to\\infty}\\left(\\dim \\widehat{\\mathrm{HFK}}(\\Sigma^{n}(K),\\widetilde{K};i)\\right)^{1/n},\n$$\n\nwhere $\\widetilde{K}$ is the branch locus in the $n$-fold cyclic branched cover $\\Sigma^{n}(K)$,\nand $i$ denotes the Alexander grading (in $\\mathbb{Z}$) with respect to the fiber surface.\nHere, $\\widehat{\\mathrm{HFK}}$ denotes the knot Floer homology of $K$ [OS04c, Ras03].\n(c) With notation as in the previous question, what is\n\n$$\n\\limsup_{n\\to\\infty}\\left(\\dim \\widehat{\\mathrm{HF}}(\\Sigma^{n}(K))\\right)^{1/n}\n$$\n\nfor the Heegaard Floer invariant $\\widehat{\\mathrm{HF}}$ [OS04e] of the closed 3-manifold\n$\\Sigma^{n}(K)$?\n(d) Given a closed surface $S$ and a pseudo-Anosov automorphism $\\psi$, let $M_{\\psi}$\ndenote the mapping torus of $\\psi$, and $\\widehat{\\mathrm{HF}}(M_{\\psi};i)$ the summand of\n$\\widehat{\\mathrm{HF}}(M_{\\psi})$\nspanned by the spin $^{c}$-structures $\\mathfrak{s}$ with $\\langle c_{1}(\\mathfrak{s}), [F]\\rangle = i$. What is\n\n$$\n\\limsup_{n\\to\\infty}\\left(\\dim \\widehat{\\mathrm{HF}}(M_{\\psi^{n}},i)\\right)^{1/n}?\n$$\n\nWhat about\n\n$$\n\\limsup_{n\\to\\infty}\\left(\\dim_{\\mathbb{F}_{2}}\\mathrm{HF}^{+}(M_{\\psi^{n}},i)\\right)^{1/n}\n$$\n\nfor $i \\neq 0$ (or for twisted $\\mathrm{HF}^{+}$ for $i = 0$)?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.47.\n\nLiterature notes:\n(1) In (a), the categorical entropy is equal to\n\n$$\n\\limsup_{n\\to\\infty}\\left(\\dim H_{*}(\\widehat{\\mathrm{CFDA}}(\\psi^{n},i-g))\\right)^{1/n},\n$$\n\nwhere $\\widehat{\\mathrm{CFDA}}(\\psi)$ is the bordered Floer bimodule [LOT15] and $g$ is the\ngenus of $S$.\nIn the case $i = 1$, this entropy is equal to the dilatation\n$\\lambda(\\psi)$ [LOT13, DHKK14]. For (d), see [OS04e] for the definition of\n$\\mathrm{HF}^{+}$, which is a Floer homology relative to a divisor; $\\widehat{\\mathrm{HF}}$ is Floer homology\nin the complement of that divisor.\n(2) Properties of bordered Floer homology imply that the categorical en-\ntropy is an upper bound for the answers to the remaining questions\n(perhaps up to a constant factor) [LOT15].\nAlso, the growth rate of\n$\\bigoplus_i \\widehat{\\mathrm{HFK}}(\\Sigma^{n}(K),\\widetilde{K},i)$ is an upper bound for the growth rate of\n$\\widehat{\\mathrm{HF}}(\\Sigma^{n}(K))$.\n(3) The growth rate of $|H_{1}(\\Sigma^{n}(K))|$ is a lower bound for the growth rate\nof $\\widehat{\\mathrm{HF}}(\\Sigma^{n}(K))$.\n(4) Some results about the growth rate of Heegaard Floer homology of branched\ncovers were proved in [HM18], though these results are about linear, not\nexponential, growth.\n(5) In (d), the reason for the restriction to $\\mathrm{HF} ^{+}$ with $i \\neq 0$ is that $\\mathrm{HF} ^{+}$ is\nfinitely generated over $\\mathbb{F}[\\operatorname{U}]$ but not over $\\mathbb{F}$ for $i = 0$.\n(6) Some further observations on these questions can be found in [Cor18].\n(7) One can ask similar questions about the growth of knot invariants like knot\nFloer homology, Khovanov homology, or symplectic Khovanov homology\nwhen one inserts a high power of a braid.\n\nReferences cited:\n- [Aur10] Denis Auroux. Fukaya categories of symmetric products and bordered HeegaardFloer homology. J. Gökova Geom. Topol. GGT, 4:1–54, 2010.\n- [DHKK14] G. Dimitrov, F. Haiden, L. Katzarkov, and M. Kontsevich. Dynamical systems and categories. In The influence of Solomon Lefschetz in geometry and topology, volume 621 of Contemp. Math., pages 133–170. Amer. Math. Soc., Providence, RI, 2014. doi:10.1090/conm/621/12421.\n- [OS04c] Peter Ozsváth and Zoltán Szabó. Holomorphic disks and knot invariants. Adv. Math., 186(1):58–116, 2004. doi:10.1016/j.aim.2003.05.001.\n- [Ras03] Jacob Rasmussen. Floer homology and knot complements. PhD thesis, Harvard University, Cambridge, MA, 2003. https://arxiv.org/abs/math/0306378.\n- [OS04e] Peter Ozsváth and Zoltán Szabó. Holomorphic disks and topological invariants for closed three-manifolds. Ann. of Math. (2), 159(3):1027–1158, 2004. doi:10.4007/annals.2004.159.1027.\n- [LOT15] Robert Lipshitz, Peter Ozsváth, and Dylan Thurston. Bimodules in bordered Heegaard Floer homology. Geom. Topol., 19(2):525–724, 2015. doi:10.2140/gt.2015.19.525.\n- [LOT13] Robert Lipshitz, Peter Ozsváth, and Dylan Thurston. A faithful linear-categorical action of the mapping class group of a surface with boundary. J. Eur. Math. Soc. (JEMS), 15(4):1279–1307, 2013. doi:10.4171/JEMS/392.\n- [HM18] Matthew Hedden and Thomas E. Mark. Floer homology and fractional Dehn twists. Adv. Math., 324:1–39, 2018. doi:10.1016/j.aim.2017.11.008.\n- [Cor18] James Cornish. Growth Rate of 3-Manifold Homologies under Branched Covers. ProQuest LLC, Ann Arbor, MI, 2018. Thesis (Ph.D.)–Columbia University. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt=info: ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqm\\&rft dat=xri:pqdiss:10791025.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2796, "problem_number": "KP-2.48", "title": "Kirby Problem 2.48", "statement": "The mapping class group of a closed, orientable, genus $g$ sur-\nface $S$ acts by symplectomorphisms on the symmetric product $\\operatorname{Sym}^{g}(S)$. In partic-\nular, it acts smoothly on $\\operatorname{Sym}^{g}(S)$.\n(a) What is the kernel of the map from the mapping class group of $S$ to the\nsmooth mapping class group of $\\operatorname{Sym}^{g}(S)$? What about in the based case?\n(b) Given disjoint simple closed curves $\\alpha_{1},..., \\alpha_{g}$ in $S$, there is a correspond-\ning torus $T_{\\alpha}$ in $\\operatorname{Sym}^{g}(S)$: the image of $\\alpha_{1} \\times \\cdots \\times \\alpha_{g} \\subset S^{g}$. These are\nthe tori that appear in Heegaard Floer theory. When do $\\alpha_{1},..., \\alpha_{g}$ and\n$\\beta_{1},..., \\beta_{g}$ give smoothly isotopic tori? When are they Lagrangian isotopic\nfor an appropriate choice of symplectic form?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.48.\n\nLiterature notes:\n(1) The question relates to whether Heegaard Floer homology is truly a sym-\nplectic invariant, or should be determined by the smooth isotopy classes\nof the Heegaard tori. (Examples of the difference between smooth and\nsymplectic topology in other settings have received substantial interest,\nthough many such examples are now known; [Sei97] is particularly rele-\nvant.)\n(2) The fact that the mapping class group acts symplectically on $\\operatorname{Sym}^{g}(S)$,\nfor appropriate symplectic forms, was shown by Perutz [Per08a]. (The\nobvious map, where a diffeomorphism $\\varphi$ of the surface $S$ sends a point\n$\\{x_{1},..., x_{g}\\} \\in \\operatorname{Sym}^{g}(S)$ to $\\{\\varphi(x_{1}),..., \\varphi(x_{g})\\}$ gives a homeomorphism but\nis not typically smooth; an exception is if $\\varphi$ is holomorphic with respect\nto some complex structure on $S$ and the smooth structure on $\\operatorname{Sym}^{g}(S)$ is\ninduced from this complex structure, in which case the obvious induced\nmap of $\\operatorname{Sym}^{g}(S)$ is also holomorphic.) Clarkson used Heegaard Floer ho-\nmology to show that the map from the based mapping class group to\nthe group of symplectomorphisms of $\\operatorname{Sym}^{g}(S\\setminus\\{z\\})$ modulo the Hamilton-\nian diffeomorphisms is injective [Cla17].\n(This uses the fact that, for\nappropriate choices of symplectic forms [Per08b], the Heegaard tori are\nexact [Hen12, HLL22a], and so Heegaard Floer homology agrees with\nthe usual Lagrangian intersection Floer homology.)\n(3) The fundamental group of $\\operatorname{Sym}^{g}(S)$ (for $g > 1$) is isomorphic to $H_{1}(\\Sigma)$.\nIn particular, the Torelli group of $S$ acts trivially on $\\pi_{1}(\\operatorname{Sym}^{g}(S))$. (It also\nacts trivially on the homology of $\\operatorname{Sym}^{g}(S)$.) Also, by Perutz’s work, if $\\alpha'_{1}$\nis obtained from $\\alpha_{1}$ by a handleslide, then $\\alpha_{1} \\times \\cdots \\times \\alpha_{g}$ and\n$\\alpha'_{1} \\times \\cdots \\times \\alpha_{g}$\nare smoothly isotopic (and, in fact, Hamiltonian isotopic for appropriate\nsymplectic forms) [Per08b].\n(4) In the case $g = 2$, one might be able to use the Abel-Jacobi map (which\nin this case presents $\\operatorname{Sym}^{2}(S)$ as a blow-up of the Jacobian torus) to show\nthat the Torelli group acts trivially, since it acts trivially on the Jacobian\ntorus $H^{1}(S; \\mathbb{R})/H^{1}(\\Sigma; \\mathbb{Z})$. Perhaps this suggests that the Torelli group or\nthe Johnson kernel acts trivially in the higher-genus case as well.\n\nReferences cited:\n- [Sei97] Paul Seidel. Floer homology and the symplectic isotopy problem. PhD thesis, University of Oxford, 1997.\n- [Per08a] Tim Perutz. Lagrangian matching invariants for fibred four-manifolds. II. Geom. Topol., 12(3):1461–1542, 2008. doi:10.2140/gt.2008.12.1461.\n- [Cla17] Corrin Clarkson. Three-manifold mutations detected by Heegaard Floer homology. Algebr. Geom. Topol., 17(1):1–16, 2017. doi:10.2140/agt.2017.17.1.\n- [Per08b] Timothy Perutz. Hamiltonian handleslides for Heegaard Floer homology. In Proceedings of Gökova Geometry-Topology Conference 2007, pages 15–35. Gökova Geometry/Topology Conference (GGT), Gökova, 2008.\n- [Hen12] Kristen Hendricks. A rank inequality for the knot Floer homology of double branched covers. Algebr. Geom. Topol., 12(4):2127–2178, 2012. doi:10.2140/agt.2012.12.2127.\n- [HLL22a] Kristen Hendricks, Tye Lidman, and Robert Lipshitz. Rank inequalities for the Heegaard Floer homology of branched covers. Doc. Math., 27:581–612, 2022.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2797, "problem_number": "KP-2.49", "title": "Kirby Problem 2.49", "statement": "(AMU conjecture). Let $S$ be a surface with negative Euler char-\nacteristic. If $\\varphi \\in \\operatorname{Mod}(S)$ acts by a pseudo-Anosov on some subsurface (including\n$S$ itself), show that there exists $p_{0}(\\varphi)$ such that the Witten–Reshetikhin–Turaev\nrepresentation $\\rho_{p}(\\varphi)$ has infinite order for $p \\geq p_{0}(\\varphi)$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.49.\n\nLiterature notes:\n(1) Given a compact Lie group $G$, the Witten–Reshetikhin–Turaev $G$-quantum\nrepresentations, introduced by Witten [Wit89] and rigorously defined\nby Reshetikhin and Turaev [RT91], are families of representations of\n$\\operatorname{Mod}(S)$.\nThe most studied case is when $G = \\operatorname{SU}(2)$ or $G = \\operatorname{SO}(3)$,\nin which case these representations are indexed by an integer $p \\geq 3$. More\nprecisely we have:\n\n$$\n\\rho_{p}: \\operatorname{Mod}(S) \\to PGL(d_{p}(S), \\mathbb{C}),\n$$\n\nwhere $d_{p}(S)$ is given by the famous Verlinde formula and is a polynomial\nin $p$ of degree $3g - 3$ (where $g$ is the genus of $S$). The representation $\\rho_{p}$\nsends a Dehn twist to a torsion element, and when $S$ has negative Euler\ncharacteristic, $\\rho_{p}(\\operatorname{Mod}(S))$ is infinite for $p$ big enough. Moreover, Ander-\nsen (see [And06]) proved that if $\\varphi \\in \\operatorname{Mod}(S)$ is non-central, then $\\rho_{p}(\\varphi)$\nis nontrivial for $p$ sufficiently large. This property is called asymptotic\nfaithfulness.\n(2) Geometric properties of these representations are quite mysterious. For\ninstance, it is not known if the Nielsen-Thurston classification can be de-\ntected by quantum representations. In [AMU06], Andersen, Masbaum,\nand Ueno studied the case of the four-holed sphere. They proved that if\n$\\varphi$ is pseudo-Anosov in the mapping class group of the four-holed sphere,\nthen $\\rho_{p}(\\varphi)$ has infinite order for $p$ big enough. In the same paper, they\nstated the above problem as a conjecture; it is now known as the AMU\nconjecture.\n(3) If $\\varphi \\in \\operatorname{Mod}(S)$ does not act as a pseudo-Anosov on a subsurface, then up\nto some power, it is the product of powers of commuting Dehn twists, and\nso $\\rho_{p}(\\varphi)$ has finite order for all $p$. Also, as the quantum representations\nenjoy some nice “splitting” properties regarding subsurfaces, it is enough\nto prove this conjecture for pseudo-Anosov elements.\n(4) The AMU conjecture is only known for two surfaces: the four-holed sphere\n(proved in [AMU06]) and the one-holed torus (see [San12]). In general,\nfinding pseudo-Anosov elements satisfying the conjecture is already a dif-\nficult task. In [EJ16], it was proved that a certain class of “homological”\npseudo-Anosov elements on holed spheres satisfy the AMU conjecture.\nExamples of point pushing mapping classes satisfying the AMU conjec-\nture have been studied in [KS16] and [MS21]. See also [DK22] where\nexamples were found using exponential growth of certain quantum invari-\nants; this paper shows that the Volume Conjecture (see Problem 1.27)\nimplies the AMU conjecture.\n\nReferences cited:\n- [Wit89] Edward Witten. Quantum field theory and the Jones polynomial. Comm. Math. Phys., 121(3):351–399, 1989. http://projecteuclid.org/euclid.cmp/1104178138.\n- [RT91] N. Reshetikhin and V. G. Turaev. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math., 103(3):547–597, 1991. doi:10.1007/BF01239527.\n- [And06] Jørgen Ellegaard Andersen. Asymptotic faithfulness of the quantum $\\mathrm{SU}(n)$ representations of the mapping class groups. Ann. of Math. (2), 163(1):347–368, 2006. doi:10.4007/annals.2006.163.347.\n- [AMU06] Jørgen Ellegaard Andersen, Gregor Masbaum, and Kenji Ueno. Topological quantum field theory and the Nielsen-Thurston classification of M$(0,4)$. Math. Proc. Cambridge Philos. Soc., 141(3):477–488, 2006. doi:10.1017/S0305004106009698.\n- [San12] Ramanujan Santharoubane. Limits of the quantum SO(3) representations for the one-holed torus. J. Knot Theory Ramifications, 21(11):1250109, 13, 2012. doi: 10.1142/S021821651250109X.\n- [EJ16] Jens Kristian Egsgaard and Søren Fuglede Jørgensen. The homological content of the Jones representations at $q=-1$. J. Knot Theory Ramifications, 25(11):1650062, 25, 2016. doi:10.1142/S0218216516500620.\n- [KS16] Thomas Koberda and Ramanujan Santharoubane. Quotients of surface groups and homology of finite covers via quantum representations. Invent. Math., 206(2):269– 292, 2016. doi:10.1007/s00222-016-0652-x.\n- [MS21] Julien Marché and Ramanujan Santharoubane. Asymptotics of quantum representations of surface groups. Ann. Sci. Éc. Norm. Supér. (4), 54(5):1275–1296, 2021. doi:10.24033/asens.2481.\n- [DK22] Renaud Detcherry and Efstratia Kalfagianni. Cosets of monodromies and quantum representations. Indiana Univ. Math. J., 71(3):1101–1129, 2022.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2798, "problem_number": "KP-2.50", "title": "Kirby Problem 2.50", "statement": "(Volume conjecture for surface diffeomorphisms). Let $S$ be a\nclosed oriented surface, let $q=e^{2\\pi i/n}$ be a root of unity, and let $\\mathcal{K}^{q}(S)$ be the Kauff-\nman bracket skein algebra. Let $\\phi: S \\to S$ be a pseudo-Anosov diffeomorphism with\nmapping torus $M_{\\phi}$, and let $r: \\pi_{1}(S) \\to \\operatorname{SL}_{2}(\\mathbb{C})$ be a smooth point in the charac-\nter variety that is $\\phi$-invariant. Finally, let $L$ be the intertwiner realizing an iso-\nmorphism between the representations of $\\mathcal{K}^{q}(S)$ corresponding to $[r]$ and $[r \\circ \\varphi_{*}]$,\nnormalized so that $\\det(L) = 1$. Show that\n\n$$\n\\lim_{n\\to\\infty}\\frac{1}{n}\\log|\\operatorname{Trace} L|=\\frac{1}{4\\pi}\\operatorname{vol}_{\\mathrm{hyp}} M_{\\phi}.\n$$", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.50.\n\nLiterature notes:\n(1) For a closed oriented surface $S$ and a root of unity $q$, results of [BW17a,\nFKBL19, GJS25, KK22, FKBL25] more or less establish a one-to-\none correspondence between irreducible representations of the Kauffman\nbracket skein algebra $\\mathcal{K}^{q}(S)$ and smooth points in the $\\operatorname{SL}_{2}(\\mathbb{C})$ character\nvariety of $S$. Thus in the above setup, the representations constructed\nfrom $[r]$ and $[r \\circ \\varphi_{*}]$ must be isomorphic by some map $L$ called the in-\ntertwiner. After normalizing $L$ so that $\\det(L) = 1, |\\operatorname{Trace} L|$ is dependent\nonly on the choice of $\\varphi, r, q$, and the puncture invariants.\n(2) This conjecture appears in [BWY21] and is a toy version of the Kashaev\nVolume Conjecture [Kas97] (see Problem 1.27), as revisited by Baseilhac-\nBenedetti [BB04]. In preprints, the conjecture has been claimed for some\nexamples [BWY22, Pan24, Wan25] and for torus knots. There also\nexist versions of this conjecture for punctured surfaces, involving puncture\ninvariants on the quantum side and 3-manifolds obtained from $M_{\\phi}$ by\nDehn filling on the hyperbolic side [PW24].\n\nReferences cited:\n- [BW17a] Francis Bonahon and Helen Wong. Representations of the Kauffman bracket skein algebra II: Punctured surfaces. Algebr. Geom. Topol., 17(6):3399–3434, 2017. doi: 10.2140/agt.2017.17.3399.\n- [FKBL19] Charles Frohman, Joanna Kania-Bartoszynska, and Thang Lê. Unicity for representations of the Kauffman bracket skein algebra. Invent. Math., 215(2):609–650, 2019. doi:10.1007/s00222-018-0833-x.\n- [GJS25] Iordan Ganev, David Jordan, and Pavel Safronov. The quantum Frobenius for character varieties and multiplicative quiver varieties. J. Eur. Math. Soc. (JEMS), 27(7):3023–3084, 2025. doi:10.4171/jems/1427.\n- [KK22] Hiroaki Karuo and Julien Korinman. Azumaya loci of skein algebras, 2022. arXiv: 2211.13700.\n- [FKBL25] Charles D. Frohman, Joanna Kania-Bartoszynska, and Thang T. Q. Lê. Sliced skein algebras and geometric Kauffman bracket. Adv. Math., 463:Paper No. 110118, 65, 2025. doi:10.1016/j.aim.2025.110118.\n- [BWY21] Francis Bonahon, Helen Wong, and Tian Yang. Asymptotics of quantum invariants of surface diffeomorphisms i: conjecture and algebraic computations, 2021. arXiv: 2112.12852.\n- [Kas97] Rinat M Kashaev. The hyperbolic volume of knots from the quantum dilogarithm. Letters in mathematical physics, 39(3):269–275, 1997.\n- [BB04] Stéphane Baseilhac and Riccardo Benedetti. Quantum hyperbolic invariants of 3-manifolds with P$\\mathrm{SL}(2,\\mathbb{C})$-characters. Topology, 43(6):1373–1423, 2004. doi:10.1016/j.top.2004.02.001.\n- [BWY22] Francis Bonahon, Helen Wong, and Tian Yang. Asymptotics of quantum invariants of surface diffeomorphisms II: The figure-eight knot complement, 2022. arXiv:2203.05730.\n- [Pan24] Tushar Pandey. The Bonahon-Wong-Yang volume conjecture for the four-puncture sphere, 2024. arXiv:2311.13151.\n- [Wan25] Zhihao Wang. Kauffman bracket intertwiners and the volume conjecture. Algebr. Geom. Topol., 25(4):2143–2177, 2025. doi:10.2140/agt.2025.25.2143.\n- [PW24] Tushar Pandey and Ka Ho Wong. Generalized Bonahon-Wong-Yang volume conjecture of quantum invariants of surface diffeomorphisms i: the figure eight knot complement, 2024. arXiv:2402.04483.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2799, "problem_number": "KP-3.1", "title": "Kirby Problem 3.1", "statement": "Classify the smallest volume hyperbolic 3-manifolds of various types. In particular:\n\n(a) Determine the nonorientable closed hyperbolic 3-manifolds of least volume.\n\n(b) Determine the n-cusped hyperbolic 3-manifolds of least volume for each $n\\geq 3$.\n\n(c) Determine the smallest volume hyperbolic 3-manifolds with n orientable cusps for each n.\n\n(d) Determine the n-cusped orientable hyperbolic 3-manifolds of least volume for each n.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.1.\n\nLiterature notes:\n(1) Parts (a)--(c) appeared in [Kir97, Problem 3.60].\n\n(2) There has been a lot of progress in determining the hyperbolic 3-manifolds of low volume [GHM+21]. In particular, the orientable hyperbolic manifold of smallest volume was shown to be the Weeks manifold by Gabai, Meyerhoff, and Milley [GMM09, Mil09]; its volume is approximately 0.9427. Extensive evidence from SnapPy indicates that the hyperbolic 3-manifold with smallest volume is orientable (hence is the Weeks manifold). Closed nonorientable 3-manifolds appear to have larger volumes, and so the currently available techniques do not readily apply to them, which suggests why part (a) remains open.\n\n(3) The two smallest orientable hyperbolic 3-orbifolds were identified by Gehring, Marshall, and Martin [GM09], [MM12], as was the smallest volume nonorientable 3-orbifold.\n\n(4) The n-cusped manifolds with smallest volume are known for n = 1 and 2; in both cases, these manifolds are nonorientable. For n = 1, Adams [Ada87] showed that the Gieseking manifold is the unique smallest manifold, with volume $v_3$ $\\approx$ 1.0149, the volume of a regular ideal tetrahedron. For n = 2, Adams [Ada88] proved that the least volume is $2v_3$, and showed that there is a unique 2-cusped manifold with this volume.\n\n(5) The smallest volume orientable hyperbolic 3-manifolds with n cusps are known for n = 1, 2 and 4. For n = 1, this is due to Cao and Meyerhoff [CM01], who showed that there are two manifolds with smallest volume $2v_3$, the figure-eight knot complement and the figure-eight knot sister. For n = 2, Agol [Ago10] proved that there are two orientable 2-cusped manifolds with smallest volume, the Whitehead link complement and the $(-2,3,8)$ pretzel link complement, both of which have volume $v_8$, which is the volume of a regular ideal octahedron, approximately 3.6638. For n = 4, Yoshida [Yos13] proved that the link $8^4_2$ is the unique orientable 4-cusped manifold with smallest volume $2v_8$ $\\approx$ 7.3276. For n = 3, the minimal volume is conjectured to be realized by the 3-chain link complement, which has volume approximately 5.33 (see [Zha23] for some results in this direction).\n\n(6) Agol [Ago10] has conjectured that for $n\\leq 10$, the minimal volume of an orientable n-cusped hyperbolic manifold is realized by the minimally twisted n-chain link complement. However, for larger values of $n$ ($11\\leq n\\leq 25$ and $n\\geq 60$), it has been shown [KPR12] that these manifolds do not have minimal volume, as the $(n-1)$-fold cyclic cover over one component of the Whitehead link has smaller volume.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [GHM+21] David Gabai, Robert Haraway, Robert Meyerhoff, Nathaniel Thurston, and Andrew Yarmola. Hyperbolic 3-manifolds of low cusp volume, 2021. arXiv:2109.14570.\n- [GMM09] David Gabai, Robert Meyerhoff, and Peter Milley. Minimum volume cusped hyperbolic three-manifolds. J. Amer. Math. Soc., 22(4):1157–1215, 2009. doi:10.1090/S0894-0347-09-00639-0.\n- [Mil09] Peter Milley. Minimum volume hyperbolic 3-manifolds. J. Topol., 2(1):181–192, 2009. doi:10.1112/jtopol/jtp006.\n- [GM09] Frederick W. Gehring and Gaven J. Martin. Minimal co-volume hyperbolic lattices. I. The spherical points of a Kleinian group. Ann. of Math. (2), 170(1):123–161, 2009. doi:10.4007/annals.2009.170.123.\n- [MM12] T. H. Marshall and G. J. Martin. Minimal co-volume hyperbolic lattices, II: Simple torsion in a Kleinian group. Ann. of Math. (2), 176(1):261–301, 2012. doi:10.4007/annals.2012.176.1.4.\n- [Ada87] Colin C. Adams. The noncompact hyperbolic 3-manifold of minimal volume. Proc. Amer. Math. Soc., 100(4):601–606, 1987. doi:10.2307/2046691.\n- [Ada88] Colin C. Adams. Volumes of N-cusped hyperbolic 3-manifolds. J. London Math. Soc. (2), 38(3):555–565, 1988. doi:10.1112/jlms/s2-38.3.555.\n- [CM01] Chun Cao and G. Robert Meyerhoff. The orientable cusped hyperbolic 3-manifolds of minimum volume. Invent. Math., 146(3):451–478, 2001. doi:10.1007/s002220100167.\n- [Ago10] Ian Agol. The minimal volume orientable hyperbolic 2-cusped 3-manifolds. Proc. Amer. Math. Soc., 138(10):3723–3732, 2010. doi: 10.1090/S0002-9939-10-10364-5.\n- [Yos13] Ken’ichi Yoshida. The minimal volume orientable hyperbolic 3-manifold with 4 cusps. Pacific J. Math., 266(2):457–476, 2013. doi:10.2140/pjm.2013.266.457.\n- [Zha23] Yue Zhang. Guts and the minimal volume orientable hyperbolic 3-manifold with 3 cusps, 2023. arXiv:2304.09950.\n- [KPR12] James Kaiser, Jessica S. Purcell, and Clint Rollins. Volumes of chain links. J. Knot Theory Ramifications, 21(11):1250115, 17, 2012. doi:10.1142/S0218216512501155.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2800, "problem_number": "KP-3.2", "title": "Kirby Problem 3.2", "statement": "Show that the volumes of hyperbolic 3-manifolds are not all rationally related.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.2.\n\nLiterature notes:\n(1) This was proposed by Thurston in [Thu82]. One could interpret \"rationally related\" to mean the volumes are all rational multiples of each other, or the weaker claim that they span a finite-dimensional $\\mathbb{Q}$-vector space. (The expectation is that neither holds.)\n\n(2) By Borel [Bor81] there is a real number $v_k$ such that the volume of an arithmetic hyperbolic 3-manifold with invariant trace-field $k$ is an integral multiple of $v_k$. More generally, for any number field $k$, there are real numbers $v_1,\\ldots,v_n$ such that for any (possibly non-arithmetic) hyperbolic 3-manifold with invariant trace-field $k$, its volume is an integral linear combination of $v_1,\\ldots,v_n$.\n\n(3) A 250+ year old problem asks whether Catalan's constant $G$ is rational. Agol [Ago10] proved that the minimal volume 2-cusped hyperbolic\n\n3-manifolds has volume equal to $4G$, the volume of the regular ideal octahedron.\n\n(4) Recently, F. Calegari, Dimitrov, and Tang [CDT24] established the irrationality of $L(2,\\chi_{-3})$, which is closely related to the volume of a regular ideal 3-simplex, and hence the volume of the figure-eight knot complement.\n\n(5) There is an analogous open question concerning the rationality of the Chern--Simons invariant. One can find an extensive discussion of this and related questions about the Chern--Simons invariant in [Kir97, Problems 3.62 and 3.63]; see also Problem 3.64.\n\n(6) The volume of an ideal tetrahedron with interior angles $\\alpha$, $\\beta$, and $\\gamma$ is $\\Lambda(\\alpha)+\\Lambda(\\beta)+\\Lambda(\\gamma)$, where $\\Lambda$ is the Lobachevsky function. Milnor [Mil82] has a conjecture that specifies precisely the rational relations between values of $\\Lambda$.\n\n(7) For a more in-depth discussion of this and related problems, see [GMM10, Problem 10.34] and [Neu98].\n\nReferences cited:\n- [Thu82] William P. Thurston. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.), 6(3):357–381, 1982. doi:10.1090/S0273-0979-1982-15003-0.\n- [Bor81] A. Borel. Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8(1):1–33, 1981. URL: http://www.numdam.org/item?id=ASNSP 1981 4 8 1 1 0.\n- [Ago10] Ian Agol. The minimal volume orientable hyperbolic 2-cusped 3-manifolds. Proc. Amer. Math. Soc., 138(10):3723–3732, 2010. doi: 10.1090/S0002-9939-10-10364-5.\n- [CDT24] Frank Calegari, Vesselin Dimitrov, and Yunqing Tang. The linear independence of 1, $\\zeta(2)$, and $L(2,\\chi_{-3})$, 2024. arXiv:2408.15403.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Mil82] John Milnor. Hyperbolic geometry: the first 150 years. Bull. Amer. Math. Soc. (N.S.), 6(1):9–24, 1982. doi:10.1090/S0273-0979-1982-14958-8.\n- [GMM10] David Gabai, Robert Meyerhoff, and Peter Milley. Mom technology and hyperbolic 3-manifolds. In In the tradition of Ahlfors-Bers. V, volume 510 of Contemp. Math., pages 84–107. Amer. Math. Soc., Providence, RI, 2010.\n- [Neu98] Walter D. Neumann. Hilbert’s 3rd problem and invariants of 3-manifolds. In The Epstein birthday schrift, volume 1 of Geom. Topol. Monogr., pages 383–411. Geom. Topol. Publ., Coventry, 1998. doi:10.2140/gtm.1998.1.383.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2801, "problem_number": "KP-3.3", "title": "Kirby Problem 3.3", "statement": "Does every cusped hyperbolic 3-manifold have a geometric ideal triangulation?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.3.\n\nLiterature notes:\n(1) A geometric ideal tetrahedron is the convex hull of any four points on the 3-sphere at infinity of $\\mathbb{H}^3$ that do not all lie on a circle. Topologically, it is a tetrahedron with its vertices removed. A geometric ideal triangulation of a hyperbolic 3-manifold M is an expression of M as a union of geometric ideal tetrahedra glued along their faces. Geometric ideal triangulations are useful, for example when studying hyperbolic Dehn surgery [BP92]. The question is whether they always exist. This was probably first asked in print by Yoshida [Yos96]; see also [WYY96], [PP00].\n\n(2) Epstein and Penner [EP88] showed that any cusped hyperbolic 3-manifold can be constructed by gluing geometric ideal polyhedra along their faces. Furthermore, it is known that any geometric ideal polyhedron can be subdivided into geometric ideal tetrahedra. However, it is not known if this can be done compatibly across the manifold.\n\n(3) Luo, Schleimer, and Tillmann [LST08] showed that the question has a positive answer virtually, in the sense that any cusped hyperbolic 3-manifold is finitely covered by a manifold admitting a geometric ideal triangulation. On the other hand, Choi [Cho04] gave an example of an incomplete hyperbolic structure on the figure-eight knot complement that does not admit a geometric ideal triangulation.\n\nReferences cited:\n- [BP92] Riccardo Benedetti and Carlo Petronio. Lectures on hyperbolic geometry. Universitext. Springer-Verlag, Berlin, 1992. doi:10.1007/978-3-642-58158-8.\n- [Yos96] Han Yoshida. Ideal tetrahedral decompositions of hyperbolic 3-manifolds. Osaka J. Math., 33(1):37–46, 1996. http://projecteuclid.org/euclid.ojm/1200786689.\n- [WYY96] Masaaki Wada, Yasushi Yamashita, and Han Yoshida. An inequality for polyhedra and ideal triangulations of cusped hyperbolic 3-manifolds. Proc. Amer. Math. Soc., 124(12):3905–3911, 1996. doi:10.1090/S0002-9939-96-03563-0.\n- [PP00] Carlo Petronio and Joan Porti. Negatively oriented ideal triangulations and a proof of Thurston’s hyperbolic Dehn filling theorem. Expo. Math., 18(1):1–35, 2000.\n- [EP88] D. B. A. Epstein and R. C. Penner. Euclidean decompositions of noncompact hyperbolic manifolds. J. Differential Geom., 27(1):67–80, 1988. http://projecteuclid.org/euclid.jdg/1214441650.\n- [LST08] Feng Luo, Saul Schleimer, and Stephan Tillmann. Geodesic ideal triangulations exist virtually. Proc. Amer. Math. Soc., 136(7):2625–2630, 2008. doi:10.1090/S0002-9939-08-09387-8.\n- [Cho04] Young-Eun Choi. Positively oriented ideal triangulations on hyperbolic threemanifolds. Topology, 43(6):1345–1371, 2004. doi:10.1016/j.top.2004.02.002.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2802, "problem_number": "KP-3.4", "title": "Kirby Problem 3.4", "statement": "(Chen--Yang Volume Conjecture). (a) Prove that, for any hyperbolic 3-manifold $M$,\n\n$$\n\\lim_{\\substack{r\\to\\infty\\\\ r\\ \\mathrm{odd}}}\\frac{1}{r}\\log\\bigl(TV(M;e^{2\\pi i/r})\\bigr)\n=\\frac{1}{2\\pi}\\operatorname{Vol}(M).\n$$\n\n(b) Prove that, for any closed, oriented, hyperbolic 3-manifold $M$,\n\n$$\n\\lim_{\\substack{r\\to\\infty\\\\ r\\ \\mathrm{odd}}}\\frac{1}{r}\\log\\bigl(WRT(M;e^{2\\pi i/r})\\bigr)\n=\\frac{1}{4\\pi}\\bigl(\\operatorname{Vol}(M)-i\\operatorname{CS}(M)\\bigr).\n$$", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.4.\n\nLiterature notes:\n(1) Given a root of unity $q$ so that $q^2$ is a primitive root of unity of odd order, $TV(M;q)$ denotes the corresponding Turaev--Viro invariant [TV92]. Similarly, $WRT(M;q)$ denotes the Witten--Reshetikhin--Turaev invariant [RT91]. For a hyperbolic 3-manifold M, $\\operatorname{Vol}(M)$ denotes the volume of M and $\\operatorname{CS}(M)$ denotes the Chern--Simons invariant of M.\n\n(2) Both forms of the conjecture were formulated by Q. Chen and T. Yang, in [CY18].\n\n(3) In Part (a), M can be closed, cusped, or have totally geodesic boundary.\n\n(4) Part (a) has been verified in many cases [DKM25], including for large families of 3-manifolds with cusps. A related version concerns the Witten-- Reshetikhin--Turaev invariant $WRT(M;q)$ where M is a 3-manifold and $q$ is a primitive root of unity of odd order $r$.\n\n(5) For a generalization of Part (b) to relative WRT invariants, see [WY23].\n\nReferences cited:\n- [TV92] V. G. Turaev and O. Ya. Viro. State sum invariants of 3-manifolds and quantum 6j-symbols. Topology, 31(4):865–902, 1992. doi:10.1016/0040-9383(92)90015-A.\n- [RT91] N. Reshetikhin and V. G. Turaev. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math., 103(3):547–597, 1991. doi:10.1007/BF01239527.\n- [CY18] Qingtao Chen and Tian Yang. Volume conjectures for the Reshetikhin-Turaev and the Turaev-Viro invariants. Quantum Topol., 9(3):419–460, 2018. doi:10.4171/QT/111.\n- [DKM25] Renaud Detcherry, Efstratia Kalfagianni, and Shashini Marasinghe. Seifert cobordisms and the Chen-Yang volume conjecture, 2025. arXiv:2505.01546.\n- [WY23] Ka Ho Wong and Tian Yang. Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms I. Comm. Math. Phys., 400(2):1019– 1070, 2023. doi:10.1007/s00220-022-04613-5.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2803, "problem_number": "KP-3.5", "title": "Kirby Problem 3.5", "statement": "(a) Do there exist closed non-Haken hyperbolic 3-manifolds with arbitrarily large injectivity radius?\n\n(b) Does there exist a cofinal tower of regular covers of closed hyperbolic 3-manifolds where all of the manifolds in the tower are non-Haken?\n\n(c) Is there a tower of hyperbolic rational homology spheres for which all fundamental groups are not left-orderable, the manifolds contain no coorientable taut foliations, or the manifolds are L-spaces? (The three properties listed are connected via the L-space conjecture. See Problem 3.48)", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.5.\n\nLiterature notes:\n(1) A cofinal tower of regular covers is a sequence of covers\n\n$$\n\\cdots \\to M_2 \\to M_1 \\to M\n$$\n\nsuch that $\\bigcap_i \\pi_1(M_i)=1$ and $\\pi_1(M_i)$ is a normal subgroup of $\\pi_1(M)$ for all i.\n\n(2) Question (a) is due to Cooper and appeared as [Kir97, Problem 3.58].\n\n(3) Clearly a negative answer to part (a) implies a negative answer to (b), since the injectivity radii of the manifolds in a tower tend to infinity. Any hyperbolic 3-manifold has a cofinal tower of regular covers, and so a negative answer to (a) or (b) would give an alternative proof of the Virtual Haken Conjecture, proved by Agol [Ago13].\n\n(4) F. Calegari and Dunfield [CD06] constructed a sequence of closed, hyperbolic rational homology 3-spheres for which their injectivity radii tend to infinity. One of the examples discussed in their paper was a tower that covers the Weeks manifold. This tower is a candidate for part (b).\n\n(5) The manifolds in F. Calegari--Dunfield's towers are rational homology 3-spheres but not integral homology 3-spheres. Indeed, it is expected (see Problem 3.6) that there do not exist cofinal towers of regular covers consisting of hyperbolic integral homology 3-spheres.\n\n(6) D. Calegari and Dunfield [CD03] showed that the fundamental group of the Weeks manifold is not left-orderable. Are the fundamental groups of the tower constructed by F. Calegari--Dunfield also not left-orderable?\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Ago13] Ian Agol. The virtual Haken conjecture. Doc. Math., 18:1045–1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning, https://elibm.org/article/10000267. doi:10.4171/DM/421.\n- [CD06] Frank Calegari and Nathan M. Dunfield. Automorphic forms and rational homology 3-spheres. Geom. Topol., 10:295–329, 2006. doi:10.2140/gt.2006.10.295.\n- [CD03] Danny Calegari and Nathan M. Dunfield. Laminations and groups of homeomorphisms of the circle. Invent. Math., 152(1):149–204, 2003. doi:10.1007/s00222-002-0271-6.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2804, "problem_number": "KP-3.6", "title": "Kirby Problem 3.6", "statement": "Given a cofinal tower of covers M $\\leftarrow$ $M_1$ $\\leftarrow$ $M_2$ $\\leftarrow$ $\\cdots$, is it true that the torsion subgroups $\\operatorname{Tor}(M_{n})$ of $H_1(M_{n}, \\mathbb{Z})$ and the hyperbolic volumes $\\operatorname{vol}(M_{n})$ satisfy\n\n$$\n\\lim_{n\\to\\infty}\\frac{\\log|\\operatorname{Tor}(M_n)|}{\\operatorname{vol}(M_n)}=\\frac{1}{6\\pi}\\,?\n$$", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.6.\n\nLiterature notes:\n(1) A tower of finite covers $M\\leftarrow M_1\\leftarrow M_2\\leftarrow\\cdots$ is called cofinal if the corresponding fundamental groups satisfy $\\bigcap_n \\pi_1(M_n)=\\{1\\}$.\n\n(2) Bergeron and Venkatesh conjectured that the above limit holds whenever M is a closed arithmetic 3-manifold and $\\{M_n\\}$ is a cofinal tower of congruence covers [BV13]. See also Lê for a closely related formulation [Lê09]. In the non-arithmetic setting, Brock and Dunfield conjectured that the above limit holds for any cofinal tower of regular covers $\\{M_n\\}$ such that $b_1(M_{n})$ = 0 for all n [BD15, Conjecture 1.13].\n\n(3) Brock and Dunfield have also compiled extensive computational data that supports both the Bergeron--Venkatesh conjecture and their extension to non-arithmetic manifolds [BD15, Section 4]. See also work of Şengün [Ş11, Ş12a], where computations are given that motivated the conjecture of Bergeron and Venkatesh. The computational data suggests that the above limit may fail to hold when M is non-arithmetic and the covers $M_n$ are allowed to have positive Betti numbers. In [BcV16], a more extensive conjectural framework is developed and more data is given, that might explain the distinction between the arithmetic and non-arithmetic cases.\n\nReferences cited:\n- [BV13] Nicolas Bergeron and Akshay Venkatesh. The asymptotic growth of torsion homology for arithmetic groups. J. Inst. Math. Jussieu, 12(2):391–447, 2013. doi: 10.1017/$S^{1}$474748012000667.\n- [Lê09] Thang T. Q. Lê. Hyperbolic volume, Mahler measure, and homology growth. Talk at Columbia University, http://www.math.columbia.edu/„volconf09/notes/leconf.pdf, 2009.\n- [BD15] Jeffrey F. Brock and Nathan M. Dunfield. Injectivity radii of hyperbolic integer homology 3-spheres. Geom. Topol., 19(1):497–523, 2015. doi:10.2140/gt.2015.19.497.\n- [Ş11] Mehmet Haluk Şengün. On the integral cohomology of Bianchi groups. Exp. Math., 20(4):487–505, 2011. doi:10.1080/10586458.2011.594671.\n- [Ş12a] Mehmet Haluk Şengün. On the torsion homology of non-arithmetic hyperbolic tetrahedral groups. Int. J. Number Theory, 8(2):311–320, 2012. doi:10.1142/$S^{1}$793042112500182.\n- [BcV16] Nicolas Bergeron, Mehmet Haluk Şengün, and Akshay Venkatesh. Torsion homology growth and cycle complexity of arithmetic manifolds. Duke Math. J., 165(9):1629– 1693, 2016. doi:10.1215/00127094-3450429.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2805, "problem_number": "KP-3.7", "title": "Kirby Problem 3.7", "statement": "Does every finite-volume hyperbolic 3-manifold admit a finitesheeted cover fibering over the circle with orientable pseudo-Anosov monodromy?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.7.\n\nLiterature notes:\n(1) Agol, building on work of Wise and Kahn--Markovic, proved that every finite-volume hyperbolic 3-manifold admits a finite-sheeted covering that fibers over the circle [Ago08, Ago13, Wis21, KM12]. The monodromy of the fibration is pseudo-Anosov. It is said to be orientable pseudo-Anosov if the stable and unstable laminations are transversely orientable. An obstruction to a pseudo-Anosov being orientable is the existence of a singularity of odd order. Indeed, in the situation where there is no odd order singularity, one may pass to a finite cover of the manifold where the monodromy is orientable pseudo-Anosov [McM13].\n\n(2) One rationale for this question is that orientable pseudo-Anosovs have many nice properties. For example, the dilatation of an orientable pseudo-Anosov is the root with largest modulus of the Alexander polynomial of the fibered manifold [Thu22]. This may be useful for understanding profinite rigidity, since Alexander polynomials can be used to analyze the profinite properties of 3-manifolds (for example [Liu23]); see also\n\nProblem 3.8.\n\nReferences cited:\n- [Ago08] Ian Agol. Criteria for virtual fibering. J. Topol., 1(2):269–284, 2008. doi:10.1112/jtopol/jtn003.\n- [Ago13] Ian Agol. The virtual Haken conjecture. Doc. Math., 18:1045–1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning, https://elibm.org/article/10000267. doi:10.4171/DM/421.\n- [Wis21] Daniel T. Wise. The structure of groups with a quasiconvex hierarchy, volume 209 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, [2021] ©2021.\n- [KM12] Jeremy Kahn and Vladimir Markovic. Immersing almost geodesic surfaces in a closed hyperbolic three manifold. Ann. of Math. (2), 175(3):1127–1190, 2012. doi: 10.4007/annals.2012.175.3.4.\n- [McM13] Curtis T. McMullen. Entropy on Riemann surfaces and the Jacobians of finite covers. Comment. Math. Helv., 88(4):953–964, 2013. doi:10.4171/CMH/308.\n- [Thu22] William P. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. In Collected works of William P. Thurston with commentary. Vol. I. Foliations, surfaces and differential geometry, pages 495–509. Amer. Math. Soc., Providence, RI, [2022] ©2022. Reprint of [ 0956596].\n- [Liu23] Yi Liu. Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups. Invent. Math., 231(2):741–804, 2023. doi:10.1007/s00222-022-01155-4.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2806, "problem_number": "KP-3.8", "title": "Kirby Problem 3.8", "statement": "If $M_1$ and $M_2$ are finite-volume hyperbolic 3-manifolds whose fundamental groups have isomorphic profinite completions, must $M_1$ and $M_2$ be isometric?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.8.\n\nLiterature notes:\n(1) The problem was stated by Reid in [Rei15, Question 9].\n\n(2) Apart from its intrinsic interest, a positive answer to this problem would have some useful consequences. For example, it would provide a new solution to the homeomorphism problem for finite-volume hyperbolic 3-manifolds, since one could distinguish non-homeomorphic manifolds by finding a finite quotient of one fundamental group that is not a finite quotient of the other one.\n\n(3) It was shown by Liu [Liu23] that only finitely many finite-volume hyperbolic 3-manifolds have fundamental groups that can share the same profinite completion. Furthermore, the profinite completion of $\\pi_1(M)$ is known to contain quite a lot of information about M. For example, it controls not only $H_1(M)$ but also the first homology of any finite covering space. Jaikin--Zapirain [JZ20a] also proved that the profinite completion determines whether M fibers over the circle. However, it is not currently known whether the volume of a hyperbolic 3-manifold M is determined by the profinite completion of M.\n\n(4) Examples of distinct compact 3-manifolds whose fundamental groups have isomorphic profinite completions were provided by Funar [Fun13] (torus bundles with sol geometry) and Hempel [Hem14] (Seifert fibered spaces). So it is reasonable to restrict this problem to hyperbolic 3-manifolds.\n\n(5) Bridson, McReynolds, Reid, and Spitler [BMRS20] gave some interesting examples of hyperbolic 3-orbifolds M that are profinitely rigid in a much stronger sense. In their examples, whenever $\\pi_1(M)$ has profinite\n\ncompletion equal to the profinite completion of some finitely generated residually finite group G, then $\\pi_1(M)$ must be isomorphic to G.\n\nReferences cited:\n- [Rei15] Alan W. Reid. Profinite properties of discrete groups. In Groups St Andrews 2013, volume 422 of London Math. Soc. Lecture Note Ser., pages 73–104. Cambridge Univ. Press, Cambridge, 2015.\n- [Liu23] Yi Liu. Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups. Invent. Math., 231(2):741–804, 2023. doi:10.1007/s00222-022-01155-4.\n- [JZ20a] Andrei Jaikin-Zapirain. Recognition of being fibered for compact 3-manifolds. Geom. Topol., 24(1):409–420, 2020. doi:10.2140/gt.2020.24.409.\n- [Fun13] Louis Funar. Torus bundles not distinguished by TQFT invariants. Geom. Topol., 17(4):2289–2344, 2013. With an appendix by Funar and Andrei Rapinchuk. doi: 10.2140/gt.2013.17.2289.\n- [Hem14] John Hempel. Some 3-manifold groups with the same finite quotients, 2014. arXiv: 1409.3509.\n- [BMRS20] M. R. Bridson, D. B. McReynolds, A. W. Reid, and R. Spitler. Absolute profinite rigidity and hyperbolic geometry. Ann. of Math. (2), 192(3):679–719, 2020. doi: 10.4007/annals.2020.192.3.1.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2807, "problem_number": "KP-3.9", "title": "Kirby Problem 3.9", "statement": "Is being Haken a profinite invariant amongst 3-manifolds? That is, if $M_1$ and $M_2$ are 3-manifolds so that $\\pi_1(M_{1})$ and $\\pi_1(M_{2})$ have isomorphic profinite completions, and $M_1$ is Haken, must $M_2$ also be Haken?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.9.\n\nLiterature notes:\n(1) The key case to consider is if M is a hyperbolic Haken rational homology sphere and N is a closed hyperbolic 3-manifold. A positive answer to Problem 3.8 implies a positive answer to this problem.\n\n(2) Note that, for general finitely presented residually finite groups, being a free product with amalgamation is not a profinite property [CWLRS25].\n\nReferences cited:\n- [CWLRS25] Tamunonye Cheetham-West, Alexander Lubotzky, Alan W. Reid, and Ryan Spitler. Property FA is not a profinite property. Groups Geom. Dyn., 19(3):1081–1087, 2025. doi:10.4171/ggd/802.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2808, "problem_number": "KP-3.10", "title": "Kirby Problem 3.10", "statement": "(a) Are there infinitely many commensurability classes of arithmetic rational homology 3-spheres?\n\n(b) Are there infinitely many arithmetic integral homology 3-spheres?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.10.\n\nLiterature notes:\n(1) Recall that two hyperbolic manifolds are commensurable if they have a common finite cover. Two lattices $\\Gamma_1$ and $\\Gamma_2$ in a Lie group G are commensurable if there is some $g\\in G$ such that $\\Gamma_1\\cap g\\Gamma_2g^{-1}$ has finite index in $\\Gamma_1$ and in $g\\Gamma_2g^{-1}$.\n\n(2) The general construction of arithmetic lattices is as follows. Start with a connected semisimple Lie group G with trivial center and no compact factor, and a semisimple algebraic subgroup H of $\\operatorname{GL}(n, \\mathbb{R})$ defined by some polynomial equations with integer coefficients. Suppose that there is surjective homomorphism $\\varphi$ from the identity component $H^0$ of $H$ to G with compact kernel. Then any lattice in G that is commensurable with $\\varphi(H^0\\cap \\operatorname{GL}(n,\\mathbb{Z}))$ is arithmetic. When G is $\\operatorname{SO}(3, 1)$, then $\\mathbb{H}^3/\\Gamma$ is an arithmetic hyperbolic 3-orbifold, and when $\\Gamma$ is also torsion-free, $\\mathbb{H}^3/\\Gamma$ is an arithmetic hyperbolic 3-manifold. Arithmetic hyperbolic 3-manifolds and 3-orbifolds can alternatively be defined in terms of orders in quaternion algebras. See [MR03] for this definition, as well as a comprehensive introduction to the subject.\n\n(3) It is known that there are infinitely many arithmetic rational homology 3-spheres, by constructions of F. Calegari--Dunfield [CD06] and Boston-- Ellenberg [BE06]. However, their constructions do not immediately give infinitely many commensurability classes.\n\n(4) It appears that there are only two known arithmetic integral homology 3-spheres. These are the $1/2$ Dehn filling on the knot $5_2$ and the 3-fold cyclic branched cover of the $(-2,3,7)$-pretzel knot.\n\n(5) See [Rei07] for further discussion of this problem, as well as a potential route to find infinitely many arithmetic hyperbolic rational homology 3-spheres up to commensurability.\n\n(6) When a hyperbolic 3-manifold M has cusps, it is never a rational homology 3-sphere, but in this case one can consider the homomorphism $H^{1}(M)$ $\\to$ $H^{1}(\\partial{}M)$ induced by inclusion. The kernel of this homomorphism is the cuspidal cohomology of M. It is known that there are only finitely many commensurability classes of cusped arithmetic hyperbolic 3-manifolds that contain a manifold with trivial cuspidal cohomology [MR03, Theorem 9.3.6]. This was used by Reid [Rei91] to show that the figure-eight knot is the unique arithmetic hyperbolic knot complement.\n\n(7) An analogous question in dimension two is whether there are infinitely many commensurability classes of genus 0 hyperbolic arithmetic 2-orbifolds. This was proved to be false in [LMR06].\n\nReferences cited:\n- [MR03] Colin Maclachlan and Alan W. Reid. The arithmetic of hyperbolic 3-manifolds, volume 219 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2003. doi:10.1007/978-1-4757-6720-9.\n- [CD06] Frank Calegari and Nathan M. Dunfield. Automorphic forms and rational homology 3-spheres. Geom. Topol., 10:295–329, 2006. doi:10.2140/gt.2006.10.295.\n- [BE06] Nigel Boston and Jordan S. Ellenberg. Pro-p groups and towers of rational homology spheres. Geom. Topol., 10:331–334, 2006. doi:10.2140/gt.2006.10.331.\n- [Rei07] Alan Reid. The geometry and topology of arithmetic hyperbolic 3-manifolds. In Proc. Symposium Topology, Complex Analysis and Arithmetic of Hyperbolic Spaces, Kyoto 2006, volume 1571 of RIMS Kokyuroku Series, pages 31–58. Research Institute for Mathematical Sciences, Kyoto University, 2007.\n- [Rei91] Alan W. Reid. Arithmeticity of knot complements. J. London Math. Soc. (2), 43(1):171–184, 1991. doi:10.1112/jlms/s2-43.1.171.\n- [LMR06] D. D. Long, C. Maclachlan, and A. W. Reid. Arithmetic Fuchsian groups of genus zero. Pure Appl. Math. Q., 2(2):569–599, 2006. doi:10.4310/PAMQ.2006.v2.n2.a9.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2809, "problem_number": "KP-3.11", "title": "Kirby Problem 3.11", "statement": "Does every hyperbolic knot in the 3-sphere have meridian length at most 4?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.11.\n\nLiterature notes:\n(1) Every hyperbolic knot has a maximal cusp, the boundary of which is a Euclidean torus. The length of a slope on the knot exterior is the length of a geodesic representative of this slope in the Euclidean torus. This is a quantity that appears frequently in the theory of Dehn surgery. It was shown by Agol [Ago00] and Lackenby [Lac00] that when M is filled along a slope with length more than 6, then the resulting manifold is hyperbolic. Hence, the meridian of any knot in the 3-sphere is known to have length at most 6. The question is whether this upper bound can be improved to 4.\n\n(2) The problem is interesting for several reasons. First, it would show that the complements of knots in the 3-sphere have distinctive geometry. Secondly, it would also lead to improvements to many results about surgery on knots in the 3-sphere.\n\n(3) Examples of hyperbolic knots in the 3-sphere with meridian lengths tending to 4 from below were given by Agol [Ago00], who asked whether it is possible to find hyperbolic knots with meridian length greater than 4. Further examples were given by Purcell [Pur08]. The question is known to have a positive answer for 2-bridge knots [Ada96] and alternating knots [ACF+06].\n\n(4) Note that, in this problem, it is necessary to restrict to knots rather than links. Görner [Gör15] has examples of hyperbolic links in the 3-sphere where the shortest length of any slope on any boundary component is $\\sqrt{21}$ $\\approx$ 4.582. It would be interesting to know whether $\\sqrt{21}$ is optimal here.\n\nReferences cited:\n- [Ago00] Ian Agol. Bounds on exceptional Dehn filling. Geom. Topol., 4:431–449, 2000. doi: 10.2140/gt.2000.4.431.\n- [Lac00] Marc Lackenby. Word hyperbolic Dehn surgery. Invent. Math., 140(2):243–282, 2000. doi:10.1007/s002220000047.\n- [Pur08] Jessica S. Purcell. Slope lengths and generalized augmented links. Comm. Anal. Geom., 16(4):883–905, 2008. doi:10.4310/CAG.2008.v16.n4.a7.\n- [Ada96] Colin C. Adams. Hyperbolic 3-manifolds with two generators. Comm. Anal. Geom., 4(1-2):181–206, 1996. doi:10.4310/CAG.1996.v4.n2.a1.\n- [ACF+06] C. Adams, A. Colestock, J. Fowler, W. Gillam, and E. Katerman. Cusp size bounds from singular surfaces in hyperbolic 3-manifolds. Trans. Amer. Math. Soc., 358(2):727–741, 2006. doi:10.1090/S0002-9947-05-03662-7.\n- [Gör15] Matthias Görner. Regular tessellation link complements. Exp. Math., 24(2):225– 246, 2015. doi:10.1080/10586458.2014.986310.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2810, "problem_number": "KP-3.12", "title": "Kirby Problem 3.12", "statement": "(a) Considering all closed, orientable, $\\pi_1$-injective surfaces (possibly non-embedded) in all closed hyperbolic 3-manifolds, what is the infimum of the areas of all these surfaces?\n\n(b) One can ask the same question, but considering just closed, orientable, $\\pi_1$-injective surfaces of a fixed genus g.\n\n(c) One can also consider only closed, orientable, embedded $\\pi_1$-injective surfaces.\n\n(d) Alternatively, one can consider all closed, orientable, essential surfaces in all hyperbolic 3-manifolds M, allowing M not to be closed.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.12.\n\nLiterature notes:\n(1) A totally geodesic genus-two surface in a hyperbolic 3-manifold has area $4\\pi$. A universal lower bound for orientable surfaces is $2\\pi$ [Has95], based on unpublished work of Uhlenbeck. Hence, the answer to part (a) lies somewhere between $2\\pi$ and $4\\pi$.\n\n(2) A lower bound for the area of a closed nonorientable essential surface is $\\pi$, since such a surface is double covered by an orientable one.\n\n(3) Any closed orientable $\\pi_1$-injective surface is homotopic to a least area surface S, which is immersed [SY82, SU82]. Its sectional curvature $\\kappa$ is then everywhere at most $-1$. Gauss-Bonnet gives that the integral of $\\kappa$ over the surface is $2\\pi\\chi(S)$. On the other hand, the area is the integral of 1 over the surface. Hence, the more negative that $\\kappa$ is, on average over S, the smaller the area of S.\n\nReferences cited:\n- [Has95] Joel Hass. Acylindrical surfaces in 3-manifolds. Michigan Math. J., 42(2):357–365, 1995. doi:10.1307/mmj/1029005233.\n- [SY82] Richard Schoen and Shing Tung Yau. Complete three-dimensional manifolds with positive Ricci curvature and scalar curvature. In Seminar on Differential Geometry, volume No. 102 of Ann. of Math. Stud., pages 209–228. Princeton Univ. Press, Princeton, NJ, 1982.\n- [SU82] J. Sacks and K. Uhlenbeck. Minimal immersions of closed Riemann surfaces. Trans. Amer. Math. Soc., 271(2):639–652, 1982. doi:10.2307/1998902.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2811, "problem_number": "KP-3.13", "title": "Kirby Problem 3.13", "statement": "Does every closed hyperbolic 3-manifold admit an immersed $\\pi_1$-injective surface with only double points? More precisely, if M is a closed, connected, hyperbolic 3-manifold, is there a closed, connected surface $\\Sigma$ (other than a 2-sphere) and an immersion f : $\\Sigma$ $\\to$ M such that $f_*$ : $\\pi_1(\\Sigma)$ $\\to$ $\\pi_1(M)$ is injective and $|f^{-1}(\\{x\\})|\\leq 2$ for all $x\\in M$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.13.\n\nLiterature notes:\n(1) This is true for Haken 3-manifolds, which by definition have embedded $\\pi_1$-injective surfaces, and for small Seifert-fibered spaces with infinite fundamental group, which can easily be seen to have immersed essential tori with only double points; hence the restriction to hyperbolic manifolds (by the Geometrization Theorem).\n\n(2) This is true for finite volume orientable connected non-compact hyperbolic 3-manifolds and for most Dehn fillings by [CL01], hence should be true for all but finitely many closed, hyperbolic 3-manifolds of bounded volume. The immersed surfaces produced in closed hyperbolic 3-manifolds by [KM12] will usually have triple points of intersection.\n\nReferences cited:\n- [CL01] D. Cooper and D. D. Long. Some surface subgroups survive surgery. Geom. Topol., 5:347–367, 2001. doi:10.2140/gt.2001.5.347.\n- [KM12] Jeremy Kahn and Vladimir Markovic. Immersing almost geodesic surfaces in a closed hyperbolic three manifold. Ann. of Math. (2), 175(3):1127–1190, 2012. doi: 10.4007/annals.2012.175.3.4.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2812, "problem_number": "KP-3.14", "title": "Kirby Problem 3.14", "statement": "Can a hyperbolic knot complement in the 3-sphere contain a closed, embedded totally geodesic surface?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.14.\n\nLiterature notes:\n(1) In 1990, Menasco and Reid conjectured that the answer is no [MR92]. There is evidence for and against this conjecture. For example, as evidence that it may be false, there are embedded totally geodesic spanning surfaces [AS05]. There are closed, embedded, totally geodesic surfaces in hyperbolic link complements in the 3-sphere with just two link components [Lei06]. For any $\\epsilon>0$, there are closed, embedded surfaces in knot complements with principal curvatures no more than $\\epsilon$ [Lei06]. There are hyperbolic knots in rational homology 3-spheres with closed, embedded, totally geodesic surfaces in their complement [DeB06].\n\n(2) In favor of Menasco--Reid's conjecture, many known families of knots have been proved not to admit a closed, embedded, totally geodesic surface, such as alternating knots [MR92], knots with tunnel number 1 [MR92], Montesinos knots [Oer84], 3-bridge knots and double torus knots [IO00].\n\n(3) Related questions for other types of surfaces are also open. For example, can a hyperbolic knot complement in the 3-sphere contain a totally geodesic separating surface? What about one with meridional boundary components?\n\nReferences cited:\n- [MR92] William Menasco and Alan W. Reid. Totally geodesic surfaces in hyperbolic link complements. In Topology ’90 (Columbus, OH, 1990), volume 1 of Ohio State Univ. Math. Res. Inst. Publ., pages 215–226. de Gruyter, Berlin, 1992.\n- [AS05] Colin Adams and Eric Schoenfeld. Totally geodesic Seifert surfaces in hyperbolic knot and link complements. I. Geom. Dedicata, 116:237–247, 2005. doi:10.1007/s10711-005-9018-z.\n- [Lei06] Christopher J. Leininger. Small curvature surfaces in hyperbolic 3-manifolds. J. Knot Theory Ramifications, 15(3):379–411, 2006. doi: 10.1142/S0218216506004531.\n- [DeB06] Jason DeBlois. Totally geodesic surfaces and homology. Algebr. Geom. Topol., 6:1413–1428, 2006. doi:10.2140/agt.2006.6.1413.\n- [Oer84] Ulrich Oertel. Closed incompressible surfaces in complements of star links. Pacific J. Math., 111(1):209–230, 1984. http://projecteuclid.org/euclid.pjm/1102710789.\n- [IO00] Kazuhiro Ichihara and Makoto Ozawa. Hyperbolic knot complements without closed embedded totally geodesic surfaces. J. Austral. Math. Soc. Ser. A, 68(3):379–386, 2000.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2813, "problem_number": "KP-3.15", "title": "Kirby Problem 3.15", "statement": "Let $M$ be a closed hyperbolic 3-manifold with positive first Betti number.\n\n(a) Which elements of $H^{2}(M;\\mathbb{R})$ are realized as the Euler classes of taut foliations?\n\n(b) Which elements of $H^{2}(M;\\mathbb{R})$ are realized as the Euler classes of tight contact structures?\n\n(c) How about universally tight contact structures?\n\n(d) The question can also be asked for pseudo-Anosov flows...\n\n(e)... and for quasi-geodesic flows. (f ) How about Euler classes of representations into $\\operatorname{Homeo}^{+}(S^1)$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.15.\n\nLiterature notes:\n(1) For any compact, orientable 3-manifold M, Thurston [Thu86] defined a pseudo-norm on the second homology groups $H_2(M;\\mathbb{R})$ and $H_2(M,\\partial M;\\mathbb{R})$, now called the Thurston norm. The Thurston norm induces dual norms on $H^{2}(M;\\mathbb{R})$ and $H^{2}(M, \\partial{}M;\\mathbb{R})$. The unit ball of the Thurston dual norm is a convex polyhedron with integral vertices [Thu86].\n\n(2) Thurston showed that the Euler class $e(F)$ of any taut foliation $F$ has dual norm at most one, and if $F$ has any compact leaf, then the dual\n\nnorm of $e(F)$ is equal to one [Thu86]. Conversely, he conjectured that any integral class in $H^{2}(M;\\mathbb{R})$ of dual norm equal to one is the Euler class of a taut foliation on M [Thu86, page 129, Conjecture 3]. Thurston was aware that $e(F)$ satisfies the parity condition meaning that for every closed orientable surface $S$, the pairing $\\langle e(F),[S]\\rangle$ has the same parity as $\\chi(S)$. Gabai proved that every vertex of the dual unit ball is realized as the Euler class of a taut foliation [Gab97, Yaz20], but Gabai and Yazdi constructed counterexamples to Thurston's conjecture [GY20, Yaz20]. It is therefore not clear which elements of $H^{2}(M;\\mathbb{R})$ are realized as the Euler classes of taut foliations, which is the first of the above questions.\n\n(3) Tight contact structures also have an Euler class. Eliashberg proved that the Euler class of a tight contact structure has dual norm at most one [Eli92]. Every $C^0$ taut foliation (other than the product foliation on $S^2$ $\\times$ $S^1$ ) can be $C^0$-approximated by a universally tight contact structure, by Eliashberg--Thurston [ET98], Kazez--Roberts [KR17], and Bowden [Bow16]. So, for any closed, orientable, irreducible 3-manifold M, the set of Euler classes of universally tight contact structures on M includes Euler classes of taut foliations. It is possible that every integral class in $H^{2}(M;\\mathbb{R})$ of dual norm one and satisfying the parity condition is realized as the Euler class of a tight contact structure on M. Sivek and Yazdi [SY23] have shown that, in the counterexamples given by Gabai and Yazdi, the classes in $H^{2}(M;\\mathbb{R})$ that were proved not to be realized by taut foliations are realized by tight contact structures.\n\n(4) Pseudo-Anosov flows and quasi-geodesic flows also admit Euler classes, so one can ask the analogous question for them. Similarly, a representation $\\pi_1(M)\\to \\operatorname{Homeo}^{+}(S^1)$ defines a circle bundle over M, and hence an Euler class. This satisfies the Milnor--Wood inequality [Mil58, Woo71], and hence has dual norm at most one. It does not necessarily satisfy the parity condition. It is possible that every integral class in $H^{2}(M;\\mathbb{R})$ of dual norm one is realized by a representation to $\\operatorname{Homeo}^{+}(S^1)$.\n\nReferences cited:\n- [Thu86] William P. Thurston. A norm for the homology of 3-manifolds. Mem. Amer. Math. Soc., 59(339):i–vi and 99–130, 1986.\n- [Gab97] David Gabai. Problems in foliations and laminations. In Geometric topology (Athens, GA, 1993), volume 2 of AMS/IP Stud. Adv. Math., pages 1–33. Amer. Math. Soc., Providence, RI, 1997. doi:10.1090/amsip/002.2/01.\n- [Yaz20] Mehdi Yazdi. On Thurston’s Euler class-one conjecture. Acta Math., 225(2):313– 368, 2020. doi:10.4310/acta.2020.v225.n2.a3.\n- [GY20] David Gabai and Mehdi Yazdi. The fully marked surface theorem. Acta Math., 225(2):369–413, 2020. doi:10.4310/acta.2020.v225.n2.a4.\n- [Eli92] Yakov Eliashberg. Contact 3-manifolds twenty years since J. Martinet’s work. Ann. Inst. Fourier (Grenoble), 42(1-2):165–192, 1992. URL: http://www.numdam.org/item?id=AIF 1992 42 1-2 165 0.\n- [ET98] Yakov M. Eliashberg and William P. Thurston. Confoliations, volume 13 of University Lecture Series. American Mathematical Society, Providence, RI, 1998. doi:10.1090/ulect/013.\n- [KR17] William H. Kazez and Rachel Roberts. C0 approximations of foliations. Geom. Topol., 21(6):3601–3657, 2017. doi:10.2140/gt.2017.21.3601.\n- [Bow16] Jonathan Bowden. Approximating C0-foliations by contact structures. Geom. Funct. Anal., 26(5):1255–1296, 2016. doi:10.1007/s00039-016-0387-2.\n- [SY23] Steven Sivek and Mehdi Yazdi. Thurston norm and Euler classes of tight contact structures. Bull. Lond. Math. Soc., 55(6):2976–2990, 2023. doi:10.1112/blms.12905.\n- [Mil58] John W Milnor. On the existence of a connection with curvature zero. Comment. Math. Helv., 32(1):215–223, 1958.\n- [Woo71] John W. Wood. Bundles with totally disconnected structure group. Comment. Math. Helv., 46:257–273, 1971. doi:10.1007/BF02566843.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2814, "problem_number": "KP-3.16", "title": "Kirby Problem 3.16", "statement": "Does every finite-volume hyperbolic 3-manifold contain infinitely many simple closed geodesics?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.16.\n\nLiterature notes:\nIt seems reasonable to suppose that, in some sense, a typical closed geodesic in a finite-volume hyperbolic 3-manifold does not self-intersect, and so one would expect the question to have a positive answer. Kuhlmann [Kuh06] proved that the answer is positive for cusped hyperbolic 3-manifolds. Moreover, she gave geometric conditions that imply that a closed hyperbolic 3-manifold contains infinitely many simple closed geodesics. Among the first 200 closed hyperbolic 3-manifolds in the census, she was able to verify these conditions for 178 of them [Kuh08]. It is straightforward to prove that every closed hyperbolic 3-manifold contains at least one simple geodesic, since a shortest geodesic is always simple.\n\nReferences cited:\n- [Kuh06] Sally M. Kuhlmann. Geodesic knots in cusped hyperbolic 3-manifolds. Algebr. Geom. Topol., 6:2151–2162, 2006. doi:10.2140/agt.2006.6.2151.\n- [Kuh08] Sally Kuhlmann. Geodesic knots in closed hyperbolic 3-manifolds. Geom. Dedicata, 131:181–211, 2008. doi:10.1007/s10711-007-9227-8.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2815, "problem_number": "KP-3.17", "title": "Kirby Problem 3.17", "statement": "Let $M_1$ and $M_2$ be finite-volume hyperbolic n--manifolds. If the length spectra of $M_1$ and $M_2$ coincide, must the two manifolds be commensurable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.17.\n\nLiterature notes:\n(1) For a hyperbolic manifold M, the length spectrum $L(M)$ is the ordered tuple of all lengths of closed geodesics in M, listed with multiplicity. The length spectrum is closely related to the eigenvalue spectrum $E(M)$ of the Laplace--Beltrami operator. Specifically, $L(M)$ determines $E(M)$, and $E(M)$ determines $L(M)$, the set of all lengths of closed geodesics, stripped of multiplicity information.\n\n(2) In all dimensions $n\\geq 2$, there are well-known constructions of isospectral twins: pairs of hyperbolic manifolds $M_1$ and $M_2$ for which the length spectra coincide, but that are not isometric. Vignéras constructed the first isospectral twins, using arithmetic methods [Vig80]. Subsequently, Sunada described a flexible group-theoretic construction in which the $M_i$ are covers of a fixed manifold or orbifold [Sun85]. In all known constructions of isospectral twins, the manifolds $M_1$ and $M_2$ are commensurable, meaning they share a common finite-sheeted cover. This prompted Reid to pose Problem 3.17 [Rei14, Question 2.1].\n\n(3) The question in this problem is known to have a positive answer for arithmetic manifolds in dimensions $n\\not\\equiv 1\\pmod{4}$. See Reid [Rei92]; Chinburg--Hamilton--Long--Reid [CHLR08]; and Prasad--Rapinchuk [PR09, PR15]. Outside the arithmetic setting, Futer and Millichap have constructed incommensurable pairs $M_1$ and $M_2$ that share an arbitrarily large finite portion of their length spectra, up to a cutoff that grows linearly with volume [FM17].\n\nReferences cited:\n- [Vig80] Marie-France Vignéras. Variétés riemanniennes isospectrales et non isométriques. Ann. of Math. (2), 112(1):21–32, 1980. doi:10.2307/1971319.\n- [Sun85] Toshikazu Sunada. Riemannian coverings and isospectral manifolds. Ann. of Math. (2), 121(1):169–186, 1985. doi:10.2307/1971195.\n- [Rei14] Alan W. Reid. Traces, lengths, axes and commensurability. Ann. Fac. Sci. Toulouse Math. (6), 23(5):1103–1118, 2014. doi:10.5802/afst.1438.\n- [Rei92] Alan W. Reid. Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds. Duke Math. J., 65(2):215–228, 1992. doi:10.1215/S0012-7094-92-06508-2.\n- [CHLR08] Ted Chinburg, Emily Hamilton, Darren D. Long, and Alan W. Reid. Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds. Duke Math. J., 145(1):25–44, 2008. doi:10.1215/00127094-2008-045.\n- [PR09] Gopal Prasad and Andrei S. Rapinchuk. Weakly commensurable arithmetic groups and isospectral locally symmetric spaces. Publ. Math. Inst. Hautes Études Sci., 109:113–184, 2009. doi:10.1007/s10240-009-0019-6.\n- [PR15] Gopal Prasad and Andrei S. Rapinchuk. Weakly commensurable groups, with applications to differential geometry. In Handbook of group actions. Vol. I, volume 31 of Adv. Lect. Math. (ALM), pages 495–524. Int. Press, Somerville, MA, 2015.\n- [FM17] David Futer and Christian Millichap. Spectrally similar incommensurable 3-manifolds. Proc. Lond. Math. Soc. (3), 115(2):411–447, 2017. doi:10.1112/plms.12045.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2816, "problem_number": "KP-3.18", "title": "Kirby Problem 3.18", "statement": "Is there a closed hyperbolic 3-manifold that is foliated with minimal leaves?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.18.\n\nLiterature notes:\nThis problem was raised by Thurston and Uhlenbeck around 1980; see [Uhl83, Page 154]. Each leaf of such a foliation is an area minimizing surface, that is, any compact subsurface of a leaf minimizes area among all homologous surfaces with the same boundary. Taut foliations can always be made minimal in some (but perhaps not the hyperbolic) Riemannian metric [Sul79a, Has86]. Hass and Thurston, at the 1984 Durham Symposium on Kleinian groups, 3-Manifolds, and Hyperbolic Geometry, gave examples of foliations in a hyperbolic manifold that cannot be made minimal [Has15]. See also [HW19]. Additional geometric\n\nconditions on a foliation can rule out the possibility of each leaf being minimal [WW20]. There is no local obstruction, since there are many foliations of $\\mathbb{H}^3$ by minimal planes.\n\nReferences cited:\n- [Uhl83] Karen K. Uhlenbeck. Closed minimal surfaces in hyperbolic 3-manifolds. In Semi-nar on minimal submanifolds, volume 103 of Ann. of Math. Stud., pages 147–168. Princeton Univ. Press, Princeton, NJ, 1983.\n- [Sul79a] Dennis Sullivan. A homological characterization of foliations consisting of minimal surfaces. Comment. Math. Helv., 54(2):218–223, 1979. doi:10.1007/BF02566269.\n- [Has86] Joel Hass. Minimal surfaces in foliated manifolds. Comment. Math. Helv., 61(1):1– 32, 1986. doi:10.1007/BF02621899.\n- [Has15] Joel Hass. Minimal fibrations of hyperbolic 3-manifolds, 2015. arXiv:1512.04145.\n- [HW19] Zheng Huang and Biao Wang. Complex length of short curves and minimal fibrations of hyperbolic three-manifolds fibering over the circle. Proceedings of the London Mathematical Society, 118(6):1305–1327, 2019.\n- [WW20] Michael Wolf and Yunhui Wu. Non-existence of geometric minimal foliations in hyperbolic three-manifolds. Comment. Math. Helv., 95(1):167–182, 2020. doi:10.4171/cmh/484.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2817, "problem_number": "KP-3.19", "title": "Kirby Problem 3.19", "statement": "(a) Does every closed hyperbolic 3-manifold have a nowhere zero vector field whose lift to the universal cover has proper flow lines?\n\n(b) Can one ensure that the leaf space of the foliation by flow lines in the universal cover is Hausdorff ?\n\n(c) If so, the leaf space in the universal cover is homeomorphic to a plane and one can then ask: can the induced action of $\\pi_1(M)$ on a plane be compactified to an action on the closed disk?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.19.\n\nLiterature notes:\n(1) The basic form of this question is due to W. Thurston.\n\n(2) If the answer to part (c) is \"yes,\" it seems likely one can show the action on the boundary must be faithful, in which case there are non-examples, since the fundamental group of the Weeks manifold is known (by Calegari-- Dunfield [CD03]) not to act faithfully on a circle.\n\n(3) Quasigeodesic flows are proper in this sense. However it is known that there are examples of hyperbolic 3-manifolds that do not admit quasigeodesic flows, since such flows give rise to a faithful action on a universal circle, as in the previous remark, by [Cal06b].\n\nReferences cited:\n- [CD03] Danny Calegari and Nathan M. Dunfield. Laminations and groups of homeomorphisms of the circle. Invent. Math., 152(1):149–204, 2003. doi:10.1007/s00222-002-0271-6.\n- [Cal06b] Danny Calegari. Universal circles for quasigeodesic flows. Geom. Topol., 10:2271– 2298, 2006. doi:10.2140/gt.2006.10.2271.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2818, "problem_number": "KP-3.20", "title": "Kirby Problem 3.20", "statement": "Let $M$ be a closed hyperbolic 3-manifold with a faithful homomorphism $\\rho:\\pi_1(M)\\to \\operatorname{Homeo}^{+}(\\mathbb{R})$. Prove that $M$ supports a dual transversely orientable, essential lamination $\\lambda$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.20.\n\nLiterature notes:\n(1) Let $\\rho$ be a faithful homomorphism as above. Suppose $\\lambda$ is a lamination on $M$, and let $\\widetilde{\\lambda}$ be its lift to the universal cover. Then $\\lambda$ is said to be dual to $\\rho$ if there is a map from the leaf space of $\\widetilde{\\lambda}$ to $\\mathbb{R}$ which is equivariant with respect to the actions of $\\pi_1(M)$. It is well known that $\\rho$ gives rise to a dual lamination that is transversely orientable; the problem asks whether there is one that is also essential.\n\n(2) There is a faithful homomorphism $\\rho$ as above if and only if $\\pi_1(M)$ is left-orderable, by [Con59] and [CR16]. In this case, the L-space Conjecture (Problem 3.48) asserts that $M$ supports a transversely orientable taut foliation. By contrast, this problem asks merely for a transversely orientable essential lamination, but one that is also dual to $\\rho$.\n\n(3) Morgan and Shalen [MS88] nearly solved the analogous problem where $\\pi_1(M)$ acts isometrically on an $\\mathbb{R}$-tree. They showed that $M$ supports an incompressible measured lamination, but they did not show that this lamination is dual to $\\rho$ in the corresponding sense.\n\n(4) There is a faithful homomorphism $\\rho:\\pi_1(M)\\to \\operatorname{Homeo}^{+}(\\mathbb{R})$ if and only if there is one with nontrivial image, by [BRW05].\n\nReferences cited:\n- [Con59] Paul Conrad. Right-ordered groups. Michigan Math. J., 6:267–275, 1959. http: //projecteuclid.org/euclid.mmj/1028998233.\n- [CR16] Adam Clay and Dale Rolfsen. Ordered groups and topology, volume 176 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2016. doi:10.1090/gsm/176.\n- [MS88] John W. Morgan and Peter B. Shalen. Degenerations of hyperbolic structures. III. Actions of 3-manifold groups on trees and Thurston’s compactness theorem. Ann. of Math. (2), 127(3):457–519, 1988. doi:10.2307/2007003.\n- [BRW05] Steven Boyer, Dale Rolfsen, and Bert Wiest. Orderable 3-manifold groups. Ann. Inst. Fourier (Grenoble), 55(1):243–288, 2005. URL: http://aif.cedram.org/item? id=AIF 2005 55 1 243 0.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2819, "problem_number": "KP-3.21", "title": "Kirby Problem 3.21", "statement": "In this problem, all 3-manifolds are orientable, while all flows are considered up to orbit equivalence and are assumed to be transitive.\n\n(a) Are there only finitely many Anosov flows on a given closed 3-manifold?\n\n(b) Are there only finitely many pseudo-Anosov flows on a given closed 3-manifold?\n\n(c) Are there only finitely many quasigeodesic pseudo-Anosov flows on a given closed 3-manifold?\n\n(d) Are there only finitely many pseudo-Anosov flows without perfect fits on a given closed 3-manifold?\n\n(e) Are there only finitely many veering triangulations on a given finite volume cusped hyperbolic 3-manifold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.21.\n\nLiterature notes:\n(1) Two flows on a 3-manifold are orbit equivalent if there is an orientation-preserving homeomorphism of the manifold sending orbits of one flow to those of the other. A flow is transitive if it has a dense orbit; every pseudo-Anosov flow on a closed, atoroidal (in particular, hyperbolic) 3-manifold is automatically transitive [Mos92].\n\n(2) Note that (an affirmative answer to) (b) implies (c) implies [Fen16] (d), and also that (b) implies (a).\n\n(3) Barthelme, Mann, and Bowden have made partial progress on (a), showing that there are finitely many contact Anosov flows on a closed 3-manifold [BM24b]. Towards (b), Zung has shown that there are only finitely many transitive pseudo-Anosov flows with positive Birkhoff sections on a given rational homology 3-sphere [Zun25]; see also [BSZ25].\n\n(4) A veering triangulation on a 3-manifold is an ideal triangulation with a particular combinatorial structure. These were first introduced by Agol on punctured mapping tori of pseudo-Anosov surface homeomorphisms [Ago11]. Agol and Guéritaud later showed more generally that if a 3-manifold admits a pseudo-Anosov flow without perfect fits, then there is a canonical veering triangulation in the complement of the singular orbits (see [LMT23]). Frankel, Schleimer, and Segerman have outlined an inverse to this construction, which should provide a one-to-one correspondence between veering triangulations and pseudo-Anosov flows without perfect fits; see [FSS22], [SS24a]. Tsang independently proved such a correspondence in his thesis [Tsa23] via a different approach based on work in [AT24].\n\n(5) Given the correspondence between pseudo-Anosov flows and veering triangulations, one can show that an affirmative answer to (e) would imply an affirmative answer to (d).\n\nReferences cited:\n- [Mos92] Lee Mosher. Dynamical systems and the homology norm of a 3-manifold. I. Efficient intersection of surfaces and flows. Duke Math. J., 65(3):449–500, 1992. doi:10.1215/S0012-7094-92-06518-5.\n- [Fen16] Sérgio R. Fenley. Quasigeodesic pseudo-Anosov flows in hyperbolic 3-manifolds and connections with large scale geometry. Adv. Math., 303:192–278, 2016. doi:10.1016/j.aim.2016.05.015.\n- [BM24b] Thomas Barthelmé and Kathryn Mann. Orbit equivalences of R-covered Anosov flows and hyperbolic-like actions on the line. Geom. Topol., 28(2):867–899, 2024. Appendix written jointly with Jonathan Bowden. doi:10.2140/gt.2024.28.867.\n- [Zun25] Jonathan Zung. Pseudo-Anosov representatives of stable Hamiltonian structures. J. Fixed Point Theory Appl., 27(4):Paper No. 87, 29, 2025. doi:10.1007/s11784-025-01238-8.\n- [BSZ25] John A. Baldwin, Steven Sivek, and Jonathan Zung. Pseudo-Anosov flows on hyperbolic l-spaces, 2025. arXiv:2505.21113.\n- [Ago11] Ian Agol. Ideal triangulations of pseudo-Anosov mapping tori. In Topology and geometry in dimension three, volume 560 of Contemp. Math., pages 1–17. Amer. Math. Soc., Providence, RI, 2011. doi:10.1090/conm/560/11087.\n- [LMT23] Michael P. Landry, Yair N. Minsky, and Samuel J. Taylor. Flows, growth rates, and the veering polynomial. Ergodic Theory Dynam. Systems, 43(9):3026–3107, 2023. doi:10.1017/etds.2022.63.\n- [FSS22] Steven Frankel, Saul Schleimer, and Henry Segerman. From veering triangulations to link spaces and back again, 2022. arXiv:1911.00006.\n- [SS24a] Saul Schleimer and Henry Segerman. From loom spaces to veering triangulations. Groups Geom. Dyn., 18(2):419–462, 2024. doi:10.4171/ggd/742.\n- [Tsa23] Chi Cheuk Tsang. Veering Triangulations and Pseudo-Anosov Flows. ProQuest LLC, Ann Arbor, MI, 2023. Thesis (Ph.D.)–University of California, Berkeley. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt=info: ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqm\\&rft dat=xri:pqdiss:30487016.\n- [AT24] Ian Agol and Chi Cheuk Tsang. Dynamics of veering triangulations: infinitesimal components of their flow graphs and applications. Algebr. Geom. Topol., 24(6):3401–3453, 2024. doi:10.2140/agt.2024.24.3401.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2820, "problem_number": "KP-3.22", "title": "Kirby Problem 3.22", "statement": "Let $G=\\pi_1(M)$ be the fundamental group of a finite-volume hyperbolic 3-manifold $M$. What is the regularity of the smoothest (virtual) action of $G$ on $S^1$? On the interval?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.22.\n\nLiterature notes:\n(1) Not all hyperbolic 3--manifold groups can act faithfully on the interval, as this is equivalent to left orderability. Calegari--Dunfield showed that the Weeks manifold group cannot act faithfully on the interval [CD03], and having finite first homology is an obstruction for a finitely generated group to act by diffeomorphisms on the interval by Thurston Stability [Thu74b].\n\n(2) Every hyperbolic 3--manifold group has a finite index subgroup that embeds into a right-angled Artin group and therefore admits a faithful $C^1$ action on the circle by [FF03, Jor12]. It is unclear whether higher levels of regularity can be achieved, or whether the existence of such an action persists in the ambient supergroup; moreover, it is interesting to consider whether or not the $C^1$ actions of finite index subgroups are conjugate to higher regularity actions, or if there are other actions of higher regularity not arising from right-angled Artin groups.\n\n(3) One can ask similar questions about other geometric 3--manifold groups, or indeed general 3--manifold groups. In some cases, the answers are known or partially known (e.g. Euclidean manifold groups virtually admit $C^\\infty$ actions whereas nilmanifold fundamental groups cannot admit faithful $C^2$ actions).\n\n(4) Some nonexistence results are known for actions of fibered 3--manifold groups which lie in a sufficiently small neighborhood of the identity; [BKKT20].\n\n(5) Actions of fundamental groups on the interval and the circle are closely related to further topological structure that a 3--manifold may have, such as certain taut foliations, Anosov flows, and so on [CR16]. Are the actions arising from these kinds of structures conjugate to smoother actions?\n\nReferences cited:\n- [CD03] Danny Calegari and Nathan M. Dunfield. Laminations and groups of homeomorphisms of the circle. Invent. Math., 152(1):149–204, 2003. doi:10.1007/s00222-002-0271-6.\n- [Thu74b] William P. Thurston. A generalization of the Reeb stability theorem. Topology, 13:347–352, 1974. doi:10.1016/0040-9383(74)90025-1.\n- [FF03] Benson Farb and John Franks. Groups of homeomorphisms of one-manifolds. III. Nilpotent subgroups. Ergodic Theory Dynam. Systems, 23(5):1467–1484, 2003. doi: 10.1017/S0143385702001712.\n- [Jor12] Eduardo Jorquera. A universal nilpotent group of C1 diffeomorphisms of the interval. Topology Appl., 159(8):2115–2126, 2012. doi:10.1016/j.topol.2012.02.003.\n- [BKKT20] Christian Bonatti, Sang-hyun Kim, Thomas Koberda, and Michele Triestino. Small C1 actions of semidirect products on compact manifolds. Algebr. Geom. Topol., 20(6):3183–3203, 2020. doi:10.2140/agt.2020.20.3183.\n- [CR16] Adam Clay and Dale Rolfsen. Ordered groups and topology, volume 176 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2016. doi:10.1090/gsm/176.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2821, "problem_number": "KP-3.23", "title": "Kirby Problem 3.23", "statement": "What is the Margulis constant in dimension 3? Is it realized uniquely by the Weeks manifold W, where $\\mu(W)$ = 0.77442...?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.23.\n\nLiterature notes:\n(1) The fundamental Margulis lemma in dimension 3 asserts that there is an $\\epsilon>0$ so that for any discrete subgroup $\\Gamma$ of $\\operatorname{Isom}(\\mathbb{H}^3)$, if $f,g\\in\\Gamma$ and\n\neach move some point $x\\in\\mathbb{H}^3$ less than $\\epsilon$, then the subgroup generated by f and g has an abelian subgroup of finite index. If $\\Gamma=\\pi_1(N)$ for N a complete, orientable hyperbolic 3-manifold, then $\\langle f,g\\rangle$ is abelian. The Margulis lemma is the foundation of the thick-thin decomposition of hyperbolic 3-manifolds; see [Thu97, §5]. For a given $\\Gamma$, call the supremum $\\mu(\\Gamma)$ of such $\\epsilon$ the Margulis number of $\\Gamma$. For a class of discrete subgroups of $\\operatorname{Isom}(\\mathbb{H}^3)$ one can define the Margulis constant as the infimum of the $\\mu(\\Gamma)$ over all groups $\\Gamma$ in that class. The problem asks for the Margulis constant $\\mu_3$, the infimum over the fundamental groups of all complete, finite volume, orientable hyperbolic 3-manifolds.\n\n(2) There are many partial results in this direction, e.g. [Mey87], [Sha11], [Sha13], [CS92], [CS12b], [FPS22].\n\nReferences cited:\n- [Thu97] William P. Thurston. Three-dimensional geometry and topology. Vol. 1, volume 35 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy.\n- [Mey87] Robert Meyerhoff. A lower bound for the volume of hyperbolic 3-manifolds. Canad. J. Math., 39(5):1038–1056, 1987. doi:10.4153/CJM-1987-053-6.\n- [Sha11] Peter B. Shalen. A generic Margulis number for hyperbolic 3-manifolds. In Topology and geometry in dimension three, volume 560 of Contemp. Math., pages 103–109. Amer. Math. Soc., Providence, RI, 2011. doi:10.1090/conm/560/11094.\n- [Sha13] Peter B. Shalen. Small optimal Margulis numbers force upper volume bounds. Trans. Amer. Math. Soc., 365(2):973–999, 2013. doi:10.1090/S0002-9947-2012-05657-1.\n- [CS92] Marc Culler and Peter B. Shalen. Paradoxical decompositions, 2-generator Kleinian groups, and volumes of hyperbolic 3-manifolds. J. Amer. Math. Soc., 5(2):231–288, 1992. doi:10.2307/2152768.\n- [CS12b] Marc Culler and Peter B. Shalen. Margulis numbers for Haken manifolds. Israel J. Math., 190:445–475, 2012. doi:10.1007/s11856-011-0189-z.\n- [FPS22] David Futer, Jessica S. Purcell, and Saul Schleimer. Effective bilipschitz bounds on drilling and filling. Geom. Topol., 26(3):1077–1188, 2022. doi:10.2140/gt.2022.26.1077.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2822, "problem_number": "KP-3.24", "title": "Kirby Problem 3.24", "statement": "(a) (Cannon Conjecture) If G is a finitely presented, Gromov hyperbolic group with space at infinity equal to the 2-sphere, must G be a cocompact Kleinian group?\n\n(b) More generally, if $G$ is a Gromov hyperbolic group whose boundary is planar, is G virtually a convex cocompact Kleinian group?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.24.\n\nLiterature notes:\n(1) The first of these questions is Cannon's conjecture [Can94, CS98].\n\n(2) Recall that a Gromov hyperbolic group G has a space at infinity $\\partial_\\infty G$, which is the space of geodesic rays based at some basepoint, up to a suitable notion of equivalence. A group is Kleinian if it acts properly discontinuously on $\\mathbb{H}^3$. Cocompact Kleinian groups have, as a finite-index subgroup, the fundamental group of a closed hyperbolic 3-manifold.\n\n(3) It was shown by Bestvina and Mess [BM91] that a Gromov hyperbolic group is a PD$_3$-group if and only if it is torsion free and has space at infinity the 2-sphere. Hence, Cannon's conjecture is equivalent to Problem 3.38 restricted to PD$_3$-groups that are Gromov hyperbolic.\n\n(4) Any group G acts on itself by isometries, and in the case of Gromov hyperbolic groups, this extends to an action of $\\partial_\\infty G$. So, in the setting of this problem, G comes equipped with an action on the 2-sphere, and the question is whether one can identify the 2-sphere with the Riemann sphere in such a way that the action is by Möbius transformations. A good deal of work has been done on the analytic quality of the action of $G$ on $\\partial_\\infty G$ (see, for example, [BK05, BK02b, BK02a]).\n\n(5) Markovic has shown that Cannon's conjecture holds for a group G if it has, in a certain sense, enough surface subgroups [Mar13]. More precisely, the conjecture is true if every pair of points in $\\partial_\\infty G$ are separated by the circle at infinity of a quasi-convex surface subgroup. By work of Kahn and Markovic [KM12], Kleinian groups always satisfy this hypothesis.\n\n(6) A higher-dimensional version of Cannon's conjecture is known to be true, by work of Bartels, Lück, and Weinberger [BLW10]: if $G$ is a Gromov\n\nhyperbolic group whose boundary is homeomorphic to $S^{n-1}$ with $n\\geq 7$, then $G$ is the fundamental group of an aspherical closed n-dimensional manifold.\n\n(7) Question (b) in the case where the boundary is homeomorphic to the limit set of a convex cocompact Kleinian group was asked by Walsh in [DHM15]. When $\\partial G$ is homeomorphic to $S^1$ the answer to (b) is yes; this is due to Tukia [Tuk88], Gabai [Gab92] and Casson--Jungreis [CJ94]. When $\\partial G$ contains no Sierpinski carpet and G has no 2-torsion, the answer to (b) is again yes, due to Haissinsky [Haï15]. Question (b) in the case where $\\partial G$ is a Sierpinski carpet is a conjecture of Kapovich--Kleiner [KK00].\n\nReferences cited:\n- [Can94] James W. Cannon. The combinatorial Riemann mapping theorem. Acta Math., 173(2):155–234, 1994. doi:10.1007/BF02398434.\n- [CS98] J. W. Cannon and E. L. Swenson. Recognizing constant curvature discrete groups in dimension 3. Trans. Amer. Math. Soc., 350(2):809–849, 1998. doi:10.1090/S0002-9947-98-02107-2.\n- [BM91] Mladen Bestvina and Geoffrey Mess. The boundary of negatively curved groups. J. Amer. Math. Soc., 4(3):469–481, 1991. doi:10.2307/2939264.\n- [BK05] Mario Bonk and Bruce Kleiner. Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary. Geom. Topol., 9:219–246, 2005. doi:10.2140/gt.2005.9.219.\n- [BK02b] Mario Bonk and Bruce Kleiner. Rigidity for quasi-Möbius group actions. J. Differential Geom., 61(1):81–106, 2002. URL: http://projecteuclid.org/euclid.jdg/1090351321.\n- [BK02a] Mario Bonk and Bruce Kleiner. Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math., 150(1):127–183, 2002. doi:10.1007/s00222-002-0233-z.\n- [Mar13] Vladimir Markovic. Criterion for Cannon’s conjecture. Geom. Funct. Anal., 23(3):1035–1061, 2013. doi:10.1007/s00039-013-0228-5.\n- [KM12] Jeremy Kahn and Vladimir Markovic. Immersing almost geodesic surfaces in a closed hyperbolic three manifold. Ann. of Math. (2), 175(3):1127–1190, 2012. doi: 10.4007/annals.2012.175.3.4.\n- [BLW10] Arthur Bartels, Wolfgang Lück, and Shmuel Weinberger. On hyperbolic groups with spheres as boundary. J. Differential Geom., 86(1):1–16, 2010. http://projecteuclid.org/euclid.jdg/1299766682.\n- [DHM15] Kelly Delp, Diane Hoffoss, and Jason Fox Manning. Problems in groups, geometry, and three-manifolds, 2015. arXiv:1512.04620.\n- [Tuk88] Pekka Tukia. Homeomorphic conjugates of Fuchsian groups. J. Reine Angew. Math., 391:1–54, 1988. doi:10.1515/crll.1988.391.1.\n- [Gab92] David Gabai. Convergence groups are Fuchsian groups. Ann. of Math. (2), 136(3):447–510, 1992. doi:10.2307/2946597.\n- [CJ94] Andrew Casson and Douglas Jungreis. Convergence groups and Seifert fibered 3-manifolds. Invent. Math., 118(3):441–456, 1994. doi:10.1007/BF01231540.\n- [Haï15] Peter Haı̈ssinsky. Hyperbolic groups with planar boundaries. Invent. Math., 201(1):239–307, 2015. doi:10.1007/s00222-014-0552-x.\n- [KK00] Michael Kapovich and Bruce Kleiner. Hyperbolic groups with low-dimensional boundary. Ann. Sci. École Norm. Sup. (4), 33(5):647–669, 2000. doi:10.1016/S0012-9593(00)01049-1.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2823, "problem_number": "KP-3.25", "title": "Kirby Problem 3.25", "statement": "(Bending Conjecture). (a) Is a quasi-Fuchsian group determined by the hyperbolic metric on the boundary of its convex core?\n\n(b) Is a quasi-Fuchsian group with parabolics, other than a Fuchsian group, determined by the bending measure on the boundary of its convex core?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.25.\n\nLiterature notes:\nLet $S$ be a closed surface (possibly with punctures) and let $\\Gamma=\\pi_1(S)$. Consider the space $QH(S)$ of quasi-Fuchsian representations, which corresponds to quasi-conformal deformations of Fuchsian representations, that is to the interior points of the space $AH(S)$ of hyperbolic structures on $S\\times\\mathbb{R}$ (such that the punctures of $S$ are mapped to parabolic elements). Quasi-Fuchsian groups are important examples of Kleinian groups with infinite covolume.\n\nLet $\\rho\\in QH(S)$ and let $\\Gamma_\\rho=\\rho(\\Gamma)$. The manifold $M_\\rho:=\\mathbb{H}^3/\\rho(\\Gamma)$ is homeomorphic to $S\\times\\mathbb{R}$. The limit set $\\Lambda_\\rho$ is a Jordan curve and its complement $\\Omega_\\rho=\\mathbb{CP}^1\\setminus\\Lambda_\\rho$ is a disjoint union of two topological disks: $\\Omega_\\rho=\\Omega_\\rho^+\\sqcup\\Omega_\\rho^-$. Bers' Simultaneous Uniformization Theorem proves that $QH(S)$ can be parametrized by the conformal structures of $\\Omega_\\rho/\\Gamma_\\rho$, that is by two copies of Teichmüller spaces $\\mathcal{T}(S)$.\n\nThe manifold $M_\\rho$ has a convex core $C(\\rho)$, which corresponds to the smallest convex subset of $M_\\rho$ such that the inclusion is a homotopy equivalence. This convex core can be defined as the quotient by $\\rho$ of the convex hull of the limit set of $\\rho$. In particular, provided $\\rho$ is not Fuchsian, $C(\\rho)$ is homeomorphic to $S\\times[-1,1]$. The boundary of $C(\\rho)$ is the disjoint union of two embedded pleated surfaces, which are surfaces totally geodesic almost everywhere, and bent along a measured geodesic lamination. In particular, from the pleated surfaces $\\partial C(\\rho)^+$ and $\\partial C(\\rho)^-$ one can consider the induced metric (that is, a point in $\\mathcal{T}(S)$) and the amount of bending, which is quantified by the bending laminations (that is, a point in the space $ML(S)$ of measured laminations on $S$). This defines two maps:\n\n$$\n\\mu:QH(S)\\to\\mathcal{T}(S)\\times\\mathcal{T}(S),\n$$\n\nand\n\n$$\n\\beta:QH(S)\\setminus F(S)\\to ML(S)\\times ML(S),\n$$\n\nwhere $F(S)$ is the space of Fuchsian representations. W. Thurston conjectured that the maps $\\mu$ and $\\beta$ are homeomorphisms onto their images.\n\nAs steps in proving Thurston's conjecture, Sullivan [Sul85] proved that the map $\\mu$ is surjective, and Bonahon and Otal [BO04] described the image of the map $\\beta$, which consists of pairs of laminations on $S$ that fill and such that the measure on each isolated leaf is less than $\\pi$. Dular and Schlenker [DS24b] in 2024 showed the injectivity of the map $\\beta$ when $S$ has no punctures. Unfortunately, their proof does not show that quasi-Fuchsian manifolds are infinitesimally rigid with respect to the measured bending lamination on the boundary of their convex core. As Bonahon [Bon96] proved, this infinitesimal rigidity is equivalent to the infinitesimal rigidity with respect to the induced metric on the boundary of the convex core.\n\nWe can then state the following still open questions. The first two questions are (a) and (b) above:\n\n(i) When $S$ is closed, is the map $\\mu$ a homeomorphism?\n\n(ii) When $S$ has punctures, are the maps $\\mu$ and $\\beta$ homeomorphisms?\n\n(iii) Are the maps $\\mu$ and $\\beta$ infinitesimally rigid?\n\nPartial results for the map $\\beta$ are due to Series [Ser06] in the case of the once-punctured torus $S_{1,1}$, Bonahon [Bon05] in a neighborhood of the Fuchsian locus, and Hodgson--Kerckhoff [HK98] for quasi-Fuchsian manifolds bent along rational laminations. (Guéritaud [Gué09] gave an alternative and independent proof of Series' result.)\n\nAnti-de Sitter geometry is a Lorentzian analogue of hyperbolic geometry and the space of globally hyperbolic maximal compact (GHMC) AdS manifolds on $S\\times\\mathbb{R}$ has many similarities with the space of quasi-Fuchsian manifolds discussed above. In particular, the conjectures discussed above have an analogue also in this context. In [BDMS21] you can see a discussion for the universal version of the same problem both in hyperbolic and anti-de Sitter space.\n\nReferences cited:\n- [Sul85] Dennis Sullivan. Quasiconformal homeomorphisms and dynamics. II. Structural stability implies hyperbolicity for Kleinian groups. Acta Math., 155(3-4):243–260, 1985. doi:10.1007/BF02392543.\n- [BO04] Francis Bonahon and Jean-Pierre Otal. Laminations measurées de plissage des variétés hyperboliques de dimension 3. Ann. of Math. (2), 160(3):1013–1055, 2004. doi:10.4007/annals.2004.160.1013.\n- [DS24b] Bruno Dular and Jean-Marc Schlenker. Convex co-compact hyperbolic manifolds are determined by their pleating lamination, 2024. arXiv:2403.10090.\n- [Bon96] Francis Bonahon. Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form. Ann. Fac. Sci. Toulouse Math. (6), 5(2):233–297, 1996. URL: http://www.numdam.org/item?id=AFST 1996 6 5 2 233 0.\n- [Ser06] Caroline Series. Thurston’s bending measure conjecture for once punctured torus groups. In Spaces of Kleinian groups, volume 329 of London Math. Soc. Lecture Note Ser., pages 75–89. Cambridge Univ. Press, Cambridge, 2006.\n- [Bon05] Francis Bonahon. Kleinian groups which are almost Fuchsian. J. Reine Angew. Math., 587:1–15, 2005. doi:10.1515/crll.2005.2005.587.1.\n- [HK98] Craig D. Hodgson and Steven P. Kerckhoff. Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. J. Differential Geom., 48(1):1–59, 1998. http://projecteuclid.org/euclid.jdg/1214460606.\n- [Gué09] François Guéritaud. Triangulated cores of punctured-torus groups. J. Differential Geom., 81(1):91–142, 2009. http://projecteuclid.org/euclid.jdg/1228400629.\n- [BDMS21] Francesco Bonsante, Jeffrey Danciger, Sara Maloni, and Jean-Marc Schlenker. The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti–de Sitter geometry. Geom. Topol., 25(6):2827–2911, 2021. With an appendix by Boubacar Diallo. doi:10.2140/gt.2021.25.2827.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2824, "problem_number": "KP-3.26", "title": "Kirby Problem 3.26", "statement": "Let $M$ be a finite-volume hyperbolic 3-manifold, and let $M^1$ be a minimal-index finite cover of $M$ such that $\\pi_1(M^1)$ embeds in a right-angled Artin group. Can one obtain good control over the complexity of the smallest right-angled Artin group containing $\\pi_1(M^1)$ from the data of $M$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.26.\n\nLiterature notes:\n(1) Work of Agol and Wise [Ago13, Wis21] established that there is always such a finite cover $M^1$.\n\n(2) The complexity of a right-angled Artin group could be interpreted in many different ways, including:\n\n(i) Number of vertices in the defining graph; (ii) Cohomological dimension; (iii) Chromatic number of the defining graph; (iv) Diameter of the defining graph.\n\n(3) This question is related to, but different from questions such as finding the minimal index of a subgroup contained in a right-angled Artin group, the minimal index Haken cover, etc.\n\nReferences cited:\n- [Ago13] Ian Agol. The virtual Haken conjecture. Doc. Math., 18:1045–1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning, https://elibm.org/article/10000267. doi:10.4171/DM/421.\n- [Wis21] Daniel T. Wise. The structure of groups with a quasiconvex hierarchy, volume 209 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, [2021] ©2021.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2825, "problem_number": "KP-3.27", "title": "Kirby Problem 3.27", "statement": "(a) What is the computational complexity of the homeomorphism problem for compact, orientable 3-manifolds?\n\n(b) Is there a polynomial-time algorithm to recognize the 3-sphere?\n\n(c) What is the computational complexity of the recognition problem for other 3-manifolds?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.27.\n\nLiterature notes:\n(1) The homeomorphism problem for 3-manifolds is as follows. One is given triangulations of two 3-manifolds, M and N, and the problem asks whether N is homeomorphic to M.\n\n(2) For a fixed 3-manifold M, its recognition problem is as follows. One is given a triangulation of a 3-manifold N, and the problem asks whether N is homeomorphic to M. It is a restricted version of the homeomorphism problem, where one of the manifolds is fixed.\n\n(3) The homeomorphism problem for compact orientable 3-manifolds has been solved [Kup19a], [SS14]. However, there is currently a very large gap between known upper bounds on its complexity and known lower bounds. If we are given two 3-manifolds triangulated with $t_1$ and $t_2$ tetrahedra, the fastest known running time for an algorithm to determine whether they are homeomorphic is\n\n$$\n2^{2^{\\cdot^{\\cdot^{\\cdot^{t_1+t_2}}}}}\n$$\n\nwhere the tower of exponentials has some fixed (but currently unknown) height [Kup19a]. An explicit bound was given by Scull [Scu21] in the\n\ncase where the manifolds are known to be hyperbolic (but the hyperbolic structure is not part of the input).\n\n(4) The best known lower bound on the computational complexity of the homeomorphism problem is rather weak. Lackenby [Lac17b] showed the homeomorphism problem is at least as hard as the graph isomorphism problem. The graph isomorphism problem is widely believed not be solvable in polynomial time (partly because of [Sch88]). However, it is solvable in quasi-polynomial time [Bab16].\n\n(5) The recognition problem is significantly more tractable than the homeomorphism problem. For example, in the case of the 3-sphere, the recognition problem is known to lie in NP [Sch11], [Iva08], and co-NP, assuming the Generalized Riemann Hypothesis, [Zen18]. The first solution to 3-sphere recognition was given by Rubinstein [Rub95, Rub97] and Thompson [Tho94].\n\n(6) There are now many manifolds for which the recognition problem is known to be in NP, including the solid torus [Iva08], a genus g handlebody [Iva08], and the product of a compact orientable surface and an interval [Lac21a]. None of these problems is known to be solvable in polynomial time, however.\n\n(7) An efficient solution to 3-sphere recognition would give an efficient solution to the unknot recognition problem (Problem 1.76). This is because one could perform $\\pm1$ Dehn surgery on the knot and, by work of Gordon and Luecke [GL89], the result is the 3-sphere if and only if the given knot is the unknot. (This assumes that the knot is given by means of a diagram, rather than via a triangulation of its exterior, so that it is possible to build a triangulation of the filled-in manifold efficiently.)\n\nReferences cited:\n- [Kup19a] Greg Kuperberg. Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization. Pacific J. Math., 301(1):189–241, 2019. doi:10.2140/pjm.2019.301.189.\n- [SS14] Peter Scott and Hamish Short. The homeomorphism problem for closed 3-manifolds. Algebr. Geom. Topol., 14(4):2431–2444, 2014. doi:10.2140/agt.2014.14.2431.\n- [Scu21] Joe Scull. The homeomorphism problem for hyperbolic manifolds I, 2021. arXiv: 2108.00779.\n- [Lac17b] Marc Lackenby. Some conditionally hard problems on links and 3-manifolds. Discrete Comput. Geom., 58(3):580–595, 2017. doi:10.1007/s00454-017-9905-8.\n- [Sch88] Uwe Schöning. Graph isomorphism is in the low hierarchy. J. Comput. System Sci., 37(3):312–323, 1988. doi:10.1016/0022-0000(88)90010-4.\n- [Bab16] László Babai. Graph isomorphism in quasipolynomial time [extended abstract]. In STOC’16—Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, pages 684–697. ACM, New York, 2016. doi:10.1145/2897518.2897542.\n- [Sch11] Saul Schleimer. Sphere recognition lies in NP. In Low-dimensional and symplectic topology, volume 82 of Proc. Sympos. Pure Math., pages 183–213. Amer. Math. Soc., Providence, RI, 2011. doi:10.1090/pspum/082/2768660.\n- [Iva08] S. V. Ivanov. The computational complexity of basic decision problems in 3-dimensional topology. Geom. Dedicata, 131:1–26, 2008. doi:10.1007/s10711-007-9210-4.\n- [Zen18] Raphael Zentner. Integer homology 3-spheres admit irreducible representations in $\\mathrm{SL}(2,\\mathbb{C})$. Duke Math. J., 167(9):1643–1712, 2018. doi:10.1215/00127094-2018-0004.\n- [Rub95] Joachim H. Rubinstein. An algorithm to recognize the 3-sphere. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 601–611. Birkhäuser, Basel, 1995.\n- [Rub97] J. H. Rubinstein. Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3-dimensional manifolds. In Geometric topology (Athens, GA, 1993), volume 2.1 of AMS/IP Stud. Adv. Math., pages 1–20. Amer. Math. Soc., Providence, RI, 1997. doi:10.1090/amsip/002.1/01.\n- [Tho94] Abigail Thompson. Thin position and the recognition problem for $S^{3}$. Math. Res. Lett., 1(5):613–630, 1994. doi:10.4310/MRL.1994.v1.n5.a9.\n- [Lac21a] Marc Lackenby. The efficient certification of knottedness and Thurston norm. Adv. Math., 387:Paper No. 107796, 142, 2021. doi:10.1016/j.aim.2021.107796.\n- [GL89] C. McA. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc., 2(2):371–415, 1989. doi:10.2307/1990979.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2826, "problem_number": "KP-3.28", "title": "Kirby Problem 3.28", "statement": "How many Pachner moves are needed to pass between two triangulations of a compact 3-manifold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.28.\n\nLiterature notes:\n(1) Pachner [Pac91] proved that any two combinatorial triangulations of a compact piecewise-linear $n$-manifold differ by a finite sequence of modifications that are now called Pachner moves. In the case of closed $n$-manifolds, these take the following form: pick a subcomplex of the triangulation that is combinatorially isomorphic to a union $B$ of facets in the boundary of an $(n+1)$-simplex $\\Delta^{n+1}$, and replace the copy of $B$ by $\\operatorname{cl}(\\partial\\Delta^{n+1}\\setminus B)$. When $n=3$, there are four types of such move: a $(1,4)$-move, which replaces a single tetrahedron with four tetrahedra arranged around a new vertex; a $(4,1)$-move, which is the reverse of this procedure; a $(2,3)$-move, which replaces two tetrahedra that share a common facet with three tetrahedra arranged around a new edge; and a $(3,2)$-move, which is the reverse of the above move. For $n$-manifolds with boundary, there is a further type of Pachner move, which involves attaching an $n$-simplex onto a union of facets in the boundary of the manifold, or the reverse of this.\n\n(2) One can ask this question for general compact 3-manifolds, or for a fixed 3-manifold, or for a class of 3-manifolds (such as all Seifert fibered spaces).\n\n(3) The question can be formulated in terms of the Pachner function, which is defined as follows. For a compact 3-manifold M and positive integers $t_1$ and $t_2$, consider all pairs of triangulations of M with $t_1$ and $t_2$ tetrahedra and let $P_M(t_1,t_2)$ be the maximal number of moves needed to pass between any two such triangulations. Let $P(t_1,t_2)$ be the maximum of $P_M(t_1,t_2)$ over all compact 3-manifolds M. The question asks how $P(t_1,t_2)$ grows as a function of $t_1$ and $t_2$.\n\n(4) Any computable upper bound on $P$ leads to a solution to the homeomorphism problem for compact 3-manifolds, as follows. Given two triangulations of two 3-manifolds $M_1$ and $M_2$ with $t_1$ and $t_2$ tetrahedra, the task is to decide whether they are homeomorphic. Pachner's theorem [Pac91] states that if they are homeomorphic, then the triangulations differ by a sequence of Pachner moves, with length at most $P(t_1,t_2)$. Thus if $P$ has a computable upper bound, then one need only try all sequences of Pachner moves with at most this length, starting from one triangulation. If none of the resulting triangulations is combinatorially isomorphic to the second triangulation, then one can deduce that the manifolds are not homeomorphic.\n\n(5) Mijatovic [Mij05a] proved that for all knot exteriors M, $P_M(t_1,t_2)$ is at most\n\n$$\n2^{2^{\\cdot^{\\cdot^{\\cdot^{t_1+t_2}}}}}\n$$\n\nwhere the height of the tower is $c^{t_1}+c^{t_2}$ and where $c=2^{200}$. The same bound holds for all `fibre-free' Haken 3-manifolds [Mij05b]. When M is the 3-sphere, Mijatovic [Mij03] and King [Kin01] found upper bounds on $P_M$ of exponential type (more precisely, of the form $k^{t_1^2}+k^{t_2^2}$ for some constant $k$). However, an explicit bound is not known for $P$ in general, or even for $P_M$ for a general hyperbolic 3-manifold M.\n\n(6) Very few nontrivial lower bounds on $P$ are known (for example [Bur11] and [JRST20, Remark 15]).\n\nReferences cited:\n- [Pac91] Udo Pachner. P.L. homeomorphic manifolds are equivalent by elementary shellings. European J. Combin., 12(2):129–145, 1991. doi:10.1016/S0195-6698(13)80080-7.\n- [Mij05a] Aleksandar Mijatović. Simplical structures of knot complements. Math. Res. Lett., 12(5-6):843–856, 2005. doi:10.4310/MRL.2005.v12.n6.a6.\n- [Mij05b] Aleksandar Mijatović. Triangulations of fibre-free Haken 3-manifolds. Pacific J. Math., 219(1):139–186, 2005. doi:10.2140/pjm.2005.219.139.\n- [Mij03] Aleksandar Mijatović. Simplifying triangulations of $S^{3}$. Pacific J. Math., 208(2):291–324, 2003. doi:10.2140/pjm.2003.208.291.\n- [Kin01] Simon A. King. The size of triangulations supporting a given link. Geom. Topol., 5:369–398, 2001. doi:10.2140/gt.2001.5.369.\n- [Bur11] Benjamin A. Burton. The Pachner graph and the simplification of 3-sphere triangulations. In Computational geometry (SCG’11), pages 153–162. ACM, New York, 2011. doi:10.1145/1998196.1998220.\n- [JRST20] William Jaco, J. Hyam Rubinstein, Jonathan Spreer, and Stephan Tillmann. $\\mathbb{Z}_2$-Thurston norm and complexity of 3-manifolds, II. Algebr. Geom. Topol., 20(1):503– 529, 2020. doi:10.2140/agt.2020.20.503.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2827, "problem_number": "KP-3.29", "title": "Kirby Problem 3.29", "statement": "(a) Given a closed hyperbolic 3-manifold $M$, can one find an explicit bound on the degree of a finite cover $\\widetilde M$ having $b_1(\\widetilde M)>0$?\n\n(b) For example, is there such a finite cover with degree at most a polynomial function of the number of tetrahedra in some triangulation of $M$?\n\n(c) Similarly, can one find an explicit upper bound on the degree of a finite cover $\\widetilde M_1$ that fibers over the circle?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.29.\n\nLiterature notes:\n(1) Agol and Wise proved that $M$ has a finite cover $\\widetilde M$ with $b_1(\\widetilde M)>0$, and indeed a finite cover $\\widetilde M_1$ that fibers over the circle [Ago13, Wis21]. However, the proof is non-constructive, and it seems hard to extract an upper bound on the covering degree.\n\n(2) Given $M$, one can compute the degrees of covers $\\widetilde M$ and $\\widetilde M_1$ as above, since one can start building all the finite covers of $M$, ordered by covering degree, and for each one, one can determine whether $b_1>0$ or whether it fibers over the circle.\n\n(3) Agol showed that an irreducible 3-manifold whose fundamental group satisfies a certain group-theoretic property called RFRS (residually finite $\\mathbb{Q}$-solvable) is virtually fibered [Ago08]. One approach to the above problem would be to find an explicit finite cover with RFRS fundamental group, at which point it seems more likely that one could get good bounds needed for a fibered cover. Currently, explicit towers of RFRS covers are only known in a few specific examples, including for certain Bianchi groups due to Agol--Stover [AS23].\n\nReferences cited:\n- [Ago13] Ian Agol. The virtual Haken conjecture. Doc. Math., 18:1045–1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning, https://elibm.org/article/10000267. doi:10.4171/DM/421.\n- [Wis21] Daniel T. Wise. The structure of groups with a quasiconvex hierarchy, volume 209 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, [2021] ©2021.\n- [Ago08] Ian Agol. Criteria for virtual fibering. J. Topol., 1(2):269–284, 2008. doi:10.1112/jtopol/jtn003.\n- [AS23] Ian Agol and Matthew Stover. Congruence RFRS towers. Ann. Inst. Fourier (Grenoble), 73(1):307–333, 2023. With an appendix by Mehmet Haluk Şengün. doi:10.5802/aif.3532.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2828, "problem_number": "KP-3.30", "title": "Kirby Problem 3.30", "statement": "(a) What is the computational complexity of determining whether a compact 3-manifold admits a hyperbolic structure?\n\n(b) If a compact 3-manifold does admit a hyperbolic structure, what is the computational complexity of finding it?\n\n(c) In particular, do these problems lie in the complexity classes NP and FNP respectively?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.30.\n\nLiterature notes:\n(1) The most efficient known algorithm to find a hyperbolic structure on a 3-manifold if one exists is due to Scull [Scu21], based on work of Kuperberg [Kup19a]. If the 3-manifold is given as a triangulation with $t$ tetrahedra, it runs in time\n\n$$\n2^{2^{t^{O(t)}}}.\n$$\n\nHowever, in practice, SnapPy [CDGW] is good at finding hyperbolic structures if they exist, and hence, it seems quite plausible that hyperbolicity can be efficiently certified, as the question proposes.\n\n(2) The problem of deciding whether a 3-manifold is hyperbolic, without actually finding a hyperbolic structure, may be much easier. To achieve this, one might establish the existence of an alternative structure that is equivalent to hyperbolicity. For example, one might show that its fundamental group is infinite and has no $\\mathbb{Z}\\times\\mathbb{Z}$ subgroup. Alternatively, one might give a metric triangulation that is CAT(-1). Another possibility is to exhibit a finite cover that fibers over the circle with pseudo-Anosov monodromy, which exists by work of Agol [Ago13]. However, none of these approaches seems straightforward.\n\n(3) The problem of deciding hyperbolicity is known to be in the complexity class co-NP assuming the Generalized Riemann Hypothesis, by work of Hass and Kuperberg [HK12a]. In the case of manifolds with nonempty toroidal boundary, this was established by Haraway and Hoffman [HIH22],\n\nwithout requiring GRH. Hence, assuming GRH, the problem of deciding whether a compact 3-manifold is hyperbolic runs in exponential time.\n\nReferences cited:\n- [Scu21] Joe Scull. The homeomorphism problem for hyperbolic manifolds I, 2021. arXiv: 2108.00779.\n- [Kup19a] Greg Kuperberg. Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization. Pacific J. Math., 301(1):189–241, 2019. doi:10.2140/pjm.2019.301.189.\n- [CDGW] Marc Culler, Nathan M. Dunfield, Matthias Goerner, and Jeffrey R. Weeks. SnapPy, a computer program for studying the geometry and topology of 3-manifolds. Available at http://snappy.computop.org.\n- [Ago13] Ian Agol. The virtual Haken conjecture. Doc. Math., 18:1045–1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning, https://elibm.org/article/10000267. doi:10.4171/DM/421.\n- [HK12a] Joel Hass and Greg Kuperberg. The complexity of recognizing the 3-sphere. Oberwolfach Reports, 9(2), 2012. https://ems.press/content/serial-article-files/46393.\n- [HIH22] Robert Haraway III and Neil R. Hoffman. On the complexity of cusped non-hyperbolicity, 2022. arXiv:1907.01675.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2829, "problem_number": "KP-3.31", "title": "Kirby Problem 3.31", "statement": "Suppose M is a closed 3-manifold.\n\n(a) Can one decide if the fundamental group of M is left-orderable?\n\n(b) What is the complexity of a certificate of left-orderability, or of non-left-orderability?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.31.\n\nLiterature notes:\n(1) A nontrivial group G is left-orderable if it admits a left-invariant total order.\n\n(2) There is no algorithm to determine in general, from a finite presentation of a group G, whether G is left-orderable.\n\n(3) It is known that $\\pi_1(M)$ is left-orderable if and only if there is a surjective homomorphism from $\\pi_1(M)$ to a nontrivial left-orderable group. Thus, one may ask: among all such $\\rho:\\pi_1(M)\\to G$, what is the least complexity of a machine that will recognize exactly the elements $a\\in\\pi_1(M)$ (expressed as the canonical representative in some regular prefix-closed bijective language of geodesics if one exists, say) for which $\\rho(a)>\\mathrm{id}$? For example, if $b_1(M)>0$ one may take $G=\\mathbb{Z}$ and $\\rho:\\pi_1(M)\\to G$ any nontrivial element of $H^{1}(M)$, and then the recognition may be done with a counter automaton (but no simpler machine for typical M, e.g. M hyperbolic).\n\n(4) One may also ask for the least complexity of a left order on $\\pi_1(M)$ itself. This might be considerably more complicated than the least complexity of a left order on an infinite quotient.\n\n(5) The L-space Conjecture (Problem 3.48) asserts that for an irreducible 3-manifold M the following conditions are equivalent:\n\n(i) $M$ has left-orderable fundamental group; (ii) $M$ admits a coorientable taut foliation; (iii) $M$ is not an L-space. The second property is known to be algorithmically decidable for atoroidal manifolds by Agol--Li [AL03], who proved more generally that whether a 3-manifold has a coorientable Reebless foliation is decidable; the third is algorithmically decidable for the elementary reason that Heegaard Floer homology can be calculated. Thus, if the property of having a left-orderable group is not decidable, the L-space Conjecture is false.\n\n(6) This problem is similar to Question 8.7 in [Cal03].\n\nReferences cited:\n- [AL03] Ian Agol and Tao Li. An algorithm to detect laminar 3-manifolds. Geom. Topol., 7:287–309, 2003. doi:10.2140/gt.2003.7.287.\n- [Cal03] Danny Calegari. Problems in foliations and laminations of 3-manifolds. In Topology and geometry of manifolds (Athens, GA, 2001), volume 71 of Proc. Sympos. Pure Math., pages 297–335. Amer. Math. Soc., Providence, RI, 2003. doi: 10.1090/pspum/071/2024640.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2830, "problem_number": "KP-3.32", "title": "Kirby Problem 3.32", "statement": "Is there an algorithm to determine whether two closed, embedded surfaces in $\\mathbb{R}^3$ are isotopic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.32.\n\nLiterature notes:\n(1) The problem of deciding whether two incompressible surfaces in a compact orientable irreducible 3-manifold are isotopic is solvable, using work of Waldhausen [Wal68b, Proposition 5.4] (see also [Bar25, Theorem 1.2]). However, surfaces in $\\mathbb{R}^3$ (other than 2-spheres) are compressible.\n\n(2) All spheres embedded in $\\mathbb{R}^3$ are isotopic by the Schoenflies theorem.\n\n(3) Any torus embedded in $\\mathbb{R}^3$ is the boundary of a regular neighbourhood of a knot, and therefore the isotopy problem for tori is basically the same as the equivalence problem for knots, which has been solved by Haken [Hak68], Hemion [Hem79] and Matveev [Mat07].\n\n(4) The case of genus two surfaces was recently solved by Baroni [Bar25]. The challenge for higher genus surfaces is as follows. Given two closed connected surfaces embedded in $\\mathbb{R}^3$, one must decide whether their exteriors are homeomorphic. If they are not, then the surfaces are not isotopic. However, if their exteriors are homeomorphic, then this is not enough to deduce that the surfaces are isotopic; one must check that there is a homeomorphism compatible with the gluing maps along the surfaces. As the exteriors might have infinite mapping class group, it is not clear that this can be checked algorithmically.\n\nReferences cited:\n- [Wal68b] Friedhelm Waldhausen. On irreducible 3-manifolds which are sufficiently large. Ann. of Math. (2), 87:56–88, 1968. doi:10.2307/1970594.\n- [Bar25] Filippo Baroni. Classification of genus-two surfaces in $S^{3}$. Algebr. Geom. Topol., 25(8):4719–4785, 2025. doi:10.2140/agt.2025.25.4719.\n- [Hak68] Wolfgang Haken. Some results on surfaces in 3-manifolds. In Studies in Modern Topology, volume Vol. 5 of Studies in Mathematics, pages 39–98. Math. Assoc. America,, 1968.\n- [Hem79] Geoffrey Hemion. On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds. Acta Math., 142(1-2):123–155, 1979. doi:10.1007/BF02395059.\n- [Mat07] Sergei Matveev. Algorithmic topology and classification of 3-manifolds, volume 9 of Algorithms and Computation in Mathematics. Springer, Berlin, second edition, 2007.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2831, "problem_number": "KP-3.33", "title": "Kirby Problem 3.33", "statement": "Let M and N be closed orientable 3-manifolds. Prove that if there is a degree-1 map $f:M\\to N$ then $g(M)\\geq g(N)$, where $g(M)$ is the Heegaard genus of $M$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.33.\n\nLiterature notes:\n(1) This is a long-standing conjecture, dating back to Haken and Waldhausen [Hak66, Wal70]. The conjecture implies the Poincaré Conjecture: suppose a closed 3-manifold N is homotopy equivalent to $S^3$. Since a homotopy equivalence is a degree-1 map, the conjecture implies that $0=g(S^3)\\geq g(N)$ and therefore that N is $S^3$.\n\n(2) A general approach to this conjecture was discussed in [RW92, Li22]. However, not much progress has been made. Let $M=W\\cup_T V$ be an amalgamation of two manifolds W and V along an incompressible torus, with W being the exterior of a knot in a homology 3-sphere. In this case, there is a canonical degree-1 map that pinches W into a solid torus. It is proved in [Li22] that the Heegaard genus of the resulting 3-manifold is at most as large as the Heegaard genus of M, verifying the conjecture in a special case.\n\nReferences cited:\n- [Hak66] Wolfgang Haken. On homotopy 3-spheres. Illinois J. Math., 10:159–178, 1966. http: //projecteuclid.org/euclid.ijm/1256055210.\n- [Wal70] F. Waldhausen. On mappings of handlebodies and Heegaard splittings. Topology of Manifolds (Markham Publishing Company), pages 205–211, 1970.\n- [RW92] Yong Wu Rong and Shi Cheng Wang. The preimages of submanifolds. Math. Proc. Cambridge Philos. Soc., 112(2):271–279, 1992. doi:10.1017/S030500410007095X.\n- [Li22] Tao Li. Heegaard genus, degree-one maps, and amalgamation of 3-manifolds. J. Topol., 15(3):1540–1579, 2022. doi:10.1112/topo.12253.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2832, "problem_number": "KP-3.34", "title": "Kirby Problem 3.34", "statement": "Do any two genus-g Heegaard splittings of a closed, orientable 3-manifold M become equivalent after at most g stabilizations?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.34.\n\nLiterature notes:\nThe answer \"yes\" was conjectured by W. Thurston. This has been studied extensively and upper bounds on the number of necessary stabilizations are known [Joh95], [RS96], but the conjectured optimal bound of g is open. It is known that g stabilizations may be needed [HTT09]; see also [Kir97, Problem 3.89].\n\nReferences cited:\n- [Joh95] Klaus Johannson. Topology and combinatorics of 3-manifolds, volume 1599 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1995. doi:10.1007/BFb0074005.\n- [RS96] Hyam Rubinstein and Martin Scharlemann. Comparing Heegaard splittings of non-Haken 3-manifolds. Topology, 35(4):1005–1026, 1996. doi:10.1016/0040-9383(95) 00055-0.\n- [HTT09] Joel Hass, Abigail Thompson, and William Thurston. Stabilization of Heegaard splittings. Geom. Topol., 13(4):2029–2050, 2009. doi:10.2140/gt.2009.13.2029.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2833, "problem_number": "KP-3.35", "title": "Kirby Problem 3.35", "statement": "Given a compact manifold M, let $r(M)$ denote the rank of its fundamental group and $g(M)$ denote its Heegaard genus.\n\n(a) Does every closed orientable hyperbolic 3-manifold M with $r(M)$ = 2 satisfy $g(M)$ = 2?\n\n(b) Is there a non-Haken manifold $M$ such that $r(M)0$ such that $g(M)\\leq k r(M)$ for every closed orientable 3-manifold $M$?\n\n(d) If $\\{M_i\\}$ is the set of congruence covers of an arithmetic hyperbolic 3-manifold M, is the infimum of $r(M_i)/\\operatorname{vol}(M_i)$ zero or positive?\n\n(e) Is there a knot $K\\subset S^3$ such that $r(S^3\\setminus\\nu(K))0$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.49.\n\nLiterature notes:\nIn proving that the existence of a coorientable taut foliation on Y implies that Y is not an L-space, Ozsváth--Szabó construct [OS04b] a weak symplectic filling of a particular contact structure on Y. This filling can be chosen to have $b_2^+>0$. Recall that for rational homology spheres, admitting a weak symplectic filling is equivalent to admitting a strong one [OO05].\n\nReferences cited:\n- [OS04b] Peter Ozsváth and Zoltán Szabó. Holomorphic disks and genus bounds. Geom. Topol., 8:311–334, 2004. doi:10.2140/gt.2004.8.311.\n- [OO05] Hiroshi Ohta and Kaoru Ono. Simple singularities and symplectic fillings. J. Differential Geom., 69(1):1–42, 2005. doi:10.4310/jdg/1121540338.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2848, "problem_number": "KP-3.50", "title": "Kirby Problem 3.50", "statement": "(The Floer Poincaré Conjecture). If Y is an integral homology sphere that is an L-space, show that Y is $S^3$ or the connected sum of some copies of the Poincaré sphere (with arbitrary orientations).", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.50.\n\nLiterature notes:\n(1) This is in the spirit of [Kir97, Problem 3.106], which asks a similar question about instanton Floer homology.\n\n(2) The statement is already known to be true for several classes of homology spheres: Dehn surgeries on knots in $S^3$ by [OS11]; Brieskorn spheres by [OS03b] in combination with [Eft09]; and manifolds with incompressible tori by [Eft18].\n\n(3) In combination with the L-space Conjecture (Problem 3.48), a solution to the problem would provide an alternative route to the Poincaré Conjecture: if Y is a homotopy 3-sphere, then $\\pi_1(Y)$ is not left-orderable, hence an L-space, hence $S^3$ or a connected sum of Poincaré spheres (but the latter possibility contradicts simple connectivity).\n\n(4) One can phrase an analogous conjecture for framed instanton homology, which would say that if $I^{\\#}(Y)\\cong\\mathbb{Z}$ then $Y=S^3$. If this were true then it would imply the Poincaré Conjecture immediately (without the L-space conjecture), because trivial $\\pi_1$ implies $I^{\\#}(Y)\\cong\\mathbb{Z}$. It would also imply a positive answer to Problem 3.52. Note that $I^{\\#}$ is somewhat different from $\\widehat{HF}$; for instance, Bhat has announced that $I^{\\#}$ of the Poincaré sphere has 2-torsion [Bha24].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [OS11] Peter Ozsváth and Zoltán Szabó. Knot Floer homology and rational surgeries. Algebr. Geom. Topol., 11(1):1–68, 2011. doi:10.2140/agt.2011.11.1.\n- [OS03b] Peter Ozsváth and Zoltán Szabó. On the Floer homology of plumbed threemanifolds. Geom. Topol., 7:185–224, 2003. doi:10.2140/gt.2003.7.185.\n- [Eft09] Eaman Eftekhary. Seifert fibered homology spheres with trivial Heegaard Floer homology, 2009. arXiv:0909.3975.\n- [Eft18] Eaman Eftekhary. Bordered Floer homology and existence of incompressible tori in homology spheres. Compos. Math., 154(6):1222–1268, 2018. doi:10.1112/s0010437x18007054.\n- [Bha24] Deeparaj Bhat. Surgery exact triangles in instanton theory, 2024. arXiv:2311.04242.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2849, "problem_number": "KP-3.51", "title": "Kirby Problem 3.51", "statement": "Suppose Y is a rational homology 3-sphere such that every homomorphism $\\pi_1(Y)$ $\\to$ $\\operatorname{SU}(2)$ has abelian image. Does it follow that\n\n$$\n\\dim_{\\mathbb{C}} I^{\\#}(Y;\\mathbb{C})=|H_1(Y;\\mathbb{Z})|,\n$$\n i.e., that Y is an instanton L-space?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.51.\n\nLiterature notes:\n(1) The conclusion follows if the corresponding set of reducible flat connections is Morse--Bott nondegenerate for the Chern--Simons functional, which is equivalent to Y being cyclically finite [BN90, BS18b]. This holds automatically, for instance, if Y is a homology 3-sphere. The problem is whether the conclusion follows without assuming that Y is cyclically finite.\n\n(2) If the L-space conjecture and the conjectured isomorphism between instanton homology and Heegaard Floer homology both hold (see Problems 3.59 and 3.48), then an affirmative answer to the question in the problem would imply the following: for Y an irreducible rational homology 3-sphere, if $\\pi_1(Y)$ is left-orderable, then $\\pi_1(Y)$ admits an irreducible $\\operatorname{SU}(2)$-representation. Proving the latter would be interesting in its own right.\n\n(3) One can ask a similar question regarding instanton Floer homology and representations from $\\pi_1(Y)$ to $\\operatorname{SU}(N)$ for $N>2$; see [DIS24] for recent work in the $N=3$ case.\n\nReferences cited:\n- [BN90] S. Boyer and A. Nicas. Varieties of group representations and Casson’s invariant for rational homology 3-spheres. Trans. Amer. Math. Soc., 322(2):507–522, 1990. doi:10.2307/2001712.\n- [BS18b] John A. Baldwin and Steven Sivek. Stein fillings and $\\mathrm{SU}(2)$ representations. Geom. Topol., 22(7):4307–4380, 2018. doi:10.2140/gt.2018.22.4307.\n- [DIS24] Aliakbar Daemi, Nobuo Iida, and Christopher Scaduto. Rank three instantons, representations and sutures, 2024. arXiv:2402.10448.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2850, "problem_number": "KP-3.52", "title": "Kirby Problem 3.52", "statement": "(a) Does every closed 3-manifold M besides the 3-sphere admit a nontrivial representation $\\pi_1(M)$ $\\to$ $\\operatorname{SU}(2)$?\n\n(b) For which M with nonabelian fundamental group does every homomorphism $\\pi_1(M)$ $\\to$ $\\operatorname{SU}(2)$ have abelian image?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.52.\n\nLiterature notes:\n(1) The question in (a) is [Kir97, Problem 3.105(A)].\n\n(2) The answer to (a) is yes if $M\\not\\cong S^3$ is Seifert fibered [FS90] or toroidal [LPCZ23] (see also [BS22]), leaving only the hyperbolic case open. It is also yes whenever M is Dehn surgery on a knot in $S^3$ [KM04], or the branched double cover of a knot in $S^3$ [CNS16, Zen17] (using results of Kronheimer and Mrowka [KM10]), or the boundary of a Stein domain with nontrivial homology [BS18b]. These ultimately rely on nonvanishing results for some form of instanton Floer homology, or closely related techniques.\n\n(3) An affirmative answer to (a) would follow from two widely believed conjectures, that (i) framed instanton homology is isomorphic to the \"hat\" version of Heegaard Floer homology over Q; and (ii) the only irreducible integer homology sphere L-spaces are $S^3$ and the Poincaré homology sphere with either orientation; see 3.59 and 3.50 for discussions of (i) and (ii).\n\n(4) The analogue of (a) for representations $\\pi_1(M)$ $\\to$ $\\operatorname{SL}(2, \\mathbb{C})$ has been answered affirmatively by Zentner [Zen18].\n\n(5) Regarding the question in (b), there are many closed 3-manifolds M such that $\\pi_1(M)$ is nonabelian but every representation $\\pi_1(M)$ $\\to$ $\\operatorname{SU}(2)$ has abelian image. This includes infinitely many graph manifolds [Mot88], built by gluing together pairs of torus knot exteriors. The Seifert fibered\n\nexamples are classified [SZ22], and only a handful of hyperbolic examples are known, including the manifold known as Vol3, as reported by Dunfield. Each of these examples has nontrivial first homology.\n\n(6) A rational homology sphere for which every representation $\\pi_1(M)$ $\\to$ $\\operatorname{SU}(2)$ has abelian image is generally (and perhaps always) an instanton L-space; see Problem 3.51.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [FS90] Ronald Fintushel and Ronald J. Stern. Instanton homology of Seifert fibred homology three spheres. Proc. London Math. Soc. (3), 61(1):109–137, 1990. doi: 10.1112/plms/s3-61.1.109.\n- [LPCZ23] Tye Lidman, Juanita Pinzón-Caicedo, and Raphael Zentner. Toroidal integer homology three-spheres have irreducible $\\mathrm{SU}(2)$-representations. J. Topol., 16(1):344– 367, 2023. doi:10.1112/topo.12275.\n- [BS22] John A. Baldwin and Steven Sivek. Instanton L-spaces and splicing. Ann. H. Lebesgue, 5:1213–1233, 2022. doi:10.5802/ahl.148.\n- [KM04] P. B. Kronheimer and T. S. Mrowka. Dehn surgery, the fundamental group and $\\mathrm{SU}(2)$. Math. Res. Lett., 11(5-6):741–754, 2004. doi:10.4310/MRL.2004.v11.n6.a3.\n- [CNS16] Christopher Cornwell, Lenhard Ng, and Steven Sivek. Obstructions to Lagrangian concordance. Algebr. Geom. Topol., 16(2):797–824, 2016. doi:10.2140/agt.2016.16.797.\n- [Zen17] Raphael Zentner. A class of knots with simple $\\mathrm{SU}(2)$-representations. Selecta Math. (N.S.), 23(3):2219–2242, 2017. doi:10.1007/s00029-017-0314-x.\n- [KM10] Peter Kronheimer and Tomasz Mrowka. Knots, sutures, and excision. J. Differential Geom., 84(2):301–364, 2010. http://projecteuclid.org/euclid.jdg/1274707316.\n- [BS18b] John A. Baldwin and Steven Sivek. Stein fillings and $\\mathrm{SU}(2)$ representations. Geom. Topol., 22(7):4307–4380, 2018. doi:10.2140/gt.2018.22.4307.\n- [Zen18] Raphael Zentner. Integer homology 3-spheres admit irreducible representations in $\\mathrm{SL}(2,\\mathbb{C})$. Duke Math. J., 167(9):1643–1712, 2018. doi:10.1215/00127094-2018-0004.\n- [Mot88] Kimihiko Motegi. Haken manifolds and representations of their fundamental groups in $\\mathrm{SL}(2,\\mathbb{C})$. Topology Appl., 29(3):207–212, 1988. doi:10.1016/0166-8641(88) 90019-3.\n- [SZ22] Steven Sivek and Raphael Zentner. A menagerie of $\\mathrm{SU}(2)$-cyclic 3-manifolds. Int. Math. Res. Not. IMRN, 2022(11):8038–8085, 2022. doi:10.1093/imrn/rnaa330.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2851, "problem_number": "KP-3.53", "title": "Kirby Problem 3.53", "statement": "Are all strong L-spaces branched double covers of alternating links in $S^3$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.53.\n\nLiterature notes:\n(1) A rational homology 3-sphere $Y$ is an L-space if\n\n$$\n\\operatorname{rank}\\widehat{HF}(Y)=|H_1(Y)|,\n$$\n\nwhere $|H_1(Y)|$ denotes the number of elements in $H_1(Y)$. A strong L-space is a rational homology 3-sphere $Y$ that admits a Heegaard diagram $H$ so that\n\n$$\n\\operatorname{rank}\\widehat{CF}(H)=|H_1(Y)|.\n$$\n\nCall such a Heegaard diagram a strong Heegaard diagram. If $H$ is strong, then the differential on $\\widehat{CF}(H)$ vanishes.\n\n(2) The notion of a strong L-space was studied by Levine and Lewallen [LL12]. They proved, for example, that the Poincaré homology 3-sphere is not a strong L-space (despite being an L-space). Greene [Gre13c] observed that the double branched cover of a non-split alternating link is a strong L-space. The question above was first asked by Greene and Levine in [GL16], who proved the case of L-spaces that admit genus 2 strong Heegaard diagrams.\n\n(3) It is open whether being a strong L-space is equivalent to admitting a Heegaard diagram $H$ for which the differential on $\\widehat{CF}(H)$ vanishes.\n\nReferences cited:\n- [LL12] Adam Simon Levine and Sam Lewallen. Strong L-spaces and left-orderability. Math. Res. Lett., 19(6):1237–1244, 2012. doi:10.4310/MRL.2012.v19.n6.a5.\n- [Gre13c] Joshua Evan Greene. A spanning tree model for the Heegaard Floer homology of a branched double-cover. J. Topol., 6(2):525–567, 2013. doi:10.1112/jtopol/jtt007.\n- [GL16] Joshua Evan Greene and Adam Simon Levine. Strong Heegaard diagrams and strong L-spaces. Algebr. Geom. Topol., 16(6):3167–3208, 2016. doi:10.2140/agt.2016.16.3167.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2852, "problem_number": "KP-3.54", "title": "Kirby Problem 3.54", "statement": "(a) Is there a closed 3-manifold $M$ whose Heegaard Floer homology $\\widehat{HF}(M;\\mathbb{Z})$ has torsion?\n\n(b) Is there a rational homology 3-sphere $M$ for which $\\widehat{HF}(M;\\mathbb{Z})$ has torsion?\n\n(c) Is there a knot $K\\subset S^3$ whose knot Floer homology $\\widehat{HFK}(S^3,K;\\mathbb{Z})$ has torsion?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.54.\n\nLiterature notes:\n(1) The Heegaard Floer homology of a closed 3-manifold $M$ with integer coefficients depends on a choice of isomorphism class of coherent orientation systems (and similarly for the knot Floer homology of links). There are $2^{b_1(M)}$ such choices in general. The Heegaard Floer group referenced in part (a) is defined using the canonical isomorphism class of orientation systems described in [OS04d]. By contrast, $\\widehat{HF}(S^1\\times S^2,o;\\mathbb{Z})\\cong \\mathbb{Z}/2\\mathbb{Z}$, with respect to the non-canonical orientation system $o$. For rational homology 3-spheres the canonical choice is the only choice.\n\n(2) Jabuka--Mark [JM08] proved that there is 2-torsion in $HF^+(\\Sigma_g\\times S^1;\\mathbb{Z})$ and $HF^{\\infty}(\\Sigma_g\\times S^1;\\mathbb{Z})$, where $\\Sigma_g$ is the genus-$g$ surface and $g\\geq 3$, but they showed there is no torsion in $\\widehat{HF}(\\Sigma_g\\times S^1;\\mathbb{Z})$ for any $g$.\n\n(3) In a related direction, Li and Ye in [LY24], and Bhat in [Bha24] announced the existence of 2-torsion in the framed instanton Floer homology $I^{\\#}(S^3_r(K);\\mathbb{Z})$ for any nontrivial knot $K$ with $r=1,1/2,1/4$, and 2-torsion in the unreduced singular instanton knot homology $I^{\\#}(S^3,K;\\mathbb{Z})$ for many knots $K$.\n\nReferences cited:\n- [OS04d] Peter Ozsváth and Zoltán Szabó. Holomorphic disks and three-manifold invariants: properties and applications. Ann. of Math. (2), 159(3):1159–1245, 2004. doi:10.4007/annals.2004.159.1159.\n- [JM08] Stanislav Jabuka and Thomas E. Mark. On the Heegaard Floer homology of a surface times a circle. Adv. Math., 218(3):728–761, 2008. doi:10.1016/j.aim.2008.01.009.\n- [LY24] Zhenkun Li and Fan Ye. 2-torsion in instanton Floer homology, 2024. arXiv:2405.16252.\n- [Bha24] Deeparaj Bhat. Surgery exact triangles in instanton theory, 2024. arXiv:2311.04242.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2853, "problem_number": "KP-3.55", "title": "Kirby Problem 3.55", "statement": "(a) For $K$ a nontrivial knot in $S^3$, does $HFK^-(K)$ always admit an $\\mathbb{F}_2$-summand, as an $\\mathbb{F}_2[U]$-module?\n\n(b) For $Y$ a rational homology sphere, if $HF_{\\mathrm{red}}(Y)$ is nontrivial, does $HF_{\\mathrm{red}}(Y)$ always admit an $\\mathbb{F}_2$-summand?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.55.\n\nLiterature notes:\n(1) The answer to Part (a) is \"yes\" for fibered knots [BVV18] and more generally for knots with $\\widehat{HFK}(K,g)$ supported in a single $\\mathbb{Z}/2\\mathbb{Z}$-grading [Ni22]. See also [HW18].\n\n(2) Regarding part (b), Lin [Lin24] proved that if a rational homology sphere $Y$ admits a taut foliation, then $HF_{\\mathrm{red}}(Y)$ admits an $\\mathbb{F}_2$-summand. In particular, if the taut foliation part of the L-space conjecture holds, then the answer to the second question is \"yes\"; and this question is implicitly raised in his paper.\n\n(3) The proofs of [BVV18] and [Ni22] use geometric perspectives (i.e., essential surfaces in knot exteriors) to deduce the existence of an $\\mathbb{F}_2$-summand, and [Lin24] uses taut foliations. Thus, it is natural to ask: does the generator of this summand hold special geometric meaning?\n\n(4) If $HF_{\\mathrm{red}}(Y)=\\mathbb{F}_2$, then [HKL19] implies the $\\mathbb{F}_2$-summand is either in grading $d$ or $d-1$, where $d$ denotes the correction term of $Y$.\n\n(5) Lastly, [AB24a, Question 1.10] asks the following refined version of the second question: If $HF_{\\mathrm{red}}(Y)$ contains an $\\mathbb{F}_2[U]/U^k$-summand, does $HF_{\\mathrm{red}}(Y)$ also contain an $\\mathbb{F}_2[U]/U^{\\ell}$-summand for all $1\\leq \\ell0$ and $S_{2,\\infty}(Y)$ of rank one; see [HP95b]. The skein module $S_{2,\\infty}(Y)\\otimes_R\\mathbb{Q}[A,A^{-1}]$ is not tame: by [HP95b], it contains $\\mathbb{Q}[A^{\\pm1}]/(\\phi_{2N})$ as a submodule, for all odd $N$.\n\n(g) The skein module of a rational homology sphere with coefficients in $\\mathbb{Q}(A)$ is at least 1-dimensional, by [DKS25].\n\nReferences cited:\n- [DKS25] Renaud Detcherry, Efstratia Kalfagianni, and Adam S. Sikora. Kauffman bracket skein modules of small 3-manifolds. Adv. Math., 467:Paper No. 110169, 45, 2025. doi:10.1016/j.aim.2025.110169.\n- [Prz00] Józef H. Przytycki. Kauffman bracket skein module of a connected sum of 3-manifolds. Manuscripta Math., 101(2):199–207, 2000. doi:10.1007/s002290050014.\n- [AM20] Mohammed Abouzaid and Ciprian Manolescu. A sheaf-theoretic model for $\\mathrm{SL}(2,\\mathbb{C})$ Floer homology. J. Eur. Math. Soc. (JEMS), 22(11):3641–3695, 2020. doi:10.4171/jems/994.\n- [Zen18] Raphael Zentner. Integer homology 3-spheres admit irreducible representations in $\\mathrm{SL}(2,\\mathbb{C})$. Duke Math. J., 167(9):1643–1712, 2018. doi:10.1215/00127094-2018-0004.\n- [Bul97] Doug Bullock. Rings of SL2pCq-characters and the Kauffman bracket skein module. Comment. Math. Helv., 72(4):521–542, 1997. doi:10.1007/s000140050032.\n- [PS00b] Józef H. Przytycki and Adam S. Sikora. On skein algebras and Sl2pCq-character varieties. Topology, 39(1):115–148, 2000. doi:10.1016/S0040-9383(98)00062-7.\n- [GJS23] Sam Gunningham, David Jordan, and Pavel Safronov. The finiteness conjecture for skein modules. Invent. Math., 232(1):301–363, 2023. doi:10.1007/s00222-022-01167-0.\n- [HP95b] Jim Hoste and Józef H. Przytycki. The Kauffman bracket skein module of $S^{1}$ $\\times$ $S^{2}$. Math. Z., 220(1):65–73, 1995. doi:10.1007/BF02572603.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2864, "problem_number": "KP-3.66", "title": "Kirby Problem 3.66", "statement": "Suppose that $Y$ is a closed, oriented 3-manifold, and let $S_{2,\\infty}(Y)$ denote the Kauffman bracket skein module over $R=\\mathbb{Z}[A,A^{-1}]$ as in Problem 3.65.\n\n(a) Is the dimension\n\n$$\n\\dim_{\\mathbb{Q}(A)} S_{2,\\infty}(Y)\\otimes_R \\mathbb{Q}(A)\n$$\n\nequal to the dimension of the degree 0 part of Abouzaid--Manolescu's sheaf-theoretic Floer homology $HP^{\\bullet}_{\\#}(Y)$ from [AM20]?\n\n(b) When is $S_{2,\\infty}(Y)\\otimes_R\\mathbb{Q}[A,A^{-1}]$ tame, as defined in Problem 3.65?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.66.\n\nLiterature notes:\n(1) An affirmative answer to the first question was conjectured in [GJS23], which also contains the proof that\n\n$$\nS_{2,\\infty}(Y)\\otimes_R \\mathbb{Q}(A)\n$$\n\nis finite dimensional over $\\mathbb{Q}(q)$.\n\n(2) When $S_{2,\\infty}(Y)\\otimes_R\\mathbb{Q}[A,A^{-1}]$ is tame, the dimension\n\n$$\n\\dim_{\\mathbb{Q}(A)} S_{2,\\infty}(Y)\\otimes_R \\mathbb{Q}(A)\n$$\n\nis bounded above by the dimension of the unreduced coordinate ring of the $SL(2,\\mathbb{C})$ character variety of $Y$, and is bounded below by the number points in this character variety [DKS25].\n\n(3) It is conjectured in [DKS25] that if $Y$ is a small 3-manifold, then\n\n$$\nS_{2,\\infty}(Y)\\otimes_R\\mathbb{Q}[A,A^{-1}]\n$$\n\nis finitely generated over $\\mathbb{Q}[A,A^{-1}]$, and hence tame. Positive progress on computing $S_{2,\\infty}(Y)$ in the case of the Cartesian product of a circle with a surface has been made by Gilmer and Masbaum [GM19] and Detcherry and Wolff [DW21]. Moreover, the skein modules of 3-manifolds obtained by surgery on the figure-eight knot and on $(2,p)$ torus knots are shown to be tame [DKS25].\n\nReferences cited:\n- [AM20] Mohammed Abouzaid and Ciprian Manolescu. A sheaf-theoretic model for $\\mathrm{SL}(2,\\mathbb{C})$ Floer homology. J. Eur. Math. Soc. (JEMS), 22(11):3641–3695, 2020. doi:10.4171/jems/994.\n- [GJS23] Sam Gunningham, David Jordan, and Pavel Safronov. The finiteness conjecture for skein modules. Invent. Math., 232(1):301–363, 2023. doi:10.1007/s00222-022-01167-0.\n- [DKS25] Renaud Detcherry, Efstratia Kalfagianni, and Adam S. Sikora. Kauffman bracket skein modules of small 3-manifolds. Adv. Math., 467:Paper No. 110169, 45, 2025. doi:10.1016/j.aim.2025.110169.\n- [GM19] Patrick M. Gilmer and Gregor Masbaum. On the skein module of the product of a surface and a circle. Proc. Amer. Math. Soc., 147(9):4091–4106, 2019. doi: 10.1090/proc/14553.\n- [DW21] Renaud Detcherry and Maxime Wolff. A basis for the Kauffman skein module of the product of a surface and a circle. Algebr. Geom. Topol., 21(6):2959–2993, 2021. doi:10.2140/agt.2021.21.2959.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2865, "problem_number": "KP-3.67", "title": "Kirby Problem 3.67", "statement": "Categorify the Witten--Reshetikhin--Turaev invariants of 3-manifolds.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.67.\n\nLiterature notes:\nThe Witten--Reshetikhin--Turaev (WRT) invariants [Wit89, RT91] are maps from the set of roots of unity to C. They are an analogue of the Jones polynomial for 3-manifolds. Khovanov [Kho00] constructed a categorification of the Jones polynomial for links in $S^3$, in the form of a bigraded abelian group whose graded Euler characteristic is the Jones polynomial. Since the WRT invariants are complex numbers, it is less clear what it means to categorify them. The first ideas in this direction appeared in the work of Crane and Frenkel [CF94]. For recent work on categorification at roots of unity, see [Kho16, EQ16, QRSW21].\n\nReferences cited:\n- [Wit89] Edward Witten. Quantum field theory and the Jones polynomial. Comm. Math. Phys., 121(3):351–399, 1989. http://projecteuclid.org/euclid.cmp/1104178138.\n- [RT91] N. Reshetikhin and V. G. Turaev. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math., 103(3):547–597, 1991. doi:10.1007/BF01239527.\n- [Kho00] Mikhail Khovanov. A categorification of the Jones polynomial. Duke Math. J., 101(3):359–426, 2000. doi:10.1215/S0012-7094-00-10131-7.\n- [CF94] Louis Crane and Igor B. Frenkel. Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. J. Math. Phys., 35(10):5136–5154, 1994. doi:10.1063/1.530746.\n- [Kho16] Mikhail Khovanov. Hopfological algebra and categorification at a root of unity: the first steps. J. Knot Theory Ramifications, 25(3):1640006, 26, 2016. doi:10.1142/S021821651640006X.\n- [EQ16] Ben Elias and You Qi. A categorification of quantum $\\mathfrak{sl}(2)$ at prime roots of unity. Adv. Math., 299:863–930, 2016. doi:10.1016/j.aim.2016.06.002.\n- [QRSW21] You Qi, Louis-Hadrien Robert, Joshua Sussan, and Emmanuel Wagner. A categorification of the colored Jones polynomial at a root of unity, 2021. arXiv:2111.13195.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2866, "problem_number": "KP-3.68", "title": "Kirby Problem 3.68", "statement": "(a) Give a mathematical definition of the $\\widehat{Z}$ invariants for all 3-manifolds.\n\n(b) Categorify the $\\widehat{Z}$ invariants.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.68.\n\nLiterature notes:\n(1) In [GPV17, GPPV20], Gukov, Pei, Putrov, and Vafa conjectured the existence of some power series with integral coefficients associated to 3-manifolds, denoted $\\widehat{Z}_a(q)$. They converge in the unit disk $|q|<1$, and their limits as $q$ approach some roots of unity should recover the Witten--Reshetikhin--Turaev invariants.\n\n(2) Rigorous definitions of the $\\widehat{Z}$ invariants exist for plumbed 3-manifolds and certain surgeries on knots; see [GPPV20, GM21, CGPS20].\n\n(3) Since the $\\widehat{Z}$ invariants have integral coefficients, they are good candidates for categorification.\n\n(4) There are many other interesting questions about $\\widehat{Z}$, as well. For example, it is interesting to study its large $N$ limit, which seems to be connected to enumerative geometry [EGG+22].\n\nReferences cited:\n- [GPV17] Sergei Gukov, Pavel Putrov, and Cumrun Vafa. Fivebranes and 3-manifold homology. J. High Energy Phys., 2017(7):071, front matter+80, 2017. doi:10.1007/JHEP07(2017)071.\n- [GPPV20] Sergei Gukov, Du Pei, Pavel Putrov, and Cumrun Vafa. BPS spectra and 3-manifold invariants. J. Knot Theory Ramifications, 29(2):2040003, 85, 2020. doi:10.1142/S0218216520400039.\n- [GM21] Sergei Gukov and Ciprian Manolescu. A two-variable series for knot complements. Quantum Topol., 12(1):1–109, 2021. doi:10.4171/qt/145.\n- [CGPS20] Sungbong Chun, Sergei Gukov, Sunghyuk Park, and Nikita Sopenko. 3d-3d correspondence for mapping tori. J. High Energy Phys., 2020(9):152, 59, 2020. doi: 10.1007/jhep09(2020)152.\n- [EGG+22] Tobias Ekholm, Angus Gruen, Sergei Gukov, Piotr Kucharski, Sunghyuk Park, Marko Stošić, and Piotr Sul kowski. Branches, quivers, and ideals for knot complements. J. Geom. Phys., 177:Paper No. 104520, 75, 2022. doi:10.1016/j.geomphys.2022.104520.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2867, "problem_number": "KP-3.69", "title": "Kirby Problem 3.69", "statement": "(a) What is the isomorphism type of $\\Theta^3_{\\mathbb{Z}}$?\n\n(b) Does there exist a torsion element $[Y]$ in $\\Theta^3_{\\mathbb{Z}}$?\n\n(c) Does there exist a torsion element $[Y]\\in\\Theta^3_{\\mathbb{Z}}$ having Rokhlin invariant $\\mu([Y])=1$?\n\n(d) Do there exist infinitely-divisible elements in $\\Theta^3_{\\mathbb{Z}}$; that is, does there exist a homology 3-sphere $Y$ such that for infinitely many $n\\in\\mathbb{N}$ there exists $[Z_n]\\in\\Theta^3_{\\mathbb{Z}}$ such that $[Y]=n[Z_n]$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.69.\n\nLiterature notes:\n(1) The integer homology cobordism group $\\Theta^3_{\\mathbb{Z}}$ is known to be a countable abelian group that has a summand isomorphic to $\\mathbb{Z}^{\\infty}$ by work of [DHST23].\n\n(2) In dimension $n=1,2$, the analogous group $\\Theta^n_{\\mathbb{Z}}$ vanishes. In dimension $n\\geq 4$, the group $\\Theta^n_{\\mathbb{Z}}$ is isomorphic to the group of homotopy spheres up to h-cobordism [GA70], and therefore finite [KM63]. However, in dimensions $n\\geq 4$, the analog of this group in the PL category vanishes; it is this\n\nlatter group that arises as an obstruction in the study of triangulations [GS80, Mat78].\n\n(3) Manolescu showed that there does not exist an element $[Y]$ of order two having Rokhlin invariant $\\mu(Y)$=1 [Man16b]. By previous work of Galewski--Stern [GS80] and Matumoto [Mat78], this was sufficient to disprove the outstanding cases of the Triangulation Conjecture, showing the existence of topological manifolds in every dimension greater than or equal to five that do not admit triangulations. Since $Y\\#(-Y)$ bounds a smooth homology 4-ball for all integer homology 3-spheres $Y$, to find a 2-torsion element, it suffices to exhibit a homology 3-sphere $Y$ with an orientation-reversing diffeomorphism, such that $[Y]=[-Y]=-[Y]$ in homology cobordism, but $[Y]\\neq [S^3]$. Noteworthy examples of 3-manifolds with such orientation-reversing diffeomorphisms include double branched covers of determinant 1, non-slice knots, such as those in [BC24b]. Various authors have given conditions that imply that a homology 3-sphere $[Y]$ must have infinite order in $\\Theta^3_{\\mathbb{Z}}$, for example [LRS18, Theorems C and D] or [HHL21, Theorem 1.13].\n\n(4) By work of Galewski--Stern, a negative answer to part (c) would imply that a topological manifold $M$ of dimension $\\geq 5$ is triangulable if and only if a particular obstruction class in $H^{5}(M;\\mathbb{Z})$ vanishes.\n\n(5) If $\\mu([Y])=1$, then $[Y]$ cannot be equal to $n[Z]$ for even $n$. Furthermore, any infinitely divisible $[Y]$ must necessarily have Heegaard Floer correction term $d([Y])=0$, and similarly for any other numerical invariant that is additive under connected sum. Any torsion element of a group is infinitely divisible, so in the presence of torsion part (d) might be modified to ask about the presence of infinitely divisible elements in the quotient of $\\Theta^3_{\\mathbb{Z}}$ by its torsion subgroup.\n\nReferences cited:\n- [DHST23] Irving Dai, Jennifer Hom, Matthew Stoffregen, and Linh Truong. An infinite-rank summand of the homology cobordism group. Duke Math. J., 172(12):2365–2432, 2023. doi:10.1215/00127094-2022-0082.\n- [GA70] Francisco Javier Gonzalez Acuna. ON HOMOLOGY SPHERES. ProQuest LLC, Ann Arbor, MI, 1970. Thesis (Ph.D.)–Princeton University. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt=info: ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri:pqdiss:7023614.\n- [KM63] Michel A. Kervaire and John W. Milnor. Groups of homotopy spheres. I. Ann. of Math. (2), 77:504–537, 1963. doi:10.2307/1970128.\n- [GS80] David E. Galewski and Ronald J. Stern. Classification of simplicial triangulations of topological manifolds. Ann. of Math. (2), 111(1):1–34, 1980. doi:10.2307/1971215.\n- [Mat78] Takao Matumoto. Triangulation of manifolds. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 3–6. Amer. Math. Soc., Providence, R.I., 1978.\n- [Man16b] Ciprian Manolescu. Pin(2)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture. J. Amer. Math. Soc., 29(1):147–176, 2016. doi:10.1090/jams829.\n- [BC24b] Keegan Boyle and Wenzhao Chen. Equivariant topological slice disks and negative amphichiral knots. Indiana Univ. Math. J., 73(5):1623–1637, 2024.\n- [LRS18] Jianfeng Lin, Daniel Ruberman, and Nikolai Saveliev. A splitting theorem for the Seiberg-Witten invariant of a homology $S^{1}$ $\\times$ $S^{3}$. Geom. Topol., 22(5):2865–2942, 2018. doi:10.2140/gt.2018.22.2865.\n- [HHL21] Kristen Hendricks, Jennifer Hom, and Tye Lidman. Applications of involutive Heegaard Floer homology. J. Inst. Math. Jussieu, 20(1):187–224, 2021. doi:10.1017/$S^{1}$47474801900015X.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2868, "problem_number": "KP-3.70", "title": "Kirby Problem 3.70", "statement": "Is $\\Theta^3_{\\mathbb{Z}}$ generated by the classes of knot surgeries $[S^3_{1/n}(K)]$, where $n$ ranges over all integers and $K$ ranges over all knots in $S^3$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.70.\n\nLiterature notes:\nThere are homology spheres that are not given by surgery on a knot in $S^3$ [GL89, Auc97, HKL16b]. Nozaki--Sato--Taniguchi used filtered instanton Floer homology to exhibit a homology 3-sphere which is not even homology cobordant to any knot surgery [NST24]. Hendricks--Hom--Lidman showed that $\\Theta^3_{\\mathbb{Z}}$ is not generated by surgeries on knots of bounded genus [HHL21].\n\nReferences cited:\n- [GL89] C. McA. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc., 2(2):371–415, 1989. doi:10.2307/1990979.\n- [Auc97] David Auckly. Surgery numbers of 3-manifolds: a hyperbolic example. In Geometric topology (Athens, GA, 1993), volume 2 of AMS/IP Stud. Adv. Math., pages 21–34. Amer. Math. Soc., Providence, RI, 1997. doi:10.1090/amsip/002.1/02.\n- [HKL16b] Jennifer Hom, Çağrı Karakurt, and Tye Lidman. Surgery obstructions and Heegaard Floer homology. Geom. Topol., 20(4):2219–2251, 2016. doi:10.2140/gt.2016.20.2219.\n- [NST24] Yuta Nozaki, Kouki Sato, and Masaki Taniguchi. Filtered instanton Floer homology and the homology cobordism group. J. Eur. Math. Soc. (JEMS), 26(12):4699–4761, 2024. doi:10.4171/jems/1371.\n- [HHL21] Kristen Hendricks, Jennifer Hom, and Tye Lidman. Applications of involutive Heegaard Floer homology. J. Inst. Math. Jussieu, 20(1):187–224, 2021. doi:10.1017/$S^{1}$47474801900015X.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2869, "problem_number": "KP-3.71", "title": "Kirby Problem 3.71", "statement": "Is there a nontrivial element in the kernel of the natural map\n\n$$\n\\Theta^3_{\\mathbb{Z}}\\longrightarrow \\Theta^3_{\\mathbb{Z}/2\\mathbb{Z}};\n$$\n\nthat is, does there exist an integer homology 3-sphere that does not bound a smooth $\\mathbb{Z}$-homology 4-ball but does bound a smooth $\\mathbb{Z}/2\\mathbb{Z}$-homology 4-ball?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.71.\n\nLiterature notes:\nThere is a natural homomorphism from the concordance group $C$ to $\\Theta^3_{\\mathbb{Z}/2\\mathbb{Z}}$ via taking double branched covers. As discussed in Problem 3.72, $\\Sigma(2,3,7)$ bounds a rational homology ball but, because its Rokhlin invariant is nontrivial, does not bound a $\\mathbb{Z}/2\\mathbb{Z}$-homology ball. See also [BL02] for some further discussion of $\\mathbb{Z}/2\\mathbb{Z}$-homology cobordism.\n\nReferences cited:\n- [BL02] Christian Bohr and Ronnie Lee. Homology cobordism and classical knot invariants. Comment. Math. Helv., 77(2):363–382, 2002. doi:10.1007/s00014-002-8344-0.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2870, "problem_number": "KP-3.72", "title": "Kirby Problem 3.72", "statement": "(a) Does the kernel of the map $\\Theta^3_{\\mathbb{Z}}\\to\\Theta^3_{\\mathbb{Q}}$ contain a subgroup that is isomorphic to $\\mathbb{Z}^{\\infty}$?\n\n(b) If so, does it contain such a subgroup as a summand?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.72.\n\nLiterature notes:\nFintushel and Stern [FS84] showed that $\\Sigma(2,3,7)=S^3_{+1}(4_1)$ bounds a smooth rational homology ball; as it has nontrivial Rokhlin invariant, it is a nontrivial element of the kernel of the map above. Indeed, as $\\Sigma(2,3,7)$ has nonvanishing Neumann--Siebenmann invariant $\\bar\\mu$ [Neu80, NR78], it is of infinite order in the homology cobordism group, implying that the kernel is infinite. This was the only known example of a Brieskorn sphere bounding a rational homology ball but not an integer homology ball until infinite families of examples were given by Akbulut and Larson [AL18], followed by further examples from Şavk [Şav20] and Simone [Sim21]. It is unknown whether these families are linearly independent.\n\nReferences cited:\n- [FS84] Ronald Fintushel and Ronald J. Stern. A µ-invariant one homology 3-sphere that bounds an orientable rational ball. In Four-manifold theory (Durham, N.H., 1982), volume 35 of Contemp. Math., pages 265–268. Amer. Math. Soc., Providence, RI, 1984. doi:10.1090/conm/035/780582.\n- [Neu80] Walter D. Neumann. An invariant of plumbed homology spheres. In Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), volume 788 of Lecture Notes in Math., pages 125–144. Springer, Berlin, 1980.\n- [NR78] Walter D. Neumann and Frank Raymond. Seifert manifolds, plumbing, µ-invariant and orientation reversing maps. In Algebraic and geometric topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977), volume 664 of Lecture Notes in Math., pages 163–196. Springer, Berlin-New York, 1978.\n- [AL18] Selman Akbulut and Kyle Larson. Brieskorn spheres bounding rational balls. Proc. Amer. Math. Soc., 146(4):1817–1824, 2018. doi:10.1090/proc/13828.\n- [Şav20] Oğuz Şavk. More Brieskorn spheres bounding rational balls. Topology Appl., 286:107400, 10, 2020. doi:10.1016/j.topol.2020.107400.\n- [Sim21] Jonathan Simone. Using rational homology circles to construct rational homology balls. Topology Appl., 291:Paper No. 107626, 16, 2021. doi:10.1016/j.topol.2021.107626.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2871, "problem_number": "KP-3.73", "title": "Kirby Problem 3.73", "statement": "(a) Calculate $\\Theta^{\\mathrm{TOP}}_{\\mathbb{Z}/p}$.\n\n(b) Calculate $\\Theta^{\\mathrm{TOP}}_{\\mathbb{Q}}$.\n\n(c) Is the linking form homomorphism $[\\operatorname{lk}]$ injective?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.73.\n\nLiterature notes:\n(1) Freedman proved that any integral homology 3-sphere bounds a contractible topological 4-manifold, unique up to homeomorphism rel boundary [FQ90, Corollary 9.3C], [Fre82, Theorem 1.41][BKK+21, Chapter 21.3.2]. This implies the topological integer homology cobordism group is trivial, $\\Theta^{\\mathrm{TOP}}_{\\mathbb{Z}}=1$.\n\n(2) Let $W(\\mathbb{Q}/\\mathbb{Z})$ denote the Witt group of non-singular bilinear forms over $\\mathbb{Q}/\\mathbb{Z}$. The $\\mathbb{Q}/\\mathbb{Z}$-valued linking form on $H_1(Y;\\mathbb{Z})$ defines a homomorphism\n\n$$\n[\\operatorname{lk}]:\\Theta^{\\mathrm{TOP}}_{\\mathbb{Q}}\\longrightarrow W(\\mathbb{Q}/\\mathbb{Z}).\n$$\n\nSee [Bre97, Chapter VI, Section 10] for a review of Poincaré--Lefschetz duality, and how one uses it to define the linking form, or [CFH16, Section 2] for a self-contained synopsis of the same. That the linking form defines a homomorphism to $W(\\mathbb{Q}/\\mathbb{Z})$ is an immediate consequence of the following fact: if a rational homology 3-sphere $Y$ bounds a rational ball, then its first homology has a subgroup of order $\\sqrt{|H_1(Y)|}$ on which the linking form vanishes. This is proved in [CG86, Theorem 1 and Theorem 2] (see also [Gil82, Lemma 1] for a self-contained treatment). One can easily show that this homomorphism is surjective; indeed, any form can be presented with an integral matrix, which can be used to build a 4-dimensional 2-handlebody whose boundary will be a 3-manifold representing the given form.\n\n(3) Homology cobordism groups are closely connected to knot concordance, by way of Dehn surgery and cyclic branched covers. For instance, both constructions define maps from the knot concordance group to $\\Theta^3_{\\mathbb{Q}}$, with each factoring through $\\Theta^3_{\\mathbb{Z}/p}$ for some $p$ depending on the surgery slope or degree of cyclic branched covering, respectively. For the latter, one should restrict to $p$ a prime power to ensure the branched covers are $\\mathbb{Z}/p$-homology spheres and, for slice knots, bound $\\mathbb{Z}/p$-homology balls [CG78]. In the case of branched coverings, these maps are actually homomorphisms. Both observations have been extensively used in the smooth setting. Understanding the problems above could have applications to the theory of topological concordance groups. For instance, if $[\\operatorname{lk}]$ fails to be injective, and refined invariants of $\\Theta^{\\mathrm{TOP}}_{\\mathbb{Q}}$ can be defined, then the latter could be used to study the topological concordance group through the surgery and branched covering maps.\n\nReferences cited:\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [BKK+21] Stefan Behrens, Boldizsár Kalmár, Min Hoon Kim, Mark Powell, and Arunima Ray, editors. The disc embedding theorem. Oxford University Press, Oxford, 2021.\n- [Bre97] Glen E. Bredon. Topology and geometry, volume 139 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. Corrected third printing of the 1993 original.\n- [CFH16] Anthony Conway, Stefan Friedl, and Gerrit Herrmann. Linking forms revisited. Pure Appl. Math. Q., 12(4):493–515, 2016. doi:10.4310/PAMQ.2016.v12.n4.a3.\n- [CG86] A. J. Casson and C. McA. Gordon. Cobordism of classical knots. In À la recherche de la topologie perdue, volume 62 of Progr. Math., pages 181–199. Birkhäuser Boston, Boston, MA, 1986.\n- [Gil82] Patrick M. Gilmer. On the slice genus of knots. Invent. Math., 66(2):191–197, 1982. doi:10.1007/BF01389390.\n- [CG78] A. J. Casson and C. McA. Gordon. On slice knots in dimension three. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, volume XXXII of Proc. Sympos. Pure Math., pages 39–53. Amer. Math. Soc., Providence, RI, 1978.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2872, "problem_number": "KP-3.74", "title": "Kirby Problem 3.74", "statement": "Let $Y$ be a rational homology sphere and $f:Y\\to Y$ be a self-diffeomorphism of $Y$. Suppose $W$ is a 4-manifold with boundary $Y$ such that $f$ extends to a self-diffeomorphism of $W$.\n\n(a) What constraints are there on the intersection form of $W$?\n\n(b) Does there exist a pair $(Y,f)$ such that $Y$ bounds a homology ball but $f:Y\\to Y$ does not extend over any definite manifold of either sign?\n\n(c) Develop methods for constraining the intersection form of $W$ that apply to indefinite, non-spin $W$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.74.\n\nLiterature notes:\n(1) It is known that the 3-dimensional bordism-with-diffeomorphism group is trivial [Mel79]; that is, if $Y$ is a 3-manifold equipped with a self-diffeomorphism $f$, then there always exists a smooth 4-manifold $W$ over which $f$ extends. It is thus natural to ask what homological restrictions occur for the class of such $W$. This is analogous to the fact that the usual 3-dimensional bordism group is trivial, but the set of intersection forms bounded by $Y$ reflects the topology of $Y$.\n\n(2) The extension question for diffeomorphisms is also motivated by the theory of corks [Akb91]. If $\\pi_1(W)$ = 1, then every self-homeomorphism of\n\n$Y$ extends as a self-homeomorphism of $W$ by work of Freedman [FQ90]. In contrast, there are many examples in which a self-diffeomorphism of $Y$ does not extend as a self-diffeomorphism over a particular contractible manifold bounded by $Y$, or even any homology ball with boundary $Y$. See for example [Akb91, LRS23a, AKS20, DHM23].\n\n(3) While it is possible to obstruct the extension of $f$ over definite manifolds of a fixed sign, it is unknown whether there are any pairs $(Y,f)$ for which $Y$ bounds a homology ball, but $f$ does not extend over any definite manifold of either sign. Disregarding the self-diffeomorphism $f$, it has only recently been shown that there are integer homology spheres $Y$ that bound no definite manifold of either sign; see [NST24]. If the action of the extension on the cohomology of $W$ is constrained, then partial results have been obtained in [ADMT23] using the Chern--Simons filtration in instanton Floer homology.\n\nReferences cited:\n- [Mel79] Paul Melvin. Bordism of diffeomorphisms. Topology, 18(2):173–175, 1979. doi:10.1016/0040-9383(79)90034-X.\n- [Akb91] Selman Akbulut. A fake compact contractible 4-manifold. J. Differential Geom., 33(2):335–356, 1991. http://projecteuclid.org/euclid.jdg/1214446320.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [LRS23a] Jianfeng Lin, Daniel Ruberman, and Nikolai Saveliev. On the Frøyshov invariant and monopole Lefschetz number. J. Differential Geom., 123(3):523–593, 2023. doi: 10.4310/jdg/1683307008.\n- [AKS20] Antonio Alfieri, Sungkyung Kang, and András I. Stipsicz. Connected Floer homology of covering involutions. Math. Ann., 377(3-4):1427–1452, 2020. doi:10.1007/s00208-020-01992-9.\n- [DHM23] Irving Dai, Matthew Hedden, and Abhishek Mallick. Corks, involutions, and Heegaard Floer homology. J. Eur. Math. Soc. (JEMS), 25(6):2319–2389, 2023. doi:10.4171/jems/1239.\n- [NST24] Yuta Nozaki, Kouki Sato, and Masaki Taniguchi. Filtered instanton Floer homology and the homology cobordism group. J. Eur. Math. Soc. (JEMS), 26(12):4699–4761, 2024. doi:10.4171/jems/1371.\n- [ADMT23] Antonio Alfieri, Irving Dai, Abhishek Mallick, and Masaki Taniguchi. Involutions and the Chern-Simons filtration in instanton Floer homology, 2023. arXiv:2309.02309.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2873, "problem_number": "KP-3.75", "title": "Kirby Problem 3.75", "statement": "Let $Y$ be a rational homology 3-sphere equipped with an action of a cyclic group $\\mathbb{Z}/p\\mathbb{Z}$. Suppose $W$ is a 4-manifold with boundary $Y$ over which the action of $\\mathbb{Z}/p\\mathbb{Z}$ extends smoothly.\n\n(a) What constraints are there on the intersection form of $W$?\n\n(b) Does there exist an integer homology sphere $Y$ bounding a homology ball such that $Y$ admits a cyclic group action that does not extend over any definite manifold of either sign?\n\n(c) Develop methods for constraining the intersection form of $W$ that apply to indefinite, non-spin $W$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.75.\n\nLiterature notes:\nWhile this is similar to Problem 3.74, the motivation and techniques for the present question are slightly different. If $f:Y\\to Y$ generates the action of a cyclic group, then obstructing the extension of $f$ as a diffeomorphism clearly obstructs the extension of $f$ as a group action. However, there are many interesting examples where the two notions differ. For instance, one may consider the standard $S^1$-action on a Brieskorn sphere $\\Sigma(p,q,r)$. Any root of unity $\\zeta\\in S^1$ generates the action of a cyclic group on $\\Sigma(p,q,r)$ whose generator $f$ is isotopic to the identity (as a diffeomorphism). In [AH16, AH21], it is shown that this never extends as a smooth group action over any contractible manifold that $\\Sigma(p,q,r)$ may bound, even though it is not hard to see that $f$ extends as a diffeomorphism. See also [Edm87, KL93, BH24a].\n\nReferences cited:\n- [AH16] Nima Anvari and Ian Hambleton. Cyclic group actions on contractible 4-manifolds. Geom. Topol., 20(2):1127–1155, 2016. doi:10.2140/gt.2016.20.1127.\n- [AH21] Nima Anvari and Ian Hambleton. Cyclic branched coverings of Brieskorn spheres bounding acyclic 4-manifolds. Glasg. Math. J., 63(2):400–413, 2021. doi:10.1017/S0017089520000269.\n- [Edm87] Allan L. Edmonds. Construction of group actions on four-manifolds. Trans. Amer. Math. Soc., 299(1):155–170, 1987. doi:10.2307/2000487.\n- [KL93] Slawomir Kwasik and Terry Lawson. Nonsmoothable $\\mathbb{Z}_p$ actions on contractible 4-manifolds. J. Reine Angew. Math., 437:29–54, 1993. doi:10.1515/crll.1993.437.29.\n- [BH24a] David Baraglia and Pedram Hekmati. Equivariant Seiberg-Witten-Floer cohomology. Algebr. Geom. Topol., 24(1):493–554, 2024. doi:10.2140/agt.2024.24.493.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2874, "problem_number": "KP-3.76", "title": "Kirby Problem 3.76", "statement": "What is the structure of the equivariant homology cobordism groups?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.76.\n\nLiterature notes:\nThere are several equivariant homology cobordism groups one can construct. One version is the $\\mathbb{Z}/p\\mathbb{Z}$-equivariant cobordism group as appropriate\n\nhomology cobordism classes of pairs consisting of integer homology spheres with an orientation-preserving $\\mathbb{Z}/p\\mathbb{Z}$ action having nonempty fixed point set. One can also define more general groups that take into account a diffeomorphism on each homology sphere and have no restriction on the fixed-point set; this is done in [DHM23]. In various cases, one can produce interesting $\\mathbb{Z}^{\\infty}$ -subgroups. For instance, it follows from [DHM23] that there exists a $\\mathbb{Z}^{\\infty}$ -subgroup spanned by cork boundaries in the case p = 2; similar techniques can be attempted for other p. The structure of these groups is otherwise poorly understood. It is well-known that there is a homomorphism from the smooth concordance group to the rational homology cobordism group given by taking branched covers. There is likewise a homomorphism from the strongly invertible concordance group to the $\\mathbb{Z}/2\\mathbb{Z}$-equivariant rational homology cobordism group; see [AB24b]. There is also a homomorphism from the usual concordance group to the equivariant cobordism group, by taking branched covers but remembering the branching action, which may be used to produce more refined sliceness or concordance obstructions; see e.g. [BH24a].\n\nReferences cited:\n- [DHM23] Irving Dai, Matthew Hedden, and Abhishek Mallick. Corks, involutions, and Heegaard Floer homology. J. Eur. Math. Soc. (JEMS), 25(6):2319–2389, 2023. doi:10.4171/jems/1239.\n- [AB24b] Antonio Alfieri and Keegan Boyle. Strongly invertible knots, invariant surfaces, and the Atiyah-Singer signature theorem. Michigan Math. J., 74(4):845–861, 2024. doi:10.1307/mmj/20226183.\n- [BH24a] David Baraglia and Pedram Hekmati. Equivariant Seiberg-Witten-Floer cohomology. Algebr. Geom. Topol., 24(1):493–554, 2024. doi:10.2140/agt.2024.24.493.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2875, "problem_number": "KP-3.77", "title": "Kirby Problem 3.77", "statement": "Does there exist a hyperbolic rational homology 3-sphere that is the totally geodesic boundary of a compact, orientable hyperbolic 4-manifold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.77.\n\nLiterature notes:\n(1) When a compact hyperbolic 3-manifold is the totally geodesic boundary of a compact hyperbolic 4-manifold, it is said to geometrically bound. Questions about geometric boundaries of hyperbolic manifolds of dimension greater than 4 are discussed in Problem 4.126(b).\n\n(2) A result of Ferrari--Kolpakov--Reid [FKR23] shows that there exist infinitely many arithmetic hyperbolic rational homology 3-spheres that geometrically bound hyperbolic 4-manifolds, but in their examples, all of the 4-manifolds are nonorientable. Prior to [FKR23], all constructed examples of hyperbolic 3-manifolds that geometrically bound had non-zero first Betti number.\n\n(3) Most constructions of hyperbolic 3-manifolds that geometrically bound (including the ones in [FKR23]) are arithmetic of simplest type and bound hyperbolic 4-manifolds cut out from ones that are arithmetic of simplest type. It would be interesting to construct non-arithmetic examples.\n\n(4) When a compact, orientable 3-manifold M geometrically bounds a compact, orientable, hyperbolic 4-manifold, this imposes restrictions on the $\\eta$-invariant of M, due to [LR00]. For example, the Weeks manifold does not geometrically bound a compact orientable hyperbolic 4-manifold.\n\nReferences cited:\n- [FKR23] L. Ferrari, A. Kolpakov, and A. W. Reid. Infinitely many arithmetic hyperbolic rational homology 3-spheres that bound geometrically. Trans. Amer. Math. Soc., 376(3):1979–1997, 2023. doi:10.1090/tran/8816.\n- [LR00] D. D. Long and A. W. Reid. On the geometric boundaries of hyperbolic 4-manifolds. Geom. Topol., 4:171–178, 2000. doi:10.2140/gt.2000.4.171.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2876, "problem_number": "KP-3.78", "title": "Kirby Problem 3.78", "statement": "(a) Is there a non-semisimple 3-TQFT whose mapping class group representation is faithful or has an element in its kernel?\n\n(b) Define a 4-manifold invariant via non-semsimple categories that can distinguish 4-manifolds that are not distinguished by classical or gauge-theoretic invariants.\n\n(c) Is there is a full 3-TQFT extension of the representations, constructed by Bonahon and Wong, of the Kauffman bracket skein algebra? In particular, do non-semisimple TQFTs recover the Bonahon--Wong representations?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.78.\n\nLiterature notes:\n(1) In 1989, Atiyah [Ati88b] gave the definition of an n-dimensional topological quantum field theory (n-TQFT for short) that assigns objects in some category to n-manifolds and morphisms to cobordisms. Many 3-TQFTs have been constructed via semisimple modular categories; this problem is asking about applications of TQFTs constructed via non-semisimple categories that do not seem accessible to semisimple TQFTs.\n\n(2) Non-semisimple 3-TQFTs are relevant to the linearity problem for the mapping class group of a surface; see Problem 2.3 for a discussion of classical approaches. For example, the 3-TQFTs of [DRGG+22] lead to mapping class group representations with the property that the action of a Dehn twist has infinite order and thus is a candidate answer to part\n\n(a). By contrast, in the usual semisimple quantum mapping class group representations, all Dehn twists have finite order and the representations are not faithful.\n\n(3) The underlying 4-manifold invariants of 4-TQFTs coming from semisimple modular categories, such as [CKY97, CY93, BB18], are conjecturally determined by the Euler characteristic, the signature and the fundamental group, see [BB18, Conjecture 8.1]. This conjecture is verified in many cases in [Reu23]. It is unknown whether the 4-TQFTs of [CGHPM23] or the invariants of [BDR24, BDR23] go beyond such classical invariants. Note that in the 3-dimensional setting, there are invariants of closed 3-manifolds underlying a 3-TQFT distinguishes diffeomorphism types of homotopically equivalent lens spaces which were not distinguished by semisimple quantum invariants; see [CGPM14].\n\n(4) In [BW16c, BW16b, BW17b, BW19] Bonahon and Wong constructed a family of finite-dimensional representations of the Kauffman bracket skein algebra of surface. Their construction is based on the theory of quantum Teichmüller space of Chekhov and Fock [FC99] and Kashaev [Kas98]. Recently, Frohman, Kania-Bartoszynska and Le showed these representation have a unicity condition, see [FKBL19]. Partial results in the direction are known: certain BW-representations arise from the nonsemisimple TQFT of [BCGPM16], see [KK22]. If a TQFT extension of the link invariants of [BGPMR20] exists it should contain the general BW-representation\n\nReferences cited:\n- [Ati88b] Michael Atiyah. Topological quantum field theories. Inst. Hautes Études Sci. Publ. Math., (68):175–186, 1988. URL: http://www.numdam.org/item?id=PMIHES 1988 68 175 0.\n- [DRGG+22] Marco De Renzi, Azat M. Gainutdinov, Nathan Geer, Bertrand PatureauMirand, and Ingo Runkel. 3-Dimensional TQFTs from non-semisimple modular categories. Selecta Math. (N.S.), 28(2):Paper No. 42, 60, 2022. doi:10.1007/s00029-021-00737-z.\n- [CKY97] Louis Crane, Louis H. Kauffman, and David N. Yetter. State-sum invariants of 4-manifolds. J. Knot Theory Ramifications, 6(2):177–234, 1997. doi:10.1142/S0218216597000145.\n- [CY93] Louis Crane and David Yetter. A categorical construction of 4d topological quantum field theories. 3:120–130, 1993. URL: https://doi.org/10.1142/9789812796387 0005, doi:10.1142/9789812796387\\\\_0005.\n- [BB18] Manuel Bärenz and John Barrett. Dichromatic state sum models for four-manifolds from pivotal functors. Comm. Math. Phys., 360(2):663–714, 2018. doi:10.1007/s00220-017-3012-9.\n- [Reu23] David Reutter. Semisimple four-dimensional topological field theories cannot detect exotic smooth structure. J. Topol., 16(2):542–566, 2023. doi:10.1112/topo.12288.\n- [CGHPM23] Francesco Costantino, Nathan Geer, Benjamin Haı̈oun, and Bertrand PatureauMirand. Skein (3+ 1)-TQFTs from non-semisimple ribbon categories, 2023. arXiv: 2306.03225.\n- [BDR24] Anna Beliakova and Marco De Renzi. Kerler-Lyubashenko functors on 4-dimensional 2-handlebodies. Int. Math. Res. Not. IMRN, 2024(13):10005–10080, 2024. doi:10.1093/imrn/rnac039.\n- [BDR23] Anna Beliakova and Marco De Renzi. Refined Bobtcheva-Messia invariants of 4-dimensional 2-handlebodies. In Essays in geometry—dedicated to Norbert A’Campo, volume 34 of IRMA Lect. Math. Theor. Phys., pages 387–431. EMS Press, Berlin, [2023] ©2023.\n- [CGPM14] Francesco Costantino, Nathan Geer, and Bertrand Patureau-Mirand. Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories. J. Topol., 7(4):1005–1053, 2014. doi:10.1112/jtopol/jtu006.\n- [BW16c] Francis Bonahon and Helen Wong. The Witten-Reshetikhin-Turaev representation of the Kauffman bracket skein algebra. Proc. Amer. Math. Soc., 144(6):2711–2724, 2016. doi:10.1090/proc/12927.\n- [BW16b] Francis Bonahon and Helen Wong. Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations. Invent. Math., 204(1):195–243, 2016. doi:10.1007/s00222-015-0611-y.\n- [BW17b] Francis Bonahon and Helen Wong. Representations of the Kauffman bracket skein algebra II: Punctured surfaces. Algebr. Geom. Topol., 17(6):3399–3434, 2017. doi: 10.2140/agt.2017.17.3399.\n- [BW19] Francis Bonahon and Helen Wong. Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality. Quantum Topol., 10(2):325–398, 2019. doi:10.4171/QT/125.\n- [FC99] Vladimir V Fock and Leonid O Chekhov. A quantum Teichmüller space. Theoretical and Mathematical Physics, 120(3):1245–1259, 1999.\n- [Kas98] Rinat M Kashaev. Quantization of Teichmüller spaces and the quantum dilogarithm. Letters in Mathematical Physics, 43:105–115, 1998.\n- [FKBL19] Charles Frohman, Joanna Kania-Bartoszynska, and Thang Lê. Unicity for representations of the Kauffman bracket skein algebra. Invent. Math., 215(2):609–650, 2019. doi:10.1007/s00222-018-0833-x.\n- [BCGPM16] Christian Blanchet, Francesco Costantino, Nathan Geer, and Bertrand PatureauMirand. Non-semi-simple TQFTs, Reidemeister torsion and Kashaev’s invariants. Adv. Math., 301:1–78, 2016. doi:10.1016/j.aim.2016.06.003.\n- [KK22] Hiroaki Karuo and Julien Korinman. Azumaya loci of skein algebras, 2022. arXiv: 2211.13700.\n- [BGPMR20] Christian Blanchet, Nathan Geer, Bertrand Patureau-Mirand, and Nicolai Reshetikhin. Holonomy braidings, biquandles and quantum invariants of links with SL2pCq flat connections. Selecta Math. (N.S.), 26(2):Paper No. 19, 58, 2020. doi:10.1007/s00029-020-0545-0.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2877, "problem_number": "KP-4.1", "title": "Kirby Problem 4.1", "statement": "(4-dimensional Poincaré conjecture). Is there a unique smooth structure on the 4-sphere?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.1.\n\nLiterature notes:\n(1) This appears as Problems 4.45 and 4.89 in [Kir97].\n\n(2) This is one of the remaining open cases of the smooth Poincaré Conjecture. In dimensions at most three, uniqueness of smooth structures follows from work of Munkres [Mun60] and Moise [Moi77a]. In higher dimensions, the only odd-dimensional spheres with unique smooth structures are $S^{5}$ and $S^{61}$ [WX17]; the situation for even-dimensional spheres is still open in general, but it is conjectured that the only even-dimensional spheres without exotic smooth structures in dimensions greater than 4 are $S^{6}, S^{12}$, and $S^{56}$ [WX17].\n\n(3) For context, we mention the history of the topological Poincaré conjecture: any manifold homotopy equivalent to $S^{n}$ is homeomorphic to $S^{n}$. This is classical for $n \\leq$ 2, follows from Perelman’s Geometrization Theorem for $n =$ 3 [Per02, Per03b, Per03a], from Freedman’s work for $n =$ 4 [Fre82], and from Newman’s work on topological engulfing for $n \\geq$ 5 [New66]. (When the manifold is smooth, the $n \\geq$ 5 case was first established as a consequence of Smale’s h-cobordism theorem [Sma62a].)\n\n(4) Some sources of potentially exotic homotopy 4-spheres are as follows.\n\n\\noindent$\\bullet$ Gluck twists on 2-knots in $S^{4}$. See Problem 4.9 for a discussion.\n\n\\noindent$\\bullet$ The Andrews–Curtis problem. See [Kir78, Problem 5.2] and Problem 5.10for a discussion.\n\n\\noindent$\\bullet$ Cappell–Shaneson homotopy spheres [CS76]. Many of their examples were proved to be standard in [Akb10] and [Gom10], but not all. More generally, one can consider the mapping torus of a 3manifold diffeomorphism, and do surgery on a section; this sometimes gives a homotopy 4-sphere. One such example was proposed in [Maz62], and was shown standard in [Zee65]. One can construct many such examples where the diffeomorphism is obtained from the identity by a point push; e.g., when the 3-manifold is made by gluing two knot complements.\n\n\\noindent$\\bullet$ Gabai’s big dot carving. See Problem 4.24.\n\n\\noindent$\\bullet$ Pairs of knots with the same 0-surgeries, such that exactly one knot is slice. See for example [Akb93], or [MP23].\n\n\\noindent$\\bullet$ Cyclic branched covers of certain 2-knots. In general, the double branched cover of a 2-knot in $S^{4}$ is a rational homology sphere that need not be simply connected, but in certain cases the cover can be arranged to be a homotopy 4-sphere. For example, the double branched cover of the roll-spun (−2,3,7)pretzel knot is a homotopy 4-sphere that is not known to be the standard 4-sphere (see [Miy23]).\n\n(5) We mention some known potential approaches to the problem.\n\n\\noindent$\\bullet$ If one wants to prove that the smooth structure on $S^{4}$ is unique, one can try to use geometric flows, such as the Ricci flow, or the mean curvature flow in $\\mathbb{R}^{5}$ [CMP15].\n\n\\noindent$\\bullet$ If one wants to put a nonstandard smooth structure on $S^{4}$ using pairs of knots with the same 0-surgeries, Rasmussen’s s-invariant [Ras10] could be used to prove one of the knots is not slice. See the remarks on Problem 1.60.\n\n\\noindent$\\bullet$ Let K be a topologically slice knot that is not smoothly slice, such as the Conway knot. If we could put a smooth structure on the complement of the interior of a tubular neighborhood of the topologically slice disk, then we could imitate the construction of an exotic $\\mathbb{R}^{4}$ using $S^{4}$ and get a nontrivial homotopy sphere.\n\n\\noindent$\\bullet$ Symplectic geometry gives a potential avenue to either proving or disproving the smooth 4-dimensional Poincaré conjecture. Given a homotopy 4-sphere X, one wants to either construct or obstruct the existence of an asymptotically standard symplectic structure on $X \\setminus$ \\{point\\}. See [Ger21] for a discussion.\n\n\\noindent$\\bullet$ Finally, it is possible to rephrase Problem 4.1 purely in terms of group theory, using trisections [AGK18].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Mun60] James Munkres. Obstructions to the smoothing of piecewise-differentiable homeomorphisms. Ann. of Math. (2), 72:521–554, 1960. doi:10.2307/1970228.\n- [Moi77a] Edwin E. Moise. Geometric topology in dimensions 2 and 3, volume Vol. 47 of Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1977.\n- [WX17] Guozhen Wang and Zhouli Xu. The triviality of the 61-stem in the stable homotopy groups of spheres. Ann. of Math. (2), 186(2):501–580, 2017. doi:10.4007/annals.2017.186.2.3.\n- [Per02] Grisha Perelman. The entropy formula for the Ricci flow and its geometric applications, 2002. arXiv:math/0211159.\n- [Per03b] Grisha Perelman. Ricci flow with surgery on three-manifolds, 2003. arXiv:math/0303109.\n- [Per03a] Grisha Perelman. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, 2003. arXiv:math/0307245.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [New66] M. H. A. Newman. The engulfing theorem for topological manifolds. Ann. of Math. (2), 84:555–571, 1966. doi:10.2307/1970460.\n- [Sma62a] S. Smale. On the structure of manifolds. Amer. J. Math., 84:387–399, 1962. doi: 10.2307/2372978.\n- [Kir78] Rob Kirby. Problems in low dimensional manifold theory. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 273–312. Amer. Math. Soc., Providence, R.I., 1978.\n- [CS76] Sylvain E. Cappell and Julius L. Shaneson. Some new four-manifolds. Ann. of Math. (2), 104(1):61–72, 1976. doi:10.2307/1971056.\n- [Akb10] Selman Akbulut. Cappell-Shaneson homotopy spheres are standard. Ann. of Math. (2), 171(3):2171–2175, 2010. doi:10.4007/annals.2010.171.2171.\n- [Gom10] Robert E. Gompf. More Cappell-Shaneson spheres are standard. Algebr. Geom. Topol., 10(3):1665–1681, 2010. doi:10.2140/agt.2010.10.1665.\n- [Maz62] Barry Mazur. Symmetric homology spheres. Illinois J. Math., 6:245–250, 1962.\n- [Zee65] E. C. Zeeman. Twisting spun knots. Trans. Amer. Math. Soc., 115:471–495, 1965. doi:10.2307/1994281.\n- [Akb93] S. Akbulut. Knots and exotic smooth structures on 4-manifolds. J. Knot Theory Ramifications, 2(1):1–10, 1993. doi:10.1142/S0218216593000027.\n- [MP23] Ciprian Manolescu and Lisa Piccirillo. From zero surgeries to candidates for exotic definite 4-manifolds. Journal of the London Mathematical Society, 108(5):2001– 2036, 2023. doi:10.1112/jlms.12800.\n- [Miy23] Jin Miyazawa. A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P2-knots, 2023. arXiv:2312.02041.\n- [CMP15] Tobias Holck Colding, William P. Minicozzi, II, and Erik Kjær Pedersen. Mean curvature flow. Bull. Amer. Math. Soc. (N.S.), 52(2):297–333, 2015. doi:10.1090/S0273-0979-2015-01468-0.\n- [Ras10] Jacob Rasmussen. Khovanov homology and the slice genus. Invent. Math., 182(2):419–447, 2010. doi:10.1007/s00222-010-0275-6.\n- [Ger21] Chris Gerig. No homotopy 4-sphere invariants using $\\mathrm{ECH}=\\mathrm{SWF}$. Algebr. Geom. Topol., 21(5):2543–2569, 2021. doi:10.2140/agt.2021.21.2543.\n- [AGK18] Aaron Abrams, David T. Gay, and Robion Kirby. Group trisections and smooth 4-manifolds. Geom. Topol., 22(3):1537–1545, 2018. doi:10.2140/gt.2018.22.1537.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2878, "problem_number": "KP-4.2", "title": "Kirby Problem 4.2", "statement": "Does every smooth, closed 4-manifold admit an exotic smooth structure? Infinitely many?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.2.\n\nLiterature notes:\n(1) The question is open for many families of 4-manifolds. See also Problems 4.3 and 4.4.\n\n(2) A version of this problem in [Kir97, Problem 4.86] raises the same question for complex algebraic surfaces. Evident families in this class, still not known to admit exotic smooth structures, are the irrational ruled surfaces $(\\Sigma_{h} \\times S^{2})\\#_{m}\\mathbb{CP}^{2}$ and $(\\Sigma_{h} \\times S^{2})\\#_{m}\\mathbb{CP}^{2}$ for any h, $m \\geq$ 1.\n\n(3) To date, on virtually any smooth, closed 4-manifold known to admit an exotic smooth structure, infinitely many distinct exotic smooth structures have been constructed. (See [BSS24] for the related discussion.) Many of the known methods for constructing infinitely many smooth structures require the presence of certain embedded tori of self-intersection zero [FS98, FS11].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [BSS24] R. Inanc Baykur, Andras I. Stipsicz, and Zoltan Szabo. Smooth structures on fourmanifolds with finite cyclic fundamental groups, 2024. arXiv:2406.09007.\n- [FS98] Ronald Fintushel and Ronald J. Stern. Knots, links, and 4-manifolds. Invent. Math., 134(2):363–400, 1998. doi:10.1007/s002220050268.\n- [FS11] Ronald Fintushel and Ronald J. Stern. Pinwheels and nullhomologous surgery on 4-manifolds with $b^+=1$. Algebr. Geom. Topol., 11(3):1649–1699, 2011. doi:10.2140/agt.2011.11.1649.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2879, "problem_number": "KP-4.3", "title": "Kirby Problem 4.3", "statement": "Are there exotic smooth structures on the following closed, simply-connected 4–manifolds?\n\n(a) $\\#_{k}\\mathbb{CP}^{2}$ for any $k \\geq$ 1.\n\n(b) $\\#_{m}(\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2})$ and $\\#_{n}(S^{2} \\times S^{2})$ for $m \\leq$ 8, $n \\leq$ 10.\n\n(c) $\\#_{p}K3 \\#_{q}(S^{2} \\times S^{2})$ for any $p \\geq$ q+5. If so, what are the smallest k, m and n? The largest p and q?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.3.\n\nLiterature notes:\n(1) These are the main families of closed, simply connected 4–manifolds, besides $S^{4}$, not known to admit exotic smooth structures. Namely, those that are (a) definite, (b) $\\sigma =$ 0 with $e <$ 20, and (c) spin with $c^{2}_{1} = 2e+3\\sigma <$ −12, respectively, where e is the Euler characteristic and $\\sigma$ is the signature. In the non-spin case, advances in the geography of minimal symplectic 4–manifolds [ABB+10, AP10], combined with blow-ups, connected sums with nontrivial Bauer–Furuta invariants, and orientation-reversals, yield exotic smooth structures on most 4-manifolds of the for $m \\#_{a}\\mathbb{CP}^{2}\\#_{b}\\overline{\\mathbb{CP}}{}^{2} withab(a-b) \\ne$ 0 and a,b not both even. Here, the gaps essentially stem from our lack of understanding related to Problem 4.17. Less is known in the spin case, but the results in the geography of spin symplectic 4–manifolds [PS00a, Par02] similarly settle the existence of exotic smooth structures on most spin 4–manifolds with $c^{2}_{1} \\geq$ 0 and $\\sigma$ <0; i.e. on $\\#_{p}K3 \\#_{q}(S^{2} \\times S^{2})$ for 0 $< p \\leq$ q+1. There are still some gaps, such as $K3 \\#(S^{2} \\times S^{2})$, which are again due to our lack of understanding related to Problem 4.17. Some exotic examples also exist in the range q+2 $\\leq p \\leq$ q+4 (e.g. see Remark 5 below for q=0), but none are known for $p \\geq$ q+5 (i.e. $2e+3\\sigma<$ −12).\n\n(2) A potential source of exotic definite 4–manifolds arise from pairs of knots with the same 0–surgeries, where only one knot bounds a disk with a given self-intersection number in $\\#_{k}\\mathbb{CP}^{2} \\setminus D^{4}$. See [MP23], [Qin25].\n\n(3) There are infinitely many exotic smooth structures on $\\#_{m}(\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2})$ and $\\#_{n}S^{2} \\times S^{2}$ for any odd $m \\geq$ 9, $n \\geq$ 11 [BH23]. A possible strategy to extend these results to cover smaller m, n is through Lefschetz pencils [Bay22] and Fintushel–Stern reverse engineering [FPS07]. Here is another potential construction for exotic $S^{2} \\times S^{2}$ or $\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2}$, which could perhaps be detected using knot Floer homology: given a pair of knots in $S^{3}$ with the samen-surgery, by gluing their traces one obtains a manifold homeomorphic to either $S^{2} \\times S^{2}$ or $\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2}$ (depending on the parity of n). See [LLP23, Remark 6.13].\n\n(4) Exotic definite 4-manifolds with finite fundamental groups can be derived from indefinite exotic 4-manifolds admitting free finite group actions. Examples with fundamental group $\\mathbb{Z}/2$ have been announced in [LLP23, SS24c] as quotients of exotic $\\mathbb{CP}^{2}\\#_{m}\\mathbb{CP}^{2}$ under free involutions, and with more general fundamental groups by Baykur, Stipsicz and Szabó in [BSS24]. The smallest definite example here has $b_{2} =$ 1. It is plausible that exotic signature zero 4-manifolds with finite fundamental groups and smaller $b_{2}$ can be derived similarly.\n\n(5) The Bauer–Furuta invariants [BF04] are sometimes able to detect exotic smooth structures on connected sums. Bauer [Bau04] used these invariants to find exotic smooth structures on $\\#_{p}K3$ for $p \\leq$ 4, but the method stops working at p=5 and is not applicable if any summand has $b^{+}_{2} \\equiv1$ mod 4.\n\nReferences cited:\n- [ABB+10] Anar Akhmedov, Scott Baldridge, R. İnanç Baykur, Paul Kirk, and B. Doug Park. Simply connected minimal symplectic 4-manifolds with signature less than -1. J. Eur. Math. Soc. (JEMS), 12(1):133–161, 2010. doi:10.4171/JEMS/192.\n- [AP10] Anar Akhmedov and B. Doug Park. Exotic smooth structures on small 4-manifolds with odd signatures. Invent. Math., 181(3):577–603, 2010. doi:10.1007/s00222-010-0254-y.\n- [PS00a] B. Doug Park and Zoltán Szabó. The geography problem for irreducible spin four-manifolds. Trans. Amer. Math. Soc., 352(8):3639–3650, 2000. doi:10.1090/S0002-9947-00-02467-3.\n- [Par02] Jongil Park. The geography of Spin symplectic 4-manifolds. Math. Z., 240(2):405– 421, 2002. doi:10.1007/s002090100390.\n- [MP23] Ciprian Manolescu and Lisa Piccirillo. From zero surgeries to candidates for exotic definite 4-manifolds. Journal of the London Mathematical Society, 108(5):2001– 2036, 2023. doi:10.1112/jlms.12800.\n- [Qin25] Qianhe Qin. An RBG construction of integral surgery homeomorphisms. Algebr. Geom. Topol., 25(6):3755–3774, 2025. doi:10.2140/agt.2025.25.3755.\n- [BH23] R. Inanc Baykur and Noriyuki Hamada. Exotic 4-manifolds with signature zero, 2023. Selecta Math., to appear. arXiv:2305.10908.\n- [Bay22] R. İnanç Baykur. Small exotic 4-manifolds and symplectic Calabi-Yau surfaces via genus-3 pencils. In Gauge theory and low-dimensional topology—progress and interaction, volume 5 of Open Book Ser., pages 185–221. Math. Sci. Publ., Berkeley, CA, 2022. https://msp.org/obs/2022/5-1/p09.xhtml.\n- [FPS07] Ronald Fintushel, B. Doug Park, and Ronald J. Stern. Reverse engineering small 4-manifolds. Algebr. Geom. Topol., 7:2103–2116, 2007. doi:10.2140/agt.2007.7.2103.\n- [LLP23] Adam Simon Levine, Tye Lidman, and Lisa Piccirillo. New constructions and invariants of closed exotic 4-manifolds, 2023. arXiv:2307.08130.\n- [SS24c] András I. Stipsicz and Zoltán Szabó. Definite four-manifolds with exotic smooth structures. J. Reine Angew. Math.), 2024(817):267–290, 2024. URL: https://doi.org/10.1515/crelle-2024-0072, doi:doi:10.1515/crelle-2024-0072.\n- [BSS24] R. Inanc Baykur, Andras I. Stipsicz, and Zoltan Szabo. Smooth structures on fourmanifolds with finite cyclic fundamental groups, 2024. arXiv:2406.09007.\n- [BF04] Stefan Bauer and Mikio Furuta. A stable cohomotopy refinement of Seiberg-Witten invariants. I. Invent. Math., 155(1):1–19, 2004. doi:10.1007/s00222-003-0288-5.\n- [Bau04] Stefan Bauer. A stable cohomotopy refinement of Seiberg-Witten invariants. II. Invent. Math., 155(1):21–40, 2004. doi:10.1007/s00222-003-0289-4.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2880, "problem_number": "KP-4.4", "title": "Kirby Problem 4.4", "statement": "Is there an exotic smooth structure on some product 4-manifold $S^{1} \\times Y^{3}$ or $\\Sigma_{g} \\times \\Sigma_{h}$? Do they all admit exotic smooth structures?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.4.\n\nLiterature notes:\n(1) A homeomorphism classification is unavailable for many of these 4–manifolds, making it challenging to determine whether a potential exotic copy falls within the desired homeomorphism class.\n\n(2) Significant special cases, besides $S^{2} \\times S^{2}$, include $S^{1} \\times S^{3}, T^{2} \\times S^{2}$, and $S^{1} \\times Y^{3}$, for any $Y^{3}$ that is $a T^{2}–bundle$ over $S^{1}$, for example $Y^{3} =T^{3}$. The first three have good fundamental groups\\{1\\}, $\\mathbb{Z}$, and $\\mathbb{Z}^{2}$ respectively, and there are homeomorphism classifications of closed 4-manifolds with these fundamental groups [FQ90, HKT09]. The remaining manifolds $S^{1} \\times Y^{3}$ are aspherical infrasolvmanifolds with good fundamental groups, for which the Borel conjecture holds [Hil02]. Thus, for $Y^{3}$ as above, any homotopy equivalence of 4-manifolds $X^{4} \\to S^{1} \\times Y^{3}$ is homotopic to a homeomorphism.\n\n(3) For $S^{1} \\times S^{3}$, see Problems 4.22and 4.65about detecting potential exotic structures.\n\n(4) For $S^{2} \\times T^{2}$, see [Bay22, Theorem 6] for a possible strategy to generate exotic smooth structures via Lefschetz pencils and some candidates.\n\nReferences cited:\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [HKT09] Ian Hambleton, Matthias Kreck, and Peter Teichner. Topological 4-manifolds with geometrically two-dimensional fundamental groups. J. Topol. Anal., 1(2):123–151, 2009. doi:10.1142/$S^{1}$793525309000084.\n- [Hil02] J. A. Hillman. Four-manifolds, geometries and knots, volume 5 of Geometry \\& Topology Monographs. Geometry \\& Topology Publications, Coventry, 2002.\n- [Bay22] R. İnanç Baykur. Small exotic 4-manifolds and symplectic Calabi-Yau surfaces via genus-3 pencils. In Gauge theory and low-dimensional topology—progress and interaction, volume 5 of Open Book Ser., pages 185–221. Math. Sci. Publ., Berkeley, CA, 2022. https://msp.org/obs/2022/5-1/p09.xhtml.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2881, "problem_number": "KP-4.5", "title": "Kirby Problem 4.5", "statement": "Does every connected, open 4-manifold admit uncountably many smooth structures?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.5.\n\nLiterature notes:\n(1) Recall that a manifold is said to be open if it has empty boundary and every component is noncompact. Any such 4-manifold is smoothable by work of Quinn [Qui82, Corollary 2.2.3] (see also [FQ90, Theorem 8.2]). However it is unknown whether every open 4-manifold has more than one smooth structure.\n\n(2) Gompf showed that $M \\setminus$ \\{x\\} for $x \\in M$, with M an arbitrary topological 4-manifold (not necessarily compact or orientable), has uncountably many smooth structures [Gom93, Theorem 2.1].\n\n(3) Euclidean 4-space has uncountably many smooth structures [Tau87], which for m a monoid acting on the smooth structures on any noncompact 4-manifold. Part of the question is therefore whether there is a 4-manifold that somehow absorbs or unwinds the Euclidean structures.\n\n(4) Recall that smooth structures on a manifold may be considered either up to diffeomorphism or up to isotopy. Isotopy implies diffeomorphism. This question is asking about smooth structures up to diffeomorphism. One could also ask the related question whether every open 4-manifold admits a smooth structure with uncountably many isotopy classes. Non-isotopic smooth structures can be produced by pulling back a smooth structure along a non-smoothable self-homeomorphism. See Problem 4.83.\n\nReferences cited:\n- [Qui82] Frank Quinn. Ends of maps. III. Dimensions 4 and 5. J. Differential Geometry, 17(3):503–521, 1982. http://projecteuclid.org/euclid.jdg/1214437139.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [Gom93] Robert E. Gompf. An exotic menagerie. J. Differential Geom., 37(1):199–223, 1993. http://projecteuclid.org/euclid.jdg/1214453429.\n- [Tau87] Clifford Henry Taubes. Gauge theory on asymptotically periodic 4-manifolds. J. Differential Geom., 25(3):363–430, 1987. http://projecteuclid.org/euclid.jdg/1214440981.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2882, "problem_number": "KP-4.6", "title": "Kirby Problem 4.6", "statement": "Does every closed, orientable 3-manifold bound an absolutely exotic pair of smooth, orientable 4-manifolds?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.6.\n\nLiterature notes:\n(1) Yasui [Yas12], Etnyre–Min–Mukherjee [EMM22], and Iida-MukherjeeTaniguchi [IMT25] gave sufficient conditions on a closed, connected, orientable 3-manifold Y to bound a compact 4-manifold X with $\\partial X = Y$ such that X admits infinitely many pairwise non-diffeomorphic smooth structures. Suppose that at least one of the following conditions are satisfied:\n\n\\noindent$\\bullet$ Y carries a contact structure $\\xi$ such that its Heegaard Floer contact invariant does not vanish e.g. if Y is Seifert fibered;\n\n\\noindent$\\bullet$ (Y, $\\xi)$ has a weak symplectic filling, e.g. if Y has a taut foliation; in particular, if Y is irreducible with $b_{1}(Y)$ >0;\n\n\\noindent$\\bullet$ Y is a rational homology sphere that embeds as a separating hypersurface in a closed definite 4-manifold. Then there exists a manifold X as above. In some of the cases, one gets smooth structures that are absolutely exotic, i.e. cannot be related by any diffeomorphism. In other cases, it is only known that they cannot be related by a diffeomorphism restricting to the identity on the boundary.\n\n(2) By arguments similar to those in [Kre84a], one can show that every closed, orientable 3-manifold bounds an exotic pair of nonorientable 4manifolds, so the second “orientable” in the question is necessary for an open question. A related open question is as follows.\n\n\\paragraph{Question.} Does every closed, nonorientable 3-manifold bound an absolutely exotic pair of smooth 4-manifolds?\n\nReferences cited:\n- [Yas12] Kouichi Yasui. Nuclei and exotic 4-manifolds, 2012. arXiv:1111.0620.\n- [EMM22] John B. Etnyre, Hyunki Min, and Anubhav Mukherjee. On 3-manifolds that are boundaries of exotic 4-manifolds. Trans. Amer. Math. Soc., 375(6):4307–4332, 2022. doi:10.1090/tran/8586.\n- [IMT25] Nobuo Iida, Anubhav Mukherjee, and Masaki Taniguchi. An adjunction inequality for the Bauer-Furuta type invariants, with applications to sliceness and 4-manifold topology. Adv. Math., 466:Paper No. 110134, 38, 2025. doi:10.1016/j.aim.2025.110134.\n- [Kre84a] M. Kreck. Some closed 4-manifolds with exotic differentiable structure. In Algebraic topology, Aarhus 1982 (Aarhus, 1982), volume 1051 of Lecture Notes in Math., pages 246–262. Springer, Berlin, 1984. doi:10.1007/BFb0075570.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2883, "problem_number": "KP-4.7", "title": "Kirby Problem 4.7", "statement": "(a) If $M_{1},M_{2}are$ two homeomorphic closed, oriented 4-manifolds, is $M_{1}\\#S^{2} \\times S^{2}$ diffeomorphic to $M_{2}\\#S^{2} \\times S^{2}$?\n\n(b) Is there a fixedn>0such that $M_{1}\\#_{n}S^{2} \\times S^{2}$ is diffeomorphic to $M_{2}\\#_{n}S^{2} \\times S^{2}$ for every pair $M_{1}, M_{2}$ as above?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.7.\n\nLiterature notes:\n(1) By Gompf’s generalization of Wall’s theorem [Wal64b, Gom84], for any given pair $M_{1}, M_{2}$ as above, there exists $k \\geq$ 0 such that $M_{1}\\#_{k}S^{2} \\times S^{2}$ is diffeomorphic to $M_{2}\\#_{k}S^{2} \\times S^{2}. A$ priorik depends on $M_{1}$ and $M_{2}$, but in most examples, $k =$ 1 is known to suffice; see [BS13] and the references therein.\n\n(2) In the case of nonempty boundary, it has been announced by S. Kang [Kan22b] that one stabilization is not enough. Kang proposes an example of two homeomorphic smooth contractible 4-manifolds $M_{1}, M_{2}$ with diffeomorphic boundaries, such that $M_{1}\\#S^{2} \\times S^{2}$ and $M_{2}\\#S^{2} \\times S^{2}$ are not diffeomorphic. The obstruction to existence of a diffeomorphism between $M_{1}\\#S^{2} \\times S^{2}$ and $M_{2}\\#S^{2} \\times S^{2}$ comes from involutive Heegaard Floer homology.\n\n(3) In the closed case, detecting an exotic pair after stabilization is difficult because most known invariants vanish for $M\\#S^{2} \\times S^{2}$. The only known exceptions are the Pin(2)-equivariant Bauer–Furuta invariant, and Fintushel–Stern’s 2-torsion instanton invariants, which can survive up to two stabilizations. To date, neither have been shown to differ for double stabilizations of homeomorphic 4-manifolds.\n\n(4) One idea for the closed case is to look at exotic surfaces in 4-manifolds that stay exotic after a weak internal stabilization, and take their double branched covers. See Problem 4.33.\n\n(5) A related but easier question is to find an exotic pair of closed simply connected 4-manifolds $X_{1}, X_{2}$ such that both manifolds contain a homologically essential square 0 sphere, and there is a homeomorphism between $X_{1}$ and $X_{2}$ (or an isomorphism between their intersection forms) that takes the homology class of one square 0 sphere to the other. The complements of the square 0 sphere are not required to be simply connected. In this setting, $X_{1}$ and $X_{2}$ would be hard to distinguish for the same reason as above: due to the presence of a square 0 sphere, most known invariants vanish.\n\n(6) A positive answer to (b) would follow from existence of a universal cork; see Problem 4.14.\n\n(7) One can ask the same questions about stabilization by the twisted $S^{2}bundle$ over $S^{2}$. Kang’s examples become diffeomorphic after one twisted stabilization [HKM23]. Twisted stabilizations are not spin, so it is harder to find invariants to distinguish them: the maps on involutive Heegaard Floer homology and the Pin(2)-equivariant Bauer–Furuta invariants do not work.\n\nReferences cited:\n- [Wal64b] C. T. C. Wall. On simply-connected 4-manifolds. J. London Math. Soc., 39:141–149, 1964. doi:10.1112/jlms/s1-39.1.141.\n- [Gom84] Robert E. Gompf. Stable diffeomorphism of compact 4-manifolds. Topology Appl., 18(2-3):115–120, 1984.\n- [BS13] R. İnanç Baykur and Nathan Sunukjian. Round handles, logarithmic transforms and smooth 4-manifolds. J. Topol., 6(1):49–63, 2013.\n- [Kan22b] Sungkyung Kang. One stabilization is not enough for contractible 4-manifolds, 2022. arXiv:2210.07510.\n- [HKM23] Kyle Hayden, Sungkyung Kang, and Anubhav Mukherjee. One stabilization is not enough for closed knotted surfaces, 2023. arXiv:2304.01504.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2884, "problem_number": "KP-4.8", "title": "Kirby Problem 4.8", "statement": "Let X be a closed, simply connected, smooth 4-manifold, and T a smoothly embedded torus in X with $\\pi_{1}(X$ −T) =1 and $[T]^{2}$ =0. Let $X_{K}$ be the result of Fintushel–Stern knot surgery on X along a knot K. If $K_{1}$ and $K_{2}$ are two prime knots such that $X_{K,1}$ and $X_{K,2}$ are diffeomorphic, does it follow that $K_{1}$ is either $K_{2}$ or its mirror?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.8.\n\nLiterature notes:\n(1) Fintushel and Stern showed that $X_{K}$ is homeomorphic to X, and $SW(X_{K}) = SW(X) \\cdot\\Delta_{K}$, where SW denotes the Seiberg-Witten series and $\\Delta_{K}$ the Alexander polynomial.\n\n(2) One can ask the question for the particular case when X is the K3 surface. In that case $SW(X) =$ 1, so the two knots need to have the same Alexander polynomial. No other constraint is known. For $X = K3$, the question in the problem was raised by Fintushel and Stern.\n\n(3) Akbulut [Akb02] showed that, if $m(K)$ denotes the mirror of K, then $X_{K}$ is diffeomorphic to $X_{m,(,K,)}$. Using work of Finashin [Fin02], Akaho [Aka06] showed that $X_{K,\\#,K}$ is diffeomorphic to $X_{K,\\#,m,(,K,)}$.\n\n(4) If one allows $\\pi_{1}(X) \\ne$ 1, then there are many examples of $X_{K,1}$ and $X_{K,2}$ that have the same Seiberg-Witten invariants (since $\\Delta_{K,1} =\\Delta_{K,2})$ but are not diffeomorphic, being distinguished by the Seiberg-Witten invariants of their finite covers [FS99a, PY15] or by invariants from Heegaard Floer theory [LLP23].\n\nReferences cited:\n- [Akb02] Selman Akbulut. Variations on Fintushel-Stern knot surgery on 4-manifolds. Turkish J. Math., 26(1):81–92, 2002.\n- [Fin02] Sergey Finashin. Knotting of algebraic curves in $\\mathbb{CP}^{2}$. Topology, 41(1):47–55, 2002. doi:10.1016/S0040-9383(00)00023-9.\n- [Aka06] Manabu Akaho. A connected sum of knots and Fintushel-Stern knot surgery on 4-manifolds. Turkish J. Math., 30(1):87–93, 2006.\n- [FS99a] Ronald Fintushel and Ronald J. Stern. Nondiffeomorphic symplectic 4-manifolds with the same Seiberg-Witten invariants. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 103–111. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.103.\n- [PY15] Jongil Park and Ki-Heon Yun. Families of nondiffeomorphic 4-manifolds with the same Seiberg-Witten invariants. J. Symplectic Geom., 13(2):279–303, 2015. doi: 10.4310/JSG.2015.v13.n2.a2.\n- [LLP23] Adam Simon Levine, Tye Lidman, and Lisa Piccirillo. New constructions and invariants of closed exotic 4-manifolds, 2023. arXiv:2307.08130.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2885, "problem_number": "KP-4.9", "title": "Kirby Problem 4.9", "statement": "Is every Gluck twist in $S^{4}$ standard?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.9.\n\nLiterature notes:\n(1) This is [Kir78, Problem 4.24]. Given an embedded 2-sphere in $S^{4}$, Gluck defined a surgery operation that yields a homotopy 4-sphere [Glu62], which is in fact homeomorphic to $S^{4}$ due to Freedman [Fre82]. However, it is unclear whether the resulting manifold is diffeomorphic to $S^{4}$.\n\n(2) The following 2-knots are known to have trivial Gluck twists i.e. the result of Gluck surgery on the 2-knot yields $S^{4}$.\n\n\\noindent$\\bullet$ Any spun knot or more generally any ribbon 2-knots [Glu62].\n\n\\noindent$\\bullet$ Any 2-knot 0-concordant to a 2-knot with trivial Gluck twist. More generally, if two 2-knots are 0-concordant, then the Gluck twist of those two 2-knots are diffeomorphic [Mel77, HMY00].\n\n\\noindent$\\bullet$ Any twist spun knot [Gor76] or branched twist spun knot [Pao78].\n\n\\noindent$\\bullet$ Any 2-knot that can be unknotted by a single finger move followed by a single Whitney move, which includes any roll spin of a classical knot of unknotting number one [NS22].\n\n\\noindent$\\bullet$ Any even degree satellite 2-knot whose pattern has trivial Gluck twist, or any satellite 2-knot (of any degree) in which both the pattern and the companion 2-knots have trivial Gluck twist [Kim20].\n\n\\noindent$\\bullet$ Any 2-knot that can be decomposed as a union of two ribbon disks, one of which has undisking number one [GNS25].\n\n\\noindent$\\bullet$ Some other examples related to the Cappell-Shaneson homotopy 4sphere (see [AK79a, Gom91a]) and any 2-knot obtained from gluin g particular ribbon disks satisfying certain conditions (see [NS12]).\n\n(3) A unit sphere in $\\mathbb{CP}^{2}$ is a smoothly embedded sphere that intersects the standard $\\mathbb{CP}^{1}$ transversely once. If a unit sphere is obtained from a 2-knot in $S^{4}$ by blowing up at a point, then we say a unit sphere is obtained from the 2-knot.\n\n\\paragraph{Question.} Is every unit sphere in $\\mathbb{CP}^{2}$ smoothly isotopic to the standard $\\mathbb{CP}^{1}$? Melvin [Mel77] showed that the Gluck twist of a 2-knot K in $S^{4}$ is trivial if and only if the pair $(\\mathbb{CP}^{2},\\Sigma)$, where $\\Sigma$ is the unit sphere obtained from K, is pairwise diffeomorphic to $(\\mathbb{CP}^{2},\\mathbb{CP}^{1})$. It is unknown whether such a pairwise diffeomorphism would imply a smooth isotopy from $\\Sigma$ to $\\mathbb{CP}^{1}$. Some unit spheres are known to be smoothly isotopic to $\\mathbb{CP}^{1}$ [HKM20].\n\nReferences cited:\n- [Kir78] Rob Kirby. Problems in low dimensional manifold theory. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 273–312. Amer. Math. Soc., Providence, R.I., 1978.\n- [Glu62] Herman Gluck. The embedding of two-spheres in the four-sphere. Trans. Amer. Math. Soc., 104:308–333, 1962. doi:10.2307/1993581.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [Mel77] Paul Melvin. Blowing up and down in 4-manifolds. PhD thesis, UC Berkeley, 1977.\n- [HMY00] Kazuo Habiro, Yoshihiko Marumoto, and Yuichi Yamada. Gluck surgery and framed links in 4-manifolds. In Knots In Hellas’ 98, pages 80–93. World Scientific, 2000.\n- [Gor76] C. McA. Gordon. Knots in the 4-sphere. Comment. Math. Helv., 51(4):585–596, 1976. doi:10.1007/BF02568175.\n- [Pao78] Peter Sie Pao. Nonlinear circle actions on the 4-sphere and twisting spun knots. Topology, 17(3):291–296, 1978. doi:10.1016/0040-9383(78)90033-2.\n- [NS22] Patrick Naylor and Hannah R. Schwartz. Gluck twisting roll spun knots. Algebr. Geom. Topol., 22(2):973–990, 2022. doi:10.2140/agt.2022.22.973.\n- [Kim20] Seungwon Kim. Gluck twist and unknotting of satellite 2-knots, 2020. arXiv:2009.07353.\n- [GNS25] David Gabai, Patrick Naylor, and Hannah Schwartz. Doubles of Gluck twists: a five-dimensional approach. Adv. Math., 480:Paper No. 110455, 29, 2025. doi:10.1016/j.aim.2025.110455.\n- [AK79a] Selman Akbulut and Robion Kirby. An exotic involution of $S^{4}$. Topology, 18(1):75– 81, 1979. doi:10.1016/0040-9383(79)90015-6.\n- [Gom91a] Robert E. Gompf. Killing the Akbulut-Kirby 4-sphere, with relevance to the Andrews-Curtis and Schoenflies problems. Topology, 30(1):97–115, 1991. doi: 10.1016/0040-9383(91)90036-4.\n- [NS12] Daniel Nash and András I. Stipsicz. Gluck twist on a certain family of 2-knots. Michigan Math. J., 61(4):703–713, 2012. doi:10.1307/mmj/1353098509.\n- [HKM20] Mark C. Hughes, Seungwon Kim, and Maggie Miller. Isotopies of surfaces in 4-manifolds via banded unlink diagrams. Geom. Topol., 24(3):1519–1569, 2020. doi: 10.2140/gt.2020.24.1519.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2886, "problem_number": "KP-4.10", "title": "Kirby Problem 4.10", "statement": "(a) Is every homotopy $B^{4}$ with boundary $S^{3}$ obtained by performing a Gluck twist on some knotted 2-sphere in $B^{4}$?\n\n(b) Suppose a homotopy 4-ball X is obtained by performing a Gluck twist on a knotted 2-sphere in $B^{4}$. Is $X \\times I$ necessarily diffeomorphic to $B^{5}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.10.\n\nLiterature notes:\n(1) Habiro, Marumoto and Yamada [HMY00] showed that if a homotopy 4-ball is obtained by performing Gluck twists on multiple 2-spheres in $B^{4}$, then it can be obtained by a Gluck construction on a single knotted 2-sphere in $B^{4}$; see [Mel77] for related ideas.\n\n(2) Part (a) of the problem could also be asked for homotopy spheres; the reason for specifying a homotopy 4-ball is that part (b)is a potential route to finding a counterexample to the Schoenflies Conjecture (Problem 4.23) as well as to the 4-dimensional Poincaré Conjecture (Problem 4.1). Part\n\n(b) is a weaker question than asking whether X is diffeomorphic to $B^{4}$ (Problem 4.9).\n\nReferences cited:\n- [HMY00] Kazuo Habiro, Yoshihiko Marumoto, and Yuichi Yamada. Gluck surgery and framed links in 4-manifolds. In Knots In Hellas’ 98, pages 80–93. World Scientific, 2000.\n- [Mel77] Paul Melvin. Blowing up and down in 4-manifolds. PhD thesis, UC Berkeley, 1977.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2887, "problem_number": "KP-4.11", "title": "Kirby Problem 4.11", "statement": "Let M be a smooth 4-manifold and letf: $S^{2} \\to M$ be a smooth embedding with trivial normal bundle. Then let $M_{f}$ denote the result of Gluck twisting on M along f.\n\n(a) Does there exist an orientable M, and an embedding f, such that M and $M_{f}$ are homeomorphic but not diffeomorphic?\n\n(b) Does there exist an orientable M, and smooth, homotopic embeddings f, $g: S^{2} \\to M$ with trivial normal bundle such that $M_{f}$ and $M_{g}$ are homeomorphic but not diffeomorphic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.11.\n\nLiterature notes:\n(1) The version of this question with $M =S^{4}$ has received considerable interest (see Problems 4.9 and4.10). It is possible that the general case might be attacked more readily and/or provide insight to the $S^{4}$ case.\n\n(2) Akbulut showed that Gluck twisting along a (homotopically essential) sphere in anonorientable 4-manifold can produce an exotic smoothing [Akb88] (see also [Tor17]).\n\n(3) Akbulut and Yasui give a simple condition on a 4-manifold and an embedded sphere f to ensure that Gluck twisting along f does not change the diffeomorphism type in [AY13]. Note that Gluck twisting may change the homeomorphism type, e.g. Gluck twisting $S^{2} \\times$ pt inside $S^{2} \\times S^{2}$ yields $S^{2} \\widetilde{\\times} S^{2}$. See [GS99, Exercise 5.2.7(a)].\n\n(4) In [KPR23, Theorem 1.2], it was shown that $M_{f}$ and $M_{g}$ are simple homotopy equivalent when f and g are homotopic. Under the stronger hypothesis thatf and g are $concordant,M_{f}$ and $M_{g}$ ares-cobordant, and therefore homeomorphic when $\\pi_{1}(M)is a$ good group (see Problem 4.46). Thus in the oriented case, finding examples where $M_{f}$ and $M_{g}$ are an exotic pair might be easier than finding examples where M and $M_{f}$ are. See [KPR23] for further discussion.\n\nReferences cited:\n- [Akb88] Selman Akbulut. Constructing a fake 4-manifold by Gluck construction to a standard 4-manifold. Topology, 27(2):239–243, 1988. doi:10.1016/0040-9383(88) 90041-9.\n- [Tor17] Rafael Torres. Smooth structures on nonorientable four-manifolds and free involutions. J. Knot Theory Ramifications, 26(13):1750085, 20, 2017. doi:10.1142/S0218216517500857.\n- [AY13] Selman Akbulut and Kouichi Yasui. Gluck twisting 4-manifolds with odd intersection form. Math. Res. Lett., 20(2):385–389, 2013. doi:10.4310/MRL.2013.v20.n2.a13.\n- [GS99] Robert E. Gompf and András I. Stipsicz. 4-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1999. doi:10.1090/gsm/020.\n- [KPR23] Daniel Kasprowski, Mark Powell, and Arunima Ray. Gluck twists on concordant or homotopic spheres, 2023. doi:10.4310/mrl.2023.v30.n6.a6.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2888, "problem_number": "KP-4.12", "title": "Kirby Problem 4.12", "statement": "For X a closed simply connected smooth 4-manifold, let $g_{X}: H_{2}(X) \\to \\mathbb{N}$ denote the genus function, which assigns to every homology class the minimal genus of a smooth embedded surface representing that class.\n\n(a) Suppose $f: X_{1} \\to X_{2}$ is a homeomorphism between simply connected, smooth, closed 4-manifolds. Let $f_{*}: H_{2}(X_{1}) \\to H_{2}(X_{2})$ be the induced map on homology. If for all x in $H_{2}(X_{1})$ we have $g_{X,1}(x) = g_{X,2}(f_{*}(x))$, is $X_{1}$ diffeomorphic to $X_{2}$?\n\n(b) Suppose $f: X_{1} \\to X_{2}$ is a homeomorphism between simply connected smooth compact 4-manifolds with the same boundary $\\partial X_{1} = \\partial X_{2}$, such thatf is the identity on the boundary. $Letf_{*}: H_{2}(X_{1},\\partial X_{1}) \\to H_{2}(X_{2},\\partial X_{2})$ be the induced map on relative homology. If for all knots K and for all x in $H_{2}(X_{1},\\partial X_{1})$ we have $g_{X,1,K}(x) =g_{X,2,K}(f_{*}(x))$, is $X_{1}$ diffeomorphic to $X_{2}$ rel. boundary?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.12.\n\nLiterature notes:\n(1) A version of this problem appeared in Stipsicz–Szabó [SS24b].\n\n(2) There are several ways one might define a genus function for a 4-manifold with boundary. For concreteness, we present one. For X a simply connected, smooth 4-manifold with boundary and K a knot in $\\partial X$, write $g_{X,K}: H_{2}(X,\\partial X) \\to \\mathbb{N}$ for the relative genus function, that assigns to every relative homology class the minimal genus of a smooth embedded surface with boundary K representing that class.\n\n(3) In the closed case, every isometry of the intersection form on $H_{2}is$ realized by a homeomorphism [FQ90, Theorem 10.1], so one could have stated the problem by starting with the isometry $f_{*}$.\n\n(4) On the other hand, it is not the case that every isometry of the intersection form that respects the genus function is realized by a diffeomorphism. For example, the isomorphism $\\varphi: H_{2}(K3) \\to H_{2}(K3)$ given by $\\varphi(a) =$ −a is not induced by a diffeomorphism; see [DK90, Corollary 9.1.4].\n\n(5) A special case of part (a) is Problem 4.1, the smooth 4-dimensional Poincaré conjecture.\n\nReferences cited:\n- [SS24b] András I. Stipsicz and Zoltán Szabó. On the minimal genus problem in fourmanifolds. In Frontiers in geometry and topology. Summer school and research conference, The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, August 1–12, 2022, pages 215–232. Providence, RI: American Mathematical Society (AMS), 2024. doi:10.1090/pspum/109/01997.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [DK90] S. K. Donaldson and P. B. Kronheimer. The geometry of four-manifolds. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2889, "problem_number": "KP-4.13", "title": "Kirby Problem 4.13", "statement": "(a) Does every large $\\mathbb{R}^{4}-homeomorph$ lie $in\\mathcal{R}_{K}$ for some Kthat is not smoothly slice?\n\n(b) Does there exist an infinite sequence of knots ${K_{i}}$ such that $\\mathcal{R}_{K,i} \\ne \\mathcal{R}_{K,j}$ whenever $i \\ne j$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.13.\n\nLiterature notes:\n(1) Given a knot K, let $X_{K}$ denote its 0-trace, which is obtained, by definition, by attaching a 0-framed 2-handle to $B^{4}$ along K and smoothing corners. Reserve the symbol $\\mathbb{R}^{4}$ to refer to 4-dimensional Euclidean space with its standard smooth structure. Following Gompf, we call a smooth manifold $\\mathcal{R}$ an $\\mathbb{R}^{4}-homeomorph$ if it is homeomorphic to $\\mathbb{R}^{4}$, but not necessarily diffeomorphic. This is to avoid the terminology ‘exotic $\\mathbb{R}^{4}’$ in cases where we want to allow the standard smooth structure.\n\n(2) An $\\mathbb{R}^{4}-homeomorph$ is said to belarge if it contains a compact subset that does not admit a smooth embedding into $\\mathbb{R}^{4}$, and small if it is not large.\n\n(3) Given a knot K let $\\mathcal{R}_{K}$ denote the set of $\\mathbb{R}^{4}-homeomorphs$ admitting a smooth embedding of $X_{K}$. Gompf [Gom85, Lemma 1.1] described a construction of elements $in\\mathcal{R}_{K}$ using Quinn’s smoothing theorem. By the trace embedding lemma, $\\mathcal{R}_{K}$ is nonempty precisely if K is topologically slice (not necessarily smoothly slice), and K is smoothly slice if and only if $\\mathbb{R}^{4} \\in \\mathcal{R}_{K}$ if and only if $\\mathcal{R}_{K}$ is the set of all $\\mathbb{R}^{4}-homeomorphs$, including all small $\\mathbb{R}^{4}-homeomorphs$.\n\n(4) Let $K_{0}$ be the unknot, $K_{1}$ be the second iterated positive Whitehead double of the right-handed trefoil, and $K_{2}$ any knot with $\\tau(K_{2})$ <0, e.g. $-K_{1}$. Then $\\mathcal{R}_{K,i} \\ne \\mathcal{R}_{K,j}$ for all $i \\ne j$.\n\n(5) An affirmative answer to the analogue of(b)for links follows from [Gom85, Lemma 1.2 and the proof of Theorem 1.3].\n\nReferences cited:\n- [Gom85] Robert E. Gompf. An infinite set of exotic $\\mathbb{R}^{4}$’s. J. Differential Geom., 21(2):283– 300, 1985. http://projecteuclid.org/euclid.jdg/1214439566.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2890, "problem_number": "KP-4.14", "title": "Kirby Problem 4.14", "statement": "Is there a universal cork? More precisely, does there exist some cork (C, f) such that given any pair W and $W^{1}$ of closed, simply connected 4-manifolds that are homeomorphic but not diffeomorphic, there exists a smooth embedding $C \\hookrightarrow W$ such that (W $\\setminus$ Int C) $\\cup _{f} C$ is diffeomorphic to $W^{1}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.14.\n\nLiterature notes:\n(1) A cork (C, f) is a pair of a compact, contractible, smooth 4-manifold C and a diffeomorphism $f: \\partial C \\to \\partial C$, such that f does not extend to a diffeomorphism of C. Some authors require f to be an involution, or C to be Stein.\n\n(2) Not all corks are universal: Ladu shows that the Akbulut cork is not universal [Lad25]. Let (C, f) denote the Akbulut cork. Briefly, the paper shows that since −C has trivial Floer homology in the relevant grading, (−C, f)cannot be used to go between two smooth 4-manifolds with differin g Seiberg-Witten invariants, such as $K3\\#\\mathbb{CP}^{2}$ and $\\#_{4}\\mathbb{CP}^{2}\\#_{19}\\overline{\\mathbb{CP}}{}^{2}.But$ then twisting by (C, f) cannot be used to go between the exotic pair $-(K3\\#\\mathbb{CP}^{2})$ and $-(\\#_{4}\\mathbb{CP}^{2}\\#_{19}\\overline{\\mathbb{CP}}{}^{2})$. This suggest an alternative definition of a universal cork, by allowing either a cork twist by C or −C. Under this definition it is still open whether the Akbulut cork is universal.\n\n(3) One could ask the question only requiring W and W1 to be compact with possibly nonempty boundary. Say a cork (C, f) is $\\partial-universal$ if any pair of compact, simply connected 4-manifolds, which are homeomorphic but not diffeomorphic, is related by a cork twist along some embedding of C. The same preprint of Ladu mentioned above shows that there is no $\\partial-universal$ cork [Lad25].\n\n(4) By Wall’s theorem on h-cobordisms [Wal64b], if (C, f) is a cork then there exists n such that f extends to a diffeomorphism of $C\\#_{n}S^{2} \\times S^{2}$. So, if there is a universal cork, then there is a fixed n so that any exotic pair $W, W^{1}$ as above are diffeomorphic after stabilization with n copies of $S^{2} \\times S^{2}$. See Problem 4.7. Note that if there is no bound on the number of $S^{2} \\times S^{2}$ summands needed to make a pair of exotic 4-manifolds diffeomorphic, there cannot be a closed universal cork. This suggests another refinement of the terminology: say a cork(C, f) is n-universal if it relates every pair of homeomorphic simply connected 4-manifolds that become diffeomorphic after connected sum with at most n copies of $S^{2} \\times S^{2}$.\n\nReferences cited:\n- [Lad25] Roberto Ladu. The Akbulut cork is not universal. Selecta Math. (N.S.), 31(4):Paper No. 74, 14, 2025. doi:10.1007/s00029-025-01061-6.\n- [Wal64b] C. T. C. Wall. On simply-connected 4-manifolds. J. London Math. Soc., 39:141–149, 1964. doi:10.1112/jlms/s1-39.1.141.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2891, "problem_number": "KP-4.15", "title": "Kirby Problem 4.15", "statement": "(11/8 Conjecture). Does every smooth, spin, closed 4-manifold X satisfy $b_{2}(X) \\geq 11|\\sigma(X)|$, where $\\sigma(X)$ is the signature of the intersection form?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.15.\n\nLiterature notes:\n(1) This is [Kir97, Problem 4.92], originally raised by Matsumoto.\n\n(2) A positive resolution would answer the Geography Question 4.54 for the smooth simply connected case. By Rokhlin’s theorem, the intersection form of a spin, smooth, closed 4-manifold is of the form $2p(\\pm E_{8})\\oplus q\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}$, for $p,q\\geq 0$ and a choice of sign $\\pm$. The 11/8 conjecture can be rephrased as $q \\geq$ 3p. If an intersection form satisfies this bound, it is realized by the manifold $p(\\pm K3)\\#(q-3p)(S^{2} \\times S^{2})$.\n\n(3) Donaldson [Don87b] used Yang-Mills theory to prove thatp $\\geq$ 1 implies $q \\geq$ 3, assuming $H_{1}(X;\\mathbb{Z})$ has no 2-torsion. Furuta [Fur01] used finite dimensional approximation of the SeibergWitten map to prove the inequality $b_{2}(X) \\geq 10|\\sigma(X)|$ +2 (i.e., $q \\geq$ 2p+1) assuming $q \\geq$ 1. Hopkins, Lin, Shi and Xu [HLSX22] refined Furuta’s method to get, for $p \\geq$ 2, $^{⎧}_{|}2p+2$ if $p\\equiv1,2,5,6$ (mod 8) $q \\geq ^{⎪}2p+3$ if $p\\equiv3,4,7$ (mod 8) $^{|}⎩2p+4$ if $p\\equiv0$ (mod 8) They also showed this is the strongest possible inequality that can be proved using that method.\n\n(4) There is a suggested strategy to prove the conjecture [Bau12, Man14]. The 11/8 conjecture would be true if one had an invariantf(Y)of oriented integral homology 3-spheres Y that satisfies the following property: Let W be a compact oriented spin smooth cobordism from an integral homology 3-sphere $Y_{0}$ to an integral homology 3-sphere $Y_{1}$. The intersection form of $W$ is of the form $rE_{8}\\oplus q\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}$, where $q\\geq 0$ and $r$ can be any integer. The required property for f is that $f(Y_{0}) \\leq f(Y_{1})$ +q+r−1.\n\n(5) There is also a proposal to approach the conjecture using the Ricci flow, by reducing it to the Hitchin-Thorpe inequality for Einstein 4-manifolds. See [Bam21b, Section 5.6].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Don87b] S. K. Donaldson. The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Differential Geom., 26(3):397–428, 1987. http://projecteuclid.org/euclid.jdg/1214441485.\n- [Fur01] M. Furuta. Monopole equation and the 11 8 -conjecture. Math. Res. Lett., 8(3):279– 291, 2001. doi:10.4310/MRL.2001.v8.n3.a5.\n- [HLSX22] Michael J. Hopkins, Jianfeng Lin, XiaoLin Danny Shi, and Zhouli Xu. Intersection forms of spin 4-manifolds and the $\\mathrm{Pin}(2)$-equivariant Mahowald invariant. Comm. Amer. Math. Soc., 2:22–132, 2022. doi:10.1090/cams/4.\n- [Bau12] Stefan A. Bauer. Intersection forms of spin four-manifolds, 2012. arXiv:1211.7092.\n- [Man14] Ciprian Manolescu. On the intersection forms of spin four-manifolds with boundary. Math. Ann., 359(3-4):695–728, 2014. doi:10.1007/s00208-014-1010-1.\n- [Bam21b] Richard H. Bamler. Recent developments in Ricci flows. Notices Amer. Math. Soc., 68(9):1486–1498, 2021. doi:10.1090/noti2343.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2892, "problem_number": "KP-4.16", "title": "Kirby Problem 4.16", "statement": "(a) Do there exist closed, oriented, smooth, irreducible 4-manifolds with $b^{+}_{2} >$ 1 and $c^{2}_{1}:=2\\chi+3\\sigma<0$?\n\n(b) Is there an irreducible exotic smooth structure on $\\mathbb{CP}^{2}\\#_{n}\\mathbb{CP}^{2}$ for n>9?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.16.\n\nLiterature notes:\n(1) $S^{2}–bundles$ over $\\Sigma_{h}$ are irreducible 4–manifolds with $b^{+}_{2} =$ 1, and for $h \\geq$ 2, they have $c^{2}_{1}<0$.\n\n(2) An exotic $\\mathbb{CP}^{2}\\#_{n}\\mathbb{CP}^{2}$ for $n >$ 9 would have $c^{2}_{1} <$ 0. All known exotica in that range are blow-ups of exotic $\\mathbb{CP}^{2}\\#_{n}\\mathbb{CP}^{2}$ for $n \\leq$ 9 (such as the Barlow or Dolgachev surface).", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2893, "problem_number": "KP-4.17", "title": "Kirby Problem 4.17", "statement": "Is there an irreducible, closed, simply connected, oriented 4– manifold with $b^{+}_{2}$ and $b^{-}_{2}$ both even?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.17.\n\nLiterature notes:\n(1) The condition $b^{+}_{2}(X)$ and $b^{-}_{2}(X)$ both being even is equivalent to the holomorphic Euler characteristic $\\chi_{h}(X):=\\frac{1}{4}(e(X)+\\sigma(X))$ not being an integer for either orientation on X. Note that $\\chi_{h}(X)$ always lies in $\\frac{1}{2}\\mathbb{Z}$ and is in $\\mathbb{Z}$ if and only if X admits an almost complex structure. All current smooth invariants, such as the Seiberg–Witten invariants used for detecting irreducibility effectively, are defined for 4–manifolds with $b_{1} =$ 0 when $\\chi_{h} \\in \\mathbb{Z}$. As a result, all known irreducible, closed, orientable 4–manifolds are almost complex under at least one orientation.\n\n(2) This problem is discussed in [Kir97, Problem 4.97], where it is conjectured that no such X exists. A notable test case, suggested by Gompf [Kir97], involves taking two copies of K3, removing the tubular neighborhood of an embedded sphere with self-intersection −2 from each, and then gluing the complements along their boundaries with an orientationreversing diffeomorphism of $\\mathbb{RP}^{3}$. The resulting 4–manifold X, which has $\\chi_{h}(X) \\notin \\mathbb{Z}$ for either orientation, is not known to be reducible.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2894, "problem_number": "KP-4.18", "title": "Kirby Problem 4.18", "statement": "(a) Does there exist a pair of smooth, closed 4-manifolds that are homotopy equivalent but not simple homotopy equivalent?\n\n(b) Does there exist a pair of smoothlyh-cobordant, smooth, closed 4-manifolds that are not smoothly s-cobordant?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.18.\n\nLiterature notes:\n(1) There exist examples of topologicallyh-cobordant 4-manifolds that are not topologically s-cobordant [KPR22], and examples of topological manifolds that are homotopy equivalent but not simple homotopy equivalent [NNP23].\n\n(2) A potential way to answer the questions is as follows. Let M be a smooth, closed 4-manifold with $\\pi:=\\pi_{1}(M)$. Let $x \\in Wh(\\pi)$ be an element of the Whitehead group of $\\pi$.\n\n\\paragraph{Question.} Does there exist a smoothh-cobordism(W;M, N)between 4-manifolds with Whitehead torsion $\\tau(W$, M) =x? Let $w: \\pi \\to C_{2}$ be the orientation character of M. The Whitehead group admits an involution, sending a representative matrix to its conjugate-transpose. Elements are conjugated by the standard involution $x \\to$ barx induced by extending $g \\to w(g)g^{-1}$ linearly. The question is of particular interest when x−barx $\\ne$ 0, because then the induced homotopy equivalence $M \\to N$ is not simple. This could lead to examples, if one can also compute the Whitehead torsions of all the homotopy self-equivalences.\n\n(3) This problem is a little different from other similar smooth realization problems. In the process of building an h-cobordism, one adds 2-handles and then tries to add 3-handles. One can in fact represent the desired homotopy classes in the middle level by smoothly embedded, framed spheres. Unfortunately this does not suffice. The challenge is to find smooth embeddings in such a way that the inclusion-induced map $\\pi_{1}(N) \\to \\pi_{1}(W)$ is an isomorphism. For every $x \\in Wh(\\pi)$, there exist $k \\in \\mathbb{N}$ such that x is smoothly realizable as the torsion $\\tau(W, M\\#_{k}S^{2} \\times S^{2})of$ some smoothh-cobordism W based on $M\\#_{k}S^{2} \\times S^{2}$ [CS71]; a preliminary construction appeared in [Sta65]. If one can find a topological h-cobordism (W;M, N) with torsion x, then it might be possible to apply smoothing theory for 5-manifolds to improve W to a smooth h-cobordism.\n\n(4) Another potential method for solving (a) was given by Kasprowski–Nicholson–Veselá in [KNV24]. Their proposed method, if it could be implemented, would give examples that are homotopy equivalent but not stably simple homotopy equivalent.\n\nReferences cited:\n- [KPR22] Daniel Kasprowski, Mark Powell, and Arunima Ray. Counterexamples in 4-manifold topology. EMS Surv. Math. Sci., 9(1):193–249, 2022. doi:10.4171/emss/56.\n- [NNP23] Csaba Nagy, John Nicholson, and Mark Powell. Simple homotopy types of even dimensional manifolds, 2023. arXiv:arXiv:2312.00322.\n- [CS71] Sylvain E. Cappell and Julius L. Shaneson. On four dimensional surgery and applications. Comment. Math. Helv., 46:500–528, 1971. doi:10.1007/BF02566862.\n- [Sta65] J. W. Stallings. On the infinite processes leading to differentiability in the complement of a point. Differ. and Combinat. Topology, Sympos. Marston Morse, Princeton, 245-254 (1965)., 1965.\n- [KNV24] Daniel Kasprowski, John Nicholson, and Simona Veselá. Stable equivalence relations on 4-manifolds, 2024. arXiv:2405.06637.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2895, "problem_number": "KP-4.19", "title": "Kirby Problem 4.19", "statement": "What are the possible Euler characteristics of closed, aspherical 4-manifolds? More specifically, we ask the following.\n\n(a) Is it always the case that $\\chi \\geq |\\sigma|$?\n\n(b) What is the smallest Euler characteristic of a closed hyperbolic 4-manifold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.19.\n\nLiterature notes:\n(1) The questions are motivated by the following conjecture.\n\n\\paragraph{Conjecture.} The Euler characteristic of X is non-negative. This is [Kir97, Problem 4.10]. One can ask the question for all aspherical, closed 4-manifolds, including non-smoothable ones. Edmonds [Edm15] has proven the conjecture for Haken 4-manifolds, which were defined by Foozwell and Rubinstein in [FR11].\n\n(2) Given that hyperbolic manifolds are aspherical, it is natural to ask the specific $subquestion(b)$. By Chern–Gauss–Bonnet, for a closed hyperbolic 4-manifold X we have $vol(X) = 4\\pi^{2}\\cdot\\chi(X)$, so we are equivalently asking for the smallest possible hyperbolic volume. The smallest known orientable hyperbolic 4-manifold was found by Conder and Maclachlan [CM05] and has $\\chi =$ 16; it covers a nonorientable hyperbolic manifold with $\\chi=8$. Notice that in the cusped case there are many known orientable examples with the minimal possible value $\\chi =$ 1 [RT00].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Edm15] Allan L. Edmonds. The Euler characteristic of a Haken 4-manifold. In Geometry, groups and dynamics, volume 639 of Contemp. Math., pages 217–234. Amer. Math. Soc., Providence, RI, 2015. doi:10.1090/conm/639/12796.\n- [FR11] Bell Foozwell and Hyam Rubinstein. Introduction to the theory of Haken nmanifolds. In Topology and geometry in dimension three, volume 560 of Contemp. Math., pages 71–84. Amer. Math. Soc., Providence, RI, 2011. doi:10.1090/conm/560/11092.\n- [CM05] Marston Conder and Colin Maclachlan. Compact hyperbolic 4-manifolds of small volume. Proc. Amer. Math. Soc., 133(8):2469–2476, 2005. doi:10.1090/S0002-9939-05-07634-3.\n- [RT00] John G. Ratcliffe and Steven T. Tschantz. The volume spectrum of hyperbolic 4-manifolds. Experiment. Math., 9(1):101–125, 2000. http://projecteuclid.org/euclid.em/1046889595.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2896, "problem_number": "KP-4.20", "title": "Kirby Problem 4.20", "statement": "Is $*\\mathbb{RP}^{4}\\#*\\mathbb{RP}^{4}$ smoothable? Is *En\\#*En smoothable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.20.\n\nLiterature notes:\n(1) The manifold $*\\mathbb{RP}^{4}$ was first constructed by Ruberman [Rub84] (see also [HKT94]). It is homotopy equivalent to $\\mathbb{RP}^{4}$ but not homeomorphic to $\\mathbb{RP}^{4}$. In particular, it has nontrivial Kirby–Siebenmann invariant. Taking the connected sum of two copies yields a 4-manifold $*\\mathbb{RP}^{4}\\#*\\mathbb{RP}^{4}$ with trivial Kirby–Siebenmann invariant. Recall that a 4-manifold with trivial Kirby–Siebenmann invariant is stably smoothable, i.e. smoothable after connected sum with $k(S^{2} \\times S^{2})$ for some $k \\geq$ 0.\n\n(2) Recall that $*\\mathbb{CP}^{2}$ denotes the Chern manifold constructed by Freedman in [Fre82]. It is homotopy equivalent to $\\mathbb{CP}^{2}$ but not homeomorphic to $\\mathbb{CP}^{2}$. In this case, we know $that*\\mathbb{CP}^{2}\\#*\\mathbb{CP}^{2} \\cong \\mathbb{CP}^{2}\\#\\mathbb{CP}^{2}$, and is therefor e smoothable. $Similarly,*\\mathbb{RP}^{4}\\#*\\mathbb{CP}^{2}$ is smoothable [RS97], answering [Kir97, Problem 4.82]. Whether $*\\mathbb{RP}^{4}\\#*\\mathbb{RP}^{4}$ is smoothable is the next natural case of a 4-manifold with trivial Kirby–Siebenmann invariant that is not obviously smoothable.\n\n(3) The smoothability of topological manifolds that are homotopy equivalent to $\\mathbb{RP}^{4}\\#\\mathbb{RP}^{4}$, arising from another construction using the action of $L_{5}$ on the structure set, is considered in Problem 4.22. By $contrast,*\\mathbb{RP}^{4}\\#*\\mathbb{RP}^{4}$ is detected in the normal invariants. See [BDK07] for the computation of the topological structure set of $\\mathbb{RP}^{4}\\#\\mathbb{RP}^{4}$.\n\n(4) We use En to denote the Enriques surface, which by definition is a quotient of the K3 surface by an involution. The manifold *En is a star partner of En, as defined by Freedman–Quinn [FQ90, Section 10.4]. Once again, *En\\#*En has trivial Kirby–Siebenmann invariant, and it would be interesting to know whether or not it is smoothable, and similarly for $*\\mathbb{RP}^{4}\\#*En$.\n\nReferences cited:\n- [Rub84] Daniel Ruberman. Invariant knots of free involutions of $S^{4}$. Topology Appl., 18(2-3):217–224, 1984. doi:10.1016/0166-8641(84)90011-7.\n- [HKT94] Ian Hambleton, Matthias Kreck, and Peter Teichner. Nonorientable 4-manifolds with fundamental group of order 2. Trans. Amer. Math. Soc., 344(2):649–665, 1994. doi:10.2307/2154500.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [RS97] Daniel Ruberman and Ronald J. Stern. A fake smooth $\\mathbb{CP}^{2}$\\#RP4. Math. Res. Lett., 4(2-3):375–378, 1997. doi:10.4310/MRL.1997.v4.n3.a6.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [BDK07] Jeremy Brookman, James F. Davis, and Qayum Khan. Manifolds homotopy equivalent to Pn\\#Pn. Math. Ann., 338(4):947–962, 2007. doi:10.1007/s00208-007-0099-x.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2897, "problem_number": "KP-4.21", "title": "Kirby Problem 4.21", "statement": "Is every topological closed 4–manifold M the union of submanifolds $Y \\cup Z$, where Y is smoothable, Z is acyclic, and $Y \\cap Z$ is their common boundary, a homology sphere $\\Sigma$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.21.\n\nLiterature notes:\n(1) This is [Kir97, Problem 4.74].\n\n(2) In the case that M is simply connected the result holds. Freedman [Fre82, Theorem 1.5] proved that the homeomorphism type of a simply connected closed 4–manifold is determined by its unimodular intersection form and its Kirby-Siebenmann invariant. The proof consists of an explicit construction of a manifold for each such for m and possible Kirby-Siebenmann invariant. The construction begins with a smooth compact 4-manifold built from $D^{4}$ by adding 2-handles and with the desired intersection form. It then modifies the handle attachments as needed to achieve the desired Rokhlin invariant for $\\Sigma$. Lastly, it caps the manifold of f with a contractible topological manifold.\n\n(3) There are compact topological manifolds that do not support such decompositions. We explain how to construct an example. Let $K \\subset S^{3}$ be a topologically slice knot that does not bound a smooth slice disk in any smooth compact acyclic 4–manifold with boundary $S^{3}$. Examples of such were first identified by Akbulut in unpublished work. For instance, the untwisted Whitehead double of the trefoil suffices. Let M denote the complement an open tubular neighborhood of a locally flat slice disk. Notice that $\\partial M \\cong S_{0}^{3}(K)$, 0–surgery on K. The claim is that M cannot have a decomposition of the desired type; an outline of a proof follows. Suppose that $\\Sigma \\subset M$ is a locally flat homology 3–sphere that splits M as the union of topological manifold Y and an acyclic manifold Z. We have that $\\partial Y \\cong \\Sigma \\cup S_{0}^{3}(K)$. Assume that there exists a smoothing A of Y; that is,A is a smooth manifold supporting a homeomorphism to Y. Attaching a smooth 2– handle to A along the meridian of K in $S_{0}^{3}(K)$, referred to as the trace of K in $S_{0}^{3}(K)$, yields a smooth homology $S^{3} \\times I$ with boundary $S^{3} \\cup \\Sigma$. Attaching a smooth 4–ball results in a smooth acyclic 4–manifold A1 with boundary $\\Sigma$. We now see that the union $B =$ A1 $\\cup _{\\Sigma} A$ is a smooth homology $S^{1} \\times B^{3}$ with boundary $S_{0}^{3}(K)$. Appropriately attaching a 2– handle to B along the trace of $K \\in S_{0}^{3}(K)$ results in a smooth acyclic 4–manifold bounded by $S^{3}$. The cocore of the 2–handle is a smooth slice disk for its boundary, which represents $K \\subset S^{3}$.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2898, "problem_number": "KP-4.22", "title": "Kirby Problem 4.22", "statement": "Let $\\pi$ be a good group, and let X be a smooth 4-manifold with $\\pi_{1}(X) = \\pi$. Does $L^{s}_{5}(\\mathbb{Z}[\\pi])$ act on the smooth structure set of X, in such a way that the action reduces under the forgetful map to the Wall realization action on the topological structure set? In particular, can this be done when $\\pi= \\mathbb{Z}, \\mathbb{Z} \\times \\mathbb{Z}_{n}$, or $\\mathbb{Z}_{2}*\\mathbb{Z}_{2}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.22.\n\nLiterature notes:\n(1) The surgery exact sequence, valid for higher dimensional manifolds, ends with $\\mathcal{N}(X \\times I,\\partial(X \\times$ I)) $\\to L^{s}_{n,1}(\\mathbb{Z}\\pi) \\to ^{\\Theta} \\to \\mathcal{S}(X) \\to \\mathcal{N}(X) \\to L^{s}_{n}(\\mathbb{Z}\\pi). + Here\\mathcal{S}(X)is$ the smooth structure set $and\\mathcal{N}(X)is$ normal bordism classes of degree one normal maps. The structure set consists of orientationpreserving homotopy equivalences f: $M \\to X$ up to diffeomorphisms F: M1 $\\to M$ commuting up to homotopy with the homotopy equivalences. What is written as a map $\\Theta$ is shorthand for an action of $L^{s}_{n,+,1}(\\mathbb{Z}\\pi)$ on $\\mathcal{S}(X)$. The action of an element $A \\in L^{s}_{n,+,1}(\\mathbb{Z}\\pi)$ on (X,Id $:X \\to$ X) produces, for n+1 $\\geq$ 6, a (normal) cobordism W from (X,Id) to another homotopy equivalencef: X1 $\\to X$. The problem asks if there is an action of $L^{s}_{5}(\\mathbb{Z}[\\pi])$ when X is a closed smooth 4-manifold, that reduces to the Wall realization action on the topological structure set $\\mathcal{S}_{TOP}(X)$.\n\n(2) Cappell and Shaneson [CS71] showed that one can realize the action of $L^{s}_{5}$ stably, i.e. after replacing X by its connected sum with some number of copies of $S^{2} \\times S^{2}$. For example, Scharlemann’s construction [Sch76] realizes the action of the generator of $L^{s}_{5}(\\mathbb{Z}[\\mathbb{Z}])$ on $(S^{1} \\times S^{3})\\#(S^{2} \\times S^{2})$ (this is also in [CS71] using an unpublished calculation of R. Lee). In general, even when if one can realize the action of an element of $L^{s}_{5}$ to produce a different element of the structure set, the resultant manifold X1 might still be diffeomorphic to X. The difference in the structure set would then be due to the resultant self homotopy equivalence of X not being homotopic to a diffeomorphism. For instance, Akbulut [Akb99] showed that Scharlemann’s manifold is diffeomorphic $to(S^{1} \\times S^{3})\\#(S^{2} \\times S^{2})$. In many cases of interest, in particular for $\\pi =\\mathbb{Z}$ and $\\pi=\\mathbb{Z}/2*\\mathbb{Z}/2$, the Whitehead group of $\\pi$ is trivial, in which case $L^{s}_{5}(\\mathbb{Z}\\pi) =L_{5}(\\mathbb{Z}\\pi)$ and one can ignore the s-decoration.\n\n(3) The heart of the question is then whether one can realize the action without stabilization. There are three cases of particular geometric interest, corresponding to fundamental groups $\\mathbb{Z},\\mathbb{Z} \\times \\mathbb{Z}/n(for$ n>1), and $\\mathbb{Z}/2*\\mathbb{Z}/2$. All of these are good groups, so that realization works in the topological category; the issue is to find smooth realizations.\n\n(i) Realizing the action of $L_{5}(\\mathbb{Z}[\\mathbb{Z}])$ on $X = S^{1} \\times S^{3}$. Every selfhomotopy equivalence from X to itself is homotopic to a diffeomorphism, so realizing the action of the generator of $L_{5}(\\mathbb{Z}[\\mathbb{Z}])would$ produce an exotic $S^{1} \\times S^{3}$. It would be detected by the Rokhlin invariant of any spin manifold carrying the generator of $H_{3}. A$ gauge-theoretic conjecture of Furuta-Ohta [FO93] would imply that no such manifold exists; see Problem 4.65 and [RS05] and [MRS11] for approaches to this conjecture.\n\n\\paragraph{Question.} Is there an exotic $S^{1} \\times S^{3}$ realizing the action of the generator of $L_{5}(\\mathbb{Z}[\\mathbb{Z}])$ on $X =S^{1} \\times S^{3}$?\n\n(ii) Realizing the action of $\\pi=\\mathbb{Z} \\times \\mathbb{Z}/n$ where $L^{s}_{5}(\\mathbb{Z}\\pi)$ is large, and one would want to produce an action on the structure set of $S^{1} \\times$ L(n, q). In this setting, the manifold would again be fake, i.e. homotopy equivalent but not homeomorphic to $S^{1} \\times$ L(n, q), detected by codimensionon e multisignature (or Atiyah-Singer invariants). In this case the topological manifold set, i.e. the structure set modulo homotopy self-equivalences of $S^{1} \\times$ L(n, q) is infinite. For a given n, only finitely many of them (corresponding to $S^{1} \\times$ L(n, $q^{1})where$ L(n, q) $\\cong$ L(n, $q^{1}))$ are known to be smoothable, but as far as we know they could all be.\n\n\\paragraph{Question.} For any $n \\geq$ 2, is there a smooth structure on a fake $S^{1} \\times$ L(n, q) realizing the action of $L^{s}_{5}(\\mathbb{Z}[\\mathbb{Z} \\times \\mathbb{Z}/n])$ on $X = S^{1} \\times$ L(n, q), that is not of the for $m S^{1} \\times$ L(n, q1)?\n\n(iii) The group $\\pi = \\mathbb{Z}/2$ * $\\mathbb{Z}/2$ is relevant to the problem of classifying manifolds homotopy equivalent to $\\mathbb{RP}^{4}\\#\\mathbb{RP}^{4}$. The group $L_{5}(\\mathbb{Z}\\pi)$ contains [Cap74] an infinitely generated subgroup $Unil_{1}(\\mathbb{Z},\\mathbb{Z}-,\\mathbb{Z}-)$. The smooth action of an element in this subgroup would produce a homotopy equivalence from a manifold X to $\\mathbb{RP}^{4}\\#\\mathbb{RP}^{4}$ that is not homotopic to $f_{1}\\#f_{2}$ where the $f_{i}$ are homotopy equivalences $X_{i} \\to \\mathbb{RP}^{4}, i =$ 1,2. A paper of Jahren-Kwasik [JK06] uses the topological realization of this Unil subgroup to construct topological 4-manifolds that are homotopy equivalent to $\\mathbb{RP}^{4}\\#\\mathbb{RP}^{4}$ that are not connected sums of homotopy $\\mathbb{RP}^{4}s$. (See [BDK07] for the full topological classification.)\n\n\\paragraph{Question.} Are the non-splittable manifolds from [JK06] smoothable? This is equivalent to asking about the smooth realizability of the corresponding Unil group elements.\n\nReferences cited:\n- [CS71] Sylvain E. Cappell and Julius L. Shaneson. On four dimensional surgery and applications. Comment. Math. Helv., 46:500–528, 1971. doi:10.1007/BF02566862.\n- [Sch76] Martin Scharlemann. Constructing strange manifolds with the dodecahedral space. Duke Math. J., 43(1):33–40, 1976.\n- [Akb99] Selman Akbulut. Scharlemann’s manifold is standard. Ann. of Math. (2), 149(2):497–510, 1999.\n- [FO93] Mikio Furuta and Hiroshi Ohta. Differentiable structures on punctured 4-manifolds. Topology Appl., 51(3):291–301, 1993. doi:10.1016/0166-8641(93)90083-P.\n- [RS05] Daniel Ruberman and Nikolai Saveliev. Casson–type invariants in dimension four. In Geometry and topology of manifolds, volume 47 of Fields Inst. Commun., pages 281–306. Amer. Math. Soc., Providence, RI, 2005. doi:10.1090/fic/047/18.\n- [MRS11] Tomasz Mrowka, Daniel Ruberman, and Nikolai Saveliev. Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin’s invariant. J. Differential Geom., 88:333–377, 2011. http://projecteuclid.org/euclid.jdg/1320067650.\n- [Cap74] Sylvain E. Cappell. Unitary nilpotent groups and Hermitian K-theory. I. Bull. Amer. Math. Soc., 80:1117–1122, 1974. doi:10.1090/S0002-9904-1974-13636-0.\n- [JK06] Bjørn Jahren and Slawomir Kwasik. Manifolds homotopy equivalent to RP4\\#RP4. Math. Proc. Cambridge Philos. Soc., 140(2):245–252, 2006. doi:10.1017/S0305004105008893.\n- [BDK07] Jeremy Brookman, James F. Davis, and Qayum Khan. Manifolds homotopy equivalent to Pn\\#Pn. Math. Ann., 338(4):947–962, 2007. doi:10.1007/s00208-007-0099-x.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2899, "problem_number": "KP-4.23", "title": "Kirby Problem 4.23", "statement": "(Schoenflies problem). If $\\Sigma$ is a smoothly embedded $S^{3}$ in $S^{4}$, then its closed complements are smooth 4-balls.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.23.\n\nLiterature notes:\n(1) This is the famous Schoenflies problem, which appears in [Kir97, Problem 4.32].\n\n(2) Progress on this old and important conjecture has, sadly, been more aspirational than concrete. Scharlemann proved this conjecture in the special case there is a smooth function $f: S^{4} \\to \\mathbb{R}$ whose restriction to the $S^{3}$ is Morse with k 0-handles and $\\leq k$ +1 1-handles (so that the middle level has genus $\\leq$ 2) [Sch84]. Following Gabai’s proof of Property R [Gab87b], Scharlemann [Sch08] used a reembedding process to extend this result in the previous update to the case of (k +2) 1-handles, so the middle level has genus $\\leq$ 3. The argument explores further connections between the Schoenflies Conjecture and the generalized Property R Conjecture (see Problem 1.10). Other reembedding approaches have been proposed by Akbulut [Akb14] (twisting corks), Agol–Freedman [AF15], and Lambert-Cole [LC21]. Lambert-Cole employs Stein trisections and proposes this generalization of the Schoenflies Conjecture: any homotopy 4-ball in a compact Stein domain of complex dimension 2 is diffeomorphic to the standard 4-ball. Gabai [Gab22] has suggested an approach via pseudo-isotopy theory. In the other direction, Gabai, Naylor and Schwartz [GNS25] exhibit potential counterexamples – Schoenflies balls that are not known to be standard.\n\n(3) This problem is equivalent to the corresponding one in the PL category. By Mazur [Maz59] both closed complements are topological 4-balls. They are also quasi-conformal [Geh67] and the Lipschitz [LV77] 4-balls. It is well known following Mazur [Maz59] and Cerf [Cer68] that the Schoenflies problem is equivalent to showing that if $f: S^{1} \\times S^{3} \\to S^{1} \\times S^{3}$ is a diffeomorphism, then after passing to a sufficiently large finite cover the lifted diffeomorphism can be isotoped to be supported in a 4-ball. By [BG19] it is also equivalent to showing that if $B \\subset S^{4}$ is a 3-ball with $\\partial B$ the standard 2-sphere S, then after passing to a finite branched cover of $S^{4}$ branched over S, B is isotopic rel $\\partial$ to the standard 3-ball. By [Gab22] it also would follow from a carving/surgery problem. It is not known whether or not a Schoenflies ball $\\times I$ is always $B^{5}$.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Sch84] Martin Scharlemann. The four-dimensional Schoenflies conjecture is true for genus two imbeddings. Topology, 23(2):211–217, 1984. doi:10.1016/0040-9383(84) 90040-5.\n- [Gab87b] David Gabai. Foliations and the topology of 3-manifolds. III. J. Differential Geom., 26(3):479–536, 1987. http://projecteuclid.org/euclid.jdg/1214441488.\n- [Sch08] Martin Scharlemann. Generalized property R and the Schoenflies conjecture. Comment. Math. Helv., 83(2):421–449, 2008. doi:10.4171/CMH/131.\n- [Akb14] Selman Akbulut. Cork twisting Schoenflies problem. J. Gökova Geom. Topol. GGT, 8:35–43, 2014. doi:10.1017/s0020269x00003418.\n- [AF15] Ian Agol and Michael Freedman. Simplifying 3-manifolds in $\\mathbb{R}^{4}$. Ann. Fac. Sci. Toulouse Math. (6), 24(5):1079–1101, 2015. doi:10.5802/afst.1476.\n- [LC21] Peter Lambert-Cole. Stein trisections and homotopy 4-balls, 2021. arXiv:2104.02003.\n- [Gab22] David Gabai. 3-spheres in the 4-sphere and pseudo-isotopies of $S^{1}$ $\\times$ $S^{3}$, 2022. arXiv:2212.02004.\n- [GNS25] David Gabai, Patrick Naylor, and Hannah Schwartz. Doubles of Gluck twists: a five-dimensional approach. Adv. Math., 480:Paper No. 110455, 29, 2025. doi:10.1016/j.aim.2025.110455.\n- [Maz59] Barry Mazur. On embeddings of spheres. Bull. Amer. Math. Soc., 65:59–65, 1959. doi:10.1090/S0002-9904-1959-10274-3.\n- [Geh67] F. W. Gehring. Extension theorems for quasiconformal mappings in n-space. J. Analyse Math., 19:149–169, 1967. doi:10.1007/BF02788713.\n- [LV77] J. Luukkainen and J. Väisälä. Elements of Lipschitz topology. Ann. Acad. Sci. Fenn. Ser. A I Math., 3(1):85–122, 1977. doi:10.5186/aasfm.1977.0315.\n- [Cer68] Jean Cerf. Sur les difféomorphismes de la sphère de dimension trois $(\\Gamma_4=0)$. Lecture Notes in Mathematics, No. 53. Springer-Verlag, Berlin-New York, 1968.\n- [BG19] Ryan Budney and David Gabai. Knotted 3-balls in $S^{4}$, 2019. arXiv:1912.09029.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2900, "problem_number": "KP-4.24", "title": "Kirby Problem 4.24", "statement": "Let K be a framed knot in $S^{3} = \\partial B^{4}$. Let U be a meridian of K. Does there exist a smoothly embedded disk D in $B^{4} \\cup _{\\nu K} h^{2}$ bounded by U that intersects the cocore of $h^{2}$ algebraically zero times, such that the 4-manifold obtained by removing an open tubular neighborhood of the disk is not diffeomorphic to the 4-ball?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.24.\n\nLiterature notes:\n(1) We represent the situation of this problem in Figure 1. Note the “big dot” placed on U. An “ordinary dot” on an unknot is an instruction to carve the standard disk (in $B^{4})$ bounded by the unknot. An important variation is to carve a specific ribbon disk with boundary a ribbon knot (see [Akb16, Section 1.4] for examples and further discussion). If the carved disk in the setting of this problem is a disk in the 0handle (i.e. intersecting the cocore of $h^{2}$ geometrically zero times) then the 4-manifold $\\mathcal{B}$ is $B^{4}$. The “big dot” indicates that we may carve along any disk bounded by U that intersects the cocore of $h^{2}$ algebraically zero times. In other words, we carve along any disk that represents the trivial class in $H_{2}(0-handle \\cup 2-handles,\\partial;\\mathbb{Z})but$ do not restrict the carving disk to be contained in the 0-handle. In the setting of this problem, the result of carving is a homotopy $4-ball\\mathcal{B}$ with boundary $S^{3}$. Thus, this construction is a natural extension of the idea of carving along nontrivial disks and yields potentially nonstandard homotopy 4-balls (see relevant discussion in Problem 4.1, i.e. the smooth 4-dimensional Poincaré Conjecture).\n\n\\begin{center}\n\\kthreefiginclude{ch4_fig1.png}{width=0.56\\linewidth}\n\\par\\small\\textbf{Figure 1.} A diagram of a family of homotopy 4-balls. The resulting 4-manifold $\\mathcal{B}$ depends on a choice of carving disk with boundary $U$. This disk is required to intersect the cocore of the pictured 2-handle algebraically zero times, causing the resulting 4-manifold $\\mathcal{B}$ to be a homotopy 4-ball.\n\\end{center}\n\n(2) The case that K is slice and $n =$ 0 is particularly interesting, for then $\\mathcal{B}$ smoothly embeds in $B^{4}$. Thus, the question of whether $\\mathcal{B}$ is necessarily a 4-ball is related to the smooth 4-dimensional Schoenflies conjecture (Problem 4.23).\n\nReferences cited:\n- [Akb16] Selman Akbulut. 4-manifolds, volume 25 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2016. doi:10.1093/acprof:oso/9780198784869.001.0001.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2901, "problem_number": "KP-4.25", "title": "Kirby Problem 4.25", "statement": "Under what conditions does a closed, orientable 3-manifold M smoothly embed in $S^{4}$? Is this question algorithmically decidable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.25.\n\nLiterature notes:\n(1) This is in [Kir78, Problem 3.20]. As noted there, every homology 3sphere embeds topologically in $S^{4}$ [Fre82]. Smoothly, even for homology spheres, there is an obstruction coming from the Rokhlin invariant, and more obstructions coming from gauge theory and its cousins, using the fact that if a homology sphere M embeds in $S^{4}$ then M bounds a homology ball, so is trivial in the homology cobordism group. (The relationship between embedding problems and homology cobordism was first noted in [GL83]; see, e.g., [DHST23] for a relatively recent summary of nontriviality results for the 3-dimensional homology cobordism group.) There are also homology spheres that bound homology balls but do not embed in $S^{4}$ [Mc D22]. For 3-manifolds with nontrivial homology, even topologically there are obstructions, starting with an obstruction coming from the homology [Han38]; see [BB22] for a relatively recent survey of such obstructions. In particular, it would also be interesting to understand the question of which non-homology spheres embed locally flatly in $S^{4}$; see [Hil24] for an extensive discussion. Another interesting special case is obstructing punctured homology 3-spheres from embedding in $S^{4}$; see\n\nProblem 4.26.\n\n(2) Related problems include Problem 4.28 (about embeddings in connected sums of $S^{2} \\times S^{2})$ and Problem 3.27(the computational complexity of the homeomorphism and recognition problems for 3-manifolds).\n\n(3) Some other related questions include which closed, orientable 3-manifolds embed (smoothly or topologically flatly) in K3; and which homology 3spheres embed in homology 4-spheres (in either the smooth or locally flat categories).\n\nReferences cited:\n- [Kir78] Rob Kirby. Problems in low dimensional manifold theory. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 273–312. Amer. Math. Soc., Providence, R.I., 1978.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [GL83] Patrick M. Gilmer and Charles Livingston. On embedding 3-manifolds in 4-space. Topology, 22(3):241–252, 1983. doi:10.1016/0040-9383(83)90011-3.\n- [DHST23] Irving Dai, Jennifer Hom, Matthew Stoffregen, and Linh Truong. An infinite-rank summand of the homology cobordism group. Duke Math. J., 172(12):2365–2432, 2023. doi:10.1215/00127094-2022-0082.\n- [McD22] Clayton McDonald. Surface slices and homology spheres, 2022. arXiv:2202.02696.\n- [Han38] W. Hantzsche. Einlagerung von Mannigfaltigkeiten in euklidische Räume. Math. Z., 43(1):38–58, 1938. doi:10.1007/BF01181085.\n- [BB22] Ryan Budney and Benjamin A. Burton. Embeddings of 3-manifolds in $S^{4}$ from the point of view of the 11-tetrahedron census. Exp. Math., 31(3):988–1013, 2022. doi:10.1080/10586458.2020.1740836.\n- [Hil24] J. A. Hillman. Locally flat embeddings of 3-manifolds in $S^{4}$, 2024. arXiv:2408.10535.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2902, "problem_number": "KP-4.26", "title": "Kirby Problem 4.26", "statement": "If Y is a homology three-sphere, does the punctured manifold $Y_{0}$ =Y $\\setminus \\operatorname{Int}(B^{3})$ smoothly embed in $S^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.26.\n\nLiterature notes:\n(1) Every homology 3-sphere embeds topologically in $S^{4}$ (see [FQ90, Corollary 9.3C]). The Poincaré homology sphere does not embed smoothly in $S^{4}$, since it has nonzero Rokhlin invariant. Every homology 3-sphere Y embeds in a homology 4-sphere, given by spinning Y. It is constructed by surgery on $S^{1} \\times$ \\{p\\} $\\subset S^{1} \\times Y$ for some $p \\in Y$.\n\n(2) This problem is related to the group-theoretic question 3.39 of whether every finitely generated perfect group has weight one, i.e. is the normal closure of a single element [KM14a, Problem 5.52]. A twisted version of the spinning construction mentioned above shows that if Y is a homology sphere whose fundamental group has weight one, then $Y_{0}$ embeds in a homotopy 4-sphere. In some cases, one can show that the homotopy sphere is actually $S^{4}$. For instance, Larson [Lar15] proved that 1/n-surgery on a knot in $S^{3}$ embeds, when punctured, into $S^{4}$.\n\n(3) Using the twist-spinning construction for knots and a result of Zeeman [Zee65, Corollary 2], one can prove that every cyclic cover of $S^{3}$ branched over a knot embeds in $S^{4}$ after puncturing. This gives a lot of examples of $\\mathbb{Z}-homology$ 3-spheres that embed, after puncturing, in $S^{4}$. For example, the 2-fold cyclic branched cover on a knot of determinant one, the p-fold cyclic cover of the torus knot T(q, r), with(p, q, r)coprime, and any finite cyclic branched cover on a knot of Alexander polynomial one.\n\nReferences cited:\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [KM14a] E. I. Khukhro and V. D. Mazurov. Unsolved Problems in Group Theory. the Kourovka Notebook, 2014. arXiv:1401.0300.\n- [Lar15] Kyle Larson. Some Constructions Involving Surgery on Surfaces in 4-manifolds. PhD thesis, University of Texas, Austin, 2015. URL: https://repositories.lib.utexas.edu/server/api/core/bitstreams/32f94088-e19b-4b49-8dd8-7b2320ef72e3/content.\n- [Zee65] E. C. Zeeman. Twisting spun knots. Trans. Amer. Math. Soc., 115:471–495, 1965. doi:10.2307/1994281.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2903, "problem_number": "KP-4.27", "title": "Kirby Problem 4.27", "statement": "Find exotic 3-balls in $S^{4}$, considered up to isotopy rel. boundary. That is, find a pair of 3-balls $B_{1}, B_{2}$ smoothly embedded in $S^{4}$ with the same boundary such that $B_{1}, B_{2}$ are topologically but not smoothly isotopic rel. boundary.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.27.\n\nLiterature notes:\n(1) In [BG19, Section 9], Budney–Gabai observe that if no such 3-balls exist, then the smooth 4-dimensional Schoenflies Conjecture (Problem 4.23) is true.\n\n(2) Budney–Gabai [BG19] find infinitely many 3-balls smoothly embedded in $S^{4}$ with the same boundary that are not topologically isotopic rel. boundary. These 3-balls are distinguished by considering corresponding automorphisms of $S^{1} \\times B^{3}$ coming from barbell diffeomorphisms. These maps are distinguished even as homeomorphisms.\n\n\\paragraph{Question.} Is there a smooth rel. boundary automorphism of $S^{1} \\times B^{3}$ that is topologically but not smoothly isotopic rel. boundary to the identity map?\n\nReferences cited:\n- [BG19] Ryan Budney and David Gabai. Knotted 3-balls in $S^{4}$, 2019. arXiv:1912.09029.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2904, "problem_number": "KP-4.28", "title": "Kirby Problem 4.28", "statement": "Every closed, orientable 3-manifold embeds smoothly in some connected sum of copies of $S^{2} \\times S^{2}$. Given a closed 3-manifold M, let $s(M) \\geq$ 0 denote the smallest integer such that M embeds smoothly (respectively locally flatly) in a connected sum of $s(M)$ copies of $S^{2} \\times S^{2}$. Let $s(M) \\geq$ 0 denote the smallest integer such that M embeds smoothly (respectively locally $flatly^{~})$ in a connected sum of $s_{~}(M)$ copies of $S^{2} \\times _{~}S^{2}$. Compute the functions $s(M)$ and $s_{~}(M)$, in either the smooth or locally flat case, for interesting classes of 3-manifolds. Here are three more specific questions.\n\n(a) What are the $functionssands_{~}for$ lens spaces, and for Brieskorn spheres?\n\n(b) Freedman [Fre82] proved that every $\\mathbb{Z}-homology$ sphere embeds topologically in $S^{4}$. Is there an integer homology 3-sphere such that (smoothly) $s(M)$ =1?\n\n(c) Are the functions s or $s_{~}algorithmically$ computable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.28.\n\nLiterature notes:\nManifolds with $s(M)$ =0 are the topic of Problem 4.25. The functionswas introduced and studied by [AGL17]. That paper includes a comprehensive summary of the literature. Note in particular $thats(M)$ =1 implies M bounds an integer homology ball, by an elementary Mayer–Vietoris argument. Edmonds proved that in the topological category, $s_{~}(L_{p,q}) \\leq$ 2 for every p, q [Edm05].\n\nReferences cited:\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [AGL17] Paolo Aceto, Marco Golla, and Kyle Larson. Embedding 3-manifolds in spin 4-manifolds. J. Topol., 10(2):301–323, 2017. doi:10.1112/topo.12010.\n- [Edm05] Allan L. Edmonds. Homology lens spaces in topological 4-manifolds. Illinois J. Math., 49(3):827–837, 2005. doi:doi.org/10.1215/ijm/1258138221.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2905, "problem_number": "KP-4.29", "title": "Kirby Problem 4.29", "statement": "Let $\\Sigma$ be a locally flat surface in $S^{4}$ with $\\pi_{1}(S^{4} \\setminus \\Sigma)$ cyclic.\n\n(a) Prove that $\\Sigma$ is topologically unknotted.\n\n(b) Assuming that $\\Sigma$ is smoothly embedded and orientable, prove that $\\Sigma$ is smoothly unknotted (or find an example that is not).", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.29.\n\nLiterature notes:\n(1) The surface $\\Sigma$ is known to be topologically unknotted in many cases.\n\n\\noindent$\\bullet$ If $\\Sigma=S^{2}$, the claim holds by work of Freedman–Quinn (see [FQ90]).\n\n\\noindent$\\bullet$ If $\\Sigma$ is orientable of genus more than 2, the claim holds by Conway– Powell [CP23].\n\n\\noindent$\\bullet$ If $\\Sigma$ is nonorientable of genus h (i.e. $\\Sigma \\cong \\#_{h}\\mathbb{RP}^{2})$, then the claim holds by Lawson [Law84] if h=1; Conway–Orson–Powell [COP23] if $h \\leq$ 3 and if h>3 and $|e(\\Sigma)|$ <2h; Pencovitch [Pen24] if h=4,5 and $|e(\\Sigma)|$ =2h. Thus, the remaining open cases of (a) (in the topological category) are the following.\n\n\\noindent$\\bullet$ $\\Sigma$ is an orientable surface of genus one or two,\n\n\\noindent$\\bullet$ $\\Sigma$ is a nonorientable surface of genus $h \\geq$ 6 and with normal Euler number equal to $\\pm$ 2h.\n\n(2) There are many instances of non-orientable surfaces in $S^{4}$ that are smoothly knotted while their complements have cyclic fundamental group. Finashin–Kreck–Viro [FKV87] constructed infinitely many genus 10 nonorientable surfaces that have complement with cyclic fundamental group (and are topologically unknotted [Kre90]) but are pairwise not smoothly isotopic. Finashin [Fin09] later produced an infinite family of genus 6 nonorientable surfaces that are pairwise not smoothly isotopic but are all topologically unknotted (via [COP23]). Matić- Öztürk-Stipsicz-ReyesUrzúa [MOR+24] have announced the existence of a single exotically knotted $\\#_{5}\\mathbb{RP}^{2}$ and Miyazawa [Miy23] has announced the existence of an infinite family of exotically knotted projective planes (both papers focus on distinguishing surface smoothly, with the topological unknotting following respectively from [COP23] and [Law84]).\n\n(3) A well-known question is the following. Question (i). Suppose that $\\Sigma$ is smooth and the radial function $h: S^{4} \\to \\mathbb{R}$ restricts to $\\Sigma$ as a Morse function with exactly one local minimum. Is $\\Sigma$ smoothly unknotted? Is $\\Sigma$ topologically unknotted? This question includes the case that $\\Sigma$ is a torus and $h|_{\\Sigma}$ has exactly four critical points, which is Problem 4.30 in [Kir78]. (Note that if $\\Sigma \\cong S^{2}$ and $h|_{\\Sigma}$ has four critical points then $\\Sigma$ is smoothly unknotted by Scharlemann [Sch85a]; if $\\Sigma \\cong \\mathbb{RP}^{2}$ and $h|_{\\Sigma}$ has three critical points then $\\Sigma$ is smoothly unknotted by Bleiler–Scharlemann [BS88a].) This question also includes the case that $\\Sigma$ is a union of two (pushedin) Seifert surfaces for a knot cross-section, or that $\\Sigma$ is a union of a ribbon surface and a Seifert surface for a knot cross-section.\n\n(4) Question (i) itself has a well-known interesting subquestion. Question (ii). Suppose $\\Sigma$ is a union of two ribbon surfaces for an unknotted cross-section. Is $\\Sigma$ smoothly unknotted? Is $\\Sigma$ topologically unknotted? The case that $\\Sigma$ is a 2-sphere appears in Suzuki’s survey of problems in 2-knot theory [Suz76].\n\nReferences cited:\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [CP23] Anthony Conway and Mark Powell. Embedded surfaces with infinite cyclic knot group. Geom. Topol., 27(2):739–821, 2023. doi:10.2140/gt.2023.27.739.\n- [Law84] Terry Lawson. Detecting the standard embedding of $\\mathbb{RP}^{2}$ in $S^{4}$. Math. Ann., 267(4):439–448, 1984. doi:10.1007/BF01455961.\n- [COP23] Anthony Conway, Patrick Orson, and Mark Powell. Unknotting nonorientable surfaces, 2023. arXiv:2306.12305.\n- [Pen24] Mark Pencovitch. Unknotting nonorientable surfaces of genus 4 and 5. Linear Algebra Appl., 702:195–217, 2024. doi:10.1016/j.laa.2024.08.014.\n- [FKV87] S. M. Finashin, M. Kreck, and O. Ya. Viro. Exotic knottings of surfaces in the 4-sphere. Bull. Amer. Math. Soc. (N.S.), 17(2):287–290, 1987. doi:10.1090/S0273-0979-1987-15562-5.\n- [Kre90] Matthias Kreck. On the homeomorphism classification of smooth knotted surfaces in the 4-sphere. In Geometry of low-dimensional manifolds, 1 (Durham, 1989), volume 150 of London Math. Soc. Lecture Note Ser., pages 63–72. Cambridge Univ. Press, Cambridge, 1990.\n- [Fin09] Sergey Finashin. Exotic embeddings of \\#6$\\mathbb{RP}^{2}$ in the 4-sphere. In Proceedings of Gökova Geometry-Topology Conference 2008, pages 151–169. Gökova Geometry/Topology Conference (GGT), Gökova, 2009.\n- [MOR+24] Gordana Matić, Ferit Oztürk, Javier Reyes, András I. Stipsicz, and Giancarlo Urzúa. An exotic 5$\\mathbb{RP}^{2}$ in the 4-sphere, 2024. arXiv:2312.03617.\n- [Miy23] Jin Miyazawa. A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P2-knots, 2023. arXiv:2312.02041.\n- [Kir78] Rob Kirby. Problems in low dimensional manifold theory. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 273–312. Amer. Math. Soc., Providence, R.I., 1978.\n- [Sch85a] Martin Scharlemann. Smooth spheres in $\\mathbb{R}^{4}$ with four critical points are standard. Invent. Math., 79(1):125–141, 1985. doi:10.1007/BF01388659.\n- [BS88a] Steven Bleiler and Martin Scharlemann. A projective plane in $\\mathbb{R}^{4}$ with three critical points is standard. Strongly invertible knots have property P. Topology, 27(4):519– 540, 1988. doi:10.1016/0040-9383(88)90030-4.\n- [Suz76] Shin’ichi Suzuki. Knotting problems of 2-spheres in 4-sphere. Math. Sem. Notes Kobe Univ., 4(3):241–371, 1976.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2906, "problem_number": "KP-4.30", "title": "Kirby Problem 4.30", "statement": "Does there exist a pair of closed, oriented surfaces in $S^{4}$ that are topologically but not smoothly isotopic? If such an exotic pair exists, does there exist an infinite family? Can the surfaces be taken to be 2-spheres? Surfaces of genus-g for any given $g \\geq$ 0?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.30.\n\nLiterature notes:\n(1) There are many examples of exotic oriented surfaces in general 4-manifolds. These include infinite families of pairwise exotic 2-spheres (existence from [Wal64a], with smooth obstruction from work of Donaldson [Don87a, Don87b]) in some simply-connected manifolds, but the usual smooth obstructions require the ambient 4-manifold to $haveb^{+}_{2}$ >0. These examples also feature surfaces that represent nontrivial homology classes. Exotic families of nullhomologous tori [HS20] and 2-spheres [Tor25] are known to exist in some simply connected 4-manifolds with $largeb_{2}$, but there are currently no known examples of exotic pairs of higher-genus surfaces in a closed 4-manifold.\n\n(2) In Problem 4.29, we discuss known constructions of pairwise exotic infinite families of nonorientable surfaces in $S^{4}$. The first such examples were given by Finashin–Kreck–Viro [FKV87]. These nonorientable surfaces are distinguished smoothly by the diffeomorphism classes of their double branched covers. The double branched cover of an orientable surface in $S^{4}$ is a signaturezero manifold, so a first step toward constructing oriented examples might be finding involutions on known constructions of exotic signature-zero manifolds; see e.g. Baykur–Hamada [BH23].\n\n(3) Compare this problem to Problem 4.29(b), which asks whether a smooth, oriented surface in $S^{4}$ whose complement has cyclic fundamental group is smoothly unknotted. Such a surface is known to be topologically unknotted if it is a 2-sphere [FQ90] or has genus greater than two [CP23], so in these cases a smoothly non-standard example would solve both this problem and Problem 4.29(b).\n\nReferences cited:\n- [Wal64a] C. T. C. Wall. Diffeomorphisms of 4-manifolds. J. London Math. Soc., 39:131–140, 1964. doi:10.1112/jlms/s1-39.1.131.\n- [Don87a] S. K. Donaldson. Irrationality and the h-cobordism conjecture. J. Differential Geom., 26(1):141–168, 1987. http://projecteuclid.org/euclid.jdg/1214441179.\n- [Don87b] S. K. Donaldson. The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Differential Geom., 26(3):397–428, 1987. http://projecteuclid.org/euclid.jdg/1214441485.\n- [HS20] Neil R. Hoffman and Nathan S. Sunukjian. Null-homologous exotic surfaces in 4-manifolds. Algebr. Geom. Topol., 20(5):2677–2685, 2020. doi:10.2140/agt.2020.20.2677.\n- [Tor25] Rafael Torres. Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds. Ann. Inst. Fourier (Grenoble), 75(6):2501–2527, 2025. doi: 10.5802/aif.3685.\n- [FKV87] S. M. Finashin, M. Kreck, and O. Ya. Viro. Exotic knottings of surfaces in the 4-sphere. Bull. Amer. Math. Soc. (N.S.), 17(2):287–290, 1987. doi:10.1090/S0273-0979-1987-15562-5.\n- [BH23] R. Inanc Baykur and Noriyuki Hamada. Exotic 4-manifolds with signature zero, 2023. Selecta Math., to appear. arXiv:2305.10908.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [CP23] Anthony Conway and Mark Powell. Embedded surfaces with infinite cyclic knot group. Geom. Topol., 27(2):739–821, 2023. doi:10.2140/gt.2023.27.739.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2907, "problem_number": "KP-4.31", "title": "Kirby Problem 4.31", "statement": "Does every knot in $S^{3}$ bound an exotic pair of orientable surfaces in $B^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.31.\n\nLiterature notes:\n(1) By an exotic pair, we mean two surfaces with the same knot as boundary, that are topologically but not smoothly isotopic rel. boundary.\n\n(2) It is known that certain knots bound exotic genus one surfaces [JMZ21] and examples bounding exotic disks are announced in [Hay21a]. Improvin g the ability to produce exotic surfaces for a fixed boundary knot can be thought of as approaching the problem of producing closed oriented exotic surfaces in $S^{4}$ (which would follow from producing exotic surfaces with boundary the unknot); see Problem 4.30. One can also ask the same question about exotic nonorientable surfaces.\n\nReferences cited:\n- [JMZ21] András Juhász, Maggie Miller, and Ian Zemke. Transverse invariants and exotic surfaces in the 4-ball. Geom. Topol., 25(6):2963–3012, 2021. doi:10.2140/gt.2021.25.2963.\n- [Hay21a] Kyle Hayden. Exotically knotted disks and complex curves, 2021. arXiv:2003.13681.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2908, "problem_number": "KP-4.32", "title": "Kirby Problem 4.32", "statement": "Does there exist a locally flat embedding $f: \\Sigma \\to S^{4}$ for some closed surface $\\Sigma$ such that f is not topologically ambiently isotopic to a smooth embedding? Particularly in the case that $\\Sigma=S^{2}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.32.\n\nLiterature notes:\n(1) There are many examples of this behavior in other 4-manifolds. In particular, suppose there is a locally flat embedding of a genus-g surface into a smooth 4-manifold X such that there is no smooth genus-g surface representing the same homology class. Then the embedding is not smoothable. As an explicit example, Rudolph [Rud84] showed that for all d>6, there is a locally flat surface in $\\mathbb{C}\\mathbb{P}^{2}$ representing the homology class $d \\in \\mathbb{Z} = H_{2}(\\mathbb{C}\\mathbb{P}^{2};\\mathbb{Z})$ of genus strictly less than (d−1)(d −2)/2. Lee–Wilczyński [LW97, Corollary 1.3] extended this to $d >$ 4 (and decreased the realized genus of these locally flat surfaces). We deduce from the Thom conjecture [KM94] that there is a nonsmoothable embedding of a surface into $\\mathbb{C}\\mathbb{P}^{2}$ representing any homology class $d \\in \\mathbb{Z}$ for $d >$ 4. For this reason, we restrict the question to the ambient manifold $S^{4}$.\n\n(2) There are examples of locally flat, non-smoothable embeddings of codimension two spheres in $S^{n}$ for manyn>4. Lashof [Las71] gives examples for n=5. In the introduction of [Bri66], Brieskorn shows that for $n\\equiv3$ (mod 8) (n>3), there is a smooth embedding of an exotic n-sphere into $S^{n}+^{2}$. Viewing this as a locally flat embedding of the standard n-sphere, the embedding is not smoothable.\n\nReferences cited:\n- [Rud84] Lee Rudolph. Some topologically locally-flat surfaces in the complex projective plane. Comment. Math. Helv., 59(4):592–599, 1984. doi:10.1007/BF02566368.\n- [LW97] Ronnie Lee and Dariusz M. Wilczyński. Representing homology classes by locally flat surfaces of minimum genus. Amer. J. Math., 119(5):1119–1137, 1997. URL: http://muse.jhu.edu/journals/american journal of mathematics/v119/119.5lee.pdf.\n- [KM94] P. B. Kronheimer and T. S. Mrowka. The genus of embedded surfaces in the projective plane. Math. Res. Lett., 1(6):797–808, 1994. doi:10.4310/MRL.1994.v1.n6.a14.\n- [Las71] Richard K. Lashof. A nonsmoothable knot. Bull. Amer. Math. Soc., 77:613–614, 1971. doi:10.1090/S0002-9904-1971-12773-8.\n- [Bri66] Egbert Brieskorn. Beispiele zur Differentialtopologie von Singularitäten. Invent. Math., 2:1–14, 1966. doi:10.1007/BF01403388.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2909, "problem_number": "KP-4.33", "title": "Kirby Problem 4.33", "statement": "Let $S_{1}, S_{2}$ be two topologically isotopic, smoothly embedded closed surfaces in a closed, oriented, smooth 4-manifold X. When do $S_{1}$ and $S_{2}$ in X become smoothly isotopic after one\n\n(a) external stabilization?\n\n(b) standard internal stabilization?\n\n(c) internal stabilizations? Is there an upper bound depending on X on the number of such stabilizations required to make the surfaces smoothly isotopic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.33.\n\nLiterature notes:\n(1) We say that (X, $S_{i})\\#(S^{2} \\times S^{2},\\emptyset)$ is an external stabilization of (X, $S_{i})$, that is, we enlarge the topology of the surface complement by taking a connected sum with $S^{2} \\times S^{2}$. If $S_{1}$ and $S_{2}$ become smoothly isotopic after a finite number of external stabilizations then we say that $S_{1}$ and $S_{2}$ are external-stably smoothly isotopic.\n\n(2) Let $T^{2} \\subset S^{4}$ be an unknotted torus. We say that (X, $S_{i})\\#(S^{4}, T^{2})$ is a standard internal stabilization of $S_{i}$. If $S_{1}$ and $S_{2}$ become smoothly isotopic after a finite number of standard internal stabilizations then we say that $S_{1}$ and $S_{2}$ are strongly internal-stably smoothly isotopic.\n\n(3) If $S_{i}^{1}$ is obtained from $S_{i}$ by a surgery corresponding to attaching a 1handle to $S_{i}$ inside X, then we say that(X, $S_{i}^{1})is$ aninternal stabilization of $S_{i}$. Note that the standard internal stabilization is the case where this 1–handle is attached in a local 4–ball neighborhood. If $S_{1}$ and $S_{2}$ become smoothly isotopic after a finite number of internal stabilizations then we say that $S_{1}$ and $S_{2}$ are internal-stably smoothly isotopic.\n\n(4) A positive answer to part (b)implies a positive answer $to(c)$, and similarly for the corresponding questions on upper bounds.\n\n(5) Regarding part (a): Galvin [Gal24] showed that any two smooth, topologically isotopic surfaces in a simply connected 4-manifold X become smoothly isotopic after a sufficiently high number of external stabilizations of X. Galvin’s proof does not control the number of external stabilizations required. A priori, the number of external stabilizations needed depends on X and on the surfaces, but one stabilization suffices in all examples successfully analyzed so far. When $\\pi_{1}(X)$ is nontrivial, it is open whether or not all topologically isotopic surfaces are external-stably smoothly isotopic.\n\n(6) It was shown in $[AKM^{+}19]$ that in a smooth simply connected manifold X, two homologous connected surfaces of the same genus, and with simply connected complements, become smoothly isotopic after one external stabilization, as long as their homology class is not dual to $w_{2}(X)$, i.e. if the class is not characteristic.\n\n(7) There are examples of exotic pairs of surfaces with boundary that require more than one external stabilization when X is not closed. Lin and Mukherjee [LM25] give an example of an exotic pair of disks in a punctured K3 that remain smoothly non-isotopic after one external stabilization; the disks are distinguished via a Pin(2)-equivariant family Bauer–Furuta invariant. Hayden–Kang–Mukherjee [HKM23] also announced examples involving closed surfaces in a manifold with boundary.\n\n(8) Regarding parts (b) and (c): Baykur-Sunukjian [BS16] showed that any two homologous surfaces of the same genus become smoothly isotopic after a sufficiently high number of internal stabilizations. Baykur and Sunukjian also prove in [BS16] that the same result can be achieved with strong internal stabilizations when $S_{1}, S_{2}$ are exotically knotted surfaces with cyclic fundamental group complement, or more generally if the map $\\pi_{1}(\\partial\\nu S_{i}) \\to \\pi_{1}(X \\setminus \\nu S_{i})$ is surjective. However, even the following question, which asks for a much weaker version of (b), is open.\n\n\\paragraph{Question.} For every pair of topologically isotopic surfaces of the same genus, does there exist a sufficiently high finite number of strong internal stabilizations such that the surfaces become smoothly isotopic?\n\n(9) In [BS16] it was shown that exotically knotted surfaces that are produced using many well-known (local) operations become smoothly isotopic after a single standard internal stabilization.\n\n(10) In [Auc23], Auckly announced that for any given n, there are two homologically essential closed surfaces as above that remain non-smoothly isotopic after n strong internal stabilizations. Auckly notes that one can find examples where such surfaces are related by a diffeomorphism that is pseudo-isotopic to the identity, and one external stabilization suffices for his examples. He also argues that this behavior exists under only mild constraints on the ambient manifold. There are no such examples of null-homologous surfaces that are known to require more than one internal stabilization.\n\n(11) This relates to versions of (b) and (c) in the relative setting, i.e. when the boundary is nonempty. The first examples of two topologically isotopic, non-smoothly isotopic disks $D^{n}_{1}, D_{2}^{n}$ in $B^{4}$ that do not become smoothly isotopic after n internal stabilizations (for an arbitrarily given n) were presented by Guth [Gut22]; the disks are distinguished by Heegaard Floer techniques. Hayden–Kang–Mukherjee [HKM23] proposed examples of exotically knotted spheres in a 4-manifold with boundary that do not become smoothly isotopic after one internal or one external stabilization; their construction is based on a claim by Kang [Kan22b] about contractible 4-manifolds surviving a stabilization; see Problem 4.7. In [Hay23], Hayden announces an example of two topologically isotopic positive genus surfaces with boundary in $B^{4}$ where the absence of smooth isotopy, even after one weak internal stabilization, is detected by Khovanov homology invariants.\n\n(12) See Problem 4.7 and Problem 4.79 for other “is one stabilization enough?” questions.\n\nReferences cited:\n- [Gal24] Daniel A. P. Galvin. The Casson-Sullivan invariant for homeomorphisms of 4-manifolds, 2024. arXiv:2405.07928.\n- [AKM+19] Dave Auckly, Hee Jung Kim, Paul Melvin, Daniel Ruberman, and Hannah Schwartz. Isotopy of surfaces in 4-manifolds after a single stabilization. Adv. Math., 341:609–615, 2019. doi:10.1016/j.aim.2018.10.040.\n- [LM25] Jianfeng Lin and Anubhav Mukherjee. Family Bauer-Furuta invariant, exotic surfaces and Smale conjecture. J. Assoc. Math. Res., 3(2):237–275, 2025. doi: 10.56994/JAMR.003.002.003.\n- [HKM23] Kyle Hayden, Sungkyung Kang, and Anubhav Mukherjee. One stabilization is not enough for closed knotted surfaces, 2023. arXiv:2304.01504.\n- [BS16] R. İnanç Baykur and Nathan Sunukjian. Knotted surfaces in 4-manifolds and stabilizations. J. Topol., 9(1):215–231, 2016. doi:10.1112/jtopol/jtv039.\n- [Auc23] David Auckly. Smoothly knotted surfaces that remain distinct after many internal stabilizations, 2023. arXiv:2307.16266.\n- [Gut22] Gary Guth. For exotic surfaces with boundary, one stabilization is not enough, 2022. J. Eur. Math. Soc., to appear. https://doi.org/10.4171/jems/1541. arXiv: 2207.11847.\n- [Kan22b] Sungkyung Kang. One stabilization is not enough for contractible 4-manifolds, 2022. arXiv:2210.07510.\n- [Hay23] Kyle Hayden. An atomic approach to Wall-type stabilization problems, 2023. arXiv:2302.10127.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2910, "problem_number": "KP-4.34", "title": "Kirby Problem 4.34", "statement": "Let $\\Sigma$ be a surface embedded in $S^{4}$. Can $\\Sigma$ be unknotted by a sequence of torus surgeries in its complement, such that the ambient manifold is always $S^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.34.\n\nLiterature notes:\n(1) This problem may be addressed in the smooth or topological locally flat categories. This would be an analogue of the 3-dimensional fact that any knot in $S^{3}$ can be transformed into the unknot via $\\pm$ 1 surgery on disjoint unknots.\n\n(2) Without restricting the torus surgeries to preserve $S^{4}$, the answer to this question is “yes” by [Iwa90]. In fact, any surface $\\Sigma$ can be unknotted by at most two torus surgeries that do not preserve $S^{4}$ using Gabai’s 4dimensional light bulb theorem [Gab20], since a standard surgery along a torus centered about a meridian of $\\Sigma$ yields a surface $S^{2} \\times S^{2}\\#S^{1} \\times S^{3}$ that can be standardized using the light bulb theorem.\n\n(3) Larson observed that spun 2-knots can be unknotted by such torus surgeries [Lar18, Theorem 3.9]. This argument also applies to twist-spun knots. Larson and Meier [LM15, Theorem 1.5] produced a different set of such torus surgeries unknotting 2-knots arising from spinning fibered classical knots.\n\n(4) Note that $\\Sigma$ and the standard unknotted surface $\\Sigma_{0} \\cong \\Sigma$ become isotopic after sufficiently many ambient 1-handle surgeries [BS13], which roughly correspond to excessive generators of complementary fundamental group. One way to approach this problem would be to realize these stabilizations and destabilizations as a sequence of certain torus surgeries.\n\nReferences cited:\n- [Iwa90] Zjuñici Iwase. Dehn surgery along a torus T2-knot. II. Japan. J. Math. (N.S.), 16(2):171–196, 1990. doi:10.4099/math1924.16.171.\n- [Gab20] David Gabai. The 4-dimensional light bulb theorem. J. Amer. Math. Soc., 33(3):609–652, 2020. doi:10.1090/jams/920.\n- [Lar18] Kyle Larson. Surgery on tori in the 4-sphere. Math. Proc. Cambridge Philos. Soc., 164(1):109–124, 2018. doi:10.1017/S0305004116000876.\n- [LM15] Kyle Larson and Jeffrey Meier. Fibered ribbon disks. J. Knot Theory Ramifications, 24(14):1550066, 22, 2015. doi:10.1142/S0218216515500662.\n- [BS13] R. İnanç Baykur and Nathan Sunukjian. Round handles, logarithmic transforms and smooth 4-manifolds. J. Topol., 6(1):49–63, 2013.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2911, "problem_number": "KP-4.35", "title": "Kirby Problem 4.35", "statement": "(a) Give an algebraic classification of groups that arise as the fundamental group of the complement of a smooth or locally flat 2-sphere in $S^{4}$.\n\n(b) Let $\\Lambda = \\mathbb{Z}[t, t-^{1}]$. Classify the $\\Lambda-modules$ that arise as the Alexander module of a smooth or locally flat 2-sphere in $S^{4}$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.35.\n\nLiterature notes:\n(1) This is [Kir97, Problem 1.48]. Hillman discusses this problem in the 2022 update of [Hil02, Part III, §14.6–14.10]. The analogous problem is also open for classical knots in $S^{3}$.\n\n(2) Kervaire [Ker65b] gave necessary conditions for a group to be the fundamental group of the complement of a knotted (n−2)-sphere in $S^{n}$ and showed these conditions are also sufficient for $n \\geq$ 5; see [MW17, Section 3].\n\n(3) The answer to Problem (b) is known in dimension three when $“\\mathbb{Z}”$ is replaced by $“\\mathbb{Q};”$ see [Gor78, Section 6]. One might try to extend this to dimension four, with the extra difficulty that Alexander ideals are generally not principal in this dimension (see [Fox61, Example 12]).\n\n(4) If G is the fundamental group of $S^{4} \\setminus \\Sigma$ for $\\Sigma a$ locally flat 2-sphere, then G has weight $w(G)$ =1, $H_{1}(G) =\\mathbb{Z}$, and deficiency $d(G) \\leq$ 1. As a partial answer to Problem (a), when $w(G)$ =1, $H_{1}(G) =\\mathbb{Z}$ and $d(G) =$ 1 (rather than $d(G) \\leq$ 1), then by Kervaire [Ker65a], G is the fundamental group of the complement of a smooth 2-sphere in a homotopy 4-sphere and hence the complement of a locally flat 2-sphere in $S^{4}[FQ90]$.\n\n(5) Gonzalez-Acuña [GAn94] has given a characterization of groups arising as the fundamental group of the complement of a smooth 2-sphere in $S^{4}$ in terms of group presentations. (Specifically, a group G satisfying the Kervaire conditions is such a group if and only if it admits a Wirtinger presentation.) This problem is specifically asking for an algebraic characterization along the lines of Kervaire’s in dimension at least 5.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Hil02] J. A. Hillman. Four-manifolds, geometries and knots, volume 5 of Geometry \\& Topology Monographs. Geometry \\& Topology Publications, Coventry, 2002.\n- [Ker65b] Michel A. Kervaire. On higher dimensional knots. In Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pages 105–119. Princeton Univ. Press, Princeton, NJ, 1965.\n- [MW17] Françoise Michel and Claude Weber. Higher-dimensional knots according to Michel Kervaire. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich, 2017. doi:10.4171/180.\n- [Gor78] C. McA. Gordon. Some aspects of classical knot theory. In Knot theory (Proc. Sem., Plans-sur-Bex, 1977), volume 685 of Lecture Notes in Math., pages 1–60. Springer, Berlin-New York, 1978.\n- [Fox61] R. H. Fox. A quick trip through knot theory. In Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), pages 120–167. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1961.\n- [Ker65a] Michel A. Kervaire. Les nœuds de dimensions supérieures. Bull. Soc. Math. France, 93:225–271, 1965. URL: http://www.numdam.org/item?id=BSMF 1965 93 225 0.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [GAn94] F. González-Acuña. A characterization of 2-knot groups. Rev. Mat. Iberoamericana, 10(2):221–228, 1994. doi:10.4171/RMI/151.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2912, "problem_number": "KP-4.36", "title": "Kirby Problem 4.36", "statement": "Are homotopy types of 2-knot complements determined by their homotopy 2-types?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.36.\n\nLiterature notes:\n(1) The homotopy 2-type of a topological space X is a triple $(\\pi_{1}(X), \\pi_{2}(X)$, k(X)), where $k(X) \\in H^{3}(\\pi_{1}(X);\\pi_{2}(X))$ is the first k-invariant of X [MW50]. Martins [Mar09] gave an algorithm to compute the homotopy 2-type of a 2-knot complement. Lomonaco [Lom81] proved that if the third homology group of the universal cover of a knot complement is trivial (quasi-aspherical),then the homotopy 2-type determines the homotopy type of a 2-knot complement. However, not every 2-knot complement is quasi-aspherical [Rat81].\n\n(2) Note that 2-knots are not even determined by the homeomorphism type of their exterior, since Gordon [Gor76] showed that there exist twistspun knots K that are not equivalent to the knots K* arising from Gluck twisting.\n\n(3) Compare this question to Problem 4.55, which asks whether the homotopy type of a 4-manifold with finite fundamental group is determined by its quadratic 2-type, which includes the additional data of the equivariant intersection form. The fundamental groups of 2-knot complements are of course infinite, but not yet completely classified (see Problem 4.35).\n\nReferences cited:\n- [MW50] Saunders MacLane and J. H. C. Whitehead. On the 3-type of a complex. Proc. Nat. Acad. Sci. U.S.A., 36:41–48, 1950. doi:10.1073/pnas.36.1.41.\n- [Mar09] João Faria Martins. The fundamental crossed module of the complement of a knotted surface. Trans. Amer. Math. Soc., 361(9):4593–4630, 2009. doi:10.1090/S0002-9947-09-04576-0.\n- [Lom81] S. J. Lomonaco, Jr. The homotopy groups of knots. I. How to compute the algebraic 2-type. Pacific J. Math., 95(2):349–390, 1981. http://projecteuclid.org/euclid.pjm/1102735075.\n- [Rat81] John G. Ratcliffe. On the ends of higher-dimensional knot groups. J. Pure Appl. Algebra, 20(3):317–324, 1981. doi:10.1016/0022-4049(81)90066-9.\n- [Gor76] C. McA. Gordon. Knots in the 4-sphere. Comment. Math. Helv., 51(4):585–596, 1976. doi:10.1007/BF02568175.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2913, "problem_number": "KP-4.37", "title": "Kirby Problem 4.37", "statement": "(Kinoshita conjecture). Does every projective plane in $S^{4}$ decompose as a connected sum of a knotted 2-sphere and an unknotted projective plane?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.37.\n\nLiterature notes:\n(1) This question can be asked in either the smooth or the locally flat categories.\n\n(2) The conjecture that the answer to this question is “yes,” is known as the Kinoshita Conjecture. This is named after the work of Kinoshita [Kin61], who first constructed an infinite family of projective planes in $S^{4}$ that are pairwise not ambiently isotopic. Each of Kinoshita’s projective planes decomposes as the connected sum of a knotted 2-sphere and an unknotted projective plane. We may refer to such projective planes as being of Kinoshita type. Then the question is: is every projective plane in $S^{4}$ of Kinoshita type? The first instance of this problem appearing as “Kinoshita’s Conjecture” in the literature is likely work of Katanaga–Saeki [KS98], who said in 1997 that this problem was already well-known to topologists in Japan.\n\n(3) Kamada showed that the Kinoshita Conjecture is true for projective planes obtained by deform-spinning knots with finite summetry groups [Kam92].\n\n(4) By Yoshikawa [Yos98], not every Klein bottle in $S^{4}$ admits an unknotted projective plane summand. The examples studied by Yoshikawa all have normal Euler number zero. Question (i). Does there exist a Klein bottle $\\Sigma$ in $S^{4}$ that does not admit an unknotted projective plane summand with $|e(\\Sigma)|$ =4?\n\n(5) Yoshikawa [Yos94] constructed 2-component links of projective planes in $S^{4}$ such that neither component admits an unknotted projective plane summand in the complement of the other component. In some sense, this can be interpreted as a failure of the Kinoshita Conjecture for multiplecomponent links. Yoshikawa’s obstruction to these links decomposing was to show that the peripheral subgroup of each component was order 4, but would be order 2 if the link admitted a trivial projective plane summand at that component. This motivates the following question. Question (ii). Given a projective plane P in $S^{4}$, must the kernel of the inclusion-induced homomorphism $\\iota_{*}: \\pi_{1}(\\partial(S^{4} \\setminus \\nu(P)) \\to \\pi_{1}(S^{4} \\setminus$ P) be order 4? (If not, then P is an example of a projective plane that does not decompose as a summand of a knotted 2-sphere and unknotted projective plane.)\n\n(6) When $\\Sigma$ is a projective plane in a 4-manifold X with normal Euler number $e(\\Sigma) = \\pm$ 2, we can perform Price surgery on X along $\\Sigma$ [Pri77]. When $X =S^{4}$, there is a preferred surgery whose result is a homotopy 4-sphere. If $\\Sigma$ is obtained by tubing an unknotted projective plane to a 2-sphere S with $e(S) =$ 0, then some Price surgery on $\\Sigma$ is diffeomorphic to the result of Gluck surgery on S [KSTY99]. In particular, when $X =S^{4}$ the homotopy 4-sphere obtained from Price surgery on $\\Sigma$ is diffeomorphic to the result of Gluck surgery on S. Thus, positively answering the proposed question (showing that all projective planes in $S^{4}$ are of Kinoshita type) would imply that Price surgeries in $S^{4}$ yield no more homotopy 4-spheres than Gluck surgeries.\n\nReferences cited:\n- [Kin61] Shin’ichi Kinoshita. On the Alexander polynomials of 2-spheres in a 4-sphere. Ann. of Math. (2), 74:518–531, 1961. doi:10.2307/1970296.\n- [KS98] Atsuko Katanaga and Osamu Saeki. Embeddings of quaternion space in $S^{4}$. J. Austral. Math. Soc. Ser. A, 65(3):313–325, 1998.\n- [Kam92] Seiichi Kamada. Projective planes in 4-sphere obtained by deform-spinnings. In Knots 90 (Osaka, 1990), pages 125–132. de Gruyter, Berlin, 1992.\n- [Yos98] Katsuyuki Yoshikawa. The order of a meridian of a knotted Klein bottle. Proc. Amer. Math. Soc., 126(12):3727–3731, 1998. doi:10.1090/S0002-9939-98-04560-2.\n- [Yos94] Katsuyuki Yoshikawa. An enumeration of surfaces in four-space. Osaka J. Math., 31(3):497–522, 1994. http://projecteuclid.org/euclid.ojm/1200785461.\n- [Pri77] T. M. Price. Homeomorphisms of quaternion space and projective planes in four space. J. Austral. Math. Soc. Ser. A, 23(1):112–128, 1977. doi:10.1017/s1446788700017407.\n- [KSTY99] Atsuko Katanaga, Osamu Saeki, Masakazu Teragaito, and Yuichi Yamada. Gluck surgery along a 2-sphere in a 4-manifold is realized by surgery along a projective plane. Michigan Math. J., 46(3):555–571, 1999. doi:10.1307/mmj/1030132479.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2914, "problem_number": "KP-4.38", "title": "Kirby Problem 4.38", "statement": "Let $\\Delta \\subset B^{4}$ be a ribbon disk. Is $B^{4} \\setminus \\Delta$ aspherical, i.e. is $\\pi_{i}(B^{4} \\setminus \\Delta)$ =0 for all i>1?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.38.\n\nLiterature notes:\n(1) This is [Kir97, Problem 1.103]. Note that the Whitehead asphericity conjecture [Whi41, p. 428] (see also [Ros07a, How85] and Problem 5.11) predicts an affirmative answer. A generalized version, asking for the asphericity of ribbon concordances, was asked by Gordon in [Gor81, Conjecture 6.5].\n\n(2) Howie showed in [How85] that the answer is yes when $\\pi_{1}(B^{4} \\setminus \\Delta)is$ locally indicable.\n\n(3) One could equally well pose this question in the locally flat category, asking whether the complements of homotopy-ribbon disks are aspherical. (Recall that a homotopy-ribbon disk $\\Delta$ for some knot $K \\subset S^{3}$ is one such that the map $\\pi_{1}(S^{3} \\setminus$ K) $\\to \\pi_{1}(B^{4} \\setminus \\Delta)is$ surjective.) In this case, it is open whether the complement is homotopy equivalent to a 2-complex.\n\n(4) An affirmative answer to the question can be used to compute the algebraic 2-type of a 2-knot complement, as shown in [Lom81]. Note that the argument there is based on an incorrect proof asserting that ribbon disk complements are aspherical. See [Mar09] for an alternate approach to computing the algebraic 2-type of a 2-knot complement.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Whi41] J. H. C. Whitehead. On adding relations to homotopy groups. Ann. of Math. (2), 42:409–428, 1941. doi:10.2307/1968907.\n- [Ros07a] Stephan Rosebrock. The Whitehead conjecture—an overview. Sib. Èlektron. Mat. Izv., 4:440–449, 2007.\n- [How85] James Howie. On the asphericity of ribbon disc complements. Trans. Amer. Math. Soc., 289(1):281–302, 1985. doi:10.2307/1999700.\n- [Gor81] C. McA. Gordon. Ribbon concordance of knots in the 3-sphere. Math. Ann., 257(2):157–170, 1981. doi:10.1007/BF01458281.\n- [Lom81] S. J. Lomonaco, Jr. The homotopy groups of knots. I. How to compute the algebraic 2-type. Pacific J. Math., 95(2):349–390, 1981. http://projecteuclid.org/euclid.pjm/1102735075.\n- [Mar09] João Faria Martins. The fundamental crossed module of the complement of a knotted surface. Trans. Amer. Math. Soc., 361(9):4593–4630, 2009. doi:10.1090/S0002-9947-09-04576-0.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2915, "problem_number": "KP-4.39", "title": "Kirby Problem 4.39", "statement": "Let K be a closed surface smoothly embedded in a 4-manifold $M^{4}$. Describe the subgroup of the mapping class group $Mod(K)$ of diffeomorphisms of K that extend to diffeomorphisms of M. Conversely, is it possible to characterize when a given subgroup of the mapping class group arises as the group of extendable diffeomorphisms for some K inside some $M^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.39.\n\nLiterature notes:\n(1) The problem is due to Hirose, who has studied instances of this problem in e.g. [Hir93, Hir02, Hir05, Hir12]. Some highlights of his results on this problem are summarized in the remarks to follow.\n\n(2) In [Hir02], Hirose studies the case of an unknotted genus-g surface in $S^{4}$, showing that a mapping class is extendable in this setting if and only if it preserves the Rokhlin quadratic for m (descending to a spin structure on the surface). In particular, this subgroup is of finite index in the mapping class group of K. In [Hir02], he extended this result to nonorientable unknotted surfaces in $S^{4}$, showing that in this setting a mapping class extends exactly when it preserves the Guillou–Marin quadratic for m(descending to a Pin− structure on the surface; see [KS23] for more discussion). Again, this subgroup is of finite index in the mapping class group of K.\n\n(3) In [Hir05], Hirose classified which surface automorphisms extend from unknotted surfaces and certain complex curves to all of $\\mathbb{CP}^{2}$. The question is open for all high-degree complex curves.\n\n(4) In [Hir93], Hirose studied the case of knotted tori in $S^{4}$ arising by spinning (or turned spinning) knots in $S^{3}$. Here, the set of extendable mapping classes is always of infinite index. It would be interesting to characterize when the group of extendable mapping classes is of finite index in general.\n\n(5) In [HY08], Hirose and Yasuhara showed that in many simply connected 4-manifolds (including $\\mathbb{CP}^{2},\\mathbb{CP}^{2}, S^{2} \\times S^{2}$, E(n)) there exist smoothly embedded surfaces of every topological type with the property that any surface automorphism extends to an ambient automorphism.\n\nReferences cited:\n- [Hir93] Susumu Hirose. On diffeomorphisms over T2-knots. Proc. Amer. Math. Soc., 119(3):1009–1018, 1993. doi:10.2307/2160546.\n- [Hir02] Susumu Hirose. On diffeomorphisms over surfaces trivially embedded in the 4-sphere. Algebr. Geom. Topol., 2:791–824, 2002. doi:10.2140/agt.2002.2.791.\n- [Hir05] Susumu Hirose. Surfaces in the complex projective plane and their mapping class groups. Algebr. Geom. Topol., 5:577–613, 2005. doi:10.2140/agt.2005.5.577.\n- [Hir12] Susumu Hirose. On diffeomorphisms over nonorientable surfaces standardly embedded in the 4-sphere. Algebr. Geom. Topol., 12(1):109–130, 2012. doi:10.2140/agt.2012.12.109.\n- [KS23] Michael R. Klug and Luuk Stehouwer. Some properties of Pin˘-structures on compact surfaces. Topology Appl., 339(part B):Paper No. 108678, 9, 2023. doi: 10.1016/j.topol.2023.108678.\n- [HY08] Susumu Hirose and Akira Yasuhara. Surfaces in 4-manifolds and their mapping class groups. Topology, 47(1):41–50, 2008. doi:10.1016/j.top.2007.05.001.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2916, "problem_number": "KP-4.40", "title": "Kirby Problem 4.40", "statement": "Is every link of 2-spheres in $S^{4}$ slice?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.40.\n\nLiterature notes:\n(1) A 2-link is slice if it bounds a disjointly embedded collection of 3-balls in the 5-ball $B^{5}$.\n\n(2) Kervaire [Ker65a] proved that every 2-knot in $S^{4}$ is slice.\n\n(3) This question can be asked in either category. If a smooth link of 2-spheres is topologically slice in $B^{5}$, then it is also smoothly slice by Daher–Powell [DP23a].\n\n(4) Cochran [Coc84] gave a sufficient condition for a 2-link to be slice, generalizing the well-known fact that boundary 2-links (see Problem 1.62) are slice (following from Kervaire’s proof for 2-knots). Not every 2-link is a boundary link [Coc84, §4].\n\nReferences cited:\n- [Ker65a] Michel A. Kervaire. Les nœuds de dimensions supérieures. Bull. Soc. Math. France, 93:225–271, 1965. URL: http://www.numdam.org/item?id=BSMF 1965 93 225 0.\n- [DP23a] Michelle Daher and Mark Powell. Smoothing 3-manifolds in 5-manifolds, 2023. arXiv:2309.15962.\n- [Coc84] Tim Cochran. Slice links in $S^{4}$. Trans. Amer. Math. Soc., 285(1):389–401, 1984. doi:10.2307/1999487.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2917, "problem_number": "KP-4.41", "title": "Kirby Problem 4.41", "statement": "Let $\\Sigma$ be a compact surface and let X be a connected 4manifold. Let $f_{0}, f_{1}: \\Sigma \\to X$ be $\\pi_{1}-negligible$ embeddings that agree on $\\partial\\Sigma$ and that are homotopic rel. boundary. Give computable invariants that decide whether there is $a \\pi_{1}-negligible$ concordance in $X \\times I$ between $f_{0}$ and $f_{1}$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.41.\n\nLiterature notes:\n(1) A surface is $\\pi_{1}-negligible$ if the complement of the surface included into X induces an isomorphism on fundamental groups [FQ90]. This assumption is equivalent to asking for an immersed dual sphere.\n\n(2) The case that $\\Sigma =S^{2}$ or that the dual sphere is framed is due to Freedman– Quinn [FQ90] and Stong [Sto93]. See also the papers of Klug–Miller [KM21, KM22].\n\n(3) A strategy would be to extend the Freedman–Quinn and Stong invariants to arbitrary compact surfaces.\n\nReferences cited:\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [Sto93] Richard Stong. Uniqueness of $\\pi_1$-negligible embeddings in 4-manifolds: a correction to Theorem 10.5 of topology of 4-manifolds [Princeton Univ. Press, Princeton, NJ, 1990; MR1201584 (94b:57021)] by M. H. Freedman and F. Quinn. Topology, 32(4):677–699, 1993. doi:10.1016/0040-9383(93)90046-X.\n- [KM21] Michael R. Klug and Maggie Miller. Concordance of surfaces in 4-manifolds and the Freedman-Quinn invariant. J. Topol., 14(2):560–586, 2021. doi:10.1112/topo.12191.\n- [KM22] Michael Klug and Maggie Miller. Concordance of spheres in 4-manifolds with an immersed dual sphere, 2022. arXiv:2211.07177.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2918, "problem_number": "KP-4.42", "title": "Kirby Problem 4.42", "statement": "(a) If X is a smooth, closed, simply connected 4-manifold $withb_{2}(X) \\geq$ 2, are all knots slice in X?\n\n(b) In particular, are all knots slice in $\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2}$? In the K3 surface?\n\n(c) Can the set of slice knots detect exotic smooth structures? That is, do there exist homeomorphic 4-manifolds X and $X^{1}$ and a knot $K \\subset S^{3}$ that is slice in X but not in $X^{1}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.42.\n\nLiterature notes:\n(1) A knot K is called slice in a closed 4-manifold X if it bounds a smoothly embedded disk in $X \\setminus$ in $t(B^{4})$, where K lives on the boundary $S^{3}$. It is called H-slice if the disk is null-homologous (relative to the boundary).\n\n(2) All knots are known to be slice in $S^{2} \\times S^{2}$ and $\\mathbb{CP}^{2}\\#\\mathbb{CP}^{2}[Nor69$, Suz69], but not all knots are slice in $\\mathbb{CP}^{2}$ or in $\\mathbb{CP}^{2}$ [Yas91].\n\n(3) Part(a)is true in the topological category; see [KPRT24, Corollary 1.15].\n\n(4) Concerning part (b), Marengon and Mihajlović [MM25a] proved that all knots with unknotting number at most 21 are slice in K3.\n\n(5) The analogue of (c) for H-slice knots (instead of slice) was answered in [MMP24]: the right handed trefoil is H-slice in $\\#3 \\mathbb{CP}^{2}\\#20 \\mathbb{CP}^{2}$ but not in $K3\\#\\mathbb{CP}^{2}$.\n\n(6) A variant of part (c) is asked in Problem 4.12.\n\n(7) Lidman and Piccirillo [LP25] announced the construction of a pair of 4manifolds X and $X^{1}$ with the same integer cohomology ring, such that there exists a knot slice in X but not in $X^{1}$. (However,X and $X^{1}$ are not homeomorphic.)\n\nReferences cited:\n- [Nor69] R. A. Norman. Dehn’s lemma for certain 4-manifolds. Invent. Math., 7:143–147, 1969. doi:10.1007/BF01389797.\n- [Suz69] Shin’ichi Suzuki. Local knots of 2-spheres in 4-manifolds. Proc. Japan Acad., 45:34– 38, 1969.\n- [Yas91] Akira Yasuhara. $(2,15)$-torus knot is not slice in $\\mathbb{CP}^{2}$. Proc. Japan Acad. Ser. A Math. Sci., 67(10):353–355, 1991. http://projecteuclid.org/euclid.pja/1195511928.\n- [KPRT24] Daniel Kasprowski, Mark Powell, Arunima Ray, and Peter Teichner. Embedding surfaces in 4-manifolds. Geom. Topol., 28(5):2399–2482, 2024. doi:10.2140/gt.2024.28.2399.\n- [MM25a] Marco Marengon and Stefan Mihajlović. Unknotting number 21 knots are slice in K3. Math. Res. Lett., 32(3):939–955, 2025. doi:10.4310/mrl.250731115553.\n- [MMP24] Ciprian Manolescu, Marco Marengon, and Lisa Piccirillo. Relative genus bounds in indefinite four-manifolds. Math. Ann., 390(1):1481–1506, 2024. doi:10.1007/s00208-023-02787-4.\n- [LP25] Tye Lidman and Lisa Piccirillo. Distinguishing closed 4-manifolds by slicing, 2025. arXiv:2505.14387.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2919, "problem_number": "KP-4.43", "title": "Kirby Problem 4.43", "statement": "Let K be a knot on the boundary of $X \\setminus B^{\\circ 4}$, where X is a negative definite, smooth 4-manifold. Suppose K bounds a smoothly embedded, nullhomotopic surface $\\Sigma$ in $X \\setminus B^{\\circ 4}$. Is the following inequality $2g(\\Sigma) \\geq$ |s(K)| (8) always satisfied? Here $s(K)$ denotes Rasmussen’s s-invariant [Ras10].", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.43.\n\nLiterature notes:\n(1) For $X =S^{4}$, the inequality (8) was proved by Rasmussen [Ras10]. When $X =\\#_{r}\\mathbb{CP}^{2}$, (8) was proved by Manolescu–Marengon–Sarkar-Willis [MMSW23].\n\n(2) Freedman–Gompf–Morrison–Walker [FGMW10] proposed the following approach to disproving the smooth four-dimensional Poincaré conjecture: Find a homotopy 4-sphere X and a knot K that is slice in X (i.e. K bounds smoothly embedded disk in $X \\setminus B^{4})$. Then prove that K is not slice in $S^{4}$ by showing that $s(K) \\ne$ 0. If this approach works, then (8) must fail for X.\n\n(3) Using instanton Floer theory, Kronheimer-Mrowka [KM13b] defined a concordance invariant s7(K) that satisfies a similar inequality $2g(\\Sigma) \\geq |s^{7}(K)|$. However, it is known that $s(K) \\ne s^{7}(K)$ for many knots including the trefoil [Gon21]. One possible approach to proving (8) is to find a gauge theoretic interpretation of $s(K)$ by modifying Kronheimer-Mrowka’s definition of s7(K).\n\n(4) One may ask whether (8) holds for other variants of Rasmussen’s sinvariant, including $s\\mathbb{F}p$ (defined using Khovanov homology over the field $\\mathbb{F}_{p}[MTV07$, LS14]) and $s^{Sq,1}$ (the Lipshitz-Sarkar $Sq^{1}s-invariants$ [LS14]). It is known that their variants are different from $s(K)$ in general [LS14].\n\nReferences cited:\n- [Ras10] Jacob Rasmussen. Khovanov homology and the slice genus. Invent. Math., 182(2):419–447, 2010. doi:10.1007/s00222-010-0275-6.\n- [MMSW23] Ciprian Manolescu, Marco Marengon, Sucharit Sarkar, and Michael Willis. A generalization of Rasmussen’s invariant, with applications to surfaces in some four-manifolds. Duke Math. J., 172(2):231–311, 2023. doi:10.1215/00127094-2022-0039.\n- [FGMW10] Michael Freedman, Robert Gompf, Scott Morrison, and Kevin Walker. Man and machine thinking about the smooth 4-dimensional Poincaré conjecture. Quantum Topol., 1(2):171–208, 2010. doi:10.4171/QT/5.\n- [KM13b] P. B. Kronheimer and T. S. Mrowka. Gauge theory and Rasmussen’s invariant. J. Topol., 6(3):659–674, 2013. doi:10.1112/jtopol/jtt008.\n- [Gon21] Sherry Gong. On the Kronheimer-Mrowka concordance invariant. J. Topol., 14(1):1–28, 2021. doi:10.1112/topo.12175.\n- [MTV07] Marco Mackaay, Paul Turner, and Pedro Vaz. A remark on Rasmussen’s invariant of knots. J. Knot Theory Ramifications, 16(3):333–344, 2007. doi:10.1142/S0218216507005312.\n- [LS14] Robert Lipshitz and Sucharit Sarkar. A refinement of Rasmussen’s S-invariant. Duke Math. J., 163(5):923–952, 2014. doi:10.1215/00127094-2644466.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2920, "problem_number": "KP-4.44", "title": "Kirby Problem 4.44", "statement": "(a) Let X be a closed, simply connected 4-manifold. Let $g \\geq$ 0 and $d \\geq$ 0 be integers. Fix $x \\in H_{2}(X;\\mathbb{Z})$. Does there exist a (smooth, locally flat) generic immersion $f: \\Sigma_{g} \\to X$ such that $f_{*}[\\Sigma_{g}] = x$, with d transverse double points?\n\n(b) An important special case is $d =$ 0. Let $g: H_{2}(X;\\mathbb{Z}) \\to \\mathbb{N}_{0}$ be the genus function assigning to $x \\in H_{2}(X;\\mathbb{Z})$ the minimal genus of any (smooth, locally flat) embedded surface in X whose fundamental class represents x. The problem is to compute the locally flat genus function for some 4-manifold $X \\ne S^{4}$, and to compute the smooth genus function for new 4-manifolds.\n\n(c) For a specific example, is the class (3,2) $\\in H_{2}(\\mathbb{CP}^{2}\\#\\mathbb{CP}^{2};\\mathbb{Z})$ represented by a sphere?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.44.\n\nLiterature notes:\n(1) The smooth genus function is known for $\\mathbb{CP}^{2}$ by a celebrated result of Kronheimer–Mrowka [KM94]. The cases of $\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2}$ and $S^{2} \\times S^{2}$ were resolved by Ruberman in [Rub96]. Partial results about the smooth genus problem in $\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2}$ appear in [Bry98], [Nou14], [MMRS24], [ACM+26]. By the symplectic Thom conjecture, proved in [OS00], any symplectic submanifold minimizes genus in its homology class; this gives a useful means of computing the value of the smooth genus function for certain homology classes.\n\n(2) The locally flat genus function is not known except for $S^{4}$. In other 4manifolds, much is known aboutsimple locally flat embeddings by work of Lee-Wilczynski [LW97, LW00]. As a special case, every primitive class in a closed, simply-connected 4-manifold is represented by a locally flat torus [LW97, KPRT24]. Not much else is known.\n\n(3) In the special case when the self-intersection number Q(x, x) $=$ 0, we have the following interesting sub-problem. As usual we $let\\mathbb{N}$ denote the positive integers.\n\n\\paragraph{Question.} For a given $x \\in H_{2}(X;\\mathbb{Z})$ such that $g(x) >$ 0, consider the function $n_{x}: \\mathbb{N} \\to \\mathbb{N}$ sending $m \\mapsto g(mx)$. What is $n_{x}$ in terms of $g(x)$? Is $n_{x}(m)$ =mg(x) −m+1? Note that $ifg(x)$ =0 then $g(mx)$ =0 for all x. The formula $mg(x)$ − m+1 comes from the following observation: if $[\\Sigma]$ =xwith Q(x, x) =0, then there is a surface $\\Sigma$ that embeds in $\\Sigma_{~} \\times D^{2}$ representing mx such that the projection to $\\Sigma$ induces an m-sheeted covering. The answer is negative for some 4-manifolds with boundary. Kawauchi [Kaw09] constructed 4-manifolds homotopy equivalent to $S^{2}(so H_{2}(X;\\mathbb{Z}) = \\mathbb{Z})$, with the property that for x a generator, $g(mx) =$ 0 for m even and $g(mx) \\ne$ 0 for m odd.\n\nReferences cited:\n- [KM94] P. B. Kronheimer and T. S. Mrowka. The genus of embedded surfaces in the projective plane. Math. Res. Lett., 1(6):797–808, 1994. doi:10.4310/MRL.1994.v1.n6.a14.\n- [Rub96] Daniel Ruberman. The minimal genus of an embedded surface of non-negative square in a rational surface. Turkish J. Math., 20(1):129–133, 1996.\n- [Bry98] Jim Bryan. Seiberg-Witten theory and $\\mathbb{Z}/2^p$ actions on spin 4-manifolds. Math. Res. Lett., 5(1-2):165–183, 1998. doi:10.4310/MRL.1998.v5.n2.a3.\n- [Nou14] Mohamed Ait Nouh. The minimal genus problem in $\\mathbb{CP}^{2}$\\#$\\mathbb{CP}^{2}$. Algebr. Geom. Topol., 14(2):671–686, 2014. doi:10.2140/agt.2014.14.671.\n- [MMRS24] Marco Marengon, Allison N. Miller, Arunima Ray, and András I. Stipsicz. A note on surfaces in $\\mathbb{CP}^{2}$ and $\\mathbb{CP}^{2}$ \\#$\\mathbb{CP}^{2}$ . Proc. Amer. Math. Soc. Ser. B, 11:187–199, 2024. doi:10.1090/bproc/218.\n- [ACM+26] Paolo Aceto, Nickolas A Castro, Maggie Miller, JungHwan Park, and András Stipsicz. Slice Obstructions From Genus Bounds in Definite 4-Manifolds. Int. Math. Res. Not. IMRN, 2026(2):Paper No. rnaf377, 2026. doi:10.1093/imrn/rnaf377.\n- [OS00] Peter Ozsváth and Zoltán Szabó. The symplectic Thom conjecture. Ann. of Math. (2), 151(1):93–124, 2000. doi:10.2307/121113.\n- [LW97] Ronnie Lee and Dariusz M. Wilczyński. Representing homology classes by locally flat surfaces of minimum genus. Amer. J. Math., 119(5):1119–1137, 1997. URL: http://muse.jhu.edu/journals/american journal of mathematics/v119/119.5lee.pdf.\n- [LW00] Ronnie Lee and Dariusz M. Wilczyński. Genus inequalities and four-dimensional surgery. Topology, 39(2):311–330, 2000. doi:10.1016/S0040-9383(99)00017-8.\n- [KPRT24] Daniel Kasprowski, Mark Powell, Arunima Ray, and Peter Teichner. Embedding surfaces in 4-manifolds. Geom. Topol., 28(5):2399–2482, 2024. doi:10.2140/gt.2024.28.2399.\n- [Kaw09] Akio Kawauchi. Rational-slice knots via strongly negative-amphicheiral knots. Commun. Math. Res., 25(2):177–192, 2009.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2921, "problem_number": "KP-4.45", "title": "Kirby Problem 4.45", "statement": "Let X be a closed simply connected smooth 4-manifold and let $\\Sigma$ be a closed, orientable surface. Fix a smooth embedding of $f: \\Sigma \\hookrightarrow X$ to serve as a basepoint. For which $k\\geq 0$ does the equality\n$$\n\\ker i_{*}=\\ker s_{*}\n$$\n hold, where\n$$\n\\begin{aligned} i_{*}&:\\pi_{k}(\\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ});f)\\to \\pi_{k}(\\operatorname{Emb}^{TOP}(\\Sigma,X^{\\circ});f),\\\\ s_{*}&:\\pi_{k}(\\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ});f)\\to \\operatorname*{colim}_{N\\to\\infty}\\pi_{k}(\\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ}\\#_{N}S^{2}\\times S^{2});f). \\end{aligned}\n$$\n Here $i_*$ is induced by inclusion and $s_*$ by stabilization. To make sense of this for k=0, define $keri_{*}$ and $kers_{*}$ to be the preimages under $i_{*}$ and $s_{*}$ respectively of [f].", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.45.\n\nLiterature notes:\n(1) Here $X^{\\circ}$ denotes the punctured $X, X^{\\circ} = X \\setminus \\operatorname{Int}(D^{4}), \\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ})$ is the space of smooth embeddings, $\\operatorname{Emb}^{TOP}(\\Sigma,X^{\\circ})$ is the space of locally flat embeddings, $i: \\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ}) \\to \\operatorname{Emb}^{TOP}(\\Sigma,X^{\\circ})$ is the inclusion, and $s: \\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ}) \\to \\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ}\\#_{N}S^{2} \\times S^{2})$ is the stabilization map. Note that we consider X punctured so that the stabilization map s is defined on the space level.\n\n(2) See Problem 4.77 for the analogous question on diffeomorphism and homeomorphism groups.\n\n(3) Fork =0, the inclusion $\\subset$ holds in the equality (9). That is, Galvin [Gal24] showed that any two smoothly embedded, topologically isotopic surfaces in a smooth, simply-connected 4-manifold X become smoothly isotopic after sufficiently many external stabilizations of X by copies of $S^{2} \\times S^{2}$. Is the converse true? What about when the complements are simply connected?\n\n(4) There is considerable literature showing, under suitable conditions, that surfaces in 4-manifolds are topologically isotopic, especially those with simply connected complements, which might help. See e.g. [LW90, LW93, LW01, Boy93, HK93b, LW97, Sun15, CP23]. New techniques seem to be required for answering the question when k>0.\n\nReferences cited:\n- [Gal24] Daniel A. P. Galvin. The Casson-Sullivan invariant for homeomorphisms of 4-manifolds, 2024. arXiv:2405.07928.\n- [LW90] Ronnie Lee and Dariusz M. Wilczyński. Locally flat 2-spheres in simply connected 4-manifolds. Comment. Math. Helv., 65(3):388–412, 1990.\n- [LW93] Ronnie Lee and Dariusz M. Wilczyński. Representing homology classes by locally flat 2-spheres. K-Theory, 7(4):333–367, 1993. doi:10.1007/BF00962053.\n- [LW01] Ronnie Lee and Dariusz M. Wilczyński. Erratum to: “Genus inequalities and four-dimensional surgery” [Topology 39 (2000), no. 2, 311–330; MR1722016 (2001j:57028)]. Topology, 40(5):1123, 2001. doi:10.1016/S0040-9383(00)00008-2.\n- [Boy93] Steven Boyer. Realization of simply-connected 4-manifolds with a given boundary. Comment. Math. Helv., 68(1):20–47, 1993.\n- [HK93b] Ian Hambleton and Matthias Kreck. Cancellation of hyperbolic forms and topological four-manifolds. J. Reine Angew. Math., 443:21–47, 1993. doi:10.1515/crll.1993.443.21.\n- [LW97] Ronnie Lee and Dariusz M. Wilczyński. Representing homology classes by locally flat surfaces of minimum genus. Amer. J. Math., 119(5):1119–1137, 1997. URL: http://muse.jhu.edu/journals/american journal of mathematics/v119/119.5lee.pdf.\n- [Sun15] Nathan S. Sunukjian. Surfaces in 4-manifolds: concordance, isotopy, and surgery. Int. Math. Res. Not. IMRN, 2015(17):7950–7978, 2015. doi:10.1093/imrn/rnu187.\n- [CP23] Anthony Conway and Mark Powell. Embedded surfaces with infinite cyclic knot group. Geom. Topol., 27(2):739–821, 2023. doi:10.2140/gt.2023.27.739.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2922, "problem_number": "KP-4.46", "title": "Kirby Problem 4.46", "statement": "Are all groups good?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.46.\n\nLiterature notes:\n(1) The problem is due to Freedman and Quinn. The term ‘good’ group was first introduced in [Fre84] and [FQ90, p. 99] to refer to any group for which Freedman’s disk embedding theorem holds. In those two sources it was shown that the latter theorem holds for any group belonging to the smallest class of groups that contains $\\mathbb{Z}$ and finite groups and is closed under colimits, subgroups, extensions, and quotients. In [Kir97, Problem 4.6] a group is defined to be good if it belongs to this class of groups. The definition in terms of a capped grope was first formulated in [FT95a]. This has now become the accepted definition.\n\n(2) A group $\\Gamma$ is said to be good if for every height 1.5 disc-like capped grope G, with some choice of basepoint, and for every group homomorphism $\\varphi: \\pi_{1}(G) \\to \\Gamma$, there exists an immersed disk D in Gwhose framed boundary coincides with the attaching region of G, such that the double point loops of D, considered as fundamental group elements by making some choice of basing path, are mapped to the identity element of $\\Gamma$ by $\\varphi$. For more details on the definition, including the definition of a capped grope, see [FT95a, KOPR21a], cf. [Kir97, Problem 5.9] (these definitions differ in the height of a capped grope that one starts with, but the two definitions are equivalent by a method called grope height raising). We limit ourselves to saying that a capped grope is a smooth 4-manifold, and its attaching region is a specified solid torus in its boundary. The core of the solid torus is null-homotopic in the capped grope (regardless of height). Therefore some immersed disk D always exists – the additional fact about the double point loops is the key feature of the definition. It is also worth observing that good groups are defined by a property that can be entirely stated in the smooth category.\n\n(3) Almost every known result about topological 4-manifolds relies on the disk embedding theorem, and therefore the question of which groups are good is fundamental. Recall that any group is the fundamental group of some 4manifold, but fundamental groups of compact 4-manifolds are necessarily finitely presented. Therefore, one could choose to ask the question only for finitely presented groups. However, as we will see in item 7 below, if all finitely presented groups were good, then all groups would be good, so there is no benefit to this restriction.\n\n(4) By work of Freedman and Quinn [FQ90, Chapter 11] (see also [OPR21b]), the action of $L_{5}(\\mathbb{Z}[\\pi_{1}(X)])$ on the structure set $\\mathcal{S}(X)is$ defined, and the surgery sequence is known to be exact at $\\mathcal{S}(X)$ and $\\mathcal{N}(X)$, for every 4manifold X with $\\pi_{1}(X)$ good. Therefore an affirmative answer would give the exactness of the topological surgery sequence for 4-manifolds in full generality (cf. [Kir97, Problem 4.6]). This is known in dimensions five and higher in the smooth, piecewise-linear, and topological categories [Bro72, Nov64, Sul96, Wal99, KS77]. See Problem 4.22 for more background on the surgery sequence.\n\n(5) An affirmative answer would also imply the 5-dimensional s-cobordism theorem in full generality. The s-cobordism theorem for dimensions 6 and higher is true in both the smooth and topological settings [Sma62a, Bar63, Maz63,Sta67,KS77] (see also [Mil65, RS72]). The smooth 5dimensionals-cobordism theorem is false by work of Donaldson [Don87a]. See also Problem 4.56.\n\n(6) The exactness of the surgery sequence and the h-cobordism theorem is a key step in the classification of closed, simply connected 4-manifolds up to homeomorphism. Therefore the extension to general fundamental groups is likely to have several spectacular applications. We observe that even if all groups were good, classification results would not immediately follow. Nevertheless this would be a result of immense importance.\n\n(7) As mentioned above, it was shown in [FQ90] that finite groups and the infinite cyclic group are good. Groups of subexponential growth were shown to be good in [KQ00, FT95a]. The class of good groups is also known to be closed under subgroups, quotients, extensions, and direct limits [FT95a, KOPR21a]. It is an open question whether any nonabelian free group is good. Recall that every finitely generated group arises as a quotient of a subgroup of $\\mathbb{Z}*\\mathbb{Z}and$ every group is a colimit of its finitely generated subgroups. Therefore, all groups are good if and only if the free group on two generators is good. We state this as a question.\n\n\\paragraph{Question.} Is the group $\\mathbb{Z}*\\mathbb{Z}$ good?\n\n(8) At present it is also open whether amenable groups are good. Addressing this would be a substantial step towards the general question.\n\n\\paragraph{Question.} Are amenable groups good?\n\n(9) For further details on the relationships between the above problems and Problems 1.62and 4.47, see [KOPR21b].\n\nReferences cited:\n- [Fre84] Michael H. Freedman. The disk theorem for four-dimensional manifolds. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 647–663. PWN, Warsaw, 1984.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [FT95a] Michael H. Freedman and Peter Teichner. 4-manifold topology. I. Subexponential groups. Invent. Math., 122(3):509–529, 1995. doi:10.1007/BF01231454.\n- [KOPR21a] Min Hoon Kim, Patrick Orson, JungHwan Park, and Arunima Ray. Good groups. In The disc embedding theorem, pages 273–282. Oxford Univ. Press, Oxford, 2021.\n- [OPR21b] Patrick Orson, Mark Powell, and Arunima Ray. Surgery theory and the classification of closed, simply connected 4-manifolds. In The disc embedding theorem, pages 331–351. Oxford Univ. Press, Oxford, 2021.\n- [Bro72] William Browder. Surgery on simply-connected manifolds, volume 65 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York-Heidelberg, 1972.\n- [Nov64] S. P. Novikov. Homotopically equivalent smooth manifolds. I. Izv. Akad. Nauk SSSR Ser. Mat., 28:365–474, 1964.\n- [Sul96] D. P. Sullivan. Triangulating and smoothing homotopy equivalences and homeomorphisms. Geometric Topology Seminar Notes. In The Hauptvermutung book, volume 1 of K-Monogr. Math., pages 69–103. Kluwer Acad. Publ., Dordrecht, 1996. doi:10.1007/978-94-017-3343-4\\_3.\n- [Wal99] C. T. C. Wall. Surgery on compact manifolds, volume 69 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition, 1999. Edited and with a foreword by A. A. Ranicki. doi:10.1090/surv/069.\n- [KS77] Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations, volume 88 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J., 1977. With notes by John Milnor and Michael Atiyah.\n- [Sma62a] S. Smale. On the structure of manifolds. Amer. J. Math., 84:387–399, 1962. doi: 10.2307/2372978.\n- [Bar63] D. Barden. The structure of manifolds. PhD thesis, Cambridge University, 1963.\n- [Maz63] Barry Mazur. Relative neighborhoods and the theorems of Smale. Ann. of Math. (2), 77:232–249, 1963. doi:10.2307/1970215.\n- [Sta67] John R. Stallings. Lectures on polyhedral topology, volume 43 of Tata Institute of Fundamental Research Lectures on Mathematics. Tata Institute of Fundamental Research, Bombay, 1967. Notes by G. Ananda Swarup.\n- [Mil65] J. W. Milnor. Lectures on the h-cobordism theorem. Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow.\n- [RS72] C. P. Rourke and B. J. Sanderson. Introduction to piecewise-linear topology. Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69.\n- [Don87a] S. K. Donaldson. Irrationality and the h-cobordism conjecture. J. Differential Geom., 26(1):141–168, 1987. http://projecteuclid.org/euclid.jdg/1214441179.\n- [KQ00] Vyacheslav S. Krushkal and Frank Quinn. Subexponential groups in 4-manifold topology. Geom. Topol., 4:407–430, 2000. doi:10.2140/gt.2000.4.407.\n- [KOPR21b] Min Hoon Kim, Patrick Orson, JungHwan Park, and Arunima Ray. Open problems. In The disc embedding theorem, pages 353–382. Oxford Univ. Press, Oxford, 2021.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2923, "problem_number": "KP-4.47", "title": "Kirby Problem 4.47", "statement": "(Round handle problem). Is there a link $L \\subset S^{3}$ with vanishing pairwise linking numbers that is not round handle slice?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.47.\n\nLiterature notes:\n(1) We define the terminology in the problem. A round handle is a copy of $D^{1} \\times D^{2} \\times S^{1}$, which is attached to the boundary of a 4-manifold along $S^{0} \\times D^{2} \\times S^{1}$. (So it is a 4–dimensional round 1–handle.) Given an m-component link $L \\subset S^{3}$ we construct a 4-manifold $R(L)$ by attaching m round handles to $D^{4}$ as follows. For the i-th component $L_{i}$ of L, let $\\lambda_{i}$ denote a 0-framed longitude and let $\\mu_{i}$ denote a meridian, chosen so that the linking number $lk(\\lambda_{i}, \\mu_{i}) =$ 0. To attach the ith round handle, identify \\{−1\\} $\\times S^{1} \\times D^{2}$ to $\\lambda_{i}$ and \\{+1\\} $\\times S^{1} \\times D^{2}$ to $\\mu_{i}$, using the trivial framing in both cases. The resulting 4-manifold contains the link L in the boundary $\\partial R(L). A$ link L is said to be round handle slice if $L \\subset \\partial R(L)$ is slice in $R(L)$, that is, if L is the boundary of a collection of locally flat pairwise disjoint embedded discs in $R(L)$.\n\n(2) The round handle problem was proposed by Freedman and Krushkal in [FK16, Section 5.1], and is presented in detail in [KPT21].\n\n(3) If the 4-dimensional topological surgery and s-cobordism conjectures hold for free groups, then every link with vanishing pairwise linking numbers is round handle slice. So this problem gives a way to potentially disprove the union of these conjectures, which are central open problems in 4-manifold topology. For more on those conjectures see Problems 4.46and 1.62.\n\nReferences cited:\n- [FK16] Michael Freedman and Vyacheslav Krushkal. Engel relations in 4-manifold topology. Forum Math. Sigma, 4:Paper No. e22, 57, 2016. doi:10.1017/fms.2016.20.\n- [KPT21] Min Hoon Kim, Mark Powell, and Peter Teichner. Round handle problem. Pure Appl. Math. Q., 17(1):237–247, 2021. doi:10.4310/PAMQ.2021.v17.n1.a6.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2924, "problem_number": "KP-4.48", "title": "Kirby Problem 4.48", "statement": "Let $\\Delta$ be a contractible, compact 4-manifold. Is the space of homeomorphisms of $\\Delta$ that fix the boundary pointwise, with the compact-open topology, a contractible space?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.48.\n\nLiterature notes:\nThis is true for $\\Delta = D^{4}$, by the Alexander trick. By PerronQuinn [Per86, Qui86], the space is connected. In dimensions at least 6, the space of homeomorphisms is contractible, as proven by Galatius–Randal-Williams [GRW24].\n\nReferences cited:\n- [Per86] B. Perron. Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique. Topology, 25(4):381–397, 1986. doi:10.1016/0040-9383(86)90018-2.\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n- [GRW24] Sø ren Galatius and Oscar Randal-Williams. The Alexander trick for homology spheres. Int. Math. Res. Not. IMRN, 2024(24):14689–14703, 2024. doi:10.1093/imrn/rnae255.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2925, "problem_number": "KP-4.49", "title": "Kirby Problem 4.49", "statement": "Give an effective necessary and sufficient condition for an open 4-manifold to be homeomorphic to the interior of a compact 4-manifold.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.49.\n\nLiterature notes:\n(1) In his thesis [Sie65], Siebenmann gave necessary and sufficient conditions for an n-dimensional open PL manifold M, for $n \\geq$ 6, to be compactifiable by adding a manifold boundary (see also [BLL65]). Recall that a manifold is said to be open if it has empty boundary and only noncompact components. The proof extends to the case where M is noncompact but has compact boundary. The case of possibly noncompact boundary is addressed in [O’B83, GG20]. The foundational results of Kirby-Siebenmann [KS77] on high-dimensional topological manifolds imply that Siebenmann’s proof can also be applied in the purely topological setting.\n\n(2) The cases $n =$ 2,3 are also well understood [HP70], [Tuc74], [GG20, Section 2].\n\n(3) Note that if an n-manifold M is the interior of an n-manifold M, then there is a neighborhood of the end of M of the for m $Y \\times [0,\\infty)$ for some (n−1)-manifold Y. In this case we say that M has a collared end. Having a collared end imposes several homotopy-theoretical conditions on M. The prior results mentioned above roughly consist of showing that some collection of such homotopy-theoretic conditions is sufficient to conclude the existence of a collared end.\n\n(4) An end of a manifold M is said to betame if there is a closed neighborhood U of the end, and a proper map $U \\times [0,\\infty) \\to M$ which is the inclusion on $U \\times$ \\{0\\}. Suppose M is a manifold with a tame connected end with finitely presented fundamental group $\\pi$. Siebenmann defined an invariant $\\sigma(end(M)) \\in K_{~,0}(\\mathbb{Z}\\pi)$, related to Wall’s finiteness obstruction [Wal65]. Siebenmann’s result from [Sie65] mentioned above states that if an nmanifold M, for $n \\geq$ 6, has a tame connected end with finitely presented fundamental group $\\pi$, then $\\sigma(end(M)) \\in K_{~,0}(\\mathbb{Z}\\pi)$ vanishes if and only if M has a collared end. This result was extended by Quinn to the case that n=5 and $\\pi$ is good (see Problem 4.46). We emphasize that in the results above $\\pi$ is the fundamental group of the end of M, and not of M itself. See [FQ90, Section 11.9] for further details.\n\n(5) The direct topological analogue of Siebenmann’s theorem in dimension four fails by a result of Kwasik-Schultz [KS88, Theorem 2.1] and unpublished work of Weinberger. These latter authors constructed a class of 4-manifolds M such that the infinite cyclic cover $M_{\\infty}$ has an end with two components, each of which is tame, has good fundamental group, and for which Siebenmann’s obstruction vanishes, but such that $M_{\\infty}$is not homeomorphic to $Y \\times \\mathbb{R}$, for any compact 3-manifold Y.\n\n(6) By the result mentioned above, a characterization of when an open 4manifold has collared end must be necessarily more complicated than it is in higher dimensions. An ideal characterization would be purely homotopy-theoretic. Partial results in dimension four in the topological setting are given in [FQ90, Section 11.9B], but there is not yet a candidate for a necessary and sufficient condition.\n\nReferences cited:\n- [Sie65] Laurence Carl Siebenmann. THE OBSTRUCTION TO FINDING A BOUNDARY FOR AN OPEN MANIFOLD OF DIMENSION GREATER THAN FIVE. ProQuest LLC, Ann Arbor, MI, 1965. Thesis (Ph.D.)–Princeton University. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt=info: ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri:pqdiss:6605012.\n- [BLL65] W. Browder, J. Levine, and G. R. Livesay. Finding a boundary for an open manifold. Amer. J. Math., 87:1017–1028, 1965. doi:10.2307/2373259.\n- [O’B83] Gary O’Brien. The missing boundary problem for smooth manifolds of dimension greater than or equal to six. Topology Appl., 16(3):303–324, 1983. doi:10.1016/0166-8641(83)90027-5.\n- [GG20] Shijie Gu and Craig R. Guilbault. Compactifications of manifolds with boundary. J. Topol. Anal., 12(4):1073–1101, 2020. doi:10.1142/$S^{1}$793525319500754.\n- [KS77] Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations, volume 88 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J., 1977. With notes by John Milnor and Michael Atiyah.\n- [HP70] L. S. Husch and T. M. Price. Finding a boundary for a 3-manifold. Ann. of Math. (2), 91:223–235, 1970. doi:10.2307/1970605.\n- [Tuc74] Thomas W. Tucker. Non-compact 3-manifolds and the missing-boundary problem. Topology, 13:267–273, 1974. doi:10.1016/0040-9383(74)90019-6.\n- [Wal65] C. T. C. Wall. Finiteness conditions for CW-complexes. Ann. of Math. (2), 81:56– 69, 1965. doi:10.2307/1970382.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [KS88] Slawomir Kwasik and Reinhard Schultz. Desuspension of group actions and the ribbon theorem. Topology, 27(4):443–457, 1988. doi:10.1016/0040-9383(88)90023-7.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2926, "problem_number": "KP-4.50", "title": "Kirby Problem 4.50", "statement": "Classify closed topological 4-manifolds (orientable and not) with finite fundamental group, up to homeomorphism. The following infinite families of fundamental groups are of particular interest.\n\n(a) Dihedral groups.\n\n(b) Quaternionic groups.\n\n(c) Abelian groups with two or three generators. A significant starting point would be to complete the classification for 4-manifolds with fundamental group of order at most eight.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.50.\n\nLiterature notes:\n(1) The problem is deemed solved if the classification is reduced to algebraic invariants. An interesting subproblem is to complete the classification for smooth 4-manifolds only.\n\n(2) The classification exists for trivial fundamental group by Freedman [Fre82], for cyclic fundamental groups in the orientable case by Hambleton-Kreck [HK88, HK93a, HK93c, HK93b], and for fundamental group $\\mathbb{Z}/2\\mathbb{Z}$ in the nonorientable case by Hambleton-Kreck-Teichner [HKT94]. See also [HH23].\n\n(3) The homotopy classification is known in some cases: [HK88, KPR24, KNR22, Hil23]. In these cases one strategy is therefore to upgrade this to a homeomorphism classification using surgery theory.\n\nReferences cited:\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [HK88] Ian Hambleton and Matthias Kreck. On the classification of topological 4-manifolds with finite fundamental group. Math. Ann., 280(1):85–104, 1988. doi:10.1007/BF01474183.\n- [HK93a] Ian Hambleton and Matthias Kreck. Cancellation, elliptic surfaces and the topology of certain four-manifolds. J. Reine Angew. Math., 444:79–100, 1993. doi:10.1515/crll.1993.444.79.\n- [HK93c] Ian Hambleton and Matthias Kreck. Cancellation of lattices and finite two-complexes. J. Reine Angew. Math., 442:91–109, 1993. doi:10.1515/crll.1993.442.91.\n- [HK93b] Ian Hambleton and Matthias Kreck. Cancellation of hyperbolic forms and topological four-manifolds. J. Reine Angew. Math., 443:21–47, 1993. doi:10.1515/crll.1993.443.21.\n- [HKT94] Ian Hambleton, Matthias Kreck, and Peter Teichner. Nonorientable 4-manifolds with fundamental group of order 2. Trans. Amer. Math. Soc., 344(2):649–665, 1994. doi:10.2307/2154500.\n- [HH23] Ian Hambleton and Jonathan A. Hillman. Quotients of $S^{2}$ $\\times$ $S^{2}$. J. Lond. Math. Soc. (2), 108(4):1393–1416, 2023. doi:10.1112/jlms.12783.\n- [KPR24] Daniel Kasprowski, Mark Powell, and Benjamin Ruppik. Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groups. Math. Proc. Cambridge Philos. Soc., 177(2):263–283, 2024.\n- [KNR22] Daniel Kasprowski, John Nicholson, and Benjamin Ruppik. Homotopy classification of 4-manifolds whose fundamental group is dihedral. Algebr. Geom. Topol., 22(6):2915–2949, 2022. doi:10.2140/agt.2022.22.2915.\n- [Hil23] Jonathan A. Hillman. Homotopy types of 4-manifolds with 3-manifold fundamental groups, 2023. arXiv:2307.15292.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2927, "problem_number": "KP-4.51", "title": "Kirby Problem 4.51", "statement": "Let X be a closed, smooth 4-manifold with fundamental group isomorphic to $\\mathbb{Z}$. Is the $\\mathbb{Z}[\\mathbb{Z}]-valued$ intersection form on $\\pi_{2}(X)$ extended from $\\mathbb{Z}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.51.\n\nLiterature notes:\n(1) We say that a Hermitian for m on a free $\\mathbb{Z}[\\mathbb{Z}]-module$ isextended from $\\mathbb{Z}if$ it is represented by a matrix, with respect to some basis over $\\mathbb{Z}[\\mathbb{Z}]$, with integer entries.\n\n(2) Hambleton–Teichner [HT97] exhibited a unimodular Hermitian for m over $\\mathbb{Z}[\\mathbb{Z}]$ that is not extended over $\\mathbb{Z}$.\n\n(3) It was known from Freedman–Quinn [FQ90] that this for m can be realized as the intersection form of closed topological manifold with fundamental group $\\mathbb{Z}. A$ manifold with such a for m is not even homotopy equivalent to the connected sum of a homotopy $S^{1} \\times S^{3}$ with a simply connected 4-manifold.\n\n(4) Conversely, if the intersection form of a 4-manifold with fundamental group $\\mathbb{Z}$ is extended from $\\mathbb{Z}$, then that 4-manifold is homeomorphic to a connected sum $M\\#(S^{1} \\times S^{3})$, where $\\pi_{1}(M) =$ \\{1\\}. Thus an affirmative answer to the question would imply that every closed, smooth, oriented 4-manifold with fundamental group $\\mathbb{Z}$ splits topologically as such a connected sum.\n\n(5) Friedl–Hambleton–Melvin–Teichner [FHMT07] showed, by an application of Donaldson’s theorem [Don83] to the finite cyclic covers (of degree at least 3) of the manifold in [HT97] that no 4-manifold with this for m can be smooth. Conjecture 1.3 in [FHMT07] states that the intersection form of any closed, smooth manifold with fundamental group $\\mathbb{Z}$ is extended from the integers.\n\n(6) Fintushel–Stern [FS94] showed that there are smooth 4-manifolds with fundamental group $\\mathbb{Z}$ that do not have an $S^{1} \\times S^{3}$ connect summand smoothly. In fact, any symplectic 4-manifold will not smoothly decompose as a connected sum in this fashion.\n\nReferences cited:\n- [HT97] Ian Hambleton and Peter Teichner. A non-extended Hermitian form over ZrZs. Manuscripta Math., 93(4):435–442, 1997. doi:10.1007/BF02677483.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [FHMT07] Stefan Friedl, Ian Hambleton, Paul Melvin, and Peter Teichner. Non-smoothable four-manifolds with infinite cyclic fundamental group. Int. Math. Res. Not. IMRN, 2007(11):Art. ID rnm031, 20, 2007. doi:10.1093/imrn/rnm031.\n- [Don83] S. K. Donaldson. An application of gauge theory to four-dimensional topology. J. Differential Geom., 18(2):279–315, 1983.\n- [FS94] Ronald Fintushel and Ronald J. Stern. A fake 4-manifold with $\\pi_1=\\mathbb{Z}$ and $b^+=4$. Turkish J. Math., 18(1):1–6, 1994.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2928, "problem_number": "KP-4.52", "title": "Kirby Problem 4.52", "statement": "Let M and N be closed, orientable topological 4-manifolds with $\\pi_{1}(M) \\cong \\pi_{1}(N)a$ good group. Suppose that M and N are simple homotopy equivalent and stably homeomorphic. Is there a simple homotopy equivalence $N \\to M$ that lies in the orbit of Id $:M \\to M$ under the Wall realization action of $L^{s}_{5}(\\mathbb{Z}[\\pi_{1}(M)])$ on the simple structure set of M?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.52.\n\nLiterature notes:\n(1) We can hope to classify 4-manifolds up to simple homotopy equivalence, and up to stable homeomorphism, for many groups. We can also hope to compute $L^{s}_{5}(\\mathbb{Z}[\\pi_{1}(M)])$. If the answer were yes, this would give a somewhat satisfying picture for the classification of 4-manifolds, for good groups. If the answer were no, this would uncover interesting new phenomena. The answer is yes for simply connected 4-manifolds, and for those with fundamental group $\\mathbb{Z}$ [FQ90] and $\\mathbb{Z}/n \\mathbb{Z}[HK88$, HK93a, HK93c, HK93b].\n\n(2) In the case of spin 4-manifolds, the program suggested by this question could potentially be made easier, only requiring the simple homotopy classification, if the answer to the following question is no.\n\n\\paragraph{Question.} Do there exist closed, spin 4-manifolds that are simple homotopy equivalent but not stably homeomorphic? The latter question was part of [Kir97, Problem 4.84], due to Teichner. Note that closed, spin 4-manifolds that are homotopy equivalent have the same Kirby–Siebenmann invariant, so that invariant cannot be used here.\n\n(3) Jim Davis [Dav05] proved that if the fundamental group is $\\pi$ and the map $\\kappa_{2}: H_{2}(\\pi;\\mathbb{Z}/2\\mathbb{Z}) \\to L_{4}(\\mathbb{Z}\\pi)is$ injective, then homotopy equivalent 4manifolds with that fundamental group and the same Kirby-Siebenmann invariant are stably homeomorphic.\n\n(4) There are counterexamples known for non-spin 4-manifolds, even assumin g they have the same Kirby-Siebenmann invariant [Tei97].\n\nReferences cited:\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [HK88] Ian Hambleton and Matthias Kreck. On the classification of topological 4-manifolds with finite fundamental group. Math. Ann., 280(1):85–104, 1988. doi:10.1007/BF01474183.\n- [HK93a] Ian Hambleton and Matthias Kreck. Cancellation, elliptic surfaces and the topology of certain four-manifolds. J. Reine Angew. Math., 444:79–100, 1993. doi:10.1515/crll.1993.444.79.\n- [HK93c] Ian Hambleton and Matthias Kreck. Cancellation of lattices and finite two-complexes. J. Reine Angew. Math., 442:91–109, 1993. doi:10.1515/crll.1993.442.91.\n- [HK93b] Ian Hambleton and Matthias Kreck. Cancellation of hyperbolic forms and topological four-manifolds. J. Reine Angew. Math., 443:21–47, 1993. doi:10.1515/crll.1993.443.21.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Dav05] James F. Davis. The Borel/Novikov conjectures and stable diffeomorphisms of 4-manifolds. In Geometry and topology of manifolds. Papers from the conference held at McMaster University, Hamilton, ON, USA, May 14–18, 2004, pages 63– 76. Providence, RI: American Mathematical Society (AMS), 2005.\n- [Tei97] Peter Teichner. On the star-construction for topological 4-manifolds. In Geometric topology. 1993 Georgia international topology conference, August 2–13, 1993, Athens, GA, USA, pages 300–312. Providence, RI: American Mathematical Society; Cambridge, MA: International Press, 1997.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2929, "problem_number": "KP-4.53", "title": "Kirby Problem 4.53", "statement": "Does there exist an algorithm that takes as input a closed, triangulated 4-manifold, and outputs in finite time whether or not that 4-manifold is homeomorphic to $S^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.53.\n\nLiterature notes:\n(1) This is called the recognition problem. We say that we can recognize an n-manifold X if there exists an algorithm that takes as input a closed, triangulated n-manifold, and outputs in finite time whether or not that 4-manifold is homeomorphic to X.\n\n(2) We assume that the input 4-manifolds are represented by the finite data of a triangulation, and hence they are smooth, since triangulated 4-manifolds are smooth. We are not assured that the input 4-manifold is simplyconnected.\n\n(3) The corresponding question has a positive answer in dimensions $\\leq$ 3 [Tho94], [Rub95], and a negative answer in dimensions $\\geq$ 5 [VKF74].\n\n(4) Markov [Mar58] showed that, for some integerk, the corresponding question for the connected sum of k copies of $S^{2} \\times S^{2}$ has a negative answer. This raises the question of the minimal k for which this holds. It was shown that k could be taken to be 14 in [Sht05], 12 in [Gor22], and 9 in [Tan23]. See also the exposition in [Kir20].\n\n(5) By Markov [Mar58], there exist infinite lists of group presentations ${P_{i}}$ such that there is no algorithm taking as input one of the $P_{i}$, and outputting in finite time whether or not the group $G(P_{i})$ presented by $P_{i}$ is trivial. One possible strategy to solve the problem is to construct a corresponding list ${X_{i}}$ of closed, triangulated 4-manifolds, whose 2-skeleta give rise to the $P_{i}$, so in particular $\\pi_{1}(X_{i}) \\cong G(P_{i})$, with the property that $X_{i}$ is homeomorphic to $S^{4}$ if and only if $G(P_{i}) =$ \\{1\\}. The forward direction clearly holds, but it is not clear how to find ${P_{i}}$ and ${X_{i}}such$ that the backwards direction holds. This strategy does work for $\\#_{k}(S^{2} \\times S^{2})$ in place of $S^{4}$, and was the basis for the proofs of [Sht05, Gor22, Tan23] for $k =$ 14,12,9 respectively.\n\n(6) One can also ask the analogous smooth recognition problem with ‘diffeomorphic’ in place of ‘homeomorphic’. This is also open for $S^{4}$. As described in [Kir20], there exists aksuch that $\\#_{k}(S^{2} \\times S^{2})is$ not smoothly recognizable, but there are no known upper bounds on the minimal k for which this holds.\n\nReferences cited:\n- [Tho94] Abigail Thompson. Thin position and the recognition problem for $S^{3}$. Math. Res. Lett., 1(5):613–630, 1994. doi:10.4310/MRL.1994.v1.n5.a9.\n- [Rub95] Joachim H. Rubinstein. An algorithm to recognize the 3-sphere. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 601–611. Birkhäuser, Basel, 1995.\n- [VKF74] I. A. Volodin, V. E. Kuznecov, and A. T. Fomenko. The problem of the algorithmic discrimination of the standard three-dimensional sphere. Uspehi Mat. Nauk, 29:71– 168, 1974. Appendix by S. P. Novikov.\n- [Mar58] A. Markov. The insolubility of the problem of homeomorphy. Dokl. Akad. Nauk SSSR, 121:218–220, 1958.\n- [Sht05] M. A. Shtan’ko. On Markov’s theorem on the algorithmic nonrecognizability of manifolds. Fundam. Prikl. Mat., 11:257–259, 2005. doi:10.1007/s10958-007-0375-z.\n- [Gor22] Cameron McA. Gordon. On the homeomorphism problem for 4-manifolds. New Zealand J. Math., 52:821–826, 2021 [2021–2022]. doi:10.53733/205.\n- [Tan23] Martin Tancer. Simpler algorithmically unrecognizable 4-manifolds, 2023. arXiv: 2310.07421.\n- [Kir20] R. C. Kirby. Markov’s theorem on the nonrecognizablility of 4-manifolds: an exposition, 2020. Celebratio Mathematica: Martin Scharlemann, https://celebratio.org/Scharlemann M/article/785/.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2930, "problem_number": "KP-4.54", "title": "Kirby Problem 4.54", "statement": "The quadratic 2 type of a 4-manifold M is the data $(\\pi_{1}(M), \\pi_{2}(M), \\lambda_{M}, k_{M})$ of the fundamental group $\\pi_{1}(M)$, the second homotopy group $\\pi_{2}(M)$ considered as $a \\mathbb{Z}[\\pi_{1}(M)]-module$, the equivariant intersection for $m \\lambda_{M}$, and the k-invariant in $k_{M} \\in H^{3}(B\\pi_{1}(M);\\pi_{2}(M))$. Which quadratic 2-types are realized by closed, oriented topological 4-manifolds? Which are realized by closed, oriented, smooth 4-manifolds? Can this problem be solved for specific fundamental groups, for example for certain families of non-cyclic finite groups?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.54.\n\nLiterature notes:\n(1) This is in [Kir97, Problem 4.1].\n\n(2) We can also ask the question with the data of the Stiefel-Whitney classes $w_{1}$ and $w_{2}$. Which Stiefel-Whitney classes are realized within a given quadratic 2-type?\n\n(3) The problem builds on the geography problem for simply connected topological 4-manifolds, where it restricts to the question of which intersection forms occur. In the smooth category, this is answered by Donaldson’s diagonalizability theorem along with a positive resolution of the 11/8conjecture.\n\n(4) For the topological case, Freedman [Fre82] showed that every nonsingular symmetric bilinear for m is realized by a closed simply connected 4-manifold, so the problem is solved when $\\pi_{1} =$ 1. A similar result is known for $\\pi_{1} =\\mathbb{Z}$ [FQ90]. For other good fundamental groups, a possible strategy in the topological category was introduced by Hambleton-Kreck in [HK88, Lemma 4.1], the paper where the quadratic 2-type initially arose. First, classify the quadratic 2-types that are stably realizable, meaning that they are realizable by topological 4-manifolds after taking the orthogonal sum with the quadratic 2-type of $S^{2} \\times S^{2}$. If a quadratic 2-type is stably realizable then it is realizable unstably by a topological 4-manifold, which can be shown using the sphere embedding theorem [FQ90, $BKK^{+}21]$. Hambleton-Kreck [HK88, HK93a] used this strategy to solve the realization problem for finite cyclic groups, in the topological category.\n\n(5) For non-simply-connected, smooth, oriented 4-manifolds, Donaldson’s diagonalization theorem holds without any assumption on the fundamental group, so definite integral intersection forms must be diagonalizable. The sphere embedding theorem cannot be applied in the smooth case, so the strategy described above of stably realizing and then destabilizing is not currently viable.\n\n(6) The existence problem has a closely related uniqueness analogue. For finite cyclic groups [HK88], abelian groups with at most two generators [KPR24], dihedral groups [KNR22], and aspherical 3-manifold groups [Hil23], we know that the quadratic 2-type determines the 4manifold up to homotopy equivalence. Hence if we could also understand the image of the invariants in one of these cases, i.e. the realization problem, we would have a fairly complete homotopy classification, at least modulo the algebraic problem of being able to reliably distinguish or identify given quadratic 2-types.\n\n(7) Kirk and Livingston’s paper [KL09] contains many related references and its own list of interesting related problems.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [HK88] Ian Hambleton and Matthias Kreck. On the classification of topological 4-manifolds with finite fundamental group. Math. Ann., 280(1):85–104, 1988. doi:10.1007/BF01474183.\n- [BKK+21] Stefan Behrens, Boldizsár Kalmár, Min Hoon Kim, Mark Powell, and Arunima Ray, editors. The disc embedding theorem. Oxford University Press, Oxford, 2021.\n- [HK93a] Ian Hambleton and Matthias Kreck. Cancellation, elliptic surfaces and the topology of certain four-manifolds. J. Reine Angew. Math., 444:79–100, 1993. doi:10.1515/crll.1993.444.79.\n- [KPR24] Daniel Kasprowski, Mark Powell, and Benjamin Ruppik. Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groups. Math. Proc. Cambridge Philos. Soc., 177(2):263–283, 2024.\n- [KNR22] Daniel Kasprowski, John Nicholson, and Benjamin Ruppik. Homotopy classification of 4-manifolds whose fundamental group is dihedral. Algebr. Geom. Topol., 22(6):2915–2949, 2022. doi:10.2140/agt.2022.22.2915.\n- [Hil23] Jonathan A. Hillman. Homotopy types of 4-manifolds with 3-manifold fundamental groups, 2023. arXiv:2307.15292.\n- [KL09] Paul Kirk and Charles Livingston. The geography problem for 4-manifolds with specified fundamental group. Trans. Amer. Math. Soc., 361(8):4091–4124, 2009. doi:10.1090/S0002-9947-09-04649-2.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2931, "problem_number": "KP-4.55", "title": "Kirby Problem 4.55", "statement": "Let M and N be closed, orientable, connected 4-manifolds with isomorphic quadratic 2-types. If $\\pi_{1}(M) \\cong \\pi_{1}(N)are$ finite, are M and N homotopy equivalent?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.55.\n\nLiterature notes:\n(1) This is known for $\\pi_{1}$ trivial, finite cyclic groups [HK88], dihedral groups [KPR24], and abelian groups with at most two generators [KNR22]. It is known to be false in the nonorientable case, due to Kim-KojimaRaymond [KKR92].\n\n(2) A potentially interesting case is $\\pi_{1}(M) = \\mathbb{Z}/2 \\times \\mathbb{Z}/2 \\times \\mathbb{Z}/2$. As noted in [KPR24], in this case we can have torsion in $\\mathbb{Z}\\otimes_{\\mathbb{Z}[\\pi_{1}(M)]}\\Gamma(\\pi_{2}(M))$, which by [HK88] leads to polarized homotopically inequivalent Poincaré 4-complexes with the same quadratic 2-type. Are they homotopy equivalent? Do the homotopy types contain topological 4-manifolds?\n\n(3) Let $L_{p,q}$ and $L_{p,q,1}$ be lens spaces that are not homotopy equivalent. Then $S^{1} \\times L_{p,q}$ and $S^{1} \\times L_{p,q,1}$ are also not homotopy equivalent but they do have isomorphic quadratic 2-type. So the problem is not true in general for infinite fundamental groups.\n\nReferences cited:\n- [HK88] Ian Hambleton and Matthias Kreck. On the classification of topological 4-manifolds with finite fundamental group. Math. Ann., 280(1):85–104, 1988. doi:10.1007/BF01474183.\n- [KPR24] Daniel Kasprowski, Mark Powell, and Benjamin Ruppik. Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groups. Math. Proc. Cambridge Philos. Soc., 177(2):263–283, 2024.\n- [KNR22] Daniel Kasprowski, John Nicholson, and Benjamin Ruppik. Homotopy classification of 4-manifolds whose fundamental group is dihedral. Algebr. Geom. Topol., 22(6):2915–2949, 2022. doi:10.2140/agt.2022.22.2915.\n- [KKR92] Myung Ho Kim, Sadayoshi Kojima, and Frank Raymond. Homotopy invariants of nonorientable 4-manifolds. Trans. Amer. Math. Soc., 333(1):71–81, 1992. doi: 10.2307/2154099.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2932, "problem_number": "KP-4.56", "title": "Kirby Problem 4.56", "statement": "(4D s-cobordism conjecture). Let $(W^{4};M_{0}^{3}, M_{1}^{3})$ be a smooth 4-dimensionals-cobordism between closed 3-manifolds. Is W diffeomorphic to $M_{0} \\times$ [0,1]?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.56.\n\nLiterature notes:\n(1) Matumoto and Siebenmann found counterexamples to the topological analogue in [MS78], where both $M_{0}$ and $M_{1}$ are $\\mathbb{RP}^{2} \\times S^{1}$ and W is not known to be smoothable. Specifically that paper showed that the scobordism theorem fails either in dimension four or five, providing specific s-cobordisms that would fail to be products. Later work of Freedman and Quinn [FQ90, Theorem 7.1A] showed that the potential 5-dimensional candidate is indeed a product. So the 4-dimensional candidate of Matumoto and Siebenmann must fail to be a product. Cappell and Shaneson later provided further topological counterexamples where $M_{0}$ and $M_{1}$ are orientable. In a subsequent paper [CS87b] they asserted these examples were smoothable, but this claim was later retracted [CS87a].\n\n(2) A weaker version of the question asks whether s-cobordant, or possibly even simple homotopy equivalent, 3-manifolds are necessarily homeomorphic. Kwasik and Schultz showed, assuming geometrization, that every topological h-cobordism between closed, orientable 3-manifolds is an s-cobordism, and that simple homotopy equivalence implies homeomorphism for closed, orientable 3-manifolds [KS92, Theorem and Theorem 1.1]. The questions appear to be open in the nonorientable setting; in particular geometrization is not yet known for nonorientable 3-manifolds. Whether there exists anh-cobordism between 3-manifolds with nontrivial Whitehead torsion appeared as Problem 4.9 on [Kir97].\n\n(3) The s-cobordism theorem for dimensions 6 and higher is true in the smooth, piecewise linear, and topological settings [Sma62a, Bar63,Maz63, Sta67, KS77] (see also [Mil65, RS72]). The s-cobordism theorem in dimension five is false in the smooth (and equivalently piecewise linear) settings, by work of Donaldson [Don87a], and known to be true in the topological setting for good fundamental groups [FQ90, Theorem 7.1A] (see also [OPR21a]). See Problem 4.46.\n\nReferences cited:\n- [MS78] T. Matumoto and L. Siebenmann. The topological s-cobordism theorem fails in dimension 4 or 5. Math. Proc. Cambridge Philos. Soc., 84(1):85–87, 1978. doi: 10.1017/S0305004100054918.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [CS87b] Sylvain E. Cappell and Julius L. Shaneson. Smooth nontrivial 4-dimensional scobordisms. Bull. Amer. Math. Soc. (N.S.), 17(1):141–143, 1987. doi:10.1090/S0273-0979-1987-15542-X.\n- [CS87a] Sylvain E. Cappell and Julius L. Shaneson. Corrigendum to: “Smooth nontrivial 4-dimensional s-cobordisms”. Bull. Amer. Math. Soc. (N.S.), 17(2):401, 1987. doi: 10.1090/S0273-0979-1987-15616-3.\n- [KS92] Slawomir Kwasik and Reinhard Schultz. Vanishing of Whitehead torsion in dimension four. Topology, 31(4):735–756, 1992. doi:10.1016/0040-9383(92)90005-3.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Sma62a] S. Smale. On the structure of manifolds. Amer. J. Math., 84:387–399, 1962. doi: 10.2307/2372978.\n- [Bar63] D. Barden. The structure of manifolds. PhD thesis, Cambridge University, 1963.\n- [Maz63] Barry Mazur. Relative neighborhoods and the theorems of Smale. Ann. of Math. (2), 77:232–249, 1963. doi:10.2307/1970215.\n- [Sta67] John R. Stallings. Lectures on polyhedral topology, volume 43 of Tata Institute of Fundamental Research Lectures on Mathematics. Tata Institute of Fundamental Research, Bombay, 1967. Notes by G. Ananda Swarup.\n- [KS77] Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations, volume 88 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J., 1977. With notes by John Milnor and Michael Atiyah.\n- [Mil65] J. W. Milnor. Lectures on the h-cobordism theorem. Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow.\n- [RS72] C. P. Rourke and B. J. Sanderson. Introduction to piecewise-linear topology. Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69.\n- [Don87a] S. K. Donaldson. Irrationality and the h-cobordism conjecture. J. Differential Geom., 26(1):141–168, 1987. http://projecteuclid.org/euclid.jdg/1214441179.\n- [OPR21a] Patrick Orson, Mark Powell, and Arunima Ray. The s-cobordism theorem, the sphere embedding theorem, and the Poincaré conjecture. In The disc embedding theorem, pages 283–293. Oxford Univ. Press, Oxford, 2021.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2933, "problem_number": "KP-4.57", "title": "Kirby Problem 4.57", "statement": "Let X and Y be closed, oriented, smooth 4-manifolds with the same Euler characteristic and signature. Is there a torus link L in X with trivial normal bundle such that some choice of torus surgery along L transforms X into a manifold diffeomorphic to Y? What if we only require that the result be homeomorphic to Y?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.57.\n\nLiterature notes:\n(1) These questions were discussed in [Ste06, BS13, FS11].\n\n(2) Let T be a torus in a 4-manifold W with trivial normal bundle. We say that a 4-manifold $W^{1}is$ obtained from W by torus surgery along T if $W^{1} = W \\setminus \\nu(T) \\cup (T^{2} \\times D^{2})for$ some choice of gluing map $\\partial\\nu(T) \\to \\partial(T^{2} \\times D^{2})$.\n\n(3) It is a theorem of Iwase [Iwa90] that there is a 4-manifold Z and links of tori $L_{X} \\subset X, L_{Y} \\subset Y$ so that every component of $L_{X}, L_{Y}$ has trivial normal bundle and there exists some choice of torus surgeries on $L_{X}, L_{Y}$ transforming X, Y into manifolds equivalent to Z. Here, if X, Y are smooth, then $L_{X}, L_{Y}$ may be taken to be smooth, and “equivalent” means “diffeomorphic.” Otherwise, surfaces are locally flat, and “equivalent” means “homeomorphic.”\n\n(4) In Iwase’s paper, he actually proves that X can be transformed into $\\#_{a}\\mathbb{CP}^{2}\\#_{b}\\overline{\\mathbb{CP}}{}^{2}\\#_{c}S^{1} \\times S^{3}$ by surgery on a torus link in X for sufficiently large a with $b = a-\\sigma(X), c =$ (a+b+2 $-\\chi(X))/2$. That is, we may take Z in the above discussion to be a connected sum of copies of $\\mathbb{CP}^{2}s, \\mathbb{CP}^{2}s$, and $S^{1} \\times S^{3} s$. Iwase’s argument holds in both categories. A generalization of this result, extended over to the nonorientable 4– manifolds, is announced in a recent preprint of Baykur and Morgan [BM25], which states that any closed smooth 4–manifold is obtained by a surgery along a link of tori in a Z that is a connected sum of copies of $S^{2} \\times \\mathbb{R}\\mathbb{P}^{2}, \\mathbb{R}\\mathbb{P}^{4}, \\mathbb{CP}^{2}s, \\mathbb{CP}^{2}s$, and $S^{1} \\times S^{3} s$.\n\n(5) The problem asks whether, instead of X and Y being related by asequence of two surgeries on torus links, all necessary torus surgeries transforming X into Y can be performed simultaneously.\n\n(6) The problem seeks the 4-dimensional analog of the fact that any two closed, oriented 3-manifolds M, N are related by Dehn surgery along a link, rather than a sequence of Dehn surgeries. In dimension three, these two facts are clearly equivalent by dimensionality, but since surfaces generically intersect in 4-manifolds the situation is different.\n\n(7) Fintushel–Stern [FS11] asked the following version of this question in the case the 4–manifolds are simply connected. Question ([FS11, §9]). Can a simply-connected closed, smooth 4manifold always be obtained from torus surgery on a link of tori in a connected sum of $\\mathbb{CP}^{2}, S^{2} \\times S^{2}$ and K3 summands, taken with either orientations? See [BS13] for an approach to this question via 5-dimensional round handles. Round 2-handle attachments correspond to certain torus surgeries, and the work of [BS13] falls short at the same point as discussed above; the authors build cobordisms made out of only round 2-handles, but it is not clear if there is always a cobordism where these round 2handles can be attached independently.\n\n(8) Fintushel–Stern noted that many interesting simply connected 4-manifolds arise from a single null-homologous torus surgery, e.g. for $n =$ 2, . . . ,7,9 there are infinitely many exotic $\\mathbb{CP}^{2}\\#_{n}\\mathbb{CP}^{2}$ that each arises from a single torus surgery on a null-homologous torus in $\\mathbb{CP}^{2}\\#_{n}\\mathbb{CP}^{2}$ [FS11, Theorem 6]. Question (Fintushel and Stern). If X and Y are homeomorphic, simply-connected, smooth 4-manifolds, is it possible to obtain Y from surgery on a single torus in X? If so, can we arrange for the torus to be null-homologous?\n\nReferences cited:\n- [Ste06] Ronald J. Stern. Will we ever classify simply-connected smooth 4-manifolds? In Floer homology, gauge theory, and low-dimensional topology, volume 5 of Clay Math. Proc., pages 225–239. Amer. Math. Soc., Providence, RI, 2006.\n- [BS13] R. İnanç Baykur and Nathan Sunukjian. Round handles, logarithmic transforms and smooth 4-manifolds. J. Topol., 6(1):49–63, 2013.\n- [FS11] Ronald Fintushel and Ronald J. Stern. Pinwheels and nullhomologous surgery on 4-manifolds with $b^+=1$. Algebr. Geom. Topol., 11(3):1649–1699, 2011. doi:10.2140/agt.2011.11.1649.\n- [Iwa90] Zjuñici Iwase. Dehn surgery along a torus T2-knot. II. Japan. J. Math. (N.S.), 16(2):171–196, 1990. doi:10.4099/math1924.16.171.\n- [BM25] R. İnanç Baykur and Porter Morgan. On nonorientable 4–manifolds, 2025. Math. Res. Lett., to appear. URL: https://arxiv.org/abs/2506.20950, arXiv:2506.20950.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2934, "problem_number": "KP-4.58", "title": "Kirby Problem 4.58", "statement": "(a) Which Seifert fibered homology spheres $\\Sigma(a_{1}$, . . . , $a_{n})bound$ acyclic manifolds? Are there any examples with four or more singular fibers that bound acyclic manifolds?\n\n(b) Which Seifert fibered homology spheres bound contractible manifolds? Is there an example that bounds an acyclic manifold but not a contractible manifold?\n\n(c) Are there any Seifert fibered homology spheres that arise as cork boundaries?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.58.\n\nLiterature notes:\n(1) The second part of Problem (a) (regarding the number of fibers) is in [Kir97, Problem 4.123] in a different guise, with a different motivation.\n\n(2) Many Seifert fibered homology spheres bound acyclic (and, in fact, contractible) manifolds (see for example [AK79b, CH81, Fic84, Şav20]) but a complete characterization is unknown. All known such examples have three singular fibers. Amongst spheres $\\Sigma(p, q$, r)with three singular fibers, there is no closed characterization of which bound acyclic manifolds in terms of p, q, and r.\n\n(3) It is a longstanding conjecture that no Seifert fibered homology sphere with four or more singular fibers bounds an acyclic manifold; see [FS87b, Kol08]. This is related to the Montgomery-Yang conjecture, which states that every pseudofree action of $S^{1}$ on $S^{5}$ has at most 3 non-free orbits. One can also ask the related question of which Seifert fibered homology spheres embed in $\\mathbb{R}^{4}$. In the setting of symplectic topology, it is known that no Seifert fibered homology sphere (of any number of fibers) occurs as a hypersurface of contact type $in(\\mathbb{R}^{4}, \\omega_{std})$ [MT22]. A recent preprint [AC24] builds on this work and announces that no (standardly-oriented) Seifert fibered homology sphere bounds a Stein rational ball.\n\n(4) One can also ask about the difference between bounding an acyclic manifold and bounding a contractible manifold. In general, these notions differ: Taubes’ periodic ends theorem implies that $\\Sigma(2,3,5)\\#-\\Sigma(2,3,5)$ bounds no contractible manifold [Tau87], whereas this trivially bounds an acyclic manifold. Many other examples can be obtained through instanto n Floer theory. However, no such example consisting of an individual Seifert fibered homology sphere is known.\n\n(5) A similar question is whether or not any Seifert fibered homology sphere Y forms a cork boundary. Here, recall that Y is a cork boundary if there exists a contractible manifold W with boundary Y, together with a selfdiffeomorphism of Y that does not extend over W (as a diffeomorphism). It is known that the standard cyclic group actions on any Brieskorn sphere $\\Sigma(p, q$, r) do not extend (smoothly) as group actions to any contractible manifold with boundary $\\Sigma(p, q$, r) [AH16, AH21]. However, these do extend as diffeomorphisms. Current Floer-theoretic techniques devoted to establishing corks [AKS20, DHM23] are known to fail for Seifert fibered homology spheres. Note that several authors take W to be Stein in the definition of a cork; the notion defined here is sometimes called a loose cork.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [AK79b] Selman Akbulut and Robion Kirby. Mazur manifolds. Michigan Math. J., 26(3):259–284, 1979. http://projecteuclid.org/euclid.mmj/1029002261.\n- [CH81] Andrew J. Casson and John L. Harer. Some homology lens spaces which bound rational homology balls. Pacific J. Math., 96(1):23–36, 1981. http://projecteuclid.org/euclid.pjm/1102734944.\n- [Fic84] Henry Clay Fickle. Knots, Z-homology 3-spheres and contractible 4-manifolds. Houston J. Math., 10(4):467–493, 1984.\n- [Şav20] Oğuz Şavk. More Brieskorn spheres bounding rational balls. Topology Appl., 286:107400, 10, 2020. doi:10.1016/j.topol.2020.107400.\n- [FS87b] Ronald Fintushel and Ronald J. Stern. $O(2)$ actions on the 5-sphere. Invent. Math., 87(3):457–476, 1987. doi:10.1007/BF01389237.\n- [Kol08] János Kollár. Is there a topological Bogomolov-Miyaoka-Yau inequality? Pure Appl. Math. Q., 4(2, Special Issue: In honor of Fedor Bogomolov. Part 1):203– 236, 2008. doi:10.4310/PAMQ.2008.v4.n2.a1.\n- [MT22] Thomas E. Mark and Bülent Tosun. On contact type hypersurfaces in 4-space. Invent. Math., 228(1):493–534, 2022. doi:10.1007/s00222-021-01083-9.\n- [AC24] Antonio Alfieri and Alberto Cavallo. Holomorphic curves in Stein domains and the tau-invariant, 2024. arXiv:2310.08657.\n- [Tau87] Clifford Henry Taubes. Gauge theory on asymptotically periodic 4-manifolds. J. Differential Geom., 25(3):363–430, 1987. http://projecteuclid.org/euclid.jdg/1214440981.\n- [AH16] Nima Anvari and Ian Hambleton. Cyclic group actions on contractible 4-manifolds. Geom. Topol., 20(2):1127–1155, 2016. doi:10.2140/gt.2016.20.1127.\n- [AH21] Nima Anvari and Ian Hambleton. Cyclic branched coverings of Brieskorn spheres bounding acyclic 4-manifolds. Glasg. Math. J., 63(2):400–413, 2021. doi:10.1017/S0017089520000269.\n- [AKS20] Antonio Alfieri, Sungkyung Kang, and András I. Stipsicz. Connected Floer homology of covering involutions. Math. Ann., 377(3-4):1427–1452, 2020. doi:10.1007/s00208-020-01992-9.\n- [DHM23] Irving Dai, Matthew Hedden, and Abhishek Mallick. Corks, involutions, and Heegaard Floer homology. J. Eur. Math. Soc. (JEMS), 25(6):2319–2389, 2023. doi:10.4171/jems/1239.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2935, "problem_number": "KP-4.59", "title": "Kirby Problem 4.59", "statement": "Are lens spaces topologically homology cobordant if and only if they are homeomorphic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.59.\n\nLiterature notes:\n(1) Livingston conjectures that the statement is correct.\n\n(2) If $L(a_{1}, b_{1})$ and $L(a_{2}, b_{2})$ are homology cobordant, then $a_{1} = a_{2}$. The problem can be restated: if L(n, $b_{1})$ and L(n, $b_{2})$ are homology cobordant, then L(n, $b_{1})$ and L(n, $b_{2})$ are homeomorphic. Gilmer and Livingston [GL83] proved this in the case that n is a prime power, using Atiyah–Singer signature invariants $\\rho_{\\alpha}(L)$ associated to characters $\\alpha: \\pi_{1}(L) \\to U(1)$ of prime-power order. The simplest unknown case is the pair they identified, $L(231,53)$ and $L(231,86)$. It remains unknown whether these lens spaces are homology cobordant.\n\n(3) In the smooth category, the conjecture was proved for neven by Fintushel– Stern [FS87a], with later generalizations by Matić [Mat88] and Ruberman [Rub88]. The conjecture in the smooth setting can also be proved using Heegaard Floer theory; see [DW15].\n\n(4) For higher dimensional lens spaces, Cappell and Ruberman [CR88] showed that $the\\rho_{\\alpha}-invariants for\\alpha of$ prime-power order give the homology cobordism classification. This uses homology surgery theory [CS74, Vog82] which is known [Akb79] to fail in dimension 4, even topologically. It is conceivable to try to construct a topological homology cobordism W between non-diffeomorphic L(n, $b_{1})$ and L(n, $b_{2}$ with $n$ composite usin g ordinary surgery theory. A first step might be to find an appropriate homotopical model for such a cobordism. The Gilmer–Livingston argument implies that the n-fold cyclic $W_{\\infty}$ would have to have nontrivial $b_{1}$, so that $\\pi_{1}(W)$ would have to be large in this sense; see [AGL18] for more information on this.\n\nReferences cited:\n- [GL83] Patrick M. Gilmer and Charles Livingston. On embedding 3-manifolds in 4-space. Topology, 22(3):241–252, 1983. doi:10.1016/0040-9383(83)90011-3.\n- [FS87a] Ronald Fintushel and Ronald Stern. Rational homology cobordisms of spherical space forms. Topology, 26(3):385–393, 1987. doi:10.1016/0040-9383(87)90008-5.\n- [Mat88] Gordana Matić. $\\mathrm{SO}(3)$-connections and rational homology cobordisms. J. Differential Geom., 28(2):277–307, 1988.\n- [Rub88] Daniel Ruberman. Rational homology cobordisms of rational space forms. Topology, 27(4):401–414, 1988. doi:10.1016/0040-9383(88)90020-1.\n- [DW15] Margaret Doig and Stephan Wehrli. A combinatorial proof of the homology cobordism classification of lens spaces, 2015. arXiv:1505.06970.\n- [CR88] Sylvain Cappell and Daniel Ruberman. Imbeddings and homology cobordisms of lens spaces. Comment. Math. Helv., 63(1):75–88, 1988. doi:10.1007/BF02566753.\n- [CS74] S.E. Cappell and J. Shaneson. Homology surgery and the codimension-two placement problem. Ann. of Math., 99:277–348, 1974.\n- [Vog82] Pierre Vogel. On the obstruction group in homology surgery. Publ. Math. Inst. Hautes Études Sci., 55:165–206, 1982. http://www.numdam.org/item?id= PMIHES 1982 55 165 0.\n- [Akb79] Selman Akbulut. A note on homology surgery and the Casson-Gordon invariant. Math. Proc. Cambridge Philos. Soc., 85(2):335–344, 1979. doi:10.1017/S0305004100055754.\n- [AGL18] Paolo Aceto, Marco Golla, and Ana G. Lecuona. Handle decompositions of rational homology balls and Casson-Gordon invariants. Proc. Amer. Math. Soc., 146(9):4059–4072, 2018. doi:10.1090/proc/14035.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2936, "problem_number": "KP-4.60", "title": "Kirby Problem 4.60", "statement": "(a) Let X be an open, spin, smooth4-manifold. Does X have a proper smooth embedding in $\\mathbb{R}^{6}$?\n\n(b) By choosing a proper exhaustion function on W,(a) would follow from an affirmative answer to the following. Let (W;M, N) be a compact, smooth spin cobordism with a smooth embedding f of M in $S^{5}$. Is there a smooth embedding F: (W;M, N) $\\to (S^{5} \\times I;S^{5} \\times$ \\{0\\}, $S^{5} \\times$ \\{1\\}) whose restriction to M coincides with f?\n\n(c) Let (W;M, N) be a compact, smooth spin cobordism with smooth embeddings f of M in $S^{5}$ and g of N in $S^{5}$ such that $\\sigma(N, S^{5}) -\\sigma(M, S^{5}) = \\sigma(W)$. Is there a smooth embedding F: (W;M, N) $\\to (S^{5} \\times I;S^{5} \\times$ \\{0\\}, $S^{5} \\times$ \\{1\\}) whose restriction to M coincides with f and whose restriction to N coincides with g?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.60.\n\nLiterature notes:\n(1) The spin condition is necessary in both parts of the problem.\n\n(2) One motivation for part (a)is to understand theembedding dimension for a Stein surface (of real dimension four). By definition, this is the minimal dfor which X has a proper holomorphic embedding in $\\mathbb{C}^{d}$. General results about embedding dimension due to Eliashberg and Gromov [EG92] imply that a Stein 4-manifold has a proper holomorphic embedding into $\\mathbb{C}^{4}$. Since a 4-manifold properly embedded in $\\mathbb{C}^{3} =\\mathbb{R}^{6}$ is spin, this is the best possible result for non-spin Stein 4-manifolds, but it is conceivable that one could get embeddings in $\\mathbb{C}^{3}$ for spin Stein 4-manifolds. Part (a) is a topological version of that question.\n\n(3) Part (b) asks for a relative version of the result, announced by CappellShaneson in [CS79] and proved by Ruberman in [Rub82], that a closed spin 4-manifold embeds in $\\mathbb{R}^{6}$ if and only if its signature is 0.\n\n(4) For part (c), note that an embedding of a 3-manifold M in $S^{5}$ has a welldefined signature $\\sigma(M, S^{5})$, given by the signature of any 4-manifold that M bounds in $S^{5}$. The equality $\\sigma(N, S^{5}) -\\sigma(M, S^{5}) =\\sigma(W)$ is necessary for W to be a cobordism as in (b). It follows, in the setting of part (b), that one cannot specify the embedding of both M and N in advance. It is not clear if there are further obstructions, so part (c) represents a sharpening of part (b).\n\nReferences cited:\n- [EG92] Yakov Eliashberg and Mikhael Gromov. Embeddings of Stein manifolds of dimension n into the affine space of dimension 3n\\{2+1. Ann. of Math. (2), 136(1):123–135, 1992. doi:10.2307/2946547.\n- [CS79] Sylvain E. Cappell and Julius L. Shaneson. Embeddings and immersions of fourdimensional manifolds in R6. In Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pages 301–303. Academic Press, New York-London, 1979.\n- [Rub82] Daniel Ruberman. Imbedding four-manifolds and slicing links. Math. Proc. Cambridge Philos. Soc., 91(1):107–110, 1982. doi:10.1017/S0305004100059168.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2937, "problem_number": "KP-4.61", "title": "Kirby Problem 4.61", "statement": "What do different 4-manifold gauge theories see?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.61.\n\nLiterature notes:\n(1) It is conjectured that the Donaldson, Seiberg–Witten, and Heegaard Floer invariants of closed 4-manifolds coincide (after organizing the Donaldson invariant as a generating function [KM95]). Yet each of these theories taken more broadly seem to see different geometric and topological properties. This problem asks how to understand some of these properties proved in one theory via one of the other theories.\n\n(2) Is it possible to prove the existence of uncountable many exotic structures on $\\mathbb{R}^{4}$ using Seiberg–Witten or Heegaard Floer theory? One might try to do this by adapting Taubes’ periodic end gauge theory [Tau87] to the Seiberg–Witten setting. This can be done but to date requires rather strong hypotheses such as positive scalar curvature on the periodic end. Another possibility is to use limit invariants of ends such as the Heegaard Floer end invariant introduced by Gadgil [Gad10].\n\n(3) Is there a Seiberg–Witten proof of the Donaldson–Sullivan [DS89] results that there are 4-manifolds without quasiconformal (and hence Lipschitz) structure, and that 4-manifolds can admit more than one such structures? A seemingly fundamental difficulty here is whether Lipschitz or quasiconformal manifolds have something like a Dirac operator; see the discussion in [Sul87, Sul99] and also Problem 4.131.\n\n(4) Donaldson and Seiberg-Witten theory have parameterized versions that can be used to study invariants of diffeomorphisms and families of 4manifolds as well as families of symplectic structures [Rub98, BK22, Kon21, Kro97]. Are there family versions of Heegaard Floer theory that would be useful for such applications?\n\n(5) Seiberg-Witten invariants can be extended to give the Bauer–Furuta invariant [BF04] living in (equivariant) stable homotopy groups. Are there similar stable homotopy theoretic invariants coming from Donaldson or Heegaard Floer theory?\n\n(6) Seiberg–Witten theory can be used to show that certain 4-manifolds admit no Riemannian metric of positive scalar curvature (PSC) [Wit94] and to distinguish path components in the space of PSC metrics [Rub01]. Find a way to do this using Donaldson or Heegaard Floer theory.\n\n(7) All three theories [Frø02, Frø96, OS03a] give statements about the definite intersection forms of 4-manifolds with given boundary. Are these statements equivalent?\n\n(8) Hambleton and Lee [HL95] used an equivariant version of Donaldson’s original argument for his definite manifolds theorem to study smooth group actions on a simply connected (positive) definite 4-manifold. Among other results, they showed that a homologically trivial cyclic group action has the same fixed-point data and tangential isotropy representations as an equivariant connected sum of linear actions on $\\mathbb{CP}^{2}$. Are there Seiberg– Witten or Heegaard Floer proofs of their results?\n\nReferences cited:\n- [KM95] P.B. Kronheimer and T.S. Mrowka. Embedded surfaces and the structure of Donaldson’s polynomial invariants. J. Diff. Geo., 41:573–734, 1995.\n- [Tau87] Clifford Henry Taubes. Gauge theory on asymptotically periodic 4-manifolds. J. Differential Geom., 25(3):363–430, 1987. http://projecteuclid.org/euclid.jdg/1214440981.\n- [Gad10] Siddhartha Gadgil. Open manifolds, Ozsváth-Szabó invariants and exotic $\\mathbb{R}^{4}$’s. Expo. Math., 28(3):254–261, 2010. doi:10.1016/j.exmath.2009.09.002.\n- [DS89] SK Donaldson and DP Sullivan. Quasiconformal 4-manifolds. Acta Mathematica, 163:181–252, 1989.\n- [Sul87] Dennis Sullivan. Quasiconformal homeomorphisms in dynamics, topology, and geometry. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pages 1216–1228. Amer. Math. Soc., Providence, RI, 1987.\n- [Sul99] Dennis Sullivan. On the foundation of geometry, analysis, and the differentiable structure for manifolds. In Topics in low-dimensional topology (University Park, PA, 1996), pages 89–92. World Sci. Publ., River Edge, NJ, 1999. doi:10.1142/4202.\n- [Rub98] Daniel Ruberman. An obstruction to smooth isotopy in dimension 4. Math. Res. Lett., 5(6):743–758, 1998. doi:10.4310/MRL.1998.v5.n6.a5.\n- [BK22] David Baraglia and Hokuto Konno. On the Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds. J. Topol., 15(2):505–586, 2022. doi:10.1112/topo.12229.\n- [Kon21] Hokuto Konno. Characteristic classes via 4-dimensional gauge theory. Geom. Topol., 25(2):711–773, 2021. doi:10.2140/gt.2021.25.711.\n- [Kro97] P.B. Kronheimer. Some non-trivial families of symplectic structures. Preprint, available from www.math.harvard.edu/„kronheim/diffsymp.pdf, 1997.\n- [BF04] Stefan Bauer and Mikio Furuta. A stable cohomotopy refinement of Seiberg-Witten invariants. I. Invent. Math., 155(1):1–19, 2004. doi:10.1007/s00222-003-0288-5.\n- [Wit94] Edward Witten. Monopoles and four-manifolds. Math. Res. Lett., 1(6):769–796, 1994. doi:10.4310/MRL.1994.v1.n6.a13.\n- [Rub01] Daniel Ruberman. Positive scalar curvature, diffeomorphisms and the SeibergWitten invariants. Geom. Topol., 5:895–924, 2001. doi:10.2140/gt.2001.5.895.\n- [Frø02] Kim A. Frøyshov. Equivariant aspects of Yang-Mills Floer theory. Topology, 41(3):525–552, 2002. doi:10.1016/S0040-9383(01)00018-0.\n- [Frø96] Kim A. Frøyshov. The Seiberg-Witten equations and four-manifolds with boundary. Math. Res. Lett., 3(3):373–390, 1996. doi:10.4310/MRL.1996.v3.n3.a7.\n- [OS03a] Peter Ozsváth and Zoltán Szabó. Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary. Adv. Math., 173(2):179–261, 2003. doi:10.1016/S0001-8708(02)00030-0.\n- [HL95] Ian Hambleton and Ronnie Lee. Smooth group actions on definite 4-manifolds and moduli spaces. Duke Math. J., 78(3):715–732, 1995. doi:10.1215/S0012-7094-95-07826-0.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2938, "problem_number": "KP-4.62", "title": "Kirby Problem 4.62", "statement": "Let X be a smooth, closed, connected, oriented 4-manifold with $b^{+}_{2}(X)$ >1.\n\n(a) Does X have Donaldson simple type?\n\n(b) Does X have Seiberg–Witten simple type?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.62.\n\nLiterature notes:\n(1) The Simple Type Conjecture holds that the answer is affirmative. Part (a) is in [Kir97, Problem 4.131], and was raised by Kronheimer and Mrowka in [KM95] for simply connected 4–manifolds.\n\n(2) A 4-manifold is said to have Donaldson simple type if the Donaldson polynomials $q_{k}$ for principal SU(2)-bundles with $c_{2}=k$ satisfy $q_{k,+,1}(\\nu,\\Sigma_{1}$, . . . $,\\Sigma_{d}) =4q_{k}(\\Sigma_{1}$, . . . $,\\Sigma_{d})$ where $\\nu = \\mu(1) (\\mu: H_{0}(X;\\mathbb{Z}) \\to H_{4}(M_{k,+,1})), \\Sigma_{i} \\in H_{2}(X;\\mathbb{Z})$, and 2d $=$ dim $M_{k}$.\n\n(3) Manifolds which have Donaldson simple type [KM95] include:\n\n\\noindent$\\bullet$ complete intersections,\n\n\\noindent$\\bullet$ elliptic surfaces,\n\n\\noindent$\\bullet$ any manifold with a Gompf nucleus,\n\n\\noindent$\\bullet$ manifolds with a smoothly embedded surface F satisfying 2(genus(F))− 2 =F $\\cdot F$ >0.\n\n(4) The Kronheimer–Mrowka structure theorem from [KM95] says that, for manifolds of simple type, the Donaldson invariants are determined by finitely many basic classes $K_{1}$, . . . , $K_{s}$ and rational numbers $\\beta_{1}$, . . . , $\\beta_{s}$. When $b^{+}_{2}$ =1, some manifolds, e.g. $\\mathbb{CP}^{2}, S^{2} \\times S^{2}, \\mathbb{CP}^{2}\\#\\mathbb{CP}^{2}$, do not have Donaldson simple type.\n\n(5) A Seiberg–Wittenbasic class $\\mathfrak{s}is a spin^{c}structure\\mathfrak{s}with$ nonzero SeibergWitten invariant $SW_{X}(\\mathfrak{s}). A$ 4-manifold has Seiberg-Witten simple type if the virtual dimension of the Seiberg-Witten moduli space $d_{X}(\\mathfrak{s}) = 1(c_{1}(\\mathfrak{s})^{2}-2\\chi(\\mathfrak{s}) -3\\sigma(X))$ is zero for every basic class $\\mathfrak{s}$. Note that, $whend_{X}(\\mathfrak{s})$ =0, the Seiberg–Witten invariants are defined by a signed count of monopoles. When $d_{X}(\\mathfrak{s}) >$ 0, they are defined by evaluating a higher degree cohomology class on the moduli space of monopoles; the conjecture says that, in such cases, the evaluation is always zero.\n\n(6) We list some important progress.\n\n\\noindent$\\bullet$ Taubes’ equivalence between the Seiberg–Witten invariants and the Gromov-Taubes invariants implies that all symplectic 4-manifolds are Seiberg–Witten simple type [Tau96].\n\n\\noindent$\\bullet$ Kato-Nakamura-Yasui proved the conjecture for the mod 2 SeibergWitten invariants under a mild condition on the homology ring [KNY22].\n\n\\noindent$\\bullet$ Baraglia [Bar23a] proved that the mod 2 Seiberg–Witten simple type conjecture holds for spin structures without any extra assumptions. Just as in the Donaldson case, there are manifolds with $b^{+}_{2}$ =1 that are not of simple type, because of the wall-crossing formula.\n\n(7) Witten’s conjecture [Wit94] states that if X has Seiberg–Witten simple type, then it also has Donaldson simple type, and there is a precise relation between its Donaldson and Seiberg–Witten invariants. The conjecture was proved in many cases by Feehan and Leness; see [FL15], [FL18].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [KM95] P.B. Kronheimer and T.S. Mrowka. Embedded surfaces and the structure of Donaldson’s polynomial invariants. J. Diff. Geo., 41:573–734, 1995.\n- [Tau96] Clifford H. Taubes. SW ñ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves. J. Amer. Math. Soc., 9(3):845–918, 1996. doi:10.1090/S0894-0347-96-00211-1.\n- [KNY22] Tsuyoshi Kato, Nobuhiro Nakamura, and Kouichi Yasui. The simple type conjecture for mod 2 Seiberg–Witten invariants. Journal of the European Mathematical Society, 11 2022. doi:10.4171/JEMS/1297.\n- [Bar23a] David Baraglia. The mod 2 Seiberg-Witten invariants of spin structures and spin families, 2023. arXiv:2303.06883.\n- [Wit94] Edward Witten. Monopoles and four-manifolds. Math. Res. Lett., 1(6):769–796, 1994. doi:10.4310/MRL.1994.v1.n6.a13.\n- [FL15] Paul M. N. Feehan and Thomas G. Leness. Witten’s conjecture for many fourmanifolds of simple type. J. Eur. Math. Soc. (JEMS), 17(4):899–923, 2015. doi: 10.4171/JEMS/521.\n- [FL18] Paul M. N. Feehan and Thomas G. Leness. An $\\mathrm{SO}(3)$-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants. Mem. Amer. Math. Soc., 256(1226):xiv+234, 2018. doi:10.1090/memo/1226.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2939, "problem_number": "KP-4.63", "title": "Kirby Problem 4.63", "statement": "How many independent basic classes can a simply connected smooth 4-manifold X have, as measured $bybr(X)$, the rank of the span of the basic classes?\n\n(a) Is there an upper bound $forbr(X)$ in terms of topological invariants of X?\n\n(b) In particular, is $br(X) \\leq b^{+}_{2}(X)$ for all simply connected X with $b^{+}_{2}(X)$ odd?\n\n(c) Is there a smooth indefinite, simply connected 4-manifold X with $b_{2}(X) \\geq$ 3 for which the image of $ev_{*}: \\operatorname{Diff}(X) \\to \\operatorname{Aut}(Q_{X})$ is finite, or even trivial?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.63.\n\nLiterature notes:\n(1) Here we define $br(X)=\\operatorname{Rank}\\operatorname{Span}\\{c_{1}(\\mathfrak{s})\\mid \\mathfrak{s}\\text{ is a Seiberg-Witten basic class on }X\\}$ if X has any basic classes, and 0 otherwise.\n\n(2) Knot surgery [FS98] onndisjoint and homologically independent tori in a manifold with nontrivial Seiberg-Witten invariant would create a manifold X for which $br(X) = n$. For example, starting with an elliptic surface, the construction in [GM93] produces examples of manifolds X for which $br(X) =b^{+}_{2}(X)$.\n\n(3) Questions(a)and(b)are relevant to the study of the map $ev_{*}: \\operatorname{Diff}(X) \\to \\operatorname{Aut}(Q_{X})$ giving the action of a diffeomorphism on the intersection form. Since any diffeomorphism must permute the basic classes up to sign, the presence of many basic classes can restrict the size of the image of $ev_{*}$.\n\n(4) The restriction to indefinite manifolds $andb_{2}$ >2 is to rule out intersection forms with finite automorphism groups. If $br(X) =b_{2}(X)$, then $ev_{*}(\\operatorname{Diff})$ is contained in a finite permutation group and hence is finite.\n\nReferences cited:\n- [FS98] Ronald Fintushel and Ronald J. Stern. Knots, links, and 4-manifolds. Invent. Math., 134(2):363–400, 1998. doi:10.1007/s002220050268.\n- [GM93] R. Gompf and T. Mrowka. Irreducible 4-manifolds need not be complex. Ann. of Math., 138:61–111, 1993.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2940, "problem_number": "KP-4.64", "title": "Kirby Problem 4.64", "statement": "Find an irreducible, closed, smooth 4-manifold with nontrivial Bauer–Furuta invariant but with trivial Seiberg–Witten invariant.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.64.\n\nLiterature notes:\nThe Bauer–Furuta invariant $\\Psi$ [BF04] is a stable cohomotopy refinement of the Seiberg–Witten invariant [Wit94], building on Furuta’s proof of the 10/8-theorem [Fur01]. There are examples for which the Bauer–Furuta invariant is strictly stronger than the Seiberg–Witten invariant; for instance, $\\Psi$ can be used to distinguish between certain connected sums of homotopy K3 surfaces (which have vanishing Seiberg–Witten invariant). Such examples rely on a gluing formula due to Bauer [Bau04]; no irreducible examples are known.\n\nReferences cited:\n- [BF04] Stefan Bauer and Mikio Furuta. A stable cohomotopy refinement of Seiberg-Witten invariants. I. Invent. Math., 155(1):1–19, 2004. doi:10.1007/s00222-003-0288-5.\n- [Wit94] Edward Witten. Monopoles and four-manifolds. Math. Res. Lett., 1(6):769–796, 1994. doi:10.4310/MRL.1994.v1.n6.a13.\n- [Fur01] M. Furuta. Monopole equation and the 11 8 -conjecture. Math. Res. Lett., 8(3):279– 291, 2001. doi:10.4310/MRL.2001.v8.n3.a5.\n- [Bau04] Stefan Bauer. A stable cohomotopy refinement of Seiberg-Witten invariants. II. Invent. Math., 155(1):21–40, 2004. doi:10.1007/s00222-003-0289-4.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2941, "problem_number": "KP-4.65", "title": "Kirby Problem 4.65", "statement": "Suppose X is a smooth4-manifold with the homology of $S^{1} \\times S^{3}$ whose infinite cyclic cover $\\widetilde{X}$ has $H^{1}(\\widetilde{X})=0$. Furuta and Ohta [FO93] define an invariant $\\lambda_{FO}(X)$ as 1/4 of the signed count of irreducible flat $SU(2)$ connections on X; this requires an orientation of X and a specified generator of $H^{1}(X)$.\n\n(a) Does the following hold? The invariant $\\lambda_{FO}(X)$ is an integer, and $\\lambda_{FO}(X)\\equiv \\rho(Y,\\mathfrak{s})$, where $\\rho(Y,\\mathfrak{s})$ is the Rokhlin invariant of an oriented spin 3 manifold Y that is Poincaré dual to the generator of $H^{1}(X)$.\n\n(b) Mrowka–Ruberman–Saveliev [MRS11] give an approach, defining an invariant $\\lambda_{SW}(X)$ by counting solutions to the Seiberg-Witten equations and adding an index-theoretic correction term. Is $\\lambda_{FO}(X)=-\\lambda_{SW}(X)$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.65.\n\nLiterature notes:\n(1) Part (a) was conjectured by Furuta–Ohta.\n\n(2) A solution to (a) would imply that the Wall group $L_{5}(\\mathbb{Z}[\\mathbb{Z}])$ does not act on the smooth structure set of $S^{1} \\times S^{3}$; compare the discussion in\n\nProblem 4.22. By construction, $\\lambda_{SW}(X)$ is an integer, and it is shown in [MRS11] that it reduces mod 2 to $\\rho(Y,\\mathfrak{s})$. So a positive answer to (b) implies a positive answer to (a).\n\n(3) The invariant $\\lambda_{FO}(X)$ is defined in greater generality; one could require only that X is a homology $S^{1} \\times S^{3}$ whose twisted cohomology $H^{1}(X;\\mathbb{C}_{\\alpha})$ vanishes for any homomorphism $\\alpha: \\pi_{1}(X) \\to U(1)$. In this setting, neither part of (a) holds, but (b) is still plausible. The paper [LRS21] shows that (b) holds for mapping tori of all orientation preserving diffeomorphisms of homology spheres generating a semifree finite cyclic group action. For involutions, the result also follows from [LRS23b].\n\nReferences cited:\n- [FO93] Mikio Furuta and Hiroshi Ohta. Differentiable structures on punctured 4-manifolds. Topology Appl., 51(3):291–301, 1993. doi:10.1016/0166-8641(93)90083-P.\n- [MRS11] Tomasz Mrowka, Daniel Ruberman, and Nikolai Saveliev. Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin’s invariant. J. Differential Geom., 88:333–377, 2011. http://projecteuclid.org/euclid.jdg/1320067650.\n- [LRS21] Jianfeng Lin, Daniel Ruberman, and Nikolai Saveliev. On the monopole Lefschetz number of finite-order diffeomorphisms. Geom. Topol., 25(7):3591–3628, 2021. doi: 10.2140/gt.2021.25.3591.\n- [LRS23b] Jianfeng Lin, Daniel Ruberman, and Nikolai Saveliev. On the Frøyshov invariant and monopole Lefschetz number. Journal of Differential Geometry, 123(3):523 – 593, 2023. doi:10.4310/jdg/1683307008.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2942, "problem_number": "KP-4.66", "title": "Kirby Problem 4.66", "statement": "Can the skein lasagna module detect exotic smooth structures on closed 4-manifolds?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.66.\n\nLiterature notes:\n(1) The skein lasagna module is an extension of Khovanov homology. It is an invariant of 4-manifolds with (possibly empty) boundary and a framed link in their boundary. It was defined by Morrison, Walker and Wedrich in [MWW22]. Ren and Willis [RW24] gave examples of exotic compact 4-manifolds with boundary that are detected by the skein lasagna module. For closed 4-manifolds, the computations so far are limited; see [MN22], [MWW23], [RW24]. The invariant is nonvanishing for $S^{4}, S^{1} \\times S^{3}$ and $\\mathbb{CP}^{2}$. On the other hand, it vanishes for manifolds that contain a smoothly embedded sphere with positive self-intersection (cf. Theorem 1.3 in [RW24]); e.g. for $\\mathbb{CP}^{2}$ or −K3.\n\n(2) The invariant is multiplicative under connected sums, and it takes the value 0 on $S^{2} \\times S^{2}$. Thus, it has a chance of detecting exotic smooth structures on simply connected, closed 4-manifolds, even though such structures become standard after sufficiently many stabilizations.\n\nReferences cited:\n- [MWW22] Scott Morrison, Kevin Walker, and Paul Wedrich. Invariants of 4-manifolds from Khovanov-Rozansky link homology. Geom. Topol., 26(8):3367–3420, 2022. doi:10.2140/gt.2022.26.3367.\n- [RW24] Qiuyu Ren and Michael Willis. Khovanov homology and exotic 4-manifolds, 2024. arXiv:2402.10452.\n- [MN22] Ciprian Manolescu and Ikshu Neithalath. Skein lasagna modules for 2-handlebodies. J. Reine Angew. Math., 788:37–76, 2022. doi:10.1515/crelle-2022-0021.\n- [MWW23] Ciprian Manolescu, Kevin Walker, and Paul Wedrich. Skein lasagna modules and handle decompositions. Adv. Math., 425:Paper No. 109071, 40, 2023. doi:10.1016/j.aim.2023.109071.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2943, "problem_number": "KP-4.67", "title": "Kirby Problem 4.67", "statement": "(a) Compute $\\pi_{0}(\\operatorname{Diff}^{+}(S^{4}))$. Do we have $\\pi_{0}(\\operatorname{Diff}^{+}(S^{4})) =$ \\{1\\}?\n\n(b) In particular, does some implantation of the barbell map provide a nontrivial element in $\\pi_{0}(\\operatorname{Diff}^{+}(S^{4}))$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.67.\n\nLiterature notes:\n(1) The 4-dimensional generalized Smale conjecture asked whether the inclusion $SO(5) \\hookrightarrow \\operatorname{Diff}^{+}(S^{4})$ is a homotopy equivalence (see Problems 4.34 and 4.126 in [Kir97]). Watanabe [Wat19] disproved this conjecture, by proving that $\\pi_{k}(Diff_{\\partial}(D^{4}))$ is nontrivial for manyk, includingk=1,4,8. A fundamental issue remaining to understand is $\\pi_{0}(\\operatorname{Diff}^{+}(S^{4}))$, the orientation-preserving mapping class group of $S^{4}$, and it is presently unknown whether or not this group is trivial. Some sources of possibly nontrivial diffeomorphisms of $S^{4}$ are suggested in [Gay25], [BG19] and $[GGH^{+}23]$. The Pin(2)-equivariant family Bauer-Furuta invariant could potentially detect such nontrivial diffeomorphisms; see [LM25].\n\n\\begin{center}\n\\kthreefiginclude{ch4_fig2.png}{width=0.92\\linewidth}\n\\par\\small\\textbf{Figure 2.} Point push map on a surface.\n\\end{center}\n\n\\begin{center}\n\\kthreefiginclude{ch4_fig3.png}{width=0.70\\linewidth}\n\\par\\small\\textbf{Figure 3.} Barbell.\n\\end{center}\n\n(2) The barbell map is defined by Budney and Gabai in [BG19] (see also [BG25] for the ‘X-resolution’) using isotopy extension, via a generalization of Birman’s ‘push map’ [Bir69]. We give an exposition here. Birman’s map can be described by pushing a pointp(see Figure 2(a)) along a path $\\alpha$ in a closed surface S and back to p. This isotopy of the surface can be assumed to fix a disk D centered at p at the end of the isotopy. Removing the interior of D, we get a diffeomorphism of the punctured surface that is the identity $on\\partial D$ and also the identity outside a nice neighborhood of the $arc\\alpha$. Figure2(b) shows what happens to the loop $\\beta$ under this diffeomorphism. In general, when pushing a 0-dimensional pointpalong a 1-dimensional loop $\\alpha$ in a 2-dimensional surface, the point crosses a loop (say $\\beta)$ and bulldozes it over D. Consider a pair of 2-spheres in $\\mathbb{R}^{3}$ centered at( $\\pm$ 2,0,0)of radius one, which are joined by the arc in the x-axis [−1,1]. We can use the z-axis to define the equator $(x^{2}+y^{2}=1)$, latitudes, and the two poles. Thicken this slightly in $\\mathbb{R}^{3}$, to what might be called a (hollow) barbell. Now cross with [−3,3] to get a four-dimensional analogue called B. We will find an interesting diffeomorphism $\\beta$ of B that is the identity on $\\partial B$. Notice the two line segments ( $\\pm$ 2,0,0) $\\times$ [−3,3]. These can be thickened so that they fill in the “hollows”, the pair of $S^{2} \\times$ [−1,1]s, so that we have a solid 4-dimensional barbell, which is obviously $B^{4}$. Figure 3 may help. Now focus just on the left line segment and move a portion of it, namely (−2,0,0) $\\times$ [−1,1], around the 2-sphere on the right side, sort of like lassoing a horse’s head. We will do this with a 1-parameter family of\n\n\\begin{center}\n\\kthreefiginclude{ch4_fig4.png}{width=0.52\\linewidth}\n\\par\\small\\textbf{Figure 4.} Embedding of barbell.\n\\end{center}\n\nBirman push maps, parameterized by $t\\in[-3,3]$. The push maps occur on a 2-dimensional surface given by fixing t and z. The push map for $t =$ 0 will push the point $p =$ (−2,0,0) $\\times$ 0 over to and then around the equator (z $=$ 0) of the 2-sphere at $t =$ 0 and back to p; for $t \\in$ (−1,1), pgoes over to and around the latitude at z =t and back to p; for $t = \\pm$ 1, p goes over to a pole and then back to p; for $t \\in$ (−2,−1) $\\cup$ (1,2), pgoes partway to a pole and then back to p; finally, for $t \\in$ (−3,−2) $\\cup$ (2,3), p does not move. This isotopy of the left line segment extends to an isotopy of $B^{4}that$ is the identity $on\\partial B^{4}$ and on the two line segments. But if this isotopy fixes the two line segments, then it can also be made to fix the above thickenings of the segments. Therefore, the end of this isotopy is a diffeomorphism $\\beta$ taking the 4-dimensional hollow barbell B back to itself, and fixing its boundary. Note that the lasso can go over the horse’s head with two possible orientations. Also there is a rotation $\\rho$ switching the two balls in the barbell. It can be checked that $\\rho\\beta\\rho^{-1} =\\beta^{-1}$. The 4-dimensional barbell,B, can be embedded, by f, in a 4-manifold X in many ways, and these are called implantations and the induced barbell map can be called $f_{*}\\beta$.\n\n(3) A specific example of a barbell diffeomorphism to consider is as follows.\n\n\\paragraph{Question.} Does the following embedding f induce a nontrivial diffeomorphism of $S^{4}$, where the embedding is given by mapping the two 2spheres to separate 2-spheres in $S^{4}$, and the bar goes from the left sphere over to link the right sphere and then back again to link the first and finally attaching to the right sphere? See Figure 4. A positive answer has been announced by Gabai–Gay–Hartman [GGH25].\n\n(4) We can also ask the following structural question.\n\n\\paragraph{Question.} Is $\\operatorname{Diff}^{+}(S^{4})$ (finitely) generated up to isotopy by a composition of $f_{*}\\beta s$?\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Wat19] Tadayuki Watanabe. Some exotic nontrivial elements of the rational homotopy groups of Diffp$S^{4}$q, 2019. arXiv:1812.02448.\n- [Gay25] David T. Gay. Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins. Algebr. Geom. Topol., 25(5):2817–2849, 2025. doi:10.2140/agt.2025.25.2817.\n- [BG19] Ryan Budney and David Gabai. Knotted 3-balls in $S^{4}$, 2019. arXiv:1912.09029.\n- [GGH+23] David Gabai, David T. Gay, Daniel Hartman, Vyacheslav Krushkal, and Mark Powell. Pseudo-isotopies of simply connected 4-manifolds, 2023. arXiv:2311.11196.\n- [LM25] Jianfeng Lin and Anubhav Mukherjee. Family Bauer-Furuta invariant, exotic surfaces and Smale conjecture. J. Assoc. Math. Res., 3(2):237–275, 2025. doi: 10.56994/JAMR.003.002.003.\n- [BG25] Ryan Budney and David Gabai. On the automorphism groups of hyperbolic manifolds, 2025. doi:10.1093/imrn/rnaf083.\n- [Bir69] Joan S Birman. Mapping class groups and their relationship to braid groups. Communications on Pure and Applied Mathematics, 22(2):213–238, 1969.\n- [GGH25] David Gabai, David T. Gay, and Daniel Hartman. Pseudo-Isotopy and Diffeomorphisms of the 4-Sphere I: Loops of Spheres, 2025. arXiv:2505.12088.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2944, "problem_number": "KP-4.68", "title": "Kirby Problem 4.68", "statement": "Does every closed smooth 4-manifold admit an exotic diffeomorphism? How about the following special cases?\n\n(a) Is there a definite smooth closed 4-manifold that admits an exotic diffeomorphism?\n\n(b) Does $S^{2} \\times S^{2}$ or K3 admit an exotic diffeomorphism?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.68.\n\nLiterature notes:\n(1) A self-diffeomorphism f: $X \\to X$ of a smooth manifold X is called exotic if it is topologically but not smoothly isotopic to the identity. The first examples of exotic diffeomorphisms of 4-manifolds were given by Ruberman [Rub98]. Most known examples of closed 4-manifolds confirmed to admit exotic diffeomorphisms are of the for $m \\#_{m}\\mathbb{C}\\mathbb{P}^{2}\\#_{n}\\mathbb{C}\\mathbb{P}^{2}$ for m, n>0 [Rub98], $\\#_{m}K3\\#_{n}S^{2} \\times S^{2}$ for m, n>0 [BK20] or m=0 [AR25], and K3\\#K3 [KM20]. There exist irreducible 4-manifolds that admit exotic diffeomorphisms [BK24a], but the proof in [BK24a] does not apply to $S^{4}, S^{2} \\times S^{2}, \\mathbb{C}\\mathbb{P}^{2}$, or K3.\n\n(2) In the literature, the smallest (in term of second Betti number) closed 4manifold that is known to admit an exotic diffeomorphism is $\\#_{2}\\mathbb{C}\\mathbb{P}^{2}\\#_{10}\\mathbb{C}\\mathbb{P}^{2}$, announced in [Qiu24]. It is natural to ask how small a closed 4-manifold with an exotic diffeomorphism can be. In particular, whether such a diffeomorphism exists on $S^{4}$ is the $\\pi_{0}$ case of the Smale conjecture; see\n\nProblem 4.67. For 4-manifolds with boundary, there exists an example of a contractible (hence, definite) 4-manifold that has an exotic diffeomorphism (relative to the boundary) [KMT23a, KMPW24,KPT26,KLMME24].\n\n(3) For a simply connected closed smooth 4-manifold X, every exotic diffeomorphism of X is smoothly isotopic to the identity after sufficiently many stabilizations by $S^{2} \\times S^{2}$. This fact follows by combining work of Kreck [Kre79] and either Quinn [Qui86] (cf. [GGH+23]) or Gabai [Gab22]. It is an interesting question to determine how many stabilizations are needed to kill the exotic property of a given diffeomorphism. Many known examples of exotic diffeomorphisms, such as those in [Rub98, BK20], are smoothly isotopic to the identity after only one stabilization [AKMR15]. On the other hand, Lin [Lin23] proved that the exotic diffeomorphism of K3\\#K3 from [KM20] stays exotic after one stabilization. There is no known upper bound for how many stabilizations are needed to trivialize this diffeomorphism. See also Problem 4.79.\n\nReferences cited:\n- [Rub98] Daniel Ruberman. An obstruction to smooth isotopy in dimension 4. Math. Res. Lett., 5(6):743–758, 1998. doi:10.4310/MRL.1998.v5.n6.a5.\n- [BK20] David Baraglia and Hokuto Konno. A gluing formula for families Seiberg-Witten invariants. Geom. Topol., 24(3):1381–1456, 2020. doi:10.2140/gt.2020.24.1381.\n- [AR25] Dave Auckly and Daniel Ruberman. Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory, 2025. arXiv: 2501.11892.\n- [KM20] P. B. Kronheimer and T. S. Mrowka. The Dehn twist on a sum of two K3 surfaces. Math. Res. Lett., 27(6):1767–1783, 2020. doi:10.4310/MRL.2020.v27.n6.a8.\n- [BK24a] David Baraglia and Hokuto Konno. Irreducible 4-manifolds can admit exotic diffeomorphisms, 2024. arXiv:2412.14398.\n- [Qiu24] Haochen Qiu. Surgery formulas for Seiberg-Witten invariants and family SeibergWitten invariants, 2024. arXiv:2411.10392.\n- [KMT23a] Hokuto Konno, Abhishek Mallick, and Masaki Taniguchi. Exotic Dehn twists on 4-manifolds, 2023. arXiv:2306.08607.\n- [KMPW24] Vyacheslav Krushkal, Anubhav Mukherjee, Mark Powell, and Terrin Warren. Corks for exotic diffeomorphisms, 2024. arXiv:2407.04696.\n- [KPT26] Sungkyung Kang, JungHwan Park, and Masaki Taniguchi. Exotic Dehn twists and homotopy coherent group actions. Invent. Math., 243(1):209–241, 2026. doi:10.1007/s00222-025-01378-1.\n- [KLMME24] Hokuto Konno, Jianfeng Lin, Anubhav Mukherjee, and Juan Muñoz-Echániz. On four-dimensional Dehn twists and Milnor fibrations, 2024. arXiv:2409.11961.\n- [Kre79] M. Kreck. Isotopy classes of diffeomorphisms of $(k-1)$-connected almostparallelizable 2k-manifolds. In Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), volume 763 of Lecture Notes in Math., pages 643– 663. Springer, Berlin, 1979.\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n- [GGH+23] David Gabai, David T. Gay, Daniel Hartman, Vyacheslav Krushkal, and Mark Powell. Pseudo-isotopies of simply connected 4-manifolds, 2023. arXiv:2311.11196.\n- [Gab22] David Gabai. 3-spheres in the 4-sphere and pseudo-isotopies of $S^{1}$ $\\times$ $S^{3}$, 2022. arXiv:2212.02004.\n- [AKMR15] Dave Auckly, Hee Jung Kim, Paul Melvin, and Daniel Ruberman. Stable isotopy in four dimensions. J. Lond. Math. Soc. (2), 91(2):439–463, 2015. doi:10.1112/jlms/jdu075.\n- [Lin23] Jianfeng Lin. Isotopy of the Dehn twist on K3 \\# K3 after a single stabilization. Geom. Topol., 27(5):1987–2012, 2023. doi:10.2140/gt.2023.27.1987.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2945, "problem_number": "KP-4.69", "title": "Kirby Problem 4.69", "statement": "Does there exist a diffeomorphism of a closed3-manifoldf: $M \\to M$ such that f is topologically but not smoothly pseudo-isotopic to the identity?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.69.\n\nLiterature notes:\n(1) Friedman–Witt [FW86] constructed diffeomorphisms f that are homotopic but not isotopic to the identity. These are also topologically pseudoisotopic to the identity [KS96].\n\n\\paragraph{Question.} Are the Friedman–Witt diffeomorphisms smoothly pseudoisotopic to the identity?\n\n(2) The following observation yields a potentially useful reformulation. Consider the embedding $i_{1,/,2}: M \\to M \\times$ \\{1/2\\} $\\hookrightarrow M \\times$ [0,1] where the first map is the canonical identification. Then f is smoothly (topologically) pseudo-isotopic to the identity if and only if $i_{1,/,2} \\circ f$ and $i_{1,/,2}$ are smoothly (topologically) isotopic as embeddings. The proof of this observation, which we give next, was provided separately by Hatcher and Igusa. It applies in all dimensions. If f were pseudo-isotopic to the identity via F: $M \\times I \\to M \\times I$, then $i_{1,/,2} \\circ f$ and $i_{1,/,2}$ would isotopic as embeddings. To see this, note that by translation it suffices to show that $i_{0}$ and $i_{1} \\circ f$ are isotopic. But $g_{t}: M \\to M$ defined by $g_{t}(x)$ =F(x, t) gives such an isotopy. Conversely, if $i_{1,/,2}$ and $i_{1,/,2} \\circ f$ are isotopic as embeddings, then apply isotopy extension rel.M $\\times$ \\{0,1\\}to the isotopy, to obtain a diffeomorphism of $M \\times$ [0,1/2] that restricts to the identity on $M \\times$ \\{0\\} and to f on $M \\times$ \\{1/2\\}(and similarly in $M \\times$ [1/2,1]). Thus after rescaling we obtain a pseudo-isotopy from f to the identity.\n\nReferences cited:\n- [FW86] John L. Friedman and Donald M. Witt. Homotopy is not isotopy for homeomorphisms of 3-manifolds. Topology, 25(1):35–44, 1986. doi:10.1016/0040-9383(86) 90003-0.\n- [KS96] Slawomir Kwasik and Reinhard Schultz. Pseudo-isotopies of 3-manifolds. Topology, 35(2):363–376, 1996. doi:10.1016/0040-9383(95)00017-8.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2946, "problem_number": "KP-4.70", "title": "Kirby Problem 4.70", "statement": "Do there exist $k \\geq$ 0 and a smooth closed 4-manifold X such that the map $\\pi_{k}(\\operatorname{Diff}(X)) \\to \\pi_{k}(\\operatorname{Homeo}(X))$ induced by the inclusion $\\operatorname{Diff}(X) \\hookrightarrow \\operatorname{Homeo}(X)$ is an isomorphism?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.70.\n\nLiterature notes:\n(1) Lin–Xie [LX23] proved that, for every orientable compact smooth 4manifold X, at least one of the following holds:\n\n\\noindent$\\bullet$ $\\pi_{1}(\\operatorname{Diff}(X)) \\to \\pi_{1}(\\operatorname{Homeo}(X))is$ not injective.\n\n\\noindent$\\bullet$ $\\pi_{2}(\\operatorname{Diff}(X)) \\to \\pi_{2}(\\operatorname{Homeo}(X))is$ not surjective. They prove also that, if $\\partial X \\ne \\emptyset$ or the signature of X is non-zero, then there are many degrees k for which $\\pi_{k}(\\operatorname{Diff}(X)) \\to \\pi_{k}(\\operatorname{Homeo}(X))is$ not an isomorphism. However, it is still possible that for some 4-manifold X and some degree k, the map $\\pi_{k}(\\operatorname{Diff}(X)) \\to \\pi_{k}(\\operatorname{Homeo}(X))is$ an isomorphism. The above result of Lin–Xie is based on Watanabe’s work [Wat19] for $X = S^{4}$. Watanabe proved that nontrivial elements in the kernel of $\\pi_{k}(\\operatorname{Diff}(S^{4})) \\to \\pi_{k}(\\operatorname{Homeo}(S^{4}))$ exist whenever $\\mathcal{A}_{k,+,1} \\ne$ 0. Here $\\mathcal{A}_{k,+,1}$ is the degree (k+1)-part of a specific graph cohomology.\n\n(2) Many other negative results are obtained by (mainly family) gauge theory, such as [AR25, FM88, Don90, MS97, Rub98, BK20, KKN21b, BK22, Bar21, BK23, KN23, KM20, Lin23, KT22b, IKMT25, KMT23a, GL25].\n\n(3) As a positive result, a classical theorem by Wall [Wal64a] shows that there are many 4-manifolds X for which the map $\\pi_{0}(\\operatorname{Diff}(X)) \\to \\pi_{0}(\\operatorname{Homeo}(X))$ is surjective. However, there also exist examples where this map is not surjective; see [FM88, Don87b].\n\nReferences cited:\n- [LX23] Jianfeng Lin and Yi Xie. Configuration space integrals and formal smooth structures, 2023. arXiv:2310.14156.\n- [Wat19] Tadayuki Watanabe. Some exotic nontrivial elements of the rational homotopy groups of Diffp$S^{4}$q, 2019. arXiv:1812.02448.\n- [AR25] Dave Auckly and Daniel Ruberman. Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory, 2025. arXiv: 2501.11892.\n- [FM88] Robert Friedman and John W. Morgan. On the diffeomorphism types of certain algebraic surfaces. I. J. Differential Geom., 27(2):297–369, 1988. http://projecteuclid.org/euclid.jdg/1214441784.\n- [Don90] S. K. Donaldson. Polynomial invariants for smooth four-manifolds. Topology, 29(3):257–315, 1990. doi:10.1016/0040-9383(90)90001-Z.\n- [MS97] John W. Morgan and Zoltán Szabó. Homotopy K3 surfaces and mod 2 SeibergWitten invariants. Math. Res. Lett., 4(1):17–21, 1997. doi:10.4310/MRL.1997.v4.n1.a2.\n- [Rub98] Daniel Ruberman. An obstruction to smooth isotopy in dimension 4. Math. Res. Lett., 5(6):743–758, 1998. doi:10.4310/MRL.1998.v5.n6.a5.\n- [BK20] David Baraglia and Hokuto Konno. A gluing formula for families Seiberg-Witten invariants. Geom. Topol., 24(3):1381–1456, 2020. doi:10.2140/gt.2020.24.1381.\n- [KKN21b] Tsuyoshi Kato, Hokuto Konno, and Nobuhiro Nakamura. Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds. Compos. Math., 157(4):770–808, 2021. doi:10.1112/s0010437x2000771x.\n- [BK22] David Baraglia and Hokuto Konno. On the Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds. J. Topol., 15(2):505–586, 2022. doi:10.1112/topo.12229.\n- [Bar21] David Baraglia. Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants. Algebr. Geom. Topol., 21(1):317–349, 2021. doi:10.2140/agt.2021.21.317.\n- [BK23] David Baraglia and Hokuto Konno. A note on the Nielsen realization problem for K3 surfaces. Proc. Amer. Math. Soc., 151(9):4079–4087, 2023. doi:10.1090/proc/15544.\n- [KN23] Hokuto Konno and Nobuhiro Nakamura. Constraints on families of smooth 4-manifolds from $\\mathrm{Pin}(2)$-monopole. Algebr. Geom. Topol., 23(1):419–438, 2023. doi:10.2140/agt.2023.23.419.\n- [KM20] P. B. Kronheimer and T. S. Mrowka. The Dehn twist on a sum of two K3 surfaces. Math. Res. Lett., 27(6):1767–1783, 2020. doi:10.4310/MRL.2020.v27.n6.a8.\n- [Lin23] Jianfeng Lin. Isotopy of the Dehn twist on K3 \\# K3 after a single stabilization. Geom. Topol., 27(5):1987–2012, 2023. doi:10.2140/gt.2023.27.1987.\n- [KT22b] Hokuto Konno and Masaki Taniguchi. The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary. Adv. Math., 409:Paper No. 108627, 58, 2022. doi:10.1016/j.aim.2022.108627.\n- [IKMT25] Nobuo Iida, Hokuto Konno, Anubhav Mukherjee, and Masaki Taniguchi. Diffeomorphisms of 4-manifolds with boundary and exotic embeddings. Math. Ann., 391(2):1845–1897, 2025. doi:10.1007/s00208-024-02974-x.\n- [KMT23a] Hokuto Konno, Abhishek Mallick, and Masaki Taniguchi. Exotic Dehn twists on 4-manifolds, 2023. arXiv:2306.08607.\n- [GL25] Daniel Galvin and Roberto Ladu. Non-smoothable homeomorphisms of 4-manifolds with boundary. Adv. Math., 467:Paper No. 110191, 22, 2025. doi:10.1016/j.aim.2025.110191.\n- [Wal64a] C. T. C. Wall. Diffeomorphisms of 4-manifolds. J. London Math. Soc., 39:131–140, 1964. doi:10.1112/jlms/s1-39.1.131.\n- [Don87b] S. K. Donaldson. The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Differential Geom., 26(3):397–428, 1987. http://projecteuclid.org/euclid.jdg/1214441485.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2947, "problem_number": "KP-4.71", "title": "Kirby Problem 4.71", "statement": "(a) Do there exist $k \\geq$ 0 and a smooth closed orientable 4-manifold X such that $\\pi_{k}(\\operatorname{Diff}(X))$ is finitely generated?\n\n(b) Do there exist $k >$ 0 and a smooth closed orientable 4-manifold X such that $H_{k}(BDiff(X);\\mathbb{Z})$ is finitely generated?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.71.\n\nLiterature notes:\n(1) For each $k >$ 0, there exist simply-connected closed smooth 4-manifolds X where $H_{k}(BDiff(X);\\mathbb{Z})$ are not finitely generated [Kon24b]. Also, Auckly–Ruberman [AR25] proved that, for eachk >0, there exist simplyconnected closed smooth 4-manifolds X where $\\pi_{k}(\\operatorname{Diff}(X))$ is not finitely generated. In dimension $\\ne$ 4, there are several finiteness results. See [Kup19b, BKK24]. See also Problem 4.72 for the analogous question in the topological category.\n\n(2) If X is oriented, we can ask an analogous question for $\\operatorname{Diff}^{+}(X)$, the orientation-preserving diffeomorphism group, in place of $\\operatorname{Diff}(X)$. (Note that $\\operatorname{Diff}(X) = \\operatorname{Diff}^{+}(X)$ if X has non-zero signature.) For homotopy groups $\\pi_{k}(\\operatorname{Diff}(X))$ and $\\pi_{k}(\\operatorname{Diff}^{+}(X))$, finite generation of $\\pi_{k}(\\operatorname{Diff}(X))$ and that of $\\pi_{k}(\\operatorname{Diff}^{+}(X))$ are equivalent. However, for $H_{k}(BDiff(X);\\mathbb{Z})$ and $H_{k}(BDiff^{+}(X);\\mathbb{Z})$, the questions may not be equivalent. For example, when $\\operatorname{Diff}(X) \\ne \\operatorname{Diff}^{+}(X)$, and if we take rational coefficients, the covering map $BDiff^{+}(X)\\to BDiff(X)$ BDiff(X)induces an isomorphism $H_{k}(BDiff^{+}(X);\\mathbb{Q})^{\\mathbb{Z}/2} \\cong H_{k}(BDiff(X);\\mathbb{Q})$, where the superscript $\\mathbb{Z}/2$ indicates the monodromy invariant part. Thus finite generation of $H_{k}(BDiff^{+}(X);\\mathbb{Q})$ implies that of $H_{k}(BDiff(X);\\mathbb{Q})$, but the converse may not be true in general.\n\n(3) As sets, the homotopy and homology groups of diffeomorphism groups of compact manifolds are always countable.\n\n(4) The diffeomorphism groups of non-compact 4-manifolds can have finitely generated homotopy and homology groups, e.g. $\\operatorname{Diff}(\\mathbb{R}^{4})$, in the weak $C^{\\infty}topology$, is homotopy equivalent to $O(4)$. Nevertheless, there also exist exotic $\\mathbb{R}^{4}’s$ with infinitely generated mapping class groups [Gom18].\n\n(5) The problem is open even for $X =S^{4}$ or $D^{4}$.\n\nReferences cited:\n- [Kon24b] Hokuto Konno. The homology of moduli spaces of 4-manifolds may be infinitely generated. Forum Math. Pi, 12:Paper No. e25, 18, 2024. doi:10.1017/fmp.2024.26.\n- [AR25] Dave Auckly and Daniel Ruberman. Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory, 2025. arXiv: 2501.11892.\n- [Kup19b] Alexander Kupers. Some finiteness results for groups of automorphisms of manifolds. Geom. Topol., 23(5):2277–2333, 2019. doi:10.2140/gt.2019.23.2277.\n- [BKK24] Mauricio Bustamante, Manuel Krannich, and Alexander Kupers. Finiteness properties of automorphism spaces of manifolds with finite fundamental group. Math. Ann., 388(4):3321–3371, 2024. doi:10.1007/s00208-023-02594-x.\n- [Gom18] Robert E. Gompf. Group actions, corks and exotic smoothings of $\\mathbb{R}^{4}$. Invent. Math., 214(3):1131–1168, 2018. doi:10.1007/s00222-018-0819-8.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2948, "problem_number": "KP-4.72", "title": "Kirby Problem 4.72", "statement": "Let X be a closed orientable topological 4-manifold with finite $\\pi_{1}(X)$.\n\n(a) Is $\\pi_{k}(\\operatorname{Homeo}(X))$ finitely generated for every $k \\geq$ 0?\n\n(b) Is $H_{k}(BHomeo(X);\\mathbb{Z})$ finitely generated for every $k \\geq$ 0?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.72.\n\nLiterature notes:\n(1) For $\\pi_{1}(X)$ =1, it follows from a result by Perron [Per86] and Quinn [Qui86] (cf. $[GGH^{+}23])$ that $\\pi_{0}(\\operatorname{Homeo}(X))$ is finitely generated, which implies that $H_{1}(BHomeo(X);\\mathbb{Z})$ is also finitely generated. There is no known finiteness result on $\\pi_{k}(\\operatorname{Homeo}(X))$ for $k \\geq$ 1 and $H_{k}(BHomeo(X);\\mathbb{Z})$ for $k \\geq$ 2. See Problem 4.71 for the analogous question in the smooth category. The following question is closely related.\n\n\\paragraph{Question.} Does $Top(4)$ have finitely generated homotopy groups in each degree? Indeed, $Top(4)$ has finitely-generated homotopy groups if and only if $\\operatorname{Homeo}(S^{4})$ has finitely-generated homotopy groups, because we have a fibration $Top(4) \\to \\operatorname{Homeo}(S^{4}) \\to S^{4}$, and the homotopy groups of $S^{4}$ are finitely generated. The question on the homotopy groups of Top(4)is itself closely related to Problem 4.73 on the Morlet correspondence in dimension 4.\n\n(2) Here is another closely related question, on topological embedding spaces. Let $\\Sigma$ be a compact surface, and consider a compact 4-manifold X. Fix a locally flat $embedding\\iota: \\partial\\Sigma \\hookrightarrow \\partial X$. Let $\\operatorname{Emb}^{t}_{\\partial}(\\Sigma$, X)be the space of locally flat embeddings of $\\Sigma$ extending $\\iota$. This is defined as the geometric realization of a semi-simplicial set, where the p-simplices are locally flat embeddings $\\Sigma \\times \\Delta^{p} \\to X \\times \\Delta^{p}$ over the projection to $\\Delta^{p}$, and extending $\\iota \\times$ Id : $\\partial\\Sigma \\times \\Delta^{p} \\to X \\times \\Delta^{p}. Letf_{0}: \\Sigma \\hookrightarrow X$ be a 0-simplex.\n\n\\paragraph{Question.} Suppose that $\\pi_{1}(X \\setminus f_{0}(\\Sigma))$ is finite, and fix $k >$ 0. Is $\\pi_{k}(\\operatorname{Emb}^{t}_{\\partial}(\\Sigma$, X), $f_{0})$ finitely generated? Randal-Williams [RW] has announced that the answer to the analogous question is no, in a case where the fundamental group of the surface complement is infinite. Using [BG19, BG25], Randal-Williams deduced that $\\pi_{4}(\\operatorname{Emb}^{t}(S^{2}, S^{4})$, U) is infinitely generated, where U is the trivial 2-knot. The homotopy groups spaces of embeddings are closely related to the homotopy groups of the homeomorphism groups of both the ambient space and of the exterior of the basepoint embedding. The space of thickenings of a fixed embedding to a closed tubular neighborhood plays an important rôle as well. As demonstrated by Randal-Williams’ note, information about any of these characters often leads to information about the others.\n\nReferences cited:\n- [Per86] B. Perron. Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique. Topology, 25(4):381–397, 1986. doi:10.1016/0040-9383(86)90018-2.\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n- [GGH+23] David Gabai, David T. Gay, Daniel Hartman, Vyacheslav Krushkal, and Mark Powell. Pseudo-isotopies of simply connected 4-manifolds, 2023. arXiv:2311.11196.\n- [RW] Oscar Randal-Williams. Topological embeddings of $S^{2}$ in $S^{4}$. https://www.dpmms.cam.ac.uk/„or257/notes/Embeddings.pdf.\n- [BG19] Ryan Budney and David Gabai. Knotted 3-balls in $S^{4}$, 2019. arXiv:1912.09029.\n- [BG25] Ryan Budney and David Gabai. On the automorphism groups of hyperbolic manifolds, 2025. doi:10.1093/imrn/rnaf083.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2949, "problem_number": "KP-4.73", "title": "Kirby Problem 4.73", "statement": "Does the Morlet correspondence $BDiff_{\\partial}(D^{n}) \\cong \\Omega^{n}_{0}(Top(n)/O(n))$ (10) hold for n=4?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.73.\n\nLiterature notes:\n(1) Here $BDiff_{\\partial}(D^{n})denotes$ the classifying space of the diffeomorphism group of $D^{n}$ relative to its boundary. We write $Top(n)$ for the group of homeomorphisms on $\\mathbb{R}^{n}$ that fix the origin, and Top(n)/O(n) for the homotopy fiber of $BO(n) \\to BTop(n)$. Finally let $\\Omega^{n}_{0}(Top(n)/O(n))$ denote the unit component of the loop space $\\Omega^{n}(Top(n)/O(n))$.\n\n(2) In dimension $n \\ne$ 4, the weak equivalence (10) follows from smoothing theory [BL74, KS77]. It is known that smoothing theory fails in dimension 4. For example, the manifold $E_{8}\\#E_{8}$ has a formal smooth structure (i.e. a vector bundle structure on its tangent microbundle) but has no smooth structure. However, (10) may still hold for n=4.\n\n(3) Watanabe [Wat19] disproved the 4-dimensional Smale conjecture by showin g that $\\pi_{k}(BDiff_{\\partial}(D^{4})) \\otimes \\mathbb{Q} \\ne$ 0 for many values of k including 2,5,9. It is known that the group $\\pi_{k,+,4}(Top(n)/O(n)) \\otimes \\mathbb{Q}$ is also nonvanishing for these values of k [LX23].\n\n(4) Gauge theory can distinguish non-diffeomorphic smooth structures on a closed 4-manifold that are isomorphic as formal smooth structures. So gauge theory could potentially be used to disprove (10).\n\n(5) The question is closely related to Problem 4.67. If there is a diffeomorphism of $D^{4}$ not isotopic to the identity, and this is detected using gauge theory, then this could show that the Morlet correspondence does not hold.\n\nReferences cited:\n- [BL74] Dan Burghelea and Richard Lashof. The homotopy type of the space of diffeomorphisms. I, II. Trans. Amer. Math. Soc., 196:1–36; ibid. 196 (1974), 37–50, 1974. doi:10.2307/1997010.\n- [KS77] Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations, volume 88 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J., 1977. With notes by John Milnor and Michael Atiyah.\n- [Wat19] Tadayuki Watanabe. Some exotic nontrivial elements of the rational homotopy groups of Diffp$S^{4}$q, 2019. arXiv:1812.02448.\n- [LX23] Jianfeng Lin and Yi Xie. Configuration space integrals and formal smooth structures, 2023. arXiv:2310.14156.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2950, "problem_number": "KP-4.74", "title": "Kirby Problem 4.74", "statement": "Does there exist a closed, smooth 4-manifold X and a diffeomorphism f: $X \\to X$ such that f is smoothly pseudo-isotopic to the identity, but f is not stably smoothly isotopic to the identity?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.74.\n\nLiterature notes:\n(1) We can stabilize a diffeomorphism by making a choice of isotopy to one that is the identity on a 4-ball, connect summing with $\\#_{k}S^{2} \\times S^{2}$ using that 4-ball, for some k, and then extending by the identity on the new $\\#_{k}S^{2} \\times S^{2}$. If some stabilization of f (for some choice of isotopy and for some k) is smoothly isotopic to the identity, then we say that f is smoothly stably isotopic to Id.\n\n(2) Gabai [Gab22] proved that there is a smooth pseudo-isotopy F with vanishing Hatcher-Wagoner pseudo-isotopy obstruction $\\Sigma(F) \\in Wh_{2}(\\pi_{1}(X))$ if and only iff is smoothly stably isotopic to Id. The question is whether there exists an f such that $\\Sigma(F)$ is nontrivial for all pseudo-isotopies F from f to Id. Or perhaps, for any pair(f, F), there is always a choice of F with $\\Sigma(F) =$ 0 restricting to the same f. Singh [Sin25] proved that the Hatcher–Wagoner obstruction $\\Sigma$ can be stably realized, which could be useful if one could control the diffeomorphism produced by his realization procedure.\n\nReferences cited:\n- [Gab22] David Gabai. 3-spheres in the 4-sphere and pseudo-isotopies of $S^{1}$ $\\times$ $S^{3}$, 2022. arXiv:2212.02004.\n- [Sin25] Oliver Singh. Pseudo-isotopies and diffeomorphisms of 4-manifolds. J. Topol., 18(4):Paper No. e70043, 61, 2025. doi:10.1112/topo.70043.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2951, "problem_number": "KP-4.75", "title": "Kirby Problem 4.75", "statement": "Let X be a connected smooth 4-manifold with nonempty boundary, with finite $\\pi_{1}(X)$, and let $k \\geq$ 0. Let $Diff_{\\partial}(X)$ denote the group of diffeomorphisms of X that are the identity near $\\partial X$. Is the image of the natural map $s_{*}: \\pi_{k}(Diff_{\\partial}(X)) \\to$ colim $\\pi_{k}(Diff_{\\partial}(X\\#_{N}S^{2} \\times S^{2}))$ (11) $N \\to \\infty$ finitely generated? What about the analogous problem in the topological category?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.75.\n\nLiterature notes:\n(1) Fixing a model of the interior connected sum $X\\#S^{2} \\times S^{2}$ defined by $X\\#S^{2} \\times S^{2}$ =X $\\cup ((\\partial X \\times [0,1])\\#S^{2} \\times S^{2})$, we have a well-defined stabilization map $s: Diff_{\\partial}(X) \\to Diff_{\\partial}(X\\#S^{2} \\times S^{2})$, extending diffeomorphisms by the identity map on $(\\partial X \\times [0,1])\\#S^{2} \\times S^{2}$.\n\n(2) In the topological category, Problem 4.72 asks the analogous question without stabilization.\n\n(3) We take a closer look at the problem for k =0, $\\pi_{1}(X)$ =1, and connected boundary. A diffeomorphism f: $X \\to X$ determines a Poincaré variation [Sae06]. The group of Poincaré variations is denoted $\\mathcal{V}(H_{2}(X), \\lambda_{X})$, where $\\lambda_{X}$ is the intersection form of X. Saeki [Sae06, Theorem 3.7] proved that the stable mapping class group of X is isomorphic to the group of stable Poincaré variations $S\\mathcal{V}(H_{2}(X), \\lambda_{X})$. Let $V:=Im(\\theta: \\pi_{0}Diff_{\\partial}(X) \\to \\mathcal{V}(H_{2}(X), \\lambda_{X}))$. The image of $s_{*}$ is thus isomorphic to the image of V under the algebraic stabilization map $\\mathcal{V}(H_{2}(X), \\lambda_{X}) \\to S\\mathcal{V}(H_{2}(X), \\lambda_{X})$. This latter map is injective, so in fact $Im(s_{*}) \\cong V$. We need to decide whether V is finitely generated. By [Sae06, Section 4], the group $\\mathcal{V}(H_{2}(X), \\lambda_{X})$ sits in an exact sequence 0 $\\to \\wedge ^{2}H^{1}(\\partial X;\\mathbb{Z}) \\to \\mathcal{V}(H_{2}(X), \\lambda_{X}) \\to Aut_{\\partial}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})$. The first group is finitely generated, and the image of the second map is a finite index subgroup of $Aut_{\\partial}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})$, the automorphisms of the intersection form of X whose algebraic boundary is trivial. The latter group is arithmetic, so is finitely generated. It follows $that\\mathcal{V}(H_{2}(X), \\lambda_{X})$ is finitely generated, since finite index subgroups and extensions of finitely generated groups are again finitely generated. Is V finitely generated? In the special case that $\\partial X =S^{3} \\mathcal{V}(H_{2}(X), \\lambda_{X}) \\cong \\operatorname{Aut}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})$. By capping of f with a 4-ball we can apply Wall’s theorem [Wal64a] to see that $\\theta$ is surjective when X is of the for m $X = M\\#S^{2} \\times S^{2}$, where M is indefinite or $b_{2}(M) <$ 9. Thus in these cases the image of $s_{*}$, for $k =$ 0, is known to be finitely generated. In other cases, such as for $X = K3 \\setminus D^{\\circ 4}, V$ is finite index in $\\mathcal{V}(H_{2}(X), \\lambda_{X}) \\cong \\operatorname{Aut}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})$, so is finitely generated. However there are also examples, such as for X the punctured Dolgachev surface, where V is infinite index [FM88] in $\\operatorname{Aut}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})$. In such cases, is V finitely generated?\n\n(4) Without stabilization, it is known that $\\pi_{k}(\\operatorname{Diff}(X))$ need not be finitely generated, even for $\\pi_{1}(X)$ =1. This was proven by Ruberman [Rub99b] for k=0, with later proofs by Baraglia [Bar23c] and Konno [Kon24b]. Fork =1 it was proven by Baraglia[Bar23b] and Lin [Lin22]. Fork $\\geq$ 1 this is announced in recent work of Auckly–Ruberman [AR25]. Different 4-manifolds are used in each work. All of these results are proven using gauge theory for families.\n\n(5) For 4-manifolds with infinite fundamental group, Budney–Gabai [BG19] and Watanabe [Wat23] proved (without using gauge theory) that some 4-manifolds with infinite fundamental group $(e.g.S^{1} \\times S^{3}$ in [BG19], and hyperbolic manifolds of dimension at least 4 in [BG25]) have infinitely generated mapping class groups. However the diffeomorphisms they construct are pseudo-isotopic to the identity, and hence are stably isotopic to the identity by Gabai’s theorem [Gab22]. It would be interesting to know whether there are counterexamples when $\\pi_{1}(X)$ is infinite.\n\n(6) Since $\\pi_{k}(Diff_{\\partial}(X)) \\cong \\pi_{k,+,1}(BDiff_{\\partial}(X))$, the problem can be described in terms of $BDiff_{\\partial}(X)$. If one considers the analogous question for the homology groups of $BDiff_{\\partial}(X)$, the answer is often positive. This is because it follows from work by Galatius and Randal-Williams [GRW17] that colim $H_{k}(BDiff_{\\partial}(X\\#_{N}S^{2} \\times S^{2}))$ (12) $N \\to \\infty$ is finitely generated for allkand many X, such as for simply-connected X. In particular, any subgroup of (12) such as the image of the stabilization map is also finitely generated. However, this is not necessarily strong evidence to hope for a positive solution to the problem for homotopy groups, since there are many spaces with finitely generated homology groups but infinitely generated homotopy groups (for example, $H_{2}(S^{1} \\vee S^{2}) =\\mathbb{Z}$ but $\\pi_{2}(S^{1} \\vee S^{2}) =\\mathbb{Z}^{\\infty})$.\n\n(7) In even, higher dimensions, the answer to the analogous problem is positive, and indeed this holds without taking the image in a colimit. For a compact smooth manifold M of dimension 2n $\\geq$ 6 with finite fundamental group, $\\pi_{k}(Diff_{\\partial}(M))$ is finitely generated, due to Bustamante–Krannich– Kupers [BKK24, Theorem 6.1].\n\nReferences cited:\n- [Sae06] Osamu Saeki. Stable mapping class groups of 4-manifolds with boundary. Trans. Amer. Math. Soc., 358(5):2091–2104, 2006. doi:10.1090/S0002-9947-05-03748-7.\n- [Wal64a] C. T. C. Wall. Diffeomorphisms of 4-manifolds. J. London Math. Soc., 39:131–140, 1964. doi:10.1112/jlms/s1-39.1.131.\n- [FM88] Robert Friedman and John W. Morgan. On the diffeomorphism types of certain algebraic surfaces. I. J. Differential Geom., 27(2):297–369, 1988. http://projecteuclid.org/euclid.jdg/1214441784.\n- [Rub99b] Daniel Ruberman. A polynomial invariant of diffeomorphisms of 4-manifolds. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 473–488. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.473.\n- [Bar23c] David Baraglia. On the mapping class groups of simply-connected smooth 4-manifolds, 2023. arXiv:2310.18819.\n- [Kon24b] Hokuto Konno. The homology of moduli spaces of 4-manifolds may be infinitely generated. Forum Math. Pi, 12:Paper No. e25, 18, 2024. doi:10.1017/fmp.2024.26.\n- [Bar23b] David Baraglia. Non-trivial smooth families of K3 surfaces. Math. Ann., 387(3-4):1719–1744, 2023. doi:10.1007/s00208-022-02508-3.\n- [Lin22] Jianfeng Lin. The family Seiberg-Witten invariant and nonsymplectic loops of diffeomorphisms, 2022. arXiv:2208.12082.\n- [AR25] Dave Auckly and Daniel Ruberman. Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory, 2025. arXiv: 2501.11892.\n- [BG19] Ryan Budney and David Gabai. Knotted 3-balls in $S^{4}$, 2019. arXiv:1912.09029.\n- [Wat23] Tadayuki Watanabe. Theta-graph and diffeomorphisms of some 4-manifolds, 2023. arXiv:2005.09545.\n- [BG25] Ryan Budney and David Gabai. On the automorphism groups of hyperbolic manifolds, 2025. doi:10.1093/imrn/rnaf083.\n- [Gab22] David Gabai. 3-spheres in the 4-sphere and pseudo-isotopies of $S^{1}$ $\\times$ $S^{3}$, 2022. arXiv:2212.02004.\n- [GRW17] Søren Galatius and Oscar Randal-Williams. Homological stability for moduli spaces of high dimensional manifolds. II. Ann. of Math. (2), 186(1):127–204, 2017. doi: 10.4007/annals.2017.186.1.4.\n- [BKK24] Mauricio Bustamante, Manuel Krannich, and Alexander Kupers. Finiteness properties of automorphism spaces of manifolds with finite fundamental group. Math. Ann., 388(4):3321–3371, 2024. doi:10.1007/s00208-023-02594-x.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2952, "problem_number": "KP-4.76", "title": "Kirby Problem 4.76", "statement": "Let X be a closed, oriented, simply connected, smooth 4manifold and fix $k >$ 0. Is there $N \\geq$ 0 such that, for every $n \\geq N$, the natural map $\\pi_{k}(\\operatorname{Diff}^{+}(X\\#_{n}S^{2} \\times S^{2})) \\to \\pi_{k}(\\operatorname{Homeo}^{+}(X\\#_{n}S^{2} \\times S^{2}))$ is surjective?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.76.\n\nLiterature notes:\n(1) If $k =$ 0, the answer to the analogous question is affirmative. First, for N =2, it follows from Wall’s theorem [Wal64a] that the natural map $\\pi_{0}(\\operatorname{Diff}^{+}(X\\#_{n}S^{2} \\times S^{2})) \\to \\operatorname{Aut}(H_{2}(X\\#_{n}S^{2} \\times S^{2};\\mathbb{Z}), \\lambda_{X,\\#,n,S,2, \\times ,S,2})$ is surjective for every $n \\geq N$. Here $\\operatorname{Aut}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})$ denotes the automorphism group of the intersection form. On the other hand, work of Freedman [Fre82], Kreck, [Kre79], Perron [Per86], and Quinn [Qui86] (plus [GGH+23]), implies that the natural map $\\pi_{0}(\\operatorname{Homeo}^{+}(X\\#_{n}S^{2} \\times S^{2})) \\to \\operatorname{Aut}(H_{2}(X\\#_{n}S^{2} \\times S^{2};\\mathbb{Z}), \\lambda_{X,\\#,n,S,2, \\times ,S,2})$ is an isomorphism. Thus, considering the obvious commuting triangle, we have $\\pi_{0}(\\operatorname{Diff}^{+}(X\\#_{n}S^{2} \\times S^{2})) \\to \\pi_{0}(\\operatorname{Homeo}^{+}(X\\#_{n}S^{2} \\times S^{2}))$ is surjective.\n\n(2) For the analogous problem obtained by replacing “surjective” with “injective”, the answer is negative for k =0. Indeed, for any simply-connected closed smooth 4-manifold X, one can find a strictly increasing divergent sequence 0 $$ 0, is there a closed, simply connected, smooth 4manifold X and a nonzero homotopy class $\\alpha \\in \\pi_{k}(Diff_{\\partial}(X^{\\circ}))$ such that $\\alpha \\in \\ker(i_{*}: \\pi_{k}(Diff_{\\partial}(X^{\\circ})) \\to \\pi_{k}(Homeo_{\\partial}(X^{\\circ})))$ (14) and $\\alpha \\notin \\ker(s_{*}: \\pi_{k}(Diff_{\\partial}(X^{\\circ})) \\to \\pi_{k}(Diff_{\\partial}(X^{\\circ}\\#S^{2} \\times S^{2})))$? (15)", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.79.\n\nLiterature notes:\n(1) Here we use the notation i, and s of Problem 4.75 and the previous two problems.\n\n(2) The answer to an analogous statement for $k =$ 0 is known to be affirmative: Kronheimer–Mrowka [KM20] proved that the Dehn twist on $X =$ K3\\#K3 along $S^{3}$ gives non-zero class $\\alpha$ that lies in (14), and Lin [Lin23] proved that this $\\alpha$ satisfies (15).\n\nReferences cited:\n- [KM20] P. B. Kronheimer and T. S. Mrowka. The Dehn twist on a sum of two K3 surfaces. Math. Res. Lett., 27(6):1767–1783, 2020. doi:10.4310/MRL.2020.v27.n6.a8.\n- [Lin23] Jianfeng Lin. Isotopy of the Dehn twist on K3 \\# K3 after a single stabilization. Geom. Topol., 27(5):1987–2012, 2023. doi:10.2140/gt.2023.27.1987.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2956, "problem_number": "KP-4.80", "title": "Kirby Problem 4.80", "statement": "Is there $n >$ 2 and a smooth closed simply connected 4manifold X for which there is an element of ordernin the $subgroupker(\\pi_{0}\\operatorname{Diff}(X) \\to \\pi_{0}\\operatorname{Homeo}(X))$? More generally, is there a subgroup of order nin $\\ker(\\pi_{0}\\operatorname{Diff}(X) \\to \\pi_{0}\\operatorname{Homeo}(X))$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.80.\n\nLiterature notes:\n(1) An element of $\\ker(\\pi_{0}\\operatorname{Diff}(X) \\to \\pi_{0}\\operatorname{Homeo}(X))is$ called an exotic diffeomorphism. Examples of exotic diffeomorphisms by Ruberman [Rub98] and Baraglia–Konno [BK20] are of infinite order in $\\ker(\\pi_{0}\\operatorname{Diff}(X) \\to \\pi_{0}\\operatorname{Homeo}(X)) A$ Dehn twist on K3\\#K3 along the connected sum $S^{3}$, detected by Kronheimer–Mrowka [KM20], is of order 2. These exotic diffeomorphisms are detected by $\\mathbb{Z}$ or $\\mathbb{Z}/2-valued$ invariants defined usin g parameterized ASD Yang–Mills or Seiberg-Witten theory. It would seem that to detect an exotic diffeomorphism of order n one would need $\\mathbb{Z}/n-valued$ invariants of this type. Note that, for a simply connected, closed, oriented 4-manifold, the orientation-preserving topological mapping class group $\\pi_{0}(\\operatorname{Homeo}^{+}(X)))$ is isomorphic to $\\operatorname{Aut}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})by$ work of Freedman [Fre82], Kreck, [Kre79], Perron [Per86], and Quinn [Qui86].\n\n(2) One could consider the following special cases.\n\n\\paragraph{Question.}\n\n(i) Is there a smooth, closed, simply connected 4-manifold X with a subgroup isomorphic to $\\mathbb{Z}/2 \\times \\mathbb{Z}/2$ in $\\ker(\\pi_{0}\\operatorname{Diff}(X) \\to \\pi_{0}\\operatorname{Homeo}(X))$?\n\n(ii) In particular, for X =K3\\#K3\\#K3, do the Dehn twists on the two connected sum copies of $S^{3}$ in X generate such a subgroup?\n\n(iii) More generally, is there a copy of $(\\mathbb{Z}/2)^{n,-1}$ in $\\ker(\\pi_{0}\\operatorname{Diff}(X) \\to \\pi_{0}\\operatorname{Homeo}(X))$ when X is a connected sum of n copies of the K3 surface?\n\nReferences cited:\n- [Rub98] Daniel Ruberman. An obstruction to smooth isotopy in dimension 4. Math. Res. Lett., 5(6):743–758, 1998. doi:10.4310/MRL.1998.v5.n6.a5.\n- [BK20] David Baraglia and Hokuto Konno. A gluing formula for families Seiberg-Witten invariants. Geom. Topol., 24(3):1381–1456, 2020. doi:10.2140/gt.2020.24.1381.\n- [KM20] P. B. Kronheimer and T. S. Mrowka. The Dehn twist on a sum of two K3 surfaces. Math. Res. Lett., 27(6):1767–1783, 2020. doi:10.4310/MRL.2020.v27.n6.a8.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [Kre79] M. Kreck. Isotopy classes of diffeomorphisms of $(k-1)$-connected almostparallelizable 2k-manifolds. In Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), volume 763 of Lecture Notes in Math., pages 643– 663. Springer, Berlin, 1979.\n- [Per86] B. Perron. Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique. Topology, 25(4):381–397, 1986. doi:10.1016/0040-9383(86)90018-2.\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2957, "problem_number": "KP-4.81", "title": "Kirby Problem 4.81", "statement": "Is there a smooth, closed, simply connected 4-manifold X for which the group $\\ker(\\pi_{0}(\\operatorname{Diff}(X)) \\to \\pi_{0}(\\operatorname{Homeo}(X)))$ is finitely generated?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.81.\n\nLiterature notes:\n(1) Let TDiff(X)be the group of diffeomorphisms that act trivially on $H_{*}(X;\\mathbb{Z})$. Quinn [Qui86] and Perron [Per86] proved that, for $\\pi_{1}(X) =$ \\{1\\}, the group $\\ker(\\pi_{0}(\\operatorname{Diff}(X)) \\to \\pi_{0}(\\operatorname{Homeo}(X)))$ is isomorphic to $\\pi_{0}(TDiff(X))$, called the Torelli group.\n\n(2) Ruberman [Rub99b] proved that there exist simply connected closed 4manifolds X for which $\\pi_{0}(TDiff(X))is$ not finitely generated.\n\n(3) In dimension 2, for the closed oriented surface $\\Sigma_{g}$ of genus $g \\geq$ 2, Johnson [Joh83] proved that the Torelli group is finitely generated for g >2, and Mc Cullough–Miller [MM86] proved that the Torelli group is not finitely generated for g =2.\n\nReferences cited:\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n- [Per86] B. Perron. Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique. Topology, 25(4):381–397, 1986. doi:10.1016/0040-9383(86)90018-2.\n- [Rub99b] Daniel Ruberman. A polynomial invariant of diffeomorphisms of 4-manifolds. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 473–488. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.473.\n- [Joh83] Dennis Johnson. The structure of the Torelli group. I. A finite set of generators for I. Ann. of Math. (2), 118(3):423–442, 1983. doi:10.2307/2006977.\n- [MM86] Darryl McCullough and Andy Miller. The genus 2 Torelli group is not finitely generated. Topology Appl., 22(1):43–49, 1986. doi:10.1016/0166-8641(86)90076-3.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2958, "problem_number": "KP-4.82", "title": "Kirby Problem 4.82", "statement": "Let $\\phi$ be a self-diffeomorphism of a closed, simply-connected, smooth 4-manifold X. Suppose that for every smooth surface $\\Sigma$ in X, the surfaces $\\Sigma$ and $\\phi(\\Sigma)$ are smoothly isotopic.\n\n(a) Is $\\phi$ necessarily smoothly isotopic to the identity map?\n\n(b) Is $\\phi smoothly$ isotopic to a diffeomorphism that is supported on $a B^{4} \\subset X$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.82.\n\nLiterature notes:\n(1) In dimension three, a self-homeomorphism f of an irreducible 3-manifold $M^{3}$ that preserves free homotopy classes of loops is isotopic to the identity map [ABD+20]. The analogous result holds for simply connected 4-manifolds in the topological category by Perron [Per86] and independently by work of Quinn [Qui86] combined with that of Gabai–Gay– Hartman–Krushkal–Powell $[GGH^{+}23]$. (These authors show that in the topological category, the weaker hypothesis that $\\phi$ induces the identity map on $H_{2}(X;\\mathbb{Z})$ implies that $\\phi$ is topologically isotopic to the identity map.) This question essentially asks whether these theorems have an analogue in dimension four in the smooth category. The stronger hypothesis is necessary, as there are many examples of exotic self-diffeomorphisms of simply connected 4-manifolds, e.g. the Dehn twist $\\varphi$: K3\\#K3 $\\to$ K3\\#K3 is topologically but not smoothly isotopic to the identity map [KM20]. In this case, the induced map from $\\varphi$ clearly preserves $H_{2}(K3\\#K3;\\mathbb{Z})$, but it is not clear whether $\\varphi(\\Sigma)is$ smoothly isotopic to $\\Sigma$ for every surface $\\Sigma$ inside K3\\#K3.\n\n(2) The first question, in the special case $X = S^{4}$, would imply that every orientation-preserving diffeomorphism of $S^{4}$ is isotopic to the identity, answering Problem 4.67. To see that every diffeomorphism $\\phi: S^{4} \\to S^{4}$ satisfies the hypothesis of the problem, isotope $\\phi to$ fix some $B^{4}$, and then isotope $\\Sigma$ into that $B^{4}$. The second question decouples this problem from\n\nProblem 4.67.\n\n(3) Another variation on the problem allows the stronger hypothesis that for every $g \\geq$ 0, every smooth embedding $h: \\Sigma_{g} \\hookrightarrow X$ is smoothly isotopic to the embedding $\\phi \\circ h$.\n\n(4) Assuming instead that $\\phi$ becomes isotopic to the identity after connected sum with $S^{2} \\times S^{2}$ (which in particular implies that $\\phi$ induces the identity on $H_{2}(X;\\mathbb{Z}))$, then Krushkal-Mukherjee-Powell-Warren [KMPW24] showed that $\\phi$ can be isotoped so as to be supported on a contractible submanifold. Another point of view on the question asks whether the assumption that $\\Sigma$ and $\\phi(\\Sigma)$ are smoothly isotopic for every $\\Sigma$, enables us to show that the contractible supporting manifold can be assumed to be a 4-ball.\n\nReferences cited:\n- [ABD+20] Paolo Aceto, Corey Bregman, Christopher W. Davis, JungHwan Park, and Arunima Ray. Isotopy and equivalence of knots in 3-manifolds, 2020. arXiv:2007.05796.\n- [Per86] B. Perron. Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique. Topology, 25(4):381–397, 1986. doi:10.1016/0040-9383(86)90018-2.\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n- [GGH+23] David Gabai, David T. Gay, Daniel Hartman, Vyacheslav Krushkal, and Mark Powell. Pseudo-isotopies of simply connected 4-manifolds, 2023. arXiv:2311.11196.\n- [KM20] P. B. Kronheimer and T. S. Mrowka. The Dehn twist on a sum of two K3 surfaces. Math. Res. Lett., 27(6):1767–1783, 2020. doi:10.4310/MRL.2020.v27.n6.a8.\n- [KMPW24] Vyacheslav Krushkal, Anubhav Mukherjee, Mark Powell, and Terrin Warren. Corks for exotic diffeomorphisms, 2024. arXiv:2407.04696.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2959, "problem_number": "KP-4.83", "title": "Kirby Problem 4.83", "statement": "For which 4-manifolds does there exist a smooth structure such that there exists a non-smoothable homeomorphism with respect to that smooth structure?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.83.\n\nLiterature notes:\n(1) We say that a homeomorphism is smoothableif it is rel. boundary isotopic to a diffeomorphism. If this does not hold, we say that the homeomorphism is non-smoothable.\n\n(2) This question is interesting for closed 4-manifolds, compact 4-manifolds with boundary, and for open 4-manifolds.\n\n(3) Recent progress for compact simply connected 4-manifolds with boundary was made by Galvin–Ladu [GL25] and Konno–Taniguchi [KT22b]. Here is an open question for such 4-manifolds. Question (i). Let X be a simply connected, smooth, spin 4-manifold with $\\partial X = Y_{1}\\cup Y_{2}$ having two connected components that do not admit generalized Dehn twists. Consider a boundary-fixing homeomorphism that has trivial Poincaré variation but that acts nontrivially on the relative spin structures of X (there are two such relative spin structures). Is it isotopic to a diffeomorphism? See [Sae06, OP25] for the definition of a Poincaré variation. If the $Y_{i}$ are hyperbolic, then they do not admit generalized Dehn twists. The answer is also yes in the case that there is a separating embedding of a 3-manifold Z, with $Y_{1}$ and $Y_{2}$ in different connected components of $X \\setminus Z$, where Z admits a generalized Dehn twist.\n\n(4) One can wonder whether there is a relationship with the existence of exotic smooth structures.\n\n\\paragraph{Question.} Does there exist a 4-manifold that admits a non-smoothable homeomorphism, but no exotic smooth structure? Or vice versa, a 4manifold admitting exotic smooth structures for which every self-homeomorphism is smoothable? See e.g. the work of Donaldson [Don90], Friedman–Morgan [FM88], and Baraglia [Bar21].\n\nReferences cited:\n- [GL25] Daniel Galvin and Roberto Ladu. Non-smoothable homeomorphisms of 4-manifolds with boundary. Adv. Math., 467:Paper No. 110191, 22, 2025. doi:10.1016/j.aim.2025.110191.\n- [KT22b] Hokuto Konno and Masaki Taniguchi. The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary. Adv. Math., 409:Paper No. 108627, 58, 2022. doi:10.1016/j.aim.2022.108627.\n- [Sae06] Osamu Saeki. Stable mapping class groups of 4-manifolds with boundary. Trans. Amer. Math. Soc., 358(5):2091–2104, 2006. doi:10.1090/S0002-9947-05-03748-7.\n- [OP25] Patrick Orson and Mark Powell. Mapping class groups of simply connected 4-manifolds with boundary. J. Differential Geom., 131(1):199–275, 2025. doi:10.4310/jdg/1755544135.\n- [Don90] S. K. Donaldson. Polynomial invariants for smooth four-manifolds. Topology, 29(3):257–315, 1990. doi:10.1016/0040-9383(90)90001-Z.\n- [FM88] Robert Friedman and John W. Morgan. On the diffeomorphism types of certain algebraic surfaces. I. J. Differential Geom., 27(2):297–369, 1988. http://projecteuclid.org/euclid.jdg/1214441784.\n- [Bar21] David Baraglia. Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants. Algebr. Geom. Topol., 21(1):317–349, 2021. doi:10.2140/agt.2021.21.317.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2960, "problem_number": "KP-4.84", "title": "Kirby Problem 4.84", "statement": "Is there a closed oriented smooth 4-manifold X for which every finite subgroup Gof the mapping class group $\\pi_{0}(\\operatorname{Diff}^{+}(X))can$ be realized by a finite group action on X?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.84.\n\nLiterature notes:\n(1) This question relates to the study of the algebraic properties of diffeomorphism groups as opposed to their homotopy type. If there is a group-theoretic section $G \\to \\operatorname{Diff}^{+}(X)$ of the quotient map $\\operatorname{Diff}^{+}(X) \\to \\pi_{0}(\\operatorname{Diff}^{+}(X))$ over G, we say that G is realized by a finite group action. The question whether a given G is realized is known as the Nielsen realization problem.\n\n(2) For orientable surfaces X, Kerckhoff [Ker83] proved that every finite subgroup of $\\pi_{0}(\\operatorname{Diff}^{+}(X))$ is realized. On the other hand, in dimension 4, there are several known examples of 4-manifolds X for which there are non-realizable finite subgroups of $\\pi_{0}(\\operatorname{Diff}^{+}(X))$ [RS77, BK23, FL24a, Kon24a, KMT23b, AB25, Bar23c].\n\n(3) One may also ask the analogous questions for the extended mapping classes group of X, which also contains the orientation-reversing diffeomorphisms, and for nonorientable X.\n\n(4) The existence of asymmetric manifolds, i.e. manifolds that do not admit any effective action of a finite group, has been studied in higher dimensions. See e.g. [Pup07, CR72]. Related to the Nielsen realization problem, one may ask whether there is a smooth asymmetric 4-manifold where the mapping class group has a nontrivial finite subgroup.\n\n(5) Here is a variant of the Nielsen realization problem, and for some specific X, several non-realizability results are known. Let $\\operatorname{Aut}(H_{2}(X;\\mathbb{Z}))denote$ the automorphism group of the intersection form and set $I(X):=Im(\\operatorname{Diff}^{+}(X) \\to \\operatorname{Aut}(H_{2}(X;\\mathbb{Z})))$.\n\n\\paragraph{Question.} Which X admits $a(finite)$ group Gand a homomorphism $\\phi: G \\to I(X)$ that cannot be realized by a smooth action on X? Here we say $that\\phi isrealizedby a$ smooth action if there a homomorphism $\\tilde\\{\\phi\\}: G \\to \\operatorname{Diff}^{+}(X)$ that descends to $\\phi$.\n\nReferences cited:\n- [Ker83] Steven P. Kerckhoff. The Nielsen realization problem. Ann. of Math. (2), 117(2):235–265, 1983. doi:10.2307/2007076.\n- [RS77] Frank Raymond and Leonard L. Scott. Failure of Nielsen’s theorem in higher dimensions. Arch. Math. (Basel), 29(6):643–654, 1977. doi:10.1007/BF01220468.\n- [BK23] David Baraglia and Hokuto Konno. A note on the Nielsen realization problem for K3 surfaces. Proc. Amer. Math. Soc., 151(9):4079–4087, 2023. doi:10.1090/proc/15544.\n- [FL24a] Benson Farb and Eduard Looijenga. The Nielsen realization problem for K3 surfaces. J. Differential Geom., 127(2):505–549, 2024. doi:10.4310/jdg/1717772420.\n- [Kon24a] Hokuto Konno. Dehn twists and the Nielsen realization problem for spin 4-manifolds. Algebr. Geom. Topol., 24(3):1739–1753, 2024. doi:10.2140/agt.2024.24.1739.\n- [KMT23b] Hokuto Konno, Jin Miyazawa, and Masaki Taniguchi. Involutions, links, and Floer cohomologies, 2023. arXiv:2304.01115.\n- [AB25] Mihail Arabadji and R. İnanç Baykur. Nielsen realization in dimension four and projective twists. Adv. Math., 463:Paper No. 110112, 22, 2025. doi:10.1016/j.aim.2025.110112.\n- [Bar23c] David Baraglia. On the mapping class groups of simply-connected smooth 4-manifolds, 2023. arXiv:2310.18819.\n- [Pup07] Volker Puppe. Do manifolds have little symmetry? J. Fixed Point Theory Appl., 2(1):85–96, 2007. doi:10.1007/s11784-007-0021-x.\n- [CR72] P. E. Conner and Frank Raymond. Manifolds with few periodic homeomorphisms. In Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part II, volume Vol. 299 of Lecture Notes in Math., pages 1–75. Springer, Berlin-New York, 1972.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2961, "problem_number": "KP-4.85", "title": "Kirby Problem 4.85", "statement": "Is there a closed orientable smooth 4-manifold X for which the identity component $Diff_{0}(X)$ of the diffeomorphism group is not uniformly perfect?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.85.\n\nLiterature notes:\n(1) By Mather and Thurston [Mat71, Mat74,Thu74a], the group $Diff_{0}(X)$ is perfect for every closed orientable manifold X.\n\n(2) For a given group G, the commutator length of an element $g \\in$ [G, G] is defined to be the minimal number of factors in expressions of g as products of commutators. The commutator length has been studied for Lie groups, groups of automorphisms of (topological, smooth, or symplectic) manifolds, and mapping class groups in contexts of dynamics and bounded cohomology. It is known to be difficult to compute the commutator length of a given element in general, which leads us to consider the following qualitative property for a perfect group (a group that is equal to its commutator subgroup). Given a perfect group G, we say that G is uniformly perfectif the commutator lengths of elements of Gare uniformly bounded.\n\n(3) For dim $X \\ne$ 2,4, $Diff_{0}(X)$ is known to be uniformly perfect due to work by Burago–Ivanov–Polterovich [BIP08] and Tsuboi [Tsu08, Tsu12]. Their works also show that $Diff_{0}(X)$ is uniformly perfect for $X = S^{2}$ and $S^{4}$. On the other hand, Bowden–Hensel–Webb [BHW22] proved that, for dim $X =$ 2, $Diff_{0}(X)$ is not uniformly perfect if the genus of X is positive. Nothing is known in dimension 4 except for $X =S^{4}$.\n\nReferences cited:\n- [Mat71] John N. Mather. The vanishing of the homology of certain groups of homeomorphisms. Topology, 10:297–298, 1971. doi:10.1016/0040-9383(71)90022-X.\n- [Mat74] John N. Mather. Commutators of diffeomorphisms. Comment. Math. Helv., 49:512– 528, 1974. doi:10.1007/BF02566746.\n- [Thu74a] William Thurston. Foliations and groups of diffeomorphisms. Bull. Amer. Math. Soc., 80:304–307, 1974. doi:10.1090/S0002-9904-1974-13475-0.\n- [BIP08] Dmitri Burago, Sergei Ivanov, and Leonid Polterovich. Conjugation-invariant norms on groups of geometric origin. In Groups of diffeomorphisms, volume 52 of Adv. Stud. Pure Math., pages 221–250. Math. Soc. Japan, Tokyo, 2008. doi:10.2969/aspm/05210221.\n- [Tsu08] Takashi Tsuboi. On the uniform perfectness of diffeomorphism groups. In Groups of diffeomorphisms, volume 52 of Adv. Stud. Pure Math., pages 505–524. Math. Soc. Japan, Tokyo, 2008. doi:10.2969/aspm/05210505.\n- [Tsu12] Takashi Tsuboi. On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds. Comment. Math. Helv., 87(1):141–185, 2012. doi:10.4171/CMH/251.\n- [BHW22] Jonathan Bowden, Sebastian Wolfgang Hensel, and Richard Webb. Quasi-morphisms on surface diffeomorphism groups. J. Amer. Math. Soc., 35(1):211–231, 2022. doi:10.1090/jams/981.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2962, "problem_number": "KP-4.86", "title": "Kirby Problem 4.86", "statement": "Is it the case that for every closed, smoothable topological 4manifold X, there exists a locally linear finite group action on X, such that for every smooth structure on X, the action is non-smoothable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.86.\n\nLiterature notes:\n(1) Given a topological manifold X and a smooth structure $\\mathcal{O}$ on X, we say that a locally linear topological $action\\phi of a$ group on X isnon-smoothable with respect $to\\mathcal{O} if\\phi$ is not conjugate to a smooth action on X.\n\n(2) A variant of the problem is the following.\n\n\\paragraph{Question.} If we fix a smooth structure $\\mathcal{O}$ on X, is there a locally linear finite group action on X that is non-smoothable with respect to $\\mathcal{O}$?\n\n(3) There are many examples of locally linear finite group actions on closed 4manifolds that are non-smoothable with respect to any smooth structure, such as [KL88, KL93, HL95, Bry98, HT04, Nak09, Bar19,Bar21, Kat22].\n\nReferences cited:\n- [KL88] Slawomir Kwasik and Kyung Bai Lee. Locally linear actions on 3-manifolds. Math. Proc. Cambridge Philos. Soc., 104(2):253–260, 1988. doi:10.1017/S0305004100065427.\n- [KL93] Slawomir Kwasik and Terry Lawson. Nonsmoothable $\\mathbb{Z}_p$ actions on contractible 4-manifolds. J. Reine Angew. Math., 437:29–54, 1993. doi:10.1515/crll.1993.437.29.\n- [HL95] Ian Hambleton and Ronnie Lee. Smooth group actions on definite 4-manifolds and moduli spaces. Duke Math. J., 78(3):715–732, 1995. doi:10.1215/S0012-7094-95-07826-0.\n- [Bry98] Jim Bryan. Seiberg-Witten theory and $\\mathbb{Z}/2^p$ actions on spin 4-manifolds. Math. Res. Lett., 5(1-2):165–183, 1998. doi:10.4310/MRL.1998.v5.n2.a3.\n- [HT04] Ian Hambleton and Mihail Tanase. Permutations, isotropy and smooth cyclic group actions on definite 4-manifolds. Geom. Topol., 8:475–509, 2004. doi:10.2140/gt.2004.8.475.\n- [Nak09] Nobuhiro Nakamura. Bauer-Furuta invariants under $\\mathbb{Z}_2$-actions. Math. Z., 262(1):219–233, 2009. doi:10.1007/s00209-008-0370-1.\n- [Bar19] David Baraglia. Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory. Adv. Math., 354:106730, 32, 2019. doi:10.1016/j.aim.2019.106730.\n- [Bar21] David Baraglia. Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants. Algebr. Geom. Topol., 21(1):317–349, 2021. doi:10.2140/agt.2021.21.317.\n- [Kat22] Yuya Kato. Nonsmoothable actions of $\\mathbb{Z}_2\\times\\mathbb{Z}_2$ on spin four-manifolds. Topology Appl., 307:Paper No. 107868, 13, 2022. doi:10.1016/j.topol.2021.107868.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2963, "problem_number": "KP-4.87", "title": "Kirby Problem 4.87", "statement": "Is there an exotic action of $\\mathbb{Z}/n$ on $S^{4}$ with 0-dimensional fixed point set? 1-dimensional? 2-dimensional?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.87.\n\nLiterature notes:\n(1) In the case that the fixed set is empty, any nontrivial symmetry is orientationreversing, son=2. In this case Cappell and Shaneson [CS76] constructed an exotic smooth $\\mathbb{RP}^{4}$, whose double cover is $S^{4}$ [AK79a, Gom91b]. Another construction was given by Fintushel and Stern [FS81]. It is not known whether these examples are diffeomorphic, or more generally whether there is more than one exotic $\\mathbb{RP}^{4}$. (In the topological setting there are exactly two homeomorphism classes of 4-manifolds that are homotopy equivalent to $\\mathbb{RP}^{4}$ according to a calculation from the surgery exact sequence [Wal99, Chapter 14]). These two manifolds are distinguished by their Kirby-Siebenmann invariant. As a point of interest, we refer the reader to Ruberman’s explicit construction of the non-smoothable homotopy $\\mathbb{RP}^{4}$ [Rub84, Section 2].)\n\n(2) In the case that the fixed set is 0-dimensional or 1-dimensional, very little is known in the smooth setting. In the case that the fixed set is 2-dimensional, there exist actions with knotted fixed point set [Gif66, Gor74, Sum75], but the tools for establishing this are topological. The case that the fixed point set has dimension 3 has been extensively studied (see e.g. [Maz61]); examples arise from gluing two copies of a cork along the boundary.\n\nReferences cited:\n- [CS76] Sylvain E. Cappell and Julius L. Shaneson. Some new four-manifolds. Ann. of Math. (2), 104(1):61–72, 1976. doi:10.2307/1971056.\n- [AK79a] Selman Akbulut and Robion Kirby. An exotic involution of $S^{4}$. Topology, 18(1):75– 81, 1979. doi:10.1016/0040-9383(79)90015-6.\n- [Gom91b] Robert E. Gompf. On Cappell-Shaneson 4-spheres. Topology Appl., 38(2):123–136, 1991. doi:10.1016/0166-8641(91)90079-2.\n- [FS81] Ronald Fintushel and Ronald J. Stern. An exotic free involution on $S^{4}$. Ann. of Math. (2), 113(2):357–365, 1981. doi:10.2307/2006987.\n- [Wal99] C. T. C. Wall. Surgery on compact manifolds, volume 69 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition, 1999. Edited and with a foreword by A. A. Ranicki. doi:10.1090/surv/069.\n- [Rub84] Daniel Ruberman. Invariant knots of free involutions of $S^{4}$. Topology Appl., 18(2-3):217–224, 1984. doi:10.1016/0166-8641(84)90011-7.\n- [Gif66] Charles H. Giffen. The generalized Smith conjecture. Amer. J. Math., 88:187–198, 1966. doi:10.2307/2373054.\n- [Gor74] C. McA. Gordon. On the higher-dimensional Smith conjecture. Proc. London Math. Soc. (3), 29:98–110, 1974. doi:10.1112/plms/s3-29.1.98.\n- [Sum75] D. W. Sumners. Smooth $\\mathbb{Z}_p$-actions on spheres which leave knots pointwise fixed. Trans. Amer. Math. Soc., 205:193–203, 1975. doi:10.2307/1997199.\n- [Maz61] Barry Mazur. A note on some contractible 4-manifolds. Ann. of Math. (2), 73:221– 228, 1961. doi:10.2307/1970288.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2964, "problem_number": "KP-4.88", "title": "Kirby Problem 4.88", "statement": "Let $\\tau: S^{4} \\to S^{4}$ be a free (hence orientation-reversing) involution. Is there an embedded $S^{2} \\subset S^{4}$ that is invariant under $\\tau$? This is equivalent to the existence of an embedded $\\mathbb{RP}^{2}$ in $S^{4}/\\tau \\cong \\mathbb{RP}^{4}$, carrying the nontrivial class in $H_{2}(\\mathbb{RP}^{4};\\mathbb{Z}/2)$. This condition on the homology class will be assumed for the rest of the problem.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.88.\n\nLiterature notes:\n(1) This is an instance of the codimension two splitting problem [CS74] and may be asked in the smooth or topological category (where one would be looking for an $\\mathbb{RP}^{2}$ with a normal bundle). Another classic phrasing is that the problem is asking if the involution desuspends twice.\n\n(2) In the topological category, there is only one 4-manifold homotopy equivalent but not homeomorphic to $\\mathbb{RP}^{4}$, which is sometimes denoted $*\\mathbb{RP}^{4}. Question(i). Does*\\mathbb{RP}^{4}$ have an embedded $\\mathbb{RP}^{2}$ with a normal bundle?\n\n(3) There are two known constructions of exotic $\\mathbb{RP}^{4}s$ in the smooth category, due to Cappell–Shaneson [CS76] and Fintushel–Stern [FS81]. By construction, the Cappell–Shaneson $\\mathbb{RP}^{4}s$ all contain an embedded $\\mathbb{RP}^{2}$. Question (ii). Do the Fintushel–Stern $\\mathbb{RP}^{4}s$ contain smoothly embedded $\\mathbb{RP}^{2}s? A$ negative answer would show that the Fintushel–Stern $\\mathbb{RP}^{4}s$ are not diffeomorphic to those obtained by the Cappell–Shaneson construction; see also Problem 4.87.\n\nReferences cited:\n- [CS74] S.E. Cappell and J. Shaneson. Homology surgery and the codimension-two placement problem. Ann. of Math., 99:277–348, 1974.\n- [CS76] Sylvain E. Cappell and Julius L. Shaneson. Some new four-manifolds. Ann. of Math. (2), 104(1):61–72, 1976. doi:10.2307/1971056.\n- [FS81] Ronald Fintushel and Ronald J. Stern. An exotic free involution on $S^{4}$. Ann. of Math. (2), 113(2):357–365, 1981. doi:10.2307/2006987.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2965, "problem_number": "KP-4.89", "title": "Kirby Problem 4.89", "statement": "Classify smooth, effective circle actions on simply connected 4-manifolds with boundary.\n\n(a) Classify simply connected4-manifolds with boundary that admit circle actions.\n\n(b) For each such 4-manifold, classify the circle actions up to conjugation by diffeomorphisms.\n\n(c) Given a 4-manifold X with a circle action on its boundary, when does the action extend over X? When does it extend uniquely?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.89.\n\nLiterature notes:\n(1) Information about the smooth classification of the closed 4-manifolds admitting a circle action was obtained by Fintushel and Pao [FP77] via Pao’s replacement trick [Pao77]. (These papers assumed the truth of the 3-dimensional Poincaré conjecture, an assumption that is now known to be satisfied.) Part (a) asks for similar results in the bounded case.\n\n(2) The classification of circle actions on closed simply connected 4-manifolds was carried out in the 1970s by Fintushel [Fin77, Fin78] in terms of orbit data. Part (b)is asking for an analogous result in the bounded case. Note that in the bounded case, one necessarily has a Seifert-fibered space on the boundary, and so the orbit data is somewhat more complicated than in the closed case.\n\n(3) There are obstructions to the extension problem in part (c) coming from gauge theory. Konno–Mallick–Taniguchi [KMT23a] studied, for M equal to one of the Milnor fibers M(2,3,7)or $M(2,3,11)$, the loop in $\\pi_{1}(\\operatorname{Diff}(\\partial M))$ corresponding to the circle action coming from the Seifert-fibred structure of the boundary. This gives rise to a boundary generalized Dehn twist, a diffeomorphism of M supported in a collar neighborhood of $\\partial M$. This diffeomorphism is isotopic to the identity rel. boundary if and only if the element of $\\pi_{1}(\\operatorname{Diff}(\\partial M))$ extends to a loop in $\\pi_{1}(\\operatorname{Diff}(M))$. Konno– Mallick–Taniguchi showed that there is no such extension, and hence there is no corresponding circle action. Montague [Mon23] drew the same conclusion for the Gompf nuclei $N(2n)$ and simple plumbing $P(2n)$ with boundary $-\\Sigma(2,3,12n-5)$, by proving non-extension results for $\\mathbb{Z}/p$ actions embedded in the circle action on the boundary. All of these results continue to hold when M is replaced by $M\\#S^{2} \\times S^{2}$. Further results along these lines have been announced in [KLMME24, KPT26].\n\n\\paragraph{Question.} Does there exist a compact 4-manifold X, and a circle action on $\\partial X$, such that the latter extends to a loop of diffeomorphisms in $\\pi_{1}(\\operatorname{Diff}(X))$ but not to a circle action on X?\n\n(4) If $\\partial X$ admits a unique circle action, then an answer to (c) would follow from an answer to (a). But there could exist distinct circle actions on $\\partial X$ such that one extends over X and one does not.\n\n(5) A complementary obstruction to extending would be via a classical theorem of Atiyah–Hirzebruch [AH70] that restricts the characteristic classes of closed spin manifolds of dimension 4k admitting a circle action; they show in this case that the $A_{(}$ genus of the manifold must vanish. In dimension 4, this means that the signature vanishes.\n\n\\paragraph{Question.} Find an analogue of the Atiyah–Hirzebruch result for 4manifolds (or more generally 4k-manifolds) with nonempty boundary. It is easy to see that the signature does not necessarily vanish for spin manifolds with circle actions if the boundary is nonempty. For instance, one could take the disk bundle over $S^{2}$ with even, nonzero, Euler class. It has signature $\\pm$ 1 but supports an $S^{1}$ action. From such examples, it seems plausible that there could be some sort of boundary correction to the Atiyah–Hirzebruch argument, presumably involving an $S^{1}-equivariant \\eta-invariant$ [Don78].\n\nReferences cited:\n- [FP77] Ronald Fintushel and Peter Sie Pao. Identification of certain 4-manifolds with group actions. Proc. Amer. Math. Soc., 67(2):344–350, 1977. doi:10.2307/2041299.\n- [Pao77] Peter Sie Pao. The topological structure of 4-manifolds with effective torus actions. I. Trans. Amer. Math. Soc., 227:279–317, 1977. doi:10.2307/1997462.\n- [Fin77] Ronald Fintushel. Circle actions on simply connected 4-manifolds. Trans. Amer. Math. Soc., 230:147–171, 1977. doi:10.2307/1997715.\n- [Fin78] Ronald Fintushel. Classification of circle actions on 4-manifolds. Trans. Amer. Math. Soc., 242:377–390, 1978. doi:10.2307/1997745.\n- [KMT23a] Hokuto Konno, Abhishek Mallick, and Masaki Taniguchi. Exotic Dehn twists on 4-manifolds, 2023. arXiv:2306.08607.\n- [Mon23] Ian Montague. Non-smoothable $\\mathbb{Z}/p$-actions on nuclei, 2023. arXiv:2401.00244.\n- [KLMME24] Hokuto Konno, Jianfeng Lin, Anubhav Mukherjee, and Juan Muñoz-Echániz. On four-dimensional Dehn twists and Milnor fibrations, 2024. arXiv:2409.11961.\n- [KPT26] Sungkyung Kang, JungHwan Park, and Masaki Taniguchi. Exotic Dehn twists and homotopy coherent group actions. Invent. Math., 243(1):209–241, 2026. doi:10.1007/s00222-025-01378-1.\n- [AH70] Michael Atiyah and Friedrich Hirzebruch. Spin-manifolds and group actions. In Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), pages 18–28. Springer, New York-Berlin, 1970.\n- [Don78] Harold Donnelly. Eta invariants for G-spaces. Indiana Univ. Math. J., 27(6):889– 918, 1978. doi:10.1512/iumj.1978.27.27060.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2966, "problem_number": "KP-4.90", "title": "Kirby Problem 4.90", "statement": "Do the Chern numbers $c^{2}_{1}$ and $c_{2}$ of every closed, symplectic 4–manifold X that is not a ruled surface satisfy the following?\n\n(a) $c^{2}_{1} \\leq 3c_{2}$.\n\n(b) $c^{2}_{1} =3c_{2}$ >0 if and only if $X =\\mathbb{CP}^{2}$ or X is a complex ball quotient.\n\n(c) $c_{2} \\geq$ 0.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.90.\n\nLiterature notes:\n(1) These invariants depend only on the underlying homotopy type of the 4–manifold X, satisfying the identities $c^{2}_{1} =2\\chi+3\\sigma$ and $c_{2} =\\chi$ where $\\chi$ and $\\sigma$ are the Euler characteristic and the signature of X.\n\n(2) All hold true for Kähler surfaces [BPVd V84]; part (a)is the well-known Bogomolov–Miyaoka–Yau inequality (BMY) and part (b) follows from Yau’s celebrated solution of the Calabi conjecture [Yau78].\n\n(3) The inequality (c) was conjectured to hold by Gompf. If the non-ruled symplectic 4–manifold X is minimal, then $c^{2}_{1} \\geq$ 0 by [Tau94, Tau95, Tau96, LL95]; so part (c) is implied by part (a) in this case.\n\n(4) These geographic constraints would have strong consequences. For instance, if true, part (b)would imply that there is no symplectic 4–manifold X homeomorphic but not diffeomorphic to $\\mathbb{CP}^{2}$; part (a)or (c)would imply the same for any ruled surface over $\\Sigma_{h}$, with $h \\geq$ 2.\n\n(5) If there exist symplectic 4–manifolds with $c^{2}_{1} = 3c_{2}$ that are not diffeomorphic to any Kähler surface, a further line of inquiry for part (b)would be whether there is a symplectic analog of Yau’s theorem for them; e.g., are they always K(G,1)s? See also Problem 4.91.\n\n(6) One approach to an affirmative solution for part (a) is via branched coverings. Auroux proved that every symplectic 4-manifold admits a simple branched covering to $\\mathbb{CP}^{2}$, where the branch locus in $\\mathbb{CP}^{2}$ is an immersed symplectic surface with nodes and simple cusps [Aur00]. One can compute the Chern numbers in terms of the degree of the branch locus, the number of cusps, the genus of the branch curve, and the number of sheets of the cover (see [Aur06b, p.266]). Thus, one can translate the existence of an example on or above the BMY line into the existence of a braided singular surface in $\\mathbb{CP}^{2}$ with a coloring with certain constraints.\n\n(7) One can ask part (a) more generally as follows (see also Problem 4.16):\n\n\\paragraph{Question.} Does any closed 4–manifold X with $b^{+}_{2} >$ 1 and with nontrivial Seiberg–Witten invariants satisfy the BMY inequality? Recall that by the work of Taubes [Tau94, Tau95], if such a 4– manifold X admits a symplectic structure, then it has a Seiberg-Witten basic class. Feehan and Leness have announced a program to answer this broader question affirmatively [Fee22, FL23,FL24b]. Their approach is to adapt Hitchin’s Morse theory analysis of the moduli space of Higgs monopoles on a rank-2 Hermitian vector bundle over a Riemann surface [Hit87] to the moduli space of non-Abelian monopoles $(A,\\Phi)$ on a rank-2 Hermitian vector bundle E.\n\nReferences cited:\n- [BPVdV84] W. Barth, C. Peters, and A. Van de Ven. Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1984. doi:10.1007/978-3-642-96754-2.\n- [Yau78] Shing Tung Yau. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math., 31(3):339–411, 1978. doi:10.1002/cpa.3160310304.\n- [Tau94] Clifford Henry Taubes. The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett., 1(6):809–822, 1994. doi:10.4310/MRL.1994.v1.n6.a15.\n- [Tau95] Clifford Henry Taubes. More constraints on symplectic forms from Seiberg-Witten invariants. Math. Res. Lett., 2(1):9–13, 1995. doi:10.4310/MRL.1995.v2.n1.a2.\n- [Tau96] Clifford H. Taubes. SW ñ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves. J. Amer. Math. Soc., 9(3):845–918, 1996. doi:10.1090/S0894-0347-96-00211-1.\n- [LL95] T. J. Li and A. Liu. Symplectic structure on ruled surfaces and a generalized adjunction formula. Math. Res. Lett., 2(4):453–471, 1995. doi:10.4310/MRL.1995.v2.n4.a6.\n- [Aur00] Denis Auroux. Symplectic 4-manifolds as branched coverings of $\\mathbb{CP}^{2}$ . Invent. Math., 139(3):551–602, 2000. doi:10.1007/s002220050019.\n- [Aur06b] Denis Auroux. Symplectic 4-manifolds, singular plane curves, and isotopy problems. In Floer homology, gauge theory, and low-dimensional topology, volume 5 of Clay Math. Proc., pages 263–276. Amer. Math. Soc., Providence, RI, 2006.\n- [Fee22] Paul M. N. Feehan. Bialynicki-Birula theory, Morse-Bott theory, and resolution of singularities for analytic spaces, 2022. arXiv:2206.14710.\n- [FL23] Paul M. N. Feehan and Thomas G. Leness. Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds, 2023. arXiv:2010.15789.\n- [FL24b] Paul M. N. Feehan and Thomas G. Leness. Almost Hermitian structures on moduli spaces of non-abelian monopoles and applications to the topology of symplectic four-manifolds, 2024. arXiv:2410.13809.\n- [Hit87] N. J. Hitchin. The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3), 55(1):59–126, 1987. doi:10.1112/plms/s3-55.1.59.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2967, "problem_number": "KP-4.91", "title": "Kirby Problem 4.91", "statement": "Present a topological construction of symplectic fake projective planes. Does there exist a symplectic fake projective plane that is not a complex ball quotient?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.91.\n\nLiterature notes:\n(1) A fake projective plane (FPP) is a closed 4–manifold X with the same rational cohomology ring as the complex projective plane $\\mathbb{CP}^{2}$ but not diffeomorphic to it. By Yau’s solution of the Calabi Conjecture [Yau78], every complex FPP is a torsion-free quotient of the complex unit ball by a discrete cocompact subgroup of $P U(1,2)$. The first example of a complex FPP was given by Mumford [Mum79] (hence the alternate name Mumford surface for a complex FPP) using p-adic uniformization, with further examples later given by Ishida and Kato [IK98] and Keum [Keu06]. Notably, Prasad and Yeung [PY07], with the help of computer-assisted calculations by Cartwright and Steeger, established that there are exactly 50 diffeomorphism types for complex FPPs [PY07, PY10, CS10]. In fact, these 50 Kähler surfaces are the only known examples of symplectic FPPs to date and their constructions involve arithmetic geometry. (Keum’s work [Keu06, Keu11] may be largely reinterpreted using topological arguments but also relies on arithmetic geometric results [Ish88].) The problem asks if there are “softer” constructions via symplectic topology.\n\n(2) A reconstruction of even the known complex FPPs using topological methods may lead to the further discovery of non-Kähler, symplectic examples on the Bogomolov–Miyaoka–Yau line. See Problem 4.90.\n\n(3) Every complex FPP is smoothly irreducible [BSS24, Proposition 6.1]; this fact extends to any other potential symplectic FPP under mild assumptions on its fundamental group. Indeed, Fintushel and Stern raised the more general question of whether there exist smoothly irreducible FPPs besides the complex ones and what could be said about their $\\pi_{1}$. (For reducible examples, one can simply take a connected sum of any rational homology 4-sphere with $\\mathbb{CP}^{2}.)$ Irreducible FPPs with various fundamental groups were recently constructed in [BSS24]; for instance, any finite abelian group with $a \\mathbb{Z}_{2}$ factor is claimed to be realized as the $\\pi_{1}$ of an irreducible FPP. None of these FPPs admit symplectic structures, and in fact, it can be shown that the approach in [BSS24] falls short of producing symplectic examples.\n\n(4) The existence of an (irreducible) FPP with trivial $\\pi_{1}$, which amounts to an exotic smooth structure on $\\mathbb{CP}^{2}$, is a particularly important open question. See Problem 4.2. Let G be a finitely presented group and Q be an integral intersection form. Baldridge and Kirk conjectured that the diffeomorphism type of a closed symplectic 4–manifold that minimizes the Euler characteristic among all with $\\pi_{1} = G$ and intersection form Q is unique [BK07, Conjecture 23]. A symplectic FPP with trivial $\\pi_{1}$ would be a counter-example to a special case of this conjecture, known as the Symplectic Poincaré\n\n\\paragraph{Conjecture.}\n\nReferences cited:\n- [Yau78] Shing Tung Yau. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math., 31(3):339–411, 1978. doi:10.1002/cpa.3160310304.\n- [Mum79] D. Mumford. An algebraic surface with K ample, $K^2=9,\\ p_g=q=0$. Amer. J. Math., 101(1):233–244, 1979. doi:10.2307/2373947.\n- [IK98] Masa-Nori Ishida and Fumiharu Kato. The strong rigidity theorem for non-Archimedean uniformization. Tohoku Math. J. (2), 50(4):537–555, 1998. doi: 10.2748/tmj/1178224897.\n- [Keu06] JongHae Keum. A fake projective plane with an order 7 automorphism. Topology, 45(5):919–927, 2006. doi:10.1016/j.top.2006.06.006.\n- [PY07] Gopal Prasad and Sai-Kee Yeung. Fake projective planes. Invent. Math., 168(2):321–370, 2007. doi:10.1007/s00222-007-0034-5.\n- [PY10] Gopal Prasad and Sai-Kee Yeung. Addendum to “Fake projective planes” Invent. Math. 168, 321–370 (2007). Invent. Math., 182(1):213–227, 2010. doi:10.1007/s00222-010-0259-6.\n- [CS10] Donald I. Cartwright and Tim Steger. Enumeration of the 50 fake projective planes. C. R. Math. Acad. Sci. Paris, 348(1-2):11–13, 2010. doi:10.1016/j.crma.2009.11.016.\n- [Keu11] JongHae Keum. A fake projective plane constructed from an elliptic surface with multiplicities $(2,4)$. Sci. China Math., 54(8):1665–1678, 2011. doi:10.1007/s11425-011-4247-0.\n- [Ish88] Masa-Nori Ishida. An elliptic surface covered by Mumford’s fake projective plane. Tohoku Math. J. (2), 40(3):367–396, 1988. doi:10.2748/tmj/1178227980.\n- [BSS24] R. Inanc Baykur, Andras I. Stipsicz, and Zoltan Szabo. Smooth structures on fourmanifolds with finite cyclic fundamental groups, 2024. arXiv:2406.09007.\n- [BK07] Scott Baldridge and Paul Kirk. On symplectic 4-manifolds with prescribed fundamental group. Comment. Math. Helv., 82(4):845–875, 2007. doi:10.4171/CMH/112.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2968, "problem_number": "KP-4.92", "title": "Kirby Problem 4.92", "statement": "Is every symplectic Calabi-Yau surface diffeomorphic to either the K3 surface, the Enriques surface or $a T^{2}-bundle$ over $T^{2}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.92.\n\nLiterature notes:\n(1) Here a symplectic Calabi-Yau surface (SCY) is defined as a closed symplectic 4-manifold with torsion first Chern class.\n\n(2) To date, the only smooth 4–manifolds known to support a symplectic structure with torsion $c_{1}$ are the ones listed in the problem. The problem asks whether this list indeed provides a complete classification of the diffeomorphism types of SCYs.\n\n(3) T-J Li [Li06a, Li06b] and Bauer [Bau08] established that any SCY has the rational homology type of one of these standard 4-manifolds. (This extends the earlier results of Morgan and Szabó [MS97] $whenb_{1}$ =0 and Ruberman and Strle [RS00] when $b_{1}$ =4.) Any SCY with $b^{+}_{2}$ >1 has only one Seiberg-Witten basic class, namely the (trivial) canonical class, and its SW invariant is one. When $b^{+}_{2}$ =1, the SW invariant of an SCY is determined by the wall crossing formula, and only depends on the cohomology ring structure.\n\n(4) The list has been verified to be complete in some specific cases. Any SCY that smoothly fibers over a circle, a surface, or a 3–manifold is standard [BF15, Bay14, FV13, LN14, Ni17b]. The same holds when the SCY admits certain finite symplectic group actions [Che20a, Che20b]. Moreover, many well-known construction techniques—such as Luttinger surgery, generalized fiber sums, knot surgery, and the simplest rational blow-downs—have been shown not to yield new SCYs from standard symplectic 4-manifolds; see [Li19] for a detailed survey and references.\n\n(5) One approach towards an affirmative answer is to first detect the existence of a (possibly singular) $T^{2}$-fibration. In the case of a homology K3, if further assuming that it has a winding family of symplectic forms (a symplectic generalization of a hyperkähler family), then a parameterized version of Taubes’ $SW\\Rightarrow GW$ theorem, including a parametrized wall-crossing formula, produces a 2–dimensional family of embedded symplectic tori in the winding family [Li10, Section 7.4].\n\n(6) One may probe the existence of new SCYs via symplectic Lefschetz pencils and multisections by analyzing certain positive factorizations in mapping class groups. New constructions of SCYs realizing all possible rational homology types are given in [BH16b, Bay22, BHM23] adapting this approach. Only a handful of these SCYs are confirmed to be standard so far [HH18a, Ful23]. Due to the shortcomings of gauge-theoretical invariants in distinguishin g SCYs (up to diffeomorphism) within the same homotopy type, the possibility of detecting a new SCY hinges on identifying a new SCY group. An intriguing, not-yet-ruled-out possibility is $\\pi_{1} =\\mathbb{Z}^{2}$ [FV13, Bay22].\n\nReferences cited:\n- [Li06a] Tian-Jun Li. Quaternionic bundles and Betti numbers of symplectic 4-manifolds with Kodaira dimension zero. Int. Math. Res. Not., pages Art. ID 37385, 28, 2006. doi:10.1155/IMRN/2006/37385.\n- [Li06b] Tian-Jun Li. Symplectic 4-manifolds with Kodaira dimension zero. J. Differential Geom., 74(2):321–352, 2006. http://projecteuclid.org/euclid.jdg/1175266207.\n- [Bau08] Stefan Bauer. Almost complex 4-manifolds with vanishing first Chern class. J. Differential Geom., 79(1):25–32, 2008. http://projecteuclid.org/euclid.jdg/1207834656.\n- [MS97] John W. Morgan and Zoltán Szabó. Homotopy K3 surfaces and mod 2 SeibergWitten invariants. Math. Res. Lett., 4(1):17–21, 1997. doi:10.4310/MRL.1997.v4.n1.a2.\n- [RS00] Daniel Ruberman and Sašo Strle. Mod 2 Seiberg-Witten invariants of homology tori. Math. Res. Lett., 7(5-6):789–799, 2000. doi:10.4310/MRL.2000.v7.n6.a11.\n- [BF15] R. İnanç Baykur and Stefan Friedl. Virtually symplectic fibered 4-manifolds. Indiana Univ. Math. J., 64(4):983–999, 2015. doi:10.1512/iumj.2015.64.5591.\n- [Bay14] R. İnanç Baykur. Virtual Betti numbers and the symplectic Kodaira dimension of fibered 4-manifolds. Proc. Amer. Math. Soc., 142(12):4377–4384, 2014. doi: 10.1090/S0002-9939-2014-12151-4.\n- [FV13] Stefan Friedl and Stefano Vidussi. On the topology of symplectic Calabi-Yau 4-manifolds. J. Topol., 6(4):945–954, 2013. doi:10.1112/jtopol/jtt020.\n- [LN14] Tian-Jun Li and Yi Ni. Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori. Math. Z., 277(1-2):195–208, 2014. doi:10.1007/s00209-013-1250-x.\n- [Ni17b] Yi Ni. Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori, II. Sci. China Math., 60(9):1591–1598, 2017. doi:10.1007/s11425-016-9052-8.\n- [Che20a] Weimin Chen. Finite group actions on symplectic Calabi-Yau 4-manifolds with $b_1>0$. J. Gökova Geom. Topol. GGT, 14:1–54, 2020.\n- [Che20b] Weimin Chen. On a class of symplectic 4-orbifolds with vanishing canonical class. J. Gökova Geom. Topol. GGT, 14:55–90, 2020.\n- [Li19] Tian-Jun Li. Kodaira dimension in low dimensional topology. In Tsinghua lectures in mathematics, volume 45 of Adv. Lect. Math. (ALM), pages 265–291. Int. Press, Somerville, MA, [2019] ©2019.\n- [Li10] Tian-Jun Li. Symplectic Calabi-Yau surfaces. In Handbook of geometric analysis, No. 3, volume 14 of Adv. Lect. Math. (ALM), pages 231–356. Int. Press, Somerville, MA, 2010.\n- [BH16b] R. İnanç Baykur and Kenta Hayano. Multisections of Lefschetz fibrations and topology of symplectic 4-manifolds. Geom. Topol., 20(4):2335–2395, 2016. doi: 10.2140/gt.2016.20.2335.\n- [Bay22] R. İnanç Baykur. Small exotic 4-manifolds and symplectic Calabi-Yau surfaces via genus-3 pencils. In Gauge theory and low-dimensional topology—progress and interaction, volume 5 of Open Book Ser., pages 185–221. Math. Sci. Publ., Berkeley, CA, 2022. https://msp.org/obs/2022/5-1/p09.xhtml.\n- [BHM23] R. İnanç Baykur, Kenta Hayano, and Naoyuki Monden. Unchaining surgery and topology of symplectic 4-manifolds. Math. Z., 303(3):Paper No. 77, 32, 2023. doi: 10.1007/s00209-023-03204-x.\n- [HH18a] Noriyuki Hamada and Kenta Hayano. Topology of holomorphic Lefschetz pencils on the four-torus. Algebr. Geom. Topol., 18(3):1515–1572, 2018. doi:10.2140/agt.2018.18.1515.\n- [Ful23] Terry Fuller. Unchaining surgery, branched covers, and pencils on elliptic surfaces. Algebr. Geom. Topol., 23(6):2867–2893, 2023. doi:10.2140/agt.2023.23.2867.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2969, "problem_number": "KP-4.93", "title": "Kirby Problem 4.93", "statement": "Is every symplectic form on the standard K3 surface symplectomorphic to a Kähler form?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.93.\n\nLiterature notes:\n(1) This question has several notable generalizations. First, since every symplectic form on the K3 surface is cohomologous to a Kähler form [Li08, 4.10], this is a special case of the following broader question [Don06]: Question (Donaldson). Let (X, $\\omega_{0})$ be a closed Kähler surface. Is any other symplectic form $\\omega$ on X with $[\\omega] = [\\omega_{0}]$ and $c_{1}(X, \\omega) = c_{1}(X, \\omega_{0})$ symplectomorphic to $\\omega_{0}$? As every closed hyperkähler surface is diffeomorphic to either the K3 surface or $T^{4}$, the problem also constitutes a special case of the following conjecture.\n\n\\paragraph{Conjecture.} Let X be a closed hyperkähler surface and let $a \\in H^{2}(X;\\mathbb{R})$ be such that $a^{2} >$ 0. Then the space of symplectic forms on X representing the class a is connected. Finally, the problem is a significant instance of the more general Problem 4.96.\n\n(2) Donaldson proposed using the almost Kähler Calabi-Yau equation given in [Don06] to get a family of cohomologous symplectic forms connecting the given symplectic form $\\omega$ to the Kähler form $\\omega_{0}$. A partial a priori estimate for the almost Kähler Calabi-Yau equation was obtained in [TWY08].\n\n(3) There are also approaches to this problem via Donaldson’s geometric flow [Don00, KS19] and hypersymplectic flow [FY19].\n\nReferences cited:\n- [Li08] Tian-Jun Li. The space of symplectic structures on closed 4-manifolds. In Third International Congress of Chinese Mathematicians. Part 1, 2, volume 2 of AMS/IP Stud. Adv. Math., 42, pt. 1, pages 259–277. Amer. Math. Soc., Providence, RI, 2008.\n- [Don06] S. K. Donaldson. Two-forms on four-manifolds and elliptic equations. In Inspired by S. S. Chern, volume 11 of Nankai Tracts Math., pages 153–172. World Sci. Publ., Hackensack, NJ, 2006. URL: https://doi.org/10.1142/9789812772688 0007, doi:10.1142/9789812772688\\\\_0007.\n- [TWY08] Valentino Tosatti, Ben Weinkove, and Shing-Tung Yau. Taming symplectic forms and the Calabi-Yau equation. Proc. Lond. Math. Soc. (3), 97(2):401–424, 2008. doi:10.1112/plms/pdn008.\n- [Don00] S. K. Donaldson. Moment maps and diffeomorphisms [ MR1701920 (2001a:53122)]. In Surveys in differential geometry, volume 7 of Surv. Differ. Geom., pages 107– 127. Int. Press, Somerville, MA, 2000. doi:10.4310/SDG.2002.v7.n1.a5.\n- [KS19] Robin S. Krom and Dietmar A. Salamon. The Donaldson geometric flow for symplectic four-manifolds. J. Symplectic Geom., 17(2):381–417, 2019. doi:10.4310/JSG.2019.v17.n2.a3.\n- [FY19] Joel Fine and Chengjian Yao. A report on the hypersymplectic flow. Pure Appl. Math. Q., 15(4):1219–1260, 2019. doi:10.4310/PAMQ.2019.v15.n4.a7.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2970, "problem_number": "KP-4.94", "title": "Kirby Problem 4.94", "statement": "Are homotopy equivalent Horikawa surfaces in different deformation classes diffeomorphic as 4–manifolds? Are they symplectomorphic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.94.\n\nLiterature notes:\n(1) AHorikawa surfaceis a minimal (non-singular) complex projective surface of general type whose Chern numbers satisfy $5c^{2}_{1} = c_{2}$ −36 (i.e. it lies on the Noether line), or equivalently, $c^{2}_{1} = 2p_{g}$ −4, where $p_{g} \\geq$ 3 is the geometric genus. They are all simply connected. These surfaces were studied extensively by Horikawa in [Hor76b, Hor76a, Hor78, Hor79], who classified them up to deformation. For eachr $\\geq$ 2, there are two deformation classes of surfaces with $c^{2}_{1}$ =8r−8 and $c_{2} =$ 40r−4. The smooth 4–manifolds $H(r)$ and H1(r) in the two respective deformation classes are distinguished by their intersection form when r is even; whereas, when r is odd, they are homotopy equivalent. They have the same Donaldson polynomials and the same Seiberg-Witten invariants. Some 50 years after Horikawa, it remains unknown whether any pair of the latter surfaces are diffeomorphic.\n\n(2) The question on the number of diffeomorphism classes was originally raised by Horikawa in [Hor76b] and it is in [Kir97, Problem 4.101(A)]. If two compact complex manifolds X and $X^{1}$ are deformation equivalent, then there exists a diffeomorphism f: $X \\to X^{1}$ such that $f^{*}(c_{1}(X, \\omega)) = c_{1}(X^{1}, \\omega^{1})$. The bold speculation of the time, namely that the converse holds in complex dimension two (the refined “DEF $=$ DIFF Conjecture”), was disproved by Manetti in [Man01], and there are simply connected counterexamples as well [CW07]. A minimal complex surface of general type has a canonical symplectic structure, unique up to symplectomorphism, which is invariant under smooth deformation [Cat09]. Catanese showed that Manetti surfaces are indeed symplectomorphic [Cat09], so they also constitute counterexamples to the elusive “DEF $=$ SYMP Conjecture”. Nonetheless, it is still open whether DEF equivalence is more strict than DIFF or SYMP equivalence for the Horikawa surfaces.\n\n(3) Let $\\mathbb{F}_{2,r}$ denote the Hirzebruch surface with fiber f and sections $\\Delta_{0}$ and $\\Delta_{\\infty}$ with self-intersections $\\Delta^{2}_{0}$ =2rand $\\Delta^{2}_{\\infty} =$ −2r. The Horikawa surface $H(r)$ is the double cover of $\\mathbb{F}_{0}$ branched over a smoothing of $6\\Delta_{0}+4rf$ and $H^{1}(r)is$ the double cover $of\\mathbb{F}_{2,r}$ branched over a disconnected branch locus that is a smoothing of $5\\Delta_{0}+\\Delta_{\\infty}$. Based on this, one can describe H(r)and $H^{1}(r)as$ smooth 4–manifolds using Kirby diagrams for branched covers, and as symplectic 4–manifolds via monodromy factorizations for compatible Lefschetz fibrations/pencils [Ful98, Aur06a]. In [Aur06a], Auroux compared the canonical symplectic Lefschetz pencils of the same genus on $H(3)$ and H1(3) and observed that they are related through fibered Luttinger surgeries. As suggested by the status of the problem, there has been otherwise no success in relating such diagrams via Kirby calculus or the pencils via monodromy manipulations to prove DIFF or SYMP equivalence.\n\n(4) Another possible strategy to obtain a diffeomorphism between $H(r)$ and $H^{1}(r)$ is based on the observation of Manetti, that a common degeneration with Wahl singularities would prove that they are diffeomorphic [Man01][Theorem 1.5]. Recently in [MNU24], through an extensive study of complex surfaces with Wahl singularities, the authors invalidated this approach. On the other hand, a different common degeneration was recently found in [RR24], without any interpretation of moving from one deformation component to the other as diffeomorphism.\n\nReferences cited:\n- [Hor76b] Eiji Horikawa. Algebraic surfaces of general type with small $c_1^2$. I. Ann. of Math. (2), 104(2):357–387, 1976. doi:10.2307/1971050.\n- [Hor76a] Eiji Horikawa. Algebraic surfaces of general type with small $c_1^2$. II. Invent. Math., 37(2):121–155, 1976. doi:10.1007/BF01418966.\n- [Hor78] Eiji Horikawa. Algebraic surfaces of general type with small $c_1^2$. III. Invent. Math., 47(3):209–248, 1978. doi:10.1007/BF01579212.\n- [Hor79] Eiji Horikawa. Algebraic surfaces of general type with small $c_1^2$. IV. Invent. Math., 50(2):103–128, 1978/79. doi:10.1007/BF01390285.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Man01] Marco Manetti. On the moduli space of diffeomorphic algebraic surfaces. Invent. Math., 143(1):29–76, 2001. doi:10.1007/s002220000101.\n- [CW07] Fabrizio Catanese and Bronislaw Wajnryb. Diffeomorphism of simply connected algebraic surfaces. J. Differential Geom., 76(2):177–213, 2007. doi:10.4310/jdg/1180135677.\n- [Cat09] Fabrizio Catanese. Canonical symplectic structures and deformations of algebraic surfaces. Commun. Contemp. Math., 11(3):481–493, 2009. doi:10.1142/S0219199709003478.\n- [Ful98] Terry Fuller. Diffeomorphism types of genus 2 Lefschetz fibrations. Math. Ann., 311(1):163–176, 1998. doi:10.1007/s002080050182.\n- [Aur06a] Denis Auroux. The canonical pencils on Horikawa surfaces. Geom. Topol., 10:2173– 2217, 2006. doi:10.2140/gt.2006.10.2173.\n- [MNU24] Vicente Monreal, Jaime Negrete, and Giancarlo Urzúa. Classification of horikawa surfaces with t-singularities, 2024. arXiv:2410.02943.\n- [RR24] Julie Rana and Sönke Rollenske. Standard stable Horikawa surfaces. Algebr. Geom., 11(4):569–592, 2024.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2971, "problem_number": "KP-4.95", "title": "Kirby Problem 4.95", "statement": "(a) Is there a closed hyperbolic oriented 4-manifold that admits a symplectic structure?\n\n(b) Do the Seiberg–Witten invariants vanish on every closed hyperbolic 4manifold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.95.\n\nLiterature notes:\n(1) In a preprint [Sto25], Stover announces the construction of a hyperbolic orbifold structure on $\\mathbb{CP}^{2}$. This resolves an orbifold version of this problem.\n\n(2) By work of Taubes [Tau94], a positive answer to (a) would imply a negative answer to (b); see [Rei06, Proposition 4.5]. By Donaldson [Don99] and Gompf–Stipsicz [GS99, Theorem 10.2.18], (a) is equivalent to askin g whether there is a closed hyperbolic oriented 4-manifold that admits a smooth Lefschetz pencil. Compare this to Problem 2.23, which asks whether there exists a surface bundle over a surface whose total space is a hyperbolic 4-manifold.\n\n(3) In [Le B02, Conjecture 1.1], Le Brun conjectures that the answer to (b) is, ‘yes,’ and hence the answer to (a) is ‘no.’ Le Brun provides evidence towards the moduli spaces being empty (for suitable perturbations) and proposes that one should consider the generalization of Seiberg-Witten invariants involving the evaluation of cohomology classes in $\\Lambda^{*}H^{1}(X;\\mathbb{Z}) \\otimes \\mathbb{Z}[U]$ on higher dimensional moduli spaces. The first examples of manifolds for which all these invariants vanish were provided in [AL20], followed by more concrete examples in [BFS24]. It is worth noting that there are no known examples of closed hyperbolic 4-manifolds with $b^{+}_{2} \\leq$ 1, cf. Problem 4.19.\n\nReferences cited:\n- [Sto25] Matthew Stover. A hyperbolic 4-orbifold with underlying space P2, 2025. arXiv: 2506.11667.\n- [Tau94] Clifford Henry Taubes. The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett., 1(6):809–822, 1994. doi:10.4310/MRL.1994.v1.n6.a15.\n- [Rei06] Alan W. Reid. Surface subgroups of mapping class groups. In Problems on mapping class groups and related topics, volume 74 of Proc. Sympos. Pure Math., pages 257– 268. Amer. Math. Soc., Providence, RI, 2006. doi:10.1090/pspum/074/2264545.\n- [Don99] S. K. Donaldson. Lefschetz pencils on symplectic manifolds. J. Differential Geom., 53(2):205–236, 1999. http://projecteuclid.org/euclid.jdg/1214425535.\n- [GS99] Robert E. Gompf and András I. Stipsicz. 4-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1999. doi:10.1090/gsm/020.\n- [LeB02] Claude LeBrun. Hyperbolic manifolds, harmonic forms, and Seiberg-Witten invariants. In Proceedings of the Euroconference on Partial Differential Equations and their Applications to Geometry and Physics (Castelvecchio Pascoli, 2000), volume 91, pages 137–154, 2002. doi:10.1023/A:1016222709901.\n- [AL20] Ian Agol and Francesco Lin. Hyperbolic four-manifolds with vanishing SeibergWitten invariants. In Characters in low-dimensional topology, volume 760 of Contemp. Math., pages 1–8. Amer. Math. Soc., [Providence], RI, [2020] ©2020. doi:10.1090/conm/760/15283.\n- [BFS24] Ludovico Battista, Leonardo Ferrari, and Diego Santoro. Dodecahedral L-spaces and hyperbolic 4-manifolds. Comm. Anal. Geom., 32(8):2095–2134, 2024. doi:10.4310/cag.241212004157.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2972, "problem_number": "KP-4.96", "title": "Kirby Problem 4.96", "statement": "Does there exist a pair of symplectic 4–manifolds $(X_{1}, \\omega_{1})and (X_{2}, \\omega_{2})$, where there is a diffeomorphism $f: X_{1} \\to X_{2}$ such that $f*(c_{1}(X_{2}, \\omega_{2})) = c_{1}(X_{1}, \\omega_{1})$ and $f*([\\omega_{2}]) = [\\omega_{1}] \\in H^{2}(X_{1},\\mathbb{R})$, but $(X_{1}, \\omega_{1})$ is not symplectomorphic to $(X_{2}, \\omega_{2})$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.96.\n\nLiterature notes:\n(1) It may be difficult to detect such a difference with current invariants due to Taubes’ result relating Gromov–Witten invariants and Seiberg–Witten invariants [Tau96].\n\n(2) There is also a relative version of this problem for fillings.\n\n\\paragraph{Question.} Does there exist a pair of Stein manifolds filling the same contact3-manifold that are diffeomorphic where the diffeomorphism takes the first Chern class of one to that of the other, but such that the Stein manifolds are not symplectomorphic? One could also ask whether there is such a pair of Stein manifolds that are not Weinstein homotopic. See also Problem 5.17.\n\nReferences cited:\n- [Tau96] Clifford H. Taubes. SW ñ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves. J. Amer. Math. Soc., 9(3):845–918, 1996. doi:10.1090/S0894-0347-96-00211-1.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2973, "problem_number": "KP-4.97", "title": "Kirby Problem 4.97", "statement": "Let $\\lambda:=c^{2}_{1}/c_{2}$ be the Chern slope of a closed, almost complex 4–manifold X. What is the supremum of $\\lambda as X$ ranges over the following families?\n\n(a) Symplectic $\\Sigma_{g}–bundles$ over $\\Sigma_{h}$, with g, $h \\geq$ 2.\n\n(b) Holomorphic $\\Sigma_{g}–bundles$ over $\\Sigma_{h}$, with g, $h \\geq$ 2.\n\n(c) Symplectic Lefschetz fibrations over $S^{2}$.\n\n(d) Holomorphic Lefschetz fibrations over $S^{2}$. (Here, the Lefschetz fibrations are assumed to have critical points.)", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.97.\n\nLiterature notes:\n(1) Recall that the Chern numbers of X are determined by its homotopy type. Any $\\Sigma_{g}–bundle$ over $\\Sigma_{h}$ with $g \\geq$ 2 or a Lefschetz fibration (with nonempty critical set) over $S^{2}$ can be made symplectic, but not always holomorphic. See e.g. [Joh86, Bay12b]. In fact, the sole obstruction to making a surface bundle over a surface holomorphic is whether the smooth 4–manifold X admits a complex structure [Hil00, Kot99].\n\n(2) The first line of research in this direction, pioneered by Atiyah, Kodaira, and Hirzebruch in the late 1960s, and Endo in the late 1990s, is to understand for which pairs of $g \\geq$ 3 and $h \\geq$ 2 one can have $a \\Sigma_{g}–bundle$ over $\\Sigma_{h}$ with $\\sigma$ >0. (The signature vanishes when $g \\leq$ 2 or $h \\leq$ 1 [Mey73].) This geography problem was nearly resolved in the recent work of Baykur and Korkmaz [BK24b], who showed that for all but 19 possible pairs (g, h), there are symplectic $\\Sigma_{g}–bundles$ over $\\Sigma_{h}$ with $\\sigma >$ 0. However, the remaining few cases are quite significant for symplectic geography; see below.\n\n(3) The Bogomolov–Miyaoka–Yau inequality for complex surfaces is encoded as $\\lambda \\leq$ 3, providing an upper bound on the slopes of X in (b) and (d). Since no complex ball quotient can smoothly fiber over a surface, it is moreover known that any X in (b) satisfies $\\lambda<3$. Two of the unsettled cases in [BK24b] have direct implications on symplectic geography: any example with (g, h) $=$ (3,2) and $\\sigma >$ 0 is a symplectic X of general type violating the BMY inequality, whereas one with(g, h) $=$ (4,2)and the smallest possible positive signature would give a symplectic X on the BMY line and cannot possibly be complex. See [BK24b, Question 2] and Problem 4.90. Do such surface bundles over surfaces exist? An unpublished preprint by Hamenstädt [Ham20] announces that $\\lambda \\leq$ 3 more generally for any X in (a).\n\n(4) Presumably, these questions will all have different answers. Nonetheless, the largest currently known slope for (a) and (b) coincide: these are the examples by Catanese and Rollenske with $\\lambda =$ 8/3 [CR09]. There exist holomorphic $\\Sigma_{g}–bundles$ over $\\Sigma_{h}$ with positive signatures, for h=2 (and largeg), as shown by Bryan and Donagi in [BD02], and with g =3 (and unspecified, presumably very large h), as shown by Kazuhiro Konno (in works only available in Japanese). The situation for (c) and (d) is more mysterious; for instance, (symplectic) Lefschetz fibrations over $S^{2}$ with $\\sigma>0$ were only recently discovered in [BH24c]. There are fewer constraints on the slopes of Lefschetz fibrations over higher genera surfaces; e.g. the Cartwright-Steger surface on the BMY line admits a holomorphic Lefschetz fibration over $T^{2}$ [KY21], realizing the maximal possible slope $\\lambda=3$ in this case.\n\nReferences cited:\n- [Joh86] F. E. A. Johnson. A class of non-Kählerian manifolds. Math. Proc. Cambridge Philos. Soc., 100(3):519–521, 1986. doi:10.1017/S030500410006624X.\n- [Bay12b] R. İnanç Baykur. Non-holomorphic surface bundles and Lefschetz fibrations. Math. Res. Lett., 19(3):567–574, 2012. doi:10.4310/MRL.2012.v19.n3.a5.\n- [Hil00] Jonathan A. Hillman. Complex surfaces which are fibre bundles. Topology Appl., 100(2-3):187–191, 2000. doi:10.1016/S0166-8641(98)00085-6.\n- [Kot99] D. Kotschick. On regularly fibered complex surfaces. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 291–298. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.291.\n- [Mey73] Werner Meyer. Die Signatur von Flächenbündeln. Math. Ann., 201:239–264, 1973. doi:10.1007/BF01427946.\n- [BK24b] R. İnanç Baykur and Mustafa Korkmaz. Geography of surface bundles over surfaces. Math. Ann., 390(4):5793–5817, 2024. doi:10.1007/s00208-024-02899-5.\n- [Ham20] Ursula Hamenstädt. Signature of surface bundles and bounded cohomology, 2020. arXiv:2011.05792.\n- [CR09] Fabrizio Catanese and Sönke Rollenske. Double Kodaira fibrations. J. Reine Angew. Math., 628:205–233, 2009. doi:10.1515/CRELLE.2009.024.\n- [BD02] Jim Bryan and Ron Donagi. Surface bundles over surfaces of small genus. Geom. Topol., 6:59–67, 2002. doi:10.2140/gt.2002.6.59.\n- [BH24c] R. İnanç Baykur and Noriyuki Hamada. Lefschetz fibrations with arbitrary signature. J. Eur. Math. Soc. (JEMS), 26(8):2837–2895, 2024. doi:10.4171/jems/1326.\n- [KY21] Vincent Koziarz and Sai-Kee Yeung. Stability of the Albanese fibration on the Cartwright-Steger surface. Taiwanese J. Math., 25(2):251–256, 2021. doi:10.11650/tjm/201108.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2974, "problem_number": "KP-4.98", "title": "Kirby Problem 4.98", "statement": "Does every closed symplectic 4-manifold admit inequivalent Lefschetz pencils with the same fiber genus g, for sufficiently large g? How about infinitely many?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.98.\n\nLiterature notes:\n(1) Here Lefschetz pencils are assumed to have base points, whereas Lefschetz fibrations do not. Equivalence is defined by a diffeomorphism of the symplectic 4–manifold commuting with any pair of pencil/fibration maps.\n\n(2) Any symplectic 4–manifold can be equipped with Lefschetz pencils with arbitrarily high fiber genera by increasing the degree in Donaldson’s construction. However, the number of base points also increases, meaning the Lefschetz fibrations derived by blowing up the base points would be on birational yet different 4–manifolds. Except for rational and ruled surfaces, the number of base points in a pencil is bounded above by 2g − 2, where g is the genus of the fiber. Therefore, a positive answer to the second question would imply the existence of infinitely many genus–g Lefschetz pencils with the same number of base points. By blowing up the base points of such examples, one can obtain inequivalent Lefschetz fibrations over $S^{2}$ with the same fiber genus, but the converse is not always feasible.\n\n(3) By [Bay16], any symplectic 4–manifold that is not a rational or ruled surface, possibly after blow-ups, admits arbitrarily many non-isomorphic Lefschetz pencils with the same genus, the same number of base points, and matching topological types of singular fibers. A slightly weaker result also holds for rational and ruled surfaces [Bay19]. These constructions require blow ups, addressing the problem only up to birational equivalence. Other examples of inequivalent Lefschetz pencils and fibrations on a few specific 4–manifolds are given in [PY09, PY17, BH16b, BHM23, Ham17]. The monodromy invariants and arguments used in these works do not distinguish more than finitely many classes; for a positive resolution of the problem, one needs finer monodromy invariants.\n\nReferences cited:\n- [Bay16] R. İnanç Baykur. Inequivalent Lefschetz fibrations and surgery equivalence of symplectic 4-manifolds. J. Symplectic Geom., 14(3):671–686, 2016. doi:10.4310/JSG.2016.v14.n3.a2.\n- [Bay19] R. İnanç Baykur. Inequivalent Lefschetz fibrations on rational and ruled surfaces. In Breadth in contemporary topology, volume 102 of Proc. Sympos. Pure Math., pages 21–28. Amer. Math. Soc., Providence, RI, 2019. doi:10.1090/pspum/102/02.\n- [PY09] Jongil Park and Ki-Heon Yun. Nonisomorphic Lefschetz fibrations on knot surgery 4-manifolds. Math. Ann., 345(3):581–597, 2009. doi:10.1007/s00208-009-0366-0.\n- [PY17] Jongil Park and Ki-Heon Yun. Lefschetz fibrations on knot surgery 4-manifolds via Stallings twist. Michigan Math. J., 66(3):481–498, 2017. doi:10.1307/mmj/1497513628.\n- [BH16b] R. İnanç Baykur and Kenta Hayano. Multisections of Lefschetz fibrations and topology of symplectic 4-manifolds. Geom. Topol., 20(4):2335–2395, 2016. doi: 10.2140/gt.2016.20.2335.\n- [BHM23] R. İnanç Baykur, Kenta Hayano, and Naoyuki Monden. Unchaining surgery and topology of symplectic 4-manifolds. Math. Z., 303(3):Paper No. 77, 32, 2023. doi: 10.1007/s00209-023-03204-x.\n- [Ham17] Noriyuki Hamada. Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration, 2017. arXiv:1610.08458.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2975, "problem_number": "KP-4.99", "title": "Kirby Problem 4.99", "statement": "Let X be a closed symplectic 4-manifold. Let $T \\subset X$ be a symplectic submanifold that is diffeomorphic to a 2-dimensional torus such that $[T]^{2} =$ 0. Let $X_{K}$ be a manifold obtained by Fintushel–Stern knot surgery on T using a knot $K \\subset S^{3}$. If $X_{K}$ has a symplectic structure, must K be a fibered knot?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.99.\n\nLiterature notes:\n(1) Fintushel and Stern [FS98] showed that $X_{K}$ has a symplectic structure if K is a fibered knot, and raised the question in the problem.\n\n(2) One can ask the question for the particular case when X admits a Lefschetz fibration whose regular fibers are tori and T is a regular fiber. This particular case was explicitly stated in [Ni17a]. An interesting example is the case when X is the K3 surface, then $SW(X)$ is essentially the Alexander polynomial $\\Delta_{K}$ of K [FS98], so we know $\\Delta_{K}$ should be monic if $X_{K}$ has a symplectic structure [Tau94, Tau95]. No other constraint is known.\n\n(3) The answer to this problem is “Yes” when $X = T^{2} \\times S^{2}$ and $T = T^{2} \\times$ \\{point\\} by Friedl–Vidussi [FV11], and when X is a torus bundle over a closed surface with homologically essential fibers and T is a fiber by Ni [Ni17a].\n\nReferences cited:\n- [FS98] Ronald Fintushel and Ronald J. Stern. Knots, links, and 4-manifolds. Invent. Math., 134(2):363–400, 1998. doi:10.1007/s002220050268.\n- [Ni17a] Yi Ni. Fintushel-Stern knot surgery in torus bundles. J. Topol., 10(1):164–177, 2017. doi:10.1112/topo.12002.\n- [Tau94] Clifford Henry Taubes. The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett., 1(6):809–822, 1994. doi:10.4310/MRL.1994.v1.n6.a15.\n- [Tau95] Clifford Henry Taubes. More constraints on symplectic forms from Seiberg-Witten invariants. Math. Res. Lett., 2(1):9–13, 1995. doi:10.4310/MRL.1995.v2.n1.a2.\n- [FV11] Stefan Friedl and Stefano Vidussi. Twisted Alexander polynomials detect fibered 3-manifolds. Ann. of Math. (2), 173(3):1587–1643, 2011. doi:10.4007/annals.2011.173.3.8.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2976, "problem_number": "KP-4.100", "title": "Kirby Problem 4.100", "statement": "Given a closed, connected, symplectic 4-manifold (X, $\\omega)$ and $c \\in H_{2}(X,\\mathbb{Z})$ represented by an embedded, connected, oriented, smooth surface S such that\n\n(i) $x[\\omega]$, cy $>$ 0, and\n\n(ii) $xc_{1}(X, \\omega)$, cy $=\\chi(S) +S^{2}$, is c represented by an embedded, connected, symplectic surface?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.100.\n\nLiterature notes:\n(1) This is a special case of the question of which homology classes in a symplectic manifold are represented by symplectic surfaces. By the adjunction formula, the homology class of every embedded symplectic surface satisfies both (i) and (ii). The Symplectic Thom Conjecture [OS00] states that symplectic surfaces minimize genus in their homology classes. In their proof of this conjecture, Ozsváth and Szabó showed that every class satisfies the adjunction inequality, $-\\chi(S) \\geq S^{2}+ |xc_{1}(X, \\omega)$, cy|, provided $b^{+}_{2}(X) >$ 1. (The result extends to $b^{+}_{2}(X) =$ 1 unless X is a rational or ruled surface.) The question is whether this is a sufficient condition to conclude the existence of a symplectic representative. See also [DLW18].\n\n(2) By Donaldson [Don99], $if[\\omega] \\in H_{2}(X,\\mathbb{Z})$, then all large enough multiples $of[\\omega]are$ represented by symplectic submanifolds. $(Whenb^{+}_{2}(X)$ =1, one can say a bit more; see e.g. [Li08, Proposition 3.18].) When $b^{+}_{2}(X)$ >1, Taubes [Tau96] showed this is also true for the Poincaré dual of the canonical class $c_{1}(X, \\omega)$; also see [DS03].\n\nReferences cited:\n- [OS00] Peter Ozsváth and Zoltán Szabó. The symplectic Thom conjecture. Ann. of Math. (2), 151(1):93–124, 2000. doi:10.2307/121113.\n- [DLW18] Josef G. Dorfmeister, Tian-Jun Li, and Weiwei Wu. Stability and existence of surfaces in symplectic 4-manifolds with $b^+=1$. J. Reine Angew. Math., 742:115–155, 2018. doi:10.1515/crelle-2015-0083.\n- [Don99] S. K. Donaldson. Lefschetz pencils on symplectic manifolds. J. Differential Geom., 53(2):205–236, 1999. http://projecteuclid.org/euclid.jdg/1214425535.\n- [Li08] Tian-Jun Li. The space of symplectic structures on closed 4-manifolds. In Third International Congress of Chinese Mathematicians. Part 1, 2, volume 2 of AMS/IP Stud. Adv. Math., 42, pt. 1, pages 259–277. Amer. Math. Soc., Providence, RI, 2008.\n- [Tau96] Clifford H. Taubes. SW ñ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves. J. Amer. Math. Soc., 9(3):845–918, 1996. doi:10.1090/S0894-0347-96-00211-1.\n- [DS03] Simon Donaldson and Ivan Smith. Lefschetz pencils and the canonical class for symplectic four-manifolds. Topology, 42(4):743–785, 2003. doi:10.1016/S0040-9383(02)00024-1.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2977, "problem_number": "KP-4.101", "title": "Kirby Problem 4.101", "statement": "Is every smooth symplectic surface in $(\\mathbb{CP}^{2}, \\omega_{FS})$ symplectically isotopic to a complex curve? Equivalently, is there a unique symplectic isotopy class of smoothly embedded symplectic surfaces of degree d in $(\\mathbb{CP}^{2}, \\omega_{FS})$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.101.\n\nLiterature notes:\n(1) Here $\\omega_{FS}$ denotes the standard Fubini–Study symplectic form on $\\mathbb{CP}^{2}$. A symplectic surface in $\\mathbb{CP}^{2}$ is said to have degree d if it represents the homology class $dh\\in H_{2}(\\mathbb{CP}^{2};\\mathbb{Z})$, where $h$ is the generator of $H_{2}(\\mathbb{CP}^{2};\\mathbb{Z}) \\cong \\mathbb{Z}$ represented by a complex projective line. Note that symplectic surfaces always represent apositive multiple of h because they have positive symplectic area.\n\n(2) A complex projective algebraic plane curve of degree d is the zero set of a homogeneous polynomial of degree d in $\\mathbb{CP}^{2}$. Such a curve represents dh $\\in H_{2}(\\mathbb{CP}^{2};\\mathbb{Z})$. If the zero set is regularly cut out, the complex algebraic curve is smooth. Any two smooth complex algebraic curves of the same degree are isotopic, i.e. connected through a family of smooth complex algebraic curves. This can be seen by observing that the set of singular curves of degree d (corresponding to homogeneous polynomials that do not regularly cut out 0) is a complex subvariety of positive complex codimension (cut out by the discriminant) in the space of all degree-d curves (a high dimensional complex projective space parametrized by the coefficients of the monomials in a degree-d homogeneous polynomial). Thus the smooth curves form a path-connected space, as it is the complement of a subset of real codimension at least 2. Thus, the two questions in the problem statement are equivalent and are known as the “symplectic isotopy problem.”\n\n(3) Key progress on the symplectic isotopy problem relies on the fact that every symplectic surface can be realized as a J-holomorphic curve for some almost complex J compatible with the symplectic form. The development of pseudoholomorphic curves originates from Gromov’s seminal paper [Gro85]. It is proven in Gromov’s work that the answer to the symplectic isotopy question is yes in degrees 1 and 2. Extensive further work using Gromov’s strategy of pseudoholomorphic curves shows that the answer is yes for degrees less than or equal to 17 [She00, Sik03, ST05]. After degree 17, we have been unable to rule out the possibility that a family of $J_{t}-holomorphic$ curves may degenerate to a singular curve with unreduced components, and it is unknown whether such degenerations can produce an unavoidable codimension-1 “wall” in the moduli space of pseudoholomorphic curves.\n\n(4) Another approach to this problem is via quasipositive factorizations in the braid group. Given a smooth symplectic surface in $\\mathbb{CP}^{2}$ realized as a J-holomorphic curve, one can generate a J-holomorphic linear pencil on $\\mathbb{CP}^{2}, \\pi: \\mathbb{CP}^{2} \\setminus$ \\{p\\} $\\to \\mathbb{CP}^{1}$. Restricting the pencil to the symplectic surface gives a simple branched covering from the surface to $\\mathbb{CP}^{1}$, assuming the pencil point p is chosen sufficiently generically. The branch points correspond to places where the fibers of the pencil are tangent to the symplectic surface. Looking at the preimages under $\\pi of$ loops in $\\mathbb{CP}^{1}$ gives a braid monodromy presentation that fully encodes the symplectic surface [MT88, MT91]. Through this, the symplectic isotopy problem can be related to the following problem in the braid group.\n\n\\paragraph{Question.} Let $\\Delta^{2}$ denote the full twist on d strands in the braid group $B_{d}$, and let $\\sigma_{i}$ denote the standard generators of the braid group (a half twist of two adjacent strands). If $\\rho_{1}$. . . $\\rho_{k} =\\Delta^{2}$ and each $\\rho_{j}$ is a conjugate of some $\\sigma_{i}$, then is there a sequence of Hurwitz moves and global conjugations one can perform on $\\rho_{1}$. . . $\\rho_{k}$ to turn it into $(\\sigma_{1}$. . . $\\sigma_{d,-,1})^{d}$? For more background on this perspective and the definition of Hurwitz moves, see this survey by Auroux [Aur06b].\n\n(5) Another strategy to attack the symplectic isotopy problem is proposed in [Sta20], which uses deformations of the smooth symplectic surface to a singular surface (a symplectic line arrangement) to show that the symplectic isotopy problem is equivalent to the existence of certain Lagrangian disks with boundary on the symplectic surface. Finding or obstructing the existence of such Lagrangians could potentially be approached using Floer theoretic/Fukaya categorical techniques.\n\n(6) Note that there are smooth surfaces in $\\mathbb{CP}^{2}$ that are not smoothly isotopic to any complex curve [Fin02, Kim06]. However, one can mix the symplectic and smooth categories for the following weakening of the symplectic isotopy problem, which is currently equally open.\n\n\\paragraph{Question.} Are any two symplectic surfaces in $\\mathbb{CP}^{2}$ of the same degree smoothly isotopic? A strategy for answering this weaker version of the problem using transverse bridge trisections was proposed by Lambert-Cole [LC23].\n\n(7) There are examples of symplectic surfaces that are homologous but not symplectically isotopic in other closed symplectic 4-manifolds [FS99b]. However, the question appears to also be open more generally for rational and ruled surfaces, and their blow-ups (symplectic manifolds of Kodaira dimension $-\\infty)$. In ruled surfaces $S^{2} \\times S^{2}, \\mathbb{CP}^{2}\\#_{n}\\mathbb{CP}^{2}$, or an $S^{2}$ bundle over a highergenus surface, in a fixed homology class, is there a unique symplectic isotopy class of symplectic surfaces?\n\nReferences cited:\n- [Gro85] M. Gromov. Pseudo holomorphic curves in symplectic manifolds. Invent. Math., 82(2):307–347, 1985. doi:10.1007/BF01388806.\n- [She00] Vsevolod Shevchishin. Pseudoholomorphic curves and the symplectic isotopy problem, 2000. arXiv:math/0010262.\n- [Sik03] Jean-Claude Sikorav. The gluing construction for normally generic J-holomorphic curves. In Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001), volume 35 of Fields Inst. Commun., pages 175–199. Amer. Math. Soc., Providence, RI, 2003. doi:10.1090/fic/035/12.\n- [ST05] Bernd Siebert and Gang Tian. On the holomorphicity of genus two Lefschetz fibrations. Ann. of Math. (2), 161(2):959–1020, 2005. doi:10.4007/annals.2005.161.959.\n- [MT88] B. Moishezon and M. Teicher. Braid group technique in complex geometry. I. Line arrangements in $\\mathbb{CP}^{2}$. In Braids (Santa Cruz, CA, 1986), volume 78 of Contemp. Math., pages 425–555. Amer. Math. Soc., Providence, RI, 1988. doi:10.1090/conm/078/975093.\n- [MT91] Boris Moishezon and Mina Teicher. Braid group technique in complex geometry. II. From arrangements of lines and conics to cuspidal curves. In Algebraic geometry (Chicago, IL, 1989), volume 1479 of Lecture Notes in Math., pages 131–180. Springer, Berlin, 1991. doi:10.1007/BFb0086269.\n- [Aur06b] Denis Auroux. Symplectic 4-manifolds, singular plane curves, and isotopy problems. In Floer homology, gauge theory, and low-dimensional topology, volume 5 of Clay Math. Proc., pages 263–276. Amer. Math. Soc., Providence, RI, 2006.\n- [Sta20] Laura Starkston. A new approach to the symplectic isotopy problem. J. Symplectic Geom., 18(3):939–960, 2020. doi:10.4310/JSG.2020.v18.n3.a11.\n- [Fin02] Sergey Finashin. Knotting of algebraic curves in $\\mathbb{CP}^{2}$. Topology, 41(1):47–55, 2002. doi:10.1016/S0040-9383(00)00023-9.\n- [Kim06] Hee Jung Kim. Modifying surfaces in 4-manifolds by twist spinning. Geom. Topol., 10:27–56, 2006. doi:10.2140/gt.2006.10.27.\n- [LC23] Peter Lambert-Cole. Symplectic surfaces and bridge position. Geom. Dedicata, 217(1):Paper No. 8, 15, 2023. doi:10.1007/s10711-022-00742-2.\n- [FS99b] Ronald Fintushel and Ronald J. Stern. Symplectic surfaces in a fixed homology class. J. Differential Geom., 52(2):203–222, 1999. http://projecteuclid.org/euclid.jdg/1214425276.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2978, "problem_number": "KP-4.102", "title": "Kirby Problem 4.102", "statement": "Is every symplectic rational cuspidal curve in $(\\mathbb{CP}^{2}, \\omega_{FS})$ equisingularly symplectically isotopic to a complex curve? More generally, which types of singular symplectic surfaces in $(\\mathbb{CP}^{2}, \\omega_{FS})$ (where type is specified by the genus of each irreducible component and the topological types of the singularities), admit symplectic representatives that are not equisingularly symplectically isotopic to complex curves?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.102.\n\nLiterature notes:\n(1) This is a singular version of the symplectic isotopy problem (Problem 4.101). Note that “curve” here refers to a real 2-dimensional surface in analogy with the terminology for complex/pseudoholomorphic curves. A rational cuspidal curve is a singular curve homeomorphic to $S^{2}$.\n\n(2) We say that two singular symplectic surfaces are equisingularly symplectically isotopic if there is a family of symplectic surfaces connecting them such that the topological type of each singularity remains constant through the isotopy. (The topological type is the isotopy class of the link of the singularity.)\n\n(3) Although one could define symplectic surfaces whose singularities are a cone on any transverse knot, for there to be any chance that the symplectic curve is equisingularly isotopic to a complex curve, its singularities must be modeled on the singularities appearing in complex plane curves. By a result of Mc Duff [Mc D92] (reproved by Micallef–White [MW95] through different methods), these are precisely the singularity types which can be realized in J-holomorphic curves for some almost complex structure J compatible with the standard symplectic form on $\\mathbb{CP}^{2}$. Thus these types of singular symplectic surfaces can be studied with pseudoholomorphic techniques.\n\n(4) There are a number of works which approach this problem for certain classes of singular surfaces and are able to prove in some cases that the symplectic surfaces with certain specified singularity types are symplectically isotopic to complex curves. For example Baurraud established symplectic isotopy results for nodal symplectic spheres [Bar99] and sufficiently generic line arrangements [Bar00]. Shevchishin proved results for nodal symplectic surfaces of sufficiently low genus [She04], conjecturing this holds more generally with a positivity assumption on the Chern number. Ohta–Ono proved uniqueness up to symplectic isotopy of a cuspidal cubic [OO05]. More generally, unicuspidal single Puiseux pair families and low degree examples of symplectic rational cuspidal curves were shown to always be equisingularly symplectically isotopic to complex curves in [GS22b, GK23], motivating the plausibility of a positive answer to the first question in the problem statement.\n\n(5) There are a number of ad hoc examples of singular symplectic surfaces in $\\mathbb{CP}^{2}$ that are not equisingularly isotopic to any complex curve. Possibly the first such examples appeared in Moishezon’s work [Moi94]. These examples were high degree and contained a very large number of nodes (positive transverse double points) and simple cusps (modeled $onz_{1}^{3} =z^{2}_{2})$. An important tool in constructing and detecting such examples is braid monodromy [MT88, KK03]. An easier and lower-degree example is the “fake Pappus” line arrangement which cannot be realized by complex projective lines, but which can be realized symplectically (see [RS19]). For an irreducible example of degree 8 with locally reducible singularities see [GS22b, Section 8]. However, we do not currently have any examples where the surface is irreducible and the singularities are locally irreducible, a.k.a., “cuspidal”.\n\n(6) One could also ask whether every pair of equisingular singular symplectic surfaces in $\\mathbb{CP}^{2}$ are equisingularly symplectically isotopic to each other. In contrast to the smooth symplectic isotopy problem, this is not equivalent to asking whether every singular symplectic surface is equisingularly symplectically isotopic to a complex curve. In fact, it is well known that there are examples of equisingular complex curves in $\\mathbb{CP}^{2}$ that are not equisingularly symplectically isotopic. Additionally there are examples of singular symplectic surfaces such that there does not exist any complex curve with the same singularities as mentioned above. Equisingular complex algebraic curves that are not equisingularly isotopic are often known as “Zariski pairs” (due to the first example being a pair of sextics discovered by Zariski [Zar29]) and are of great interest in the study of complex algebraic plane curves. (Note that Zariski pair sometimes is used to refer to examples that are not related by a weaker equivalence than equisingular isotopy, such as the fundamental groups of the complements being non-isomorphic; this implies they are not symplectically isotopic.) See [ABCT08] for an in-depth survey on Zariski pairs. For singular symplectic curves, it is interesting to ask when there are further “Zariski pairs” that can be realized symplectically than can be realized complexly.\n\nReferences cited:\n- [McD92] Dusa McDuff. Singularities of J-holomorphic curves in almost complex 4-manifolds. J. Geom. Anal., 2(3):249–266, 1992. doi:10.1007/BF02921295.\n- [MW95] Mario J. Micallef and Brian White. The structure of branch points in minimal surfaces and in pseudoholomorphic curves. Ann. of Math. (2), 141(1):35–85, 1995. doi:10.2307/2118627.\n- [Bar99] Jean-François Barraud. Nodal symplectic spheres in $\\mathbb{CP}^{2}$ with positive self-intersection. Internat. Math. Res. Notices, 1999(9):495–508, 1999. doi:10.1155/$S^{1}$073792899000252.\n- [Bar00] Jean-François Barraud. Courbes pseudo-holomorphes équisingulières en dimension 4. Bull. Soc. Math. France, 128(2):179–206, 2000. URL: http://www.numdam.org/item?id=BSMF 2000 128 2 179 0.\n- [She04] Vsevolod V. Shevchishin. On the local Severi problem. Int. Math. Res. Not., 2004(5):211–237, 2004. doi:10.1155/$S^{1}$073792804211163.\n- [OO05] Hiroshi Ohta and Kaoru Ono. Simple singularities and symplectic fillings. J. Differential Geom., 69(1):1–42, 2005. doi:10.4310/jdg/1121540338.\n- [GS22b] Marco Golla and Laura Starkston. The symplectic isotopy problem for rational cuspidal curves. Compos. Math., 158(7):1595–1682, 2022. doi:10.1112/s0010437x2200762x.\n- [GK23] Marco Golla and Fabien Kütle. Symplectic isotopy of rational cuspidal sextics and septics. Int. Math. Res. Not. IMRN, 2023(8):6504–6578, 2023. doi:10.1093/imrn/rnab364.\n- [Moi94] B. Moishezon. The arithmetic of braids and a statement of Chisini. In Geometric topology (Haifa, 1992), volume 164 of Contemp. Math., pages 151–175. Amer. Math. Soc., Providence, RI, 1994. doi:10.1090/conm/164/01591.\n- [MT88] B. Moishezon and M. Teicher. Braid group technique in complex geometry. I. Line arrangements in $\\mathbb{CP}^{2}$. In Braids (Santa Cruz, CA, 1986), volume 78 of Contemp. Math., pages 425–555. Amer. Math. Soc., Providence, RI, 1988. doi:10.1090/conm/078/975093.\n- [KK03] Vik.S̃. Kulikov and V. M. Kharlamov. On braid monodromy factorizations. Izv. Ross. Akad. Nauk Ser. Mat., 67(3):79–118, 2003. doi:10.1070/IM2003v067n03ABEH000436.\n- [RS19] Daniel Ruberman and Laura Starkston. Topological realizations of line arrangements. Int. Math. Res. Not. IMRN, 2019(8):2295–2331, 2019. doi:10.1093/imrn/rnx190.\n- [Zar29] Oscar Zariski. On the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given Branch Curve. Amer. J. Math., 51(2):305–328, 1929. doi:10.2307/2370712.\n- [ABCT08] Enrique Artal Bartolo, José Ignacio Cogolludo, and Hiro-o Tokunaga. A survey on Zariski pairs. In Algebraic geometry in East Asia—Hanoi 2005, volume 50 of Adv. Stud. Pure Math., pages 1–100. Math. Soc. Japan, Tokyo, 2008. doi:10.2969/aspm/05010001.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2979, "problem_number": "KP-4.103", "title": "Kirby Problem 4.103", "statement": "(a) What polynomials can occur as the Alexander polynomials of complex plane algebraic curves?\n\n(b) More generally, what are the conditions that must be satisfied by a finitely presented group G, so that there exist a plane algebraic curve having Gas the fundamental group of its complement?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.103.\n\nLiterature notes:\n(1) Let C be an algebraic curve in $\\mathbb{C}^{2}$. The Alexander polynomial of C relative to a surjection $\\pi_{1}(\\mathbb{C}^{2} \\setminus$ C) $\\to \\mathbb{Z}$ is the characteristic polynomial the automorphism of the homology of the associated infinite cyclic cover $H1(\\mathbb{C}\\supsetneq^{2} \\setminus C,\\mathbb{C})induced$ by a generator of the group of deck transformations of the cover.\n\n(2) Since a constraint on the Alexander polynomial is also a constraint on the fundamental group, part (a) is a very special case of part (b). One of many other questions underlying the second part is: which finite groups can occur as the fundamental groups of the complements to irreducible algebraic curves in $\\mathbb{CP}^{2}$?\n\n(3) A root of the Alexander polynomial of C must be a root of the Alexander polynomial of the link of at least one of the singularities of C and also a root of the Alexander polynomial of the link at infinity, i.e., the intersection of C with a 3-sphere in $\\mathbb{C}^{2}$ of a sufficiently large radius [Lib83, Lib21]. In particular the Alexander polynomial of an algebraic curve is cyclotomic but degrees of the factors that can occur are unknown. For example, for an irreducible plane curve having only ordinary cusps and nodes as singularities (i.e., locally homeomorphic to $x^{2}+y^{3}$ =0 and $x^{2}+y^{2}$ =0 respectively) the Alexander polynomial has the for $m (t^{2}-t+1)^{r}$ and the largest known value for r is 4; see [CAL14]. Is the set of possible degrees of the Alexander polynomials of curves C of arbitrary degrees but with fixed local types of singularities bounded?\n\n(4) For a recent survey of this circle of questions and mentioned properties of the Alexander polynomials, see [Lib82]. Further developments discussed there include characteristic varieties providing a multivariable generalization of Alexander polynomials, the Alexander polynomials of the complements to ample singular divisors on projective simply connected surfaces, and the role of the Alexander polynomials in the study of Zariski pairs (see Problem 4.102).\n\n(5) It would be interesting to understand a symplectic analog of the above problems. The Alexander polynomials of the fundamental groups of the complements to pseudoholomorphic curves with isolated singularities can be similarly defined, but their characterization, and the question “are the classes of realizable polynomials different in the symplectic or algebraic context?” are open. In particular, it is unknown if the classes of the fundamental groups of the complements to plane algebraic and pseudoholomorphic curves are different. Recently, a symplectic analog of the divisibility theorem mentioned in (3) was announced in the symplectic context; see [AG24].\n\nReferences cited:\n- [Lib83] A. Libgober. Alexander invariants of plane algebraic curves. In Singularities, Part 2 (Arcata, Calif., 1981), volume 40 of Proc. Sympos. Pure Math., pages 135–143. Amer. Math. Soc., Providence, RI, 1983. doi:10.1090/pspum/040.2/713242.\n- [Lib21] Anatoly Libgober. Complements to ample divisors and singularities. In Handbook of geometry and topology of singularities II, pages 501–567. Springer, Cham, [2021] ©2021. doi:10.1007/978-3-030-78024-1\\\\_10.\n- [CAL14] Jose-Ignacio Cogolludo-Agustı́n and Anatoly Libgober. Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves. J. Reine Angew. Math., 697:15–55, 2014. doi:10.1515/crelle-2012-0096.\n- [Lib82] A. Libgober. Alexander polynomial of plane algebraic curves and cyclic multiple planes. Duke Math. J., 49(4):833–851, 1982. URL: http://projecteuclid.org/euclid.dmj/1077315533.\n- [AG24] Hanine Awada and Marco Golla. Alexander polynomials of symplectic curves and divisibility relations, 2024. arXiv:2412.15792.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2980, "problem_number": "KP-4.104", "title": "Kirby Problem 4.104", "statement": "Does there exist a transverse link $L \\subset (S^{3}, \\xi_{std})$ bounding a pair of complex curves in $B^{4} \\subset \\mathbb{C}^{2}$ that are isotopic through embedded smooth surfaces but not through complex curves? Smoothly isotopic but not symplectically? Can there be infinitely many of them?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.104.\n\nLiterature notes:\n(1) By Rudolph [Rud83] and Boileau-Orevkov [BO01], L should be a quasipositive link and the complex curves can be realized as quasipositive surfaces in $B^{4}$. The boundary braids of the latter do not necessarily need to be the same quasipositive braid representatives of K, but after Markov stabilization one can assume they are. In fact, there are transverse knots $in(S^{3}, \\xi_{std})with$ quasipositive braid representatives that bound infinitely many quasipositive surfaces that are not smoothly isotopic [BVHM18], but these are distinguished by the topology of their complements. There are also pairs that are topologically isotopic but not smoothly isotopic, announced in [Hay21a].\n\n(2) This can be regarded as a relative version of the symplectic isotopy problem; see Problem 4.101. The case of K =T(d, d), the (d, d) torus link, is equivalent to the symplectic isotopy problem. This is because the symplectic isotopy classes of a non-singular surface of degreedare in bijection with the equisingular symplectic isotopy classes of the union of the non-singular degree d curve with a generically intersecting line [GS22b, Proposition 5.1]. The complement of a neighborhood of the line is symplectomorphic to $B^{4}$ and the degree dsymplectic surface will intersect the boundary $S^{3}$ in a transverse (d, d) torus link. There may be some generalizations of this to other algebraic links besides the (d, d) torus link, by allowing the degreed surface to intersect the line tangentially or by studying degree d surfaces with singularities.\n\n(3) There are infinitely many examples of a Legendrian link $L \\subset (S^{3}, \\xi_{std})$ bounding infinitely many (exact) Lagrangians in $B^{4} \\subset \\mathbb{C}^{2}that$ are smoothly isotopic but not Hamiltonian isotopic [CG22].\n\nReferences cited:\n- [Rud83] Lee Rudolph. Algebraic functions and closed braids. Topology, 22(2):191–202, 1983. doi:10.1016/0040-9383(83)90031-9.\n- [BO01] Michel Boileau and Stepan Orevkov. Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe. C. R. Acad. Sci. Paris Sér. I Math., 332(9):825–830, 2001. doi:10.1016/S0764-4442(01)01945-0.\n- [BVHM18] R. İnanç Baykur and Jeremy Van Horn-Morris. Fillings of genus-1 open books and 4-braids. Int. Math. Res. Not. IMRN, 2018(5):1329–1346, 2018. doi:10.1093/imrn/rnw281.\n- [Hay21a] Kyle Hayden. Exotically knotted disks and complex curves, 2021. arXiv:2003.13681.\n- [GS22b] Marco Golla and Laura Starkston. The symplectic isotopy problem for rational cuspidal curves. Compos. Math., 158(7):1595–1682, 2022. doi:10.1112/s0010437x2200762x.\n- [CG22] Roger Casals and Honghao Gao. Infinitely many Lagrangian fillings. Ann. of Math. (2), 195(1):207–249, 2022. doi:10.4007/annals.2022.195.1.3.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2981, "problem_number": "KP-4.105", "title": "Kirby Problem 4.105", "statement": "Does there exist a planar contact 3-manifold that has infinitely many distinct Stein fillings?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.105.\n\nLiterature notes:\n(1) A contact 3-manifold is planar if it is supported by a planar open book decomposition.\n\n(2) There are many interesting classes of planar contact 3-manifolds, for example lens spaces and links of rational surface singularities with reduced fundamental cycle. By Wendl [Wen10], and Wendl and Niederkr¨uger [NW11], any minimal weak symplectic filling of a planar contact manifold is in fact Stein, fills any given planar open book, and hence is determined by a positive Dehn twist factorization of the monodromy of that open book. Additionally, by Plamenevskaya [Pla12] and separately Wand [Wan12], there are only finitely many ”homological types” of positive factorizations of the monodromy of a planar open book. (See [BVHM18] for an explicit statement.) Lisca [Lis08] classified the diffeomorphism types of Stein fillings of the standard contact structure on lens spaces, showing that there are only finitely many.\n\n(3) The adjective ”distinct” could be interpreted in many ways and most are interesting. One strong ”no” result would be to show there exist only finitely many Stein fillings up to diffeomorphism. Stronger would be to prove this up to symplectomorphism and deformation. Lisi and Wendl conjecture that there are only finitely many deformation classes of minimal symplectic fillings [LW21].\n\n(4) Such examples are known when the page genus is 1 and higher [OS04a] [BVHM18]. Moreover, for genus 2 and higher the Stein fillings can have arbitrarily $largeb_{2}$. This is thus related to thesupport genusof the contact manifold; see Problem 3.45.\n\nReferences cited:\n- [Wen10] Chris Wendl. Strongly fillable contact manifolds and J-holomorphic foliations. Duke Math. J., 151(3):337–384, 2010. doi:10.1215/00127094-2010-001.\n- [NW11] Klaus Niederkrüger and Chris Wendl. Weak symplectic fillings and holomorphic curves. Ann. Sci. Éc. Norm. Supér. (4), 44(5):801–853, 2011. doi:10.24033/asens.2155.\n- [Pla12] Olga Plamenevskaya. On Legendrian surgeries between lens spaces. J. Symplectic Geom., 10(2):165–181, 2012. http://projecteuclid.org/euclid.jsg/1339096433.\n- [Wan12] Andy Wand. Mapping class group relations, Stein fillings, and planar open book decompositions. J. Topol., 5(1):1–14, 2012. doi:10.1112/jtopol/jtr025.\n- [BVHM18] R. İnanç Baykur and Jeremy Van Horn-Morris. Fillings of genus-1 open books and 4-braids. Int. Math. Res. Not. IMRN, 2018(5):1329–1346, 2018. doi:10.1093/imrn/rnw281.\n- [Lis08] Paolo Lisca. On symplectic fillings of lens spaces. Trans. Amer. Math. Soc., 360(2):765–799, 2008. doi:10.1090/S0002-9947-07-04228-6.\n- [LW21] Samuel Lisi and Chris Wendl. Spine removal surgery and the geography of symplectic fillings. Michigan Math. J., 70(2):403–422, 2021. doi:10.1307/mmj/1594260053.\n- [OS04a] Burak Ozbagci and András I. Stipsicz. Contact 3-manifolds with infinitely many Stein fillings. Proc. Amer. Math. Soc., 132(5):1549–1558, 2004. doi:10.1090/S0002-9939-03-07328-3.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2982, "problem_number": "KP-4.106", "title": "Kirby Problem 4.106", "statement": "Is the exact symplectomorphism type of $T*X^{4}$ sensitive to the smooth structure on a 4-manifold X, or does it depend only on the simple-homotopy or homeomorphism type of X?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.106.\n\nLiterature notes:\n(1) Recall that the cotangent bundle $T^{*}X$ has a canonical 1-form $\\lambda_{can}$; the canonical symplectic form on $T^{*}X$ isd $\\lambda_{can}. A$ diffeomorphism $\\varphi: T^{*}X \\to T^{*}Y$ is an exact symplectomorphism if $\\varphi^{*}\\lambda_{can} =\\lambda_{can}+df$ for some function f: $X \\to \\mathbb{R}$.\n\n(2) The problem is a version of Arnol’d’s Nearby Lagrangian Conjecture, which states that any exact Lagrangian submanifold of a cotangent bundle is Hamiltonian isotopic to the 0-section. In particular, a positive solution to the Nearby Lagrangian Conjecture would imply that the exact symplectomorphism type of T*X determines the diffeomorphism type of X.\n\n(3) Evidently, the symplectomorphism type of T*X determines the homotopy type of T*X and hence of X. In fact, deep results in Floer theory show that it determines the simple homotopy type of X [AK18].\n\n(4) In higher dimensions, the symplectic structure on the cotangent bundle can detect (some) exotic smooth structures [Abo12, EKS16]. In dimension 3, a relative version of this construction—considering the unit conormal bundle to a knot—gives a strong invariant of knots [Ng05, ENS18], though of course the homotopy type of a knot complement is itself a strong knot invariant. It would be interesting to know whether this conormal construction detects exotic embeddings of surfaces in 4-manifolds.\n\nReferences cited:\n- [AK18] Mohammed Abouzaid and Thomas Kragh. Simple homotopy equivalence of nearby Lagrangians. Acta Math., 220(2):207–237, 2018. doi:10.4310/ACTA.2018.v220.n2.a1.\n- [Abo12] Mohammed Abouzaid. Framed bordism and Lagrangian embeddings of exotic spheres. Ann. of Math. (2), 175(1):71–185, 2012. doi:10.4007/annals.2012.175.1.4.\n- [EKS16] Tobias Ekholm, Thomas Kragh, and Ivan Smith. Lagrangian exotic spheres. J. Topol. Anal., 8(3):375–397, 2016. doi:10.1142/$S^{1}$793525316500199.\n- [Ng05] Lenhard Ng. Knot and braid invariants from contact homology. I. Geom. Topol., 9:247–297, 2005. doi:10.2140/gt.2005.9.247.\n- [ENS18] Tobias Ekholm, Lenhard Ng, and Vivek Shende. A complete knot invariant from contact homology. Invent. Math., 211(3):1149–1200, 2018. doi:10.1007/s00222-017-0761-1.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2983, "problem_number": "KP-4.107", "title": "Kirby Problem 4.107", "statement": "Problems about contact hypersurfaces:\n\n(a) Let (Y, $\\xi)$ be a contact manifold and $(\\mathbb{R} \\times Y, \\omega)$ its symplectization. Let f:Y $\\to \\mathbb{R} \\times Y$ be a smooth embedding such thatf induces an isomorphism on homology. When can we isotope f so that its image is a contact type hypersurface?\n\n(b) Special case: Is every embedded 3-sphere $Y \\subset (\\mathbb{R}^{4}, \\omega_{std})$ smoothly isotopic to a hypersurface of contact type?\n\n(c) Is there any contact rational homology sphere other than $(S^{3}, \\xi_{std})$ which embeds as a contact type hypersurface in $(\\mathbb{R}^{4}, \\omega_{std}))$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.107.\n\nLiterature notes:\n(1) There are some geometric obstructions to a hypersurface Y being isotopic to a contact type hypersurface in the ambient symplectic 4–manifold (M, $\\omega)$; e.g. [Sul76] provides a necessary condition formulated in terms of a characteristic foliation of $\\omega$ on Y and [Cie98] shows that the existence of certain concentric annuli is an obstruction.\n\n(2) Question (b) has a bearing on the Schoenflies conjecture. If it is possible to isotope a smooth embedding of $S^{3}$ to make it a contact hypersurface in $(\\mathbb{R}^{4}, \\xi_{std})$ then it bounds a symplectic filling that is necessarily the 4-ball by Gromov [Gro85]. A strategy towards trying to isotope a smoothly embedded $S^{3}$ to a contact type hypersurface has been proposed by Lambert-Cole [LC21].\n\n(3) It was conjectured by Gompf [Gom13] that no Brieskorn sphere with either orientation admits a pseudoconvex embedding in $\\mathbb{C}^{2}$. This has a bearing on question (c). Mark and Tosun proved half of this conjecture in [MT22] that no positively oriented Brieskorn sphere (with any number of singular fibers) embeds as a contact type hypersurface $of(\\mathbb{R}^{4}, \\omega std)$.\n\nReferences cited:\n- [Sul76] Dennis Sullivan. Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math., 36:225–255, 1976. doi:10.1007/BF01390011.\n- [Cie98] Kai Cieliebak. A geometric obstruction to the contact type property. Math. Z., 228(3):451–487, 1998. doi:10.1007/PL00004626.\n- [Gro85] M. Gromov. Pseudo holomorphic curves in symplectic manifolds. Invent. Math., 82(2):307–347, 1985. doi:10.1007/BF01388806.\n- [LC21] Peter Lambert-Cole. Stein trisections and homotopy 4-balls, 2021. arXiv:2104.02003.\n- [Gom13] Robert E. Gompf. Smooth embeddings with Stein surface images. J. Topol., 6(4):915–944, 2013. doi:10.1112/jtopol/jtt017.\n- [MT22] Thomas E. Mark and Bülent Tosun. On contact type hypersurfaces in 4-space. Invent. Math., 228(1):493–534, 2022. doi:10.1007/s00222-021-01083-9.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2984, "problem_number": "KP-4.108", "title": "Kirby Problem 4.108", "statement": "Let $W_{+}$ and $W_{-}$ be two 4-dimensional Liouville domains with a contactomorphism $\\Phi :\\partial W_{-} \\cong \\partial W_{+}$. This determines an $\\mathbb{R}-invariant$ contact structure $\\xi$ on $\\mathbb{R} \\times X$ where X is the gluing $X =W_{+} \\cup _{\\Phi}W_{-}$ Moreover, 0 $\\times X \\subset \\mathbb{R} \\times X$ is a convex hypersurface in the sense of Giroux.\n\n(a) Is there a 4–dimensional version of Giroux’s criterion in the case where $W_{+}$ and $W_{-}$ are Weinstein? That is, a necessary and sufficient topological criterion on $(\\Phi, W_{+}, W_{-})$ for $\\xi$ to be overtwisted.\n\n(b) Does every 4-manifold X admit a decomposition $W_{+} \\cup _{\\Gamma} W_{-}$ so that the corresponding contact structure $\\xi$ is tight (i.e. not overtwisted)?\n\n(c) Is there a 4-dimensional Liouville domain W and a contactomorphism $\\Phi$ : $\\partial W \\to \\partial W$ that extends to a diffeomorphism $\\Psi$ of W, but so that the contact structure corresponding to $(\\Phi, W$, W) is overtwisted?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.108.\n\nLiterature notes:\n(1) A hypersurface $\\Sigma \\subset Y$ of a contact manifold (Y, $\\xi)$ is convex if there is a contact vector-field V that is transverse to $\\Sigma. A$ contact manifold is overtwisted if there is an embedded codimension one disk $D_{ot}$ [BEM15] with characteristic foliation determining a standard overtwisted contact structure in a neighborhood.\n\n(2) Overtwistedness is essentially characterized by an h-principle, and the overtwisted-tight dichotomy is emblematic of the general “flexible-rigid” dichotomy in symplectic topology. The theory of convex surfaces in contact 3-manifolds was pioneered by Giroux [Gir02, GM03].\n\n(3) The foundational work on convex surfaces was extended to higher dimensions by several works of Honda-Huang [HH18b, HH19], Breen–Honda– Huang [BHH23] and Eliashberg–Pancholi [EP23]. When $W_{+}$ and $W_{-}$ are Weinstein, the data $(\\Phi, W_{+}, W_{-})$ can be described using a variant of the Weinstein-Kirby diagrams due to Gompf (c.f. Breen-Christian [BC24c]). In particular, many of the outstanding open problems in higher dimensional convex surface theory may be particularly interesting to study in dimension four.\n\n(4) On part (a): given a splitting $\\Sigma=W_{+} \\cup _{\\Phi}W_{-}$ of a closed, oriented, connected surface, Giroux’s criterion states that the resulting contact structure on $\\mathbb{R} \\times \\Sigma$ is overtwisted if and only if either $\\Sigma \\cong S^{2}$ and $\\Gamma$ has more than one component or $\\Sigma$ fi $S^{2}$ and $\\Gamma$ contains a contractible curve. A higher dimensional, algebraic analogue of this criterion has been obtained by Avdek [Avd23], but a true topological analogue is still unknown.\n\n(5) On Part (b): a celebrated result of Etnyre–Honda [EH01b] gives an example of a closed contact 3-manifold with no tight contact structures. Part (b) of this problem may be viewed as a 4-manifold analogue of that result.\n\n(6) On Part (c): this problem is motivated by the goal of finding diffeomorphisms fixed at the boundary that act nontrivially on the space of symplectic structures. Indeed, the extension $\\Psi$ of $\\Phi$ in part (c) would constitute such a map, since the double $W \\cup _{Id}W$ is always tight by Avdek [Avd23]. Moreover, such a map may be usable as a sort of symplectic cork twist by embedding W into a closed symplectic manifold X, removing W and regluing it by $\\Psi$ to acquire a new symplectic manifold X1, one may be able to find diffeomorphic, non-symplectomorphic closed 4-manifolds (see\n\nProblem 4.96).\n\nReferences cited:\n- [BEM15] Matthew Strom Borman, Yakov Eliashberg, and Emmy Murphy. Existence and classification of overtwisted contact structures in all dimensions. Acta Math., 215(2):281–361, 2015. doi:10.1007/s11511-016-0134-4.\n- [Gir02] Emmanuel Giroux. Géométrie de contact: de la dimension trois vers les dimensions supérieures. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 405–414. Higher Ed. Press, Beijing, 2002.\n- [GM03] E Giroux and JP Mohsen. Structures de contact et fibrations symplectiques audessus du cercle. Lecture notes, 2003.\n- [HH18b] Ko Honda and Yang Huang. Bypass attachments in higher-dimensional contact topology, 2018. arXiv:1803.09142.\n- [HH19] Ko Honda and Yang Huang. Convex hypersurface theory in contact topology, 2019. arXiv:1907.06025.\n- [BHH23] Joseph Breen, Ko Honda, and Yang Huang. The Giroux correspondence in arbitrary dimensions, 2023. arXiv:2307.02317.\n- [EP23] Yakov Eliashberg and Dishant M. Pancholi. Honda-Huang’s work on contact convexity revisited. In Essays in geometry—dedicated to Norbert A’Campo, volume 34 of IRMA Lect. Math. Theor. Phys., pages 453–492. EMS Press, Berlin, [2023] ©2023.\n- [BC24c] Joseph Breen and Austin Christian. Bypass moves in convex hypersurface theory, 2024. arXiv:2206.14710.\n- [Avd23] Russell Avdek. An algebraic generalization of Giroux’s criterion, 2023. arXiv:2206.14710.\n- [EH01b] John B. Etnyre and Ko Honda. On the nonexistence of tight contact structures. Ann. of Math. (2), 153(3):749–766, 2001. doi:10.2307/2661367.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2985, "problem_number": "KP-4.109", "title": "Kirby Problem 4.109", "statement": "If $\\Sigma \\subset$ (X, $\\omega)$ is a symplectic surface in a closed symplectic 4-manifold with $[\\Sigma] = P D(k[\\omega]), k \\in \\mathbb{Z}$, does (X $\\setminus \\Sigma, \\omega)$ support a Weinstein structure?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.109.\n\nLiterature notes:\n(1) In particular, one could ask this for surfaces in $(\\mathbb{CP}^{2}, \\omega_{std}). A$ negative answer to the question in this case would yield a counterexample to the symplectic isotopy problem (Problem 4.101).\n\n(2) Donaldson [Don96] and Giroux [Gir02, Gir17] prove that for suitably large k there is some symplectic surface for which this is true. However, there are not currently known bounds on k, or known examples where k =1 is not enough. In principle, large upper bounds could be obtained by carefully tracking the proof in Donaldson’s construction. Such bounds are likely much larger than needed, but any explicit bound in terms of the symplectic manifold would represent progress on this question. Obtainin g more effective bounds through new techniques would be particularly interesting.\n\nReferences cited:\n- [Don96] S. K. Donaldson. Symplectic submanifolds and almost-complex geometry. J. Differential Geom., 44(4):666–705, 1996. http://projecteuclid.org/euclid.jdg/1214459407.\n- [Gir02] Emmanuel Giroux. Géométrie de contact: de la dimension trois vers les dimensions supérieures. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 405–414. Higher Ed. Press, Beijing, 2002.\n- [Gir17] Emmanuel Giroux. Remarks on Donaldson’s symplectic submanifolds. Pure Appl. Math. Q., 13(3):369–388, 2017. doi:10.4310/pamq.2017.v13.n3.a1.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2986, "problem_number": "KP-4.110", "title": "Kirby Problem 4.110", "statement": "Does there exist a 2-handlebody W that admits an exact symplectic structure with convex contact boundary that does not admit a Weinstein structure filling the same contact boundary?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.110.\n\nLiterature notes:\n(1) So far, all known examples of exact symplectic manifolds that we can obstruct from having a Weinstein structure are known to have handle structures requiring 3-handles [Mc D91, Bow12].\n\n(2) We have limited tools for obstructing an exact filling from being Weinstein when it has the topology of a 2-handlebody. Bowden’s argument in [Bow12] utilized a theorem of Eliashberg which says that a Weinstein fillin g of a connected sum is necessarily a boundary sum of Weinstein fillings of the summands [Eli90]. A potential strategy to generalize this obstruction to other examples may be to use Menke’s “mixed tori” [CM19], which gives a different decomposition criterion for Weinstein fillings.\n\nReferences cited:\n- [McD91] Dusa McDuff. Symplectic manifolds with contact type boundaries. Invent. Math., 103(3):651–671, 1991. doi:10.1007/BF01239530.\n- [Bow12] Jonathan Bowden. Exactly fillable contact structures without Stein fillings. Algebr. Geom. Topol., 12(3):1803–1810, 2012. doi:10.2140/agt.2012.12.1803.\n- [Eli90] Yakov Eliashberg. Filling by holomorphic discs and its applications. In Geometry of low-dimensional manifolds, 2 (Durham, 1989), volume 151 of London Math. Soc. Lecture Note Ser., pages 45–67. Cambridge Univ. Press, Cambridge, 1990.\n- [CM19] Austin Christian and Michael Menke. Splitting symplectic fillings, 2019. arXiv: 1909.00420.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2987, "problem_number": "KP-4.111", "title": "Kirby Problem 4.111", "statement": "Is trisection genus additive? In other words, must it be the case that $g(X\\#X1)$ =g(X) +g(X1).", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.111.\n\nLiterature notes:\n(1) This is Conjecture 1.6 of [LCM22].\n\n(2) The trisection genus of a 4–manifold X is $g(X)$ =min\\{g|X admits a genus g trisection\\}.\n\n(3) $If\\mathfrak{T}and\\mathfrak{T}1$ are trisections for X and X1 of genusgandg1, respectively, then $\\mathfrak{T}\\#\\mathfrak{T}1$ is a trisection of genus g+g1 for X\\#X1. It follows that trisection genus satisfies: $g(X\\#X^{1}) \\leq g(X) +g(X^{1})$. Equality holds when the lower-bound on $g(X)$ and $g(X1)$ coming from standard algebraic topology is sharp; see Problem 4.117.\n\n(4) A positive resolution to this problem would have sweeping consequences. If trisection genus is additive, then no manifold from the following set has an exotic copy: $\\mathcal{M}= {S^{4},\\mathbb{CP}^{2}, S^{1} \\times S^{3},2 \\mathbb{CP}^{2},\\mathbb{CP}^{2}\\#\\mathbb{CP}^{2}, S^{2} \\times S^{2}}$. This follows from the classification of trisections up to genus two [GK16, MZ17b] and the observation that if trisection genus is additive, homeomorphic smooth four-manifolds have the same trisection genus; see Proposition 1.7 of [LCM22], where modest evidence for this phenomenon is given; see also Remark 5 of Problem 4.113. In particular, if trisection genus is additive, then any exotic copy of $\\mathbb{CP}^{2}\\#2 \\mathbb{CP}^{2}$ admits a genus three trisection. Such exotic copies are known to exist [AP10, FS11]. This motivates the important problem of classifying genus three trisections; see Problem 4.113.\n\n(5) There is an analogous problem for orientable surface-links in $S^{4}$ that is related to the Meridional Rank Conjecture for orientable surface-links [JP25]; see Problem 1.18. For more details and precise definitions, see [MZ17a] and [AAD+23, Question 6.1]. Given an orientable surface-link $\\mathcal{K} \\subset S^{4}, letp(\\mathcal{K})denote$ itspatch number, the minimum valuepsuch $that\\mathcal{K}admits a$ (b;p, p1, p2)–bridge trisection.\n\n\\paragraph{Question.} Is patch number(−1)–additive for orientable surface-links? In other words, must it be the case that $p(\\mathcal{K}_{1}\\#\\mathcal{K}_{2}) =p(\\mathcal{K}_{1}) +p(\\mathcal{K}_{2})$ −1 for orientable surface-links $\\mathcal{K}_{1}$ and $\\mathcal{K}_{2}$?\n\nReferences cited:\n- [LCM22] Peter Lambert-Cole and Jeffrey Meier. Bridge trisections in rational surfaces. J. Topol. Anal., 14(3):655–708, 2022. doi:10.1142/$S^{1}$793525321500047.\n- [GK16] David Gay and Robion Kirby. Trisecting 4-manifolds. Geom. Topol., 20(6):3097– 3132, 2016. doi:10.2140/gt.2016.20.3097.\n- [MZ17b] Jeffrey Meier and Alexander Zupan. Genus-two trisections are standard. Geom. Topol., 21(3):1583–1630, 2017. doi:10.2140/gt.2017.21.1583.\n- [AP10] Anar Akhmedov and B. Doug Park. Exotic smooth structures on small 4-manifolds with odd signatures. Invent. Math., 181(3):577–603, 2010. doi:10.1007/s00222-010-0254-y.\n- [FS11] Ronald Fintushel and Ronald J. Stern. Pinwheels and nullhomologous surgery on 4-manifolds with $b^+=1$. Algebr. Geom. Topol., 11(3):1649–1699, 2011. doi:10.2140/agt.2011.11.1649.\n- [JP25] Jason Joseph and Puttipong Pongtanapaisan. Meridional rank and bridge number of knotted 2-spheres. Canad. J. Math., 77(1):282–299, 2025. doi:10.4153/S0008414X23000883.\n- [MZ17a] Jeffrey Meier and Alexander Zupan. Bridge trisections of knotted surfaces in $S^{4}$. Trans. Amer. Math. Soc., 369(10):7343–7386, 2017. doi:10.1090/tran/6934.\n- [AAD+23] Wolfgang Allred, Manuel Aragón, Zack Dooley, Alexander Goldman, Yucong Lei, Isaiah Martinez, Nicholas Meyer, Devon Peters, Scott Warrander, Ana Wright, and Alexander Zupan. Tri-plane diagrams for simple surfaces in $S^{4}$. J. Knot Theory Ramifications, 32(6):Paper No. 2350041, 28, 2023. doi:10.1142/S0218216523500414.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2988, "problem_number": "KP-4.112", "title": "Kirby Problem 4.112", "statement": "Is every trisection of the 4-sphere with positive genus a stabilization of the genus zero trisection?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.112.\n\nLiterature notes:\n(1) This is Conjecture 3.11 of [MSZ16]. The analogous result regarding Heegaard splittings of the 3–sphere is Waldhausen’s Theorem [Wal68a]. Connections between this problem, the Generalized Property R Conjecture (Problem 1.10), and the Andrews–Curtis Conjecture (Problem 5.10) are detailed in [MZ18] and elaborated in [MZ22]. In particular, if every positive-genus trisection of $S^{4}$ is stabilized, then well-known potential counter-examples (dating back to [AK85]) to the two conjectures mentioned above are not, in fact, counter-examples.\n\n(2) The following related question may be more tractable and of independent interest [MZ17a]. It is a four-dimensional analog of the classical result of Otal that says the unknot has a unique bridge splitting for each bridge number [Ota82].\n\n\\paragraph{Question.} Is every b–bridge trisection of the unknotted 2–sphere in $S^{4}$ with b >1 a perturbation of the 1–bridge trisection?\n\nReferences cited:\n- [MSZ16] Jeffrey Meier, Trent Schirmer, and Alexander Zupan. Classification of trisections and the generalized property R conjecture. Proc. Amer. Math. Soc., 144(11):4983– 4997, 2016. doi:10.1090/proc/13105.\n- [Wal68a] Friedhelm Waldhausen. Heegaard-Zerlegungen der 3-Sphäre. Topology, 7:195–203, 1968. doi:10.1016/0040-9383(68)90027-X.\n- [MZ18] Jeffrey Meier and Alexander Zupan. Characterizing Dehn surgeries on links via trisections. Proc. Natl. Acad. Sci. USA, 115(43):10887–10893, 2018. doi:10.1073/pnas.1717187115.\n- [MZ22] Jeffrey Meier and Alexander Zupan. Generalized square knots and homotopy 4-spheres. J. Differential Geom., 122(1):69–129, 2022. doi:10.4310/jdg/1668186788.\n- [AK85] Selman Akbulut and Robion Kirby. A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schoenflies conjecture, and the AndrewsCurtis conjecture. Topology, 24(4):375–390, 1985. doi:10.1016/0040-9383(85) 90010-2.\n- [MZ17a] Jeffrey Meier and Alexander Zupan. Bridge trisections of knotted surfaces in $S^{4}$. Trans. Amer. Math. Soc., 369(10):7343–7386, 2017. doi:10.1090/tran/6934.\n- [Ota82] Jean-Pierre Otal. Présentations en ponts du nœud trivial. C. R. Acad. Sci. Paris Sér. I Math., 294(16):553–556, 1982.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2989, "problem_number": "KP-4.113", "title": "Kirby Problem 4.113", "statement": "Which closed, oriented, smooth 4–manifolds admit genus–3 trisections? Which ones admit genus–3 simplified trisections? How about genus–4?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.113.\n\nLiterature notes:\n(1) There is a complete classification of trisections of genus $g \\leq$ 2 and only a handful of standard 4–manifolds admit them: $S^{4}$ for g =0; $\\mathbb{CP}^{2}, \\mathbb{CP}^{2}$, or $S^{1} \\times S^{3}$ for $g =$ 1; and $S^{2} \\times S^{2}$, or connected sums of $\\mathbb{CP}^{2}, \\mathbb{CP}^{2}$ and $S^{1} \\times S^{3}$ with two summands, for g =2 by [GK16, MZ17b].\n\n(2) While the simplified trisections (see [BS23b] for a precise definition) constitute a subclass of Gay-Kirby trisections, the following question [BS18c, Question 2] is still open:\n\n\\paragraph{Question.} Is there a closed 4–manifold that admits a trisection, but not a simplified trisection of the same genus? The classification of genus $g \\leq$ 2 simplified trisections coincide with that of standard trisections [BS18c, Hay20]. So, the genus g =3 is the lowest genus where a discrepancy may appear.\n\n(3) Some partial results are known: A trisection with complexity $(3;k_{1}, k_{2}, k_{3})$ and $k_{i} \\geq$ 2 for some $i \\in \\mathbb{Z}_{3}$ is reducible [MSZ16]. Recall that a trisection that is the connected sum of smaller trisections is called reducible; otherwise, it is irreducible. Infinitely many, pairwise homotopy inequivalent 4–manifolds admit genus–3 trisections; e.g. every spun lens space admits a (simplified) genus– 3 trisection [Mei18, BS18c]. Similarly, there are genus–4 (simplified) trisections of 3–manifold bundles over $S^{1}$ with lens space or $S^{1} \\times S^{2}$ fibers [Mei18, BS18c].\n\n(4) The classification of low genera trisections is intimately related to that of simplified broken Lefschetz fibrations [BS23b, BS18c]. The classification for the latter is complete for $g \\leq$ 1 and spans a larger class of 4–manifolds [BK15a, Bay12a, Hay11, Hay14].\n\n(5) In the case of (broken) Lefschetz fibrations, the following exotic phenomenon already appears when we hit g =2: there are pairs of homeomorphic but not diffeomorphic 4–manifolds which both admit genus–2 Lefschetz fibrations; for instance, let $X_{i}:= E(1)_{K,i}, i =$ 1,2, be the knot surgered rational elliptic surface for $K_{1} a$ trefoil knot and $K_{2}$ the figure eight knot [FS04, Bay09]. There are analogous results for (simplified) trisections when g =20 [BS23b] and many other, larger genera, starting atg =23, where one of the exotic pairs is an algebraic surface [ST18, LCM22]. It is reasonable to expect this exotic phenomenon to manifest for much smaller g. (This is related to the question on the additivity of triseciton genus under connected sum; see Problem 4.111.)\n\n\\paragraph{Question.} What is the smallest g for which there is an exotic pair of closed, oriented 4–manifolds admitting genus–g trisections?\n\n(6) While the problem is formulated only for orientable 4–manifolds, the classification of small-genera (simplified) trisections and (simplified) broken Lefschetz fibrations on nonorientable 4–manifolds is also within reach, and suggests analogous classification schemes; see [MN24, ST22, BM25].\n\nReferences cited:\n- [GK16] David Gay and Robion Kirby. Trisecting 4-manifolds. Geom. Topol., 20(6):3097– 3132, 2016. doi:10.2140/gt.2016.20.3097.\n- [MZ17b] Jeffrey Meier and Alexander Zupan. Genus-two trisections are standard. Geom. Topol., 21(3):1583–1630, 2017. doi:10.2140/gt.2017.21.1583.\n- [BS23b] R. İnanç Baykur and Osamu Saeki. Simplifying indefinite fibrations on 4-manifolds. Trans. Amer. Math. Soc., 376(5):3011–3062, 2023. doi:10.1090/tran/8325.\n- [BS18c] R. İnanç Baykur and Osamu Saeki. Simplified broken Lefschetz fibrations and trisections of 4-manifolds. Proc. Natl. Acad. Sci. USA, 115(43):10894–10900, 2018. doi:10.1073/pnas.1717175115.\n- [Hay20] Kenta Hayano. On diagrams of simplified trisections and mapping class groups. Osaka J. Math., 57(1):17–37, 2020. https://projecteuclid.org/euclid.ojm/1579079109.\n- [MSZ16] Jeffrey Meier, Trent Schirmer, and Alexander Zupan. Classification of trisections and the generalized property R conjecture. Proc. Amer. Math. Soc., 144(11):4983– 4997, 2016. doi:10.1090/proc/13105.\n- [Mei18] Jeffrey Meier. Trisections and spun four-manifolds. Math. Res. Lett., 25(5):1497– 1524, 2018. doi:10.4310/MRL.2018.v25.n5.a7.\n- [BK15a] R. İnanç Baykur and Seiichi Kamada. Classification of broken Lefschetz fibrations with small fiber genera. J. Math. Soc. Japan, 67(3):877–901, 2015. doi:10.2969/jmsj/06730877.\n- [Bay12a] R. İnanç Baykur. Broken Lefschetz fibrations and smooth structures on 4-manifolds. In Proceedings of the Freedman Fest, volume 18 of Geom. Topol. Monogr., pages 9–34. Geom. Topol. Publ., Coventry, 2012. doi:10.2140/gtm.2012.18.9.\n- [Hay11] Kenta Hayano. On genus-1 simplified broken Lefschetz fibrations. Algebr. Geom. Topol., 11(3):1267–1322, 2011. doi:10.2140/agt.2011.11.1267.\n- [Hay14] Kenta Hayano. Complete classification of genus-1 simplified broken Lefschetz fibrations. Hiroshima Math. J., 44(2):223–234, 2014. http://projecteuclid.org/euclid.hmj/1408972909.\n- [FS04] Ronald Fintushel and Ronald J. Stern. Families of simply connected 4-manifolds with the same Seiberg-Witten invariants. Topology, 43(6):1449–1467, 2004. doi: 10.1016/j.top.2004.03.002.\n- [Bay09] Refik İnanç Baykur. Topology of broken Lefschetz fibrations and near-symplectic four-manifolds. Pacific J. Math., 240(2):201–230, 2009. doi:10.2140/pjm.2009.240.201.\n- [ST18] Jonathan Spreer and Stephan Tillmann. The trisection genus of standard simply connected PL 4-manifolds. In 34th International Symposium on Computational Geometry, volume 99 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 71, 13. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018.\n- [LCM22] Peter Lambert-Cole and Jeffrey Meier. Bridge trisections in rational surfaces. J. Topol. Anal., 14(3):655–708, 2022. doi:10.1142/$S^{1}$793525321500047.\n- [MN24] Maggie Miller and Patrick Naylor. Trisections of nonorientable 4-manifolds. Michigan Math. J., 74(2):403–447, 2024. doi:10.1307/mmj/20216127.\n- [ST22] Jonathan Spreer and Stephan Tillmann. Determining the trisection genus of orientable and non-orientable PL 4-manifolds through triangulations. Exp. Math., 31(3):897–907, 2022. doi:10.1080/10586458.2020.1723744.\n- [BM25] R. İnanç Baykur and Porter Morgan. On nonorientable 4–manifolds, 2025. Math. Res. Lett., to appear. URL: https://arxiv.org/abs/2506.20950, arXiv:2506.20950.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2990, "problem_number": "KP-4.114", "title": "Kirby Problem 4.114", "statement": "For a given Heegaard splitting of a closed orientable3–manifold, classify self-indexing Morse functions that give the given Heegaard splitting, up to $C\\infty$ right-left (or right) equivalence. Likewise, for a given (simplified) trisection of a closed orientable 4–manifold, classify generic maps that give the given trisection, up to right-left (or right) equivalence.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.114.\n\nLiterature notes:\n(1) Let $f_{i}: M_{i} \\to N_{i}$ be $C\\infty$ maps between smooth manifolds, $i =$ 0,1. We say that they are $C\\infty$ right-left equivalent if there exist diffeomorphisms $\\phi: M_{0} \\to M_{1}$ and $\\psi: N_{0} \\to N_{1}$ such that $f_{1} =\\psi \\circ f_{0} \\circ \\phi-^{1}$. If $N_{0} =N_{1} and\\psi$ can be taken to be the identity, we say that $f_{0} andf_{1}$ are $C\\infty$ right equivalent. \\begin{center}\n\\kthreefiginclude{ch4_fig5.png}{width=0.58\\linewidth}\n\\par\\small\\textbf{Figure 5.} Two commutative diagrams. Left: the maps $f_{0},f_{1}$ are right-left equivalent. Right: the maps $f_{0},f_{1}$ are right-equivalent.\n\\end{center}\n\nIt is known that to a self-indexing Morse function on a closed orientable 3–manifold is canonically associated a Heegaard splitting. Likewise, to a certain generic map, called Morse 2–function, on a closed orientable 4–manifold is canonically associated a trisection [GK16] or a simplified trisection [BS23b, BS18c].\n\n(2) The 4–dimensional problem can be rephrased as follows. Let M be a closed orientable 4–manifold and let f: $M \\to \\mathbb{R}^{2} a C^{\\infty}$ be a stable map that corresponds to a (simplified) trisection. If two such maps are $C^{\\infty}$ right-left equivalent, then the resulting trisections are naturally equivalent. Does the converse hold? Namely, if the resulting (simplified) trisections are equivalent, then are the original $C\\infty$ stable maps $C\\infty$ right-left equivalent? If not, how different are they? Some related results are obtained in [Hay20] and [Asa23].\n\n(3) It is known that Morse functions on closed surfaces can be classified by means of Kronrod-Reeb graphs up to $C\\infty$ right-left (or right) equivalence (for example, see [Mak05]).\n\nReferences cited:\n- [GK16] David Gay and Robion Kirby. Trisecting 4-manifolds. Geom. Topol., 20(6):3097– 3132, 2016. doi:10.2140/gt.2016.20.3097.\n- [BS23b] R. İnanç Baykur and Osamu Saeki. Simplifying indefinite fibrations on 4-manifolds. Trans. Amer. Math. Soc., 376(5):3011–3062, 2023. doi:10.1090/tran/8325.\n- [BS18c] R. İnanç Baykur and Osamu Saeki. Simplified broken Lefschetz fibrations and trisections of 4-manifolds. Proc. Natl. Acad. Sci. USA, 115(43):10894–10900, 2018. doi:10.1073/pnas.1717175115.\n- [Hay20] Kenta Hayano. On diagrams of simplified trisections and mapping class groups. Osaka J. Math., 57(1):17–37, 2020. https://projecteuclid.org/euclid.ojm/1579079109.\n- [Asa23] Nobutaka Asano. Right-left equivalent maps of simplified (2, 0)-trisections with different configurations of vanishing cycles. Kyushu J. Math., 77(2):299–317, 2023.\n- [Mak05] Sergey Maksymenko. Path-components of Morse mappings spaces of surfaces. Comment. Math. Helv., 80(3):655–690, 2005. doi:10.4171/CMH/30.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2991, "problem_number": "KP-4.115", "title": "Kirby Problem 4.115", "statement": "(a) Find two diffeomorphic but non-isotopic trisections of the same4–manifold.\n\n(b) Find two non-diffeomorphic balanced trisections of the same genus on a closed simply connected 4–manifold.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.115.\n\nLiterature notes:\n(1) Part of the difficulty of Problem (a)is that it is connected to the problem of understanding the smooth mapping class groups of 4–manifolds. In particular, if a 4–manifold X has trivial smooth mapping class group, so that all (orientation preserving) diffeomorphisms are isotopic, then all diffeomorphic trisections are obviously isotopic. On the other hand, 4– manifolds with nontrivial homology can have diffeomorphisms that act nontrivially on homology, and one might imagine that one could apply such a diffeomorphism to a given trisection to get a diffeomorphic but non-isotopic trisection. However this diffeomorphism might be isotopic to a diffeomorphism that is not the identity but which fixes the initial trisection (i.e. fixes the sectors setwise), in which case the new trisection would in fact be isotopic to the initial one.\n\n(2) Problem (b) seems to be easier than Problem (a), with the first vague approximation being to come up with examples of trisections that look at first glance like they might be diffeomorphic but which in fact are not. Examples of non-diffeomorphic same-genus trisections of a 4-manifold were first constructed by Islambouli [Isl21]. However, Islambouli’s techniques rely on the 4-manifold having nontrivial fundamental group G, as the trisections are distinguished by the Nielsen classes of associated presentations of G. This mirrors an argument in the 3–dimensional Heegaard splitting setting [Eng70]\n\n(3) In principle, invariants of trisections up to diffeomorphism should be easier to find than invariants up to isotopy. Meier and Lambert-Cole construct various genus–22 trisections of K3 that may or may not be diffeomorphic and/or isotopic by viewing K3 as a branched cover in different ways [LCM22]. See Problem 4.112 (and examples of Meier and Zupan [MZ18]) for a related problem regarding the uniqueness of trisections of $S^{4}$.\n\nReferences cited:\n- [Isl21] Gabriel Islambouli. Nielsen equivalence and trisections. Geom. Dedicata, 214:303– 317, 2021. doi:10.1007/s10711-021-00617-y.\n- [Eng70] Renate Engmann. Nicht-homöomorphe Heegaard-Zerlegungen vom Geschlecht 2 der zusammenhängenden Summe zweier Linsenräume. Abh. Math. Sem. Univ. Hamburg, 35:33–38, 1970. doi:10.1007/BF02992472.\n- [LCM22] Peter Lambert-Cole and Jeffrey Meier. Bridge trisections in rational surfaces. J. Topol. Anal., 14(3):655–708, 2022. doi:10.1142/$S^{1}$793525321500047.\n- [MZ18] Jeffrey Meier and Alexander Zupan. Characterizing Dehn surgeries on links via trisections. Proc. Natl. Acad. Sci. USA, 115(43):10887–10893, 2018. doi:10.1073/pnas.1717187115.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2992, "problem_number": "KP-4.116", "title": "Kirby Problem 4.116", "statement": "Is there an algorithm to compute ‘distance’ in the cut complex of a trisection? Is the L-invariant computable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.116.\n\nLiterature notes:\n(1) A Heegaard splitting for a 3-manifold M is defined by two “cut systems” of curves on a Heegaard surface $\\Sigma$. Various notions of ‘distance’ (e.g. the Hempel distance) for Heegaard splittings have been extensively studied and yielded valuable geometric information about M. A trisected smooth closed 4-manifold X is defined by three “cut systems” of curves on a surface $\\Sigma$. One can define various analogous notions of 3-manifold Heegaard distance for a trisected X by measuring the length of a shortest loop in the cut complex (or pants complex) that includes a representative of each of the three specified cut systems. Given a specific trisection, the first question asks whether any such distance is computable. The L-invariant [KT22a] uses this idea in a limit to define a 4-manifold invariant.\n\n(2) When L is zero and X is a homology sphere, X is diffeomorphic to the 4-sphere. Hence a positive answer to the second question is related to recognizability of the 4-sphere.\n\n(3) In a preprint, Asano–Naoe–Ogawa [ANO24] gave a lower bound on the L-invariant of a 4-manifold X in terms of the first Betti number of X.\n\nReferences cited:\n- [KT22a] Robion Kirby and Abigail Thompson. Trisections and link surgeries. New Zealand J. Math., 52:145–152, 2021 [2021–2022]. doi:10.53733/94.\n- [ANO24] Nobutaka Asano, Hironobu Naoe, and Masaki Ogawa. Some lower bounds for the Kirby-Thompson invariant. Math. Res. Lett., 31(6):1611–1637, 2024. doi:10.4310/mrl.250210225353.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2993, "problem_number": "KP-4.117", "title": "Kirby Problem 4.117", "statement": "Let X be a closed, orientable, smooth 4-manifold, with $g(X)$ the trisection genus of X. Does $g(X) =\\chi(X) -2+3rk(\\pi_{1}(X))$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.117.\n\nLiterature notes:\n(1) Chu and Tillmann proved that $\\chi(X) -2+3rk(\\pi_{1}(X))$ is a lower bound for $g(X)$ [CT19]. It is reasonable to expect there to exist manifolds for which this inequality is strict, but at present, Chu and Tillmann’s result remains the only known lower bound on trisection genus. Following the classification of trisections up to genus two [MZ17b], if the problem has an affirmative answer, then there do not exist exotic copies of any X such that $g(X) \\leq$ 2, including $S^{4}, S^{1} \\times S^{3}, \\mathbb{CP}^{2}$, and $S^{2} \\times S^{2}$. (See also\n\nProblem 4.111.)\n\n(2) Like many problems in trisections, there is an analogous problem related to Heegaard splittings of 3-manifolds, the rank-genus conjecture. The now-disproved rank-genus conjecture asks $whetherg(Y) =rk(\\pi_{1}(Y))for a$ closed, orientable 3-manifold Y. Originally posed by Waldhausen [Wal78], this conjecture was also called the Generalized Poincaré Conjecture, since it implies the 3-dimensional Poincaré conjecture [Hak70]. The first examples of a 3-manifold Y for which $g(Y) > rk(\\pi_{1}(Y))$ were exhibited by Boileau and Zieschang [BZ84], while the first hyperbolic counterexamples to the conjecture were produced by Li [Li13].\n\n(3) A related problem for bridge trisections of a knotted surface $\\mathcal{K} \\subset S^{4}$ involves themeridional rank of the group $\\pi_{1}(S^{4} \\setminus \\mathcal{K})$, denoted $mrk(\\pi_{1}(S^{4} \\setminus \\mathcal{K}))$, the smallest number of meridians needed to generate the group, and the bridge number $b(\\mathcal{K})$, the minimal b such that $\\mathcal{K}$ admits a b-bridge trisection.\n\n\\paragraph{Question.} Let $\\mathcal{K}$ be an orientable knotted surface in $S^{4}$. Does $b(\\mathcal{K}) = -\\chi(\\mathcal{K}) +3mrk(\\pi_{1}(S^{4}-\\mathcal{K}))$? As in the case of trisections, the question is suggested by a straightforward (and the only currently known) lower bound. To derive the bound, note that $if\\mathcal{K}$ admits $a(b;c_{1}, c_{2}, c_{3})-bridge$ trisection, then $\\chi(\\mathcal{K}) = c_{1}+c_{2}+c_{3}-b$ and $mrk(\\pi_{1}(S^{4} \\setminus \\mathcal{K})) \\leq c_{i}$ for all i[MZ17a, Corollary 5.3]). Thus, $b = -\\chi(\\mathcal{K}) +c_{1}+c_{2}+c_{3} \\geq -\\chi(\\mathcal{K}) +3mrk(\\pi_{1}(S^{4}-\\mathcal{K}))$. This problem can be compared to the meridional rank conjecture ([Kir97, Problem 1.11] and Problem 1.18), an unsolved problem that asks whether the bridge number of a knot K in $S^{3}$ is equal to the minimal number of meridional generators required to generate its knot group. See also\n\nProblem 4.111.\n\n(4) Meier and Zupan showed that if $\\mathcal{K}$ is the spin of a classical knot K that satisfies the meridional rank conjecture in dimension three, then $\\mathcal{K}$ satisfies the equality in the problem above [MZ17a]. Note that a positive answer to the problem would imply that any 2-sphere or torus in $S^{4}$ with infinite cyclic fundamental group has bridge number one or three, respectively, and as such is smoothly unknotted by the classification of b-bridge trisections with $b \\leq$ 3 [MZ17a]. Miyazawa’s recent construction of an exotic $\\mathbb{RP}^{2}$, announced in [Miy23], yields a negative answer to the problem in the case of nonorientable surfaces.\n\nReferences cited:\n- [CT19] Michelle Chu and Stephan Tillmann. Reflections on trisection genus. Rev. Roumaine Math. Pures Appl., 64(4):395–402, 2019.\n- [MZ17b] Jeffrey Meier and Alexander Zupan. Genus-two trisections are standard. Geom. Topol., 21(3):1583–1630, 2017. doi:10.2140/gt.2017.21.1583.\n- [Wal78] Friedhelm Waldhausen. Some problems on 3-manifolds. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, volume XXXII of Proc. Sympos. Pure Math., pages 313–322. Amer. Math. Soc., Providence, RI, 1978.\n- [Hak70] Wolfgang Haken. Various aspects of the three-dimensional Poincaré problem. In Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), pages 140–152. Markham Publishing Co., Chicago, IL, 1970.\n- [BZ84] M. Boileau and H. Zieschang. Heegaard genus of closed orientable Seifert 3-manifolds. Invent. Math., 76(3):455–468, 1984. doi:10.1007/BF01388469.\n- [Li13] Tao Li. Rank and genus of 3-manifolds. J. Amer. Math. Soc., 26(3):777–829, 2013. doi:10.1090/S0894-0347-2013-00767-5.\n- [MZ17a] Jeffrey Meier and Alexander Zupan. Bridge trisections of knotted surfaces in $S^{4}$. Trans. Amer. Math. Soc., 369(10):7343–7386, 2017. doi:10.1090/tran/6934.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Miy23] Jin Miyazawa. A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P2-knots, 2023. arXiv:2312.02041.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2994, "problem_number": "KP-4.118", "title": "Kirby Problem 4.118", "statement": "Does every simply connected, closed, smooth 4-manifold admit a handle decomposition without any 1-handles? Without 1-handles and 3-handles?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.118.\n\nLiterature notes:\n(1) A manifold is called geometrically simply connected if it admits a handle decomposition without 1-handles. Any simply connected closed manifold of any dimension other than four is geometrically simply connected; this follows from the celebrated works of Smale in higher dimensions [Sma62b] and of Perelman in dimension three [Per02, Per03b, Per03a].\n\n(2) This is [Kir97, Problem 4.18]. Some candidates for counter-examples, such as the Dolgachev surface $E(1)_{2,3}$, have since been shown to admit handle decompositions without any 1- and 3-handles [Yas08, Akb12].\n\n(3) If one allows the 4-manifold to have boundary, the answer is negative. Indeed, there are many contractible 4-manifolds that require 1-handles, by the following argument due to Casson: If a compact, contractible 4– manifold X can be built without 1-handles, then, turning the handlebody upside down, we can also build X from $\\partial X$ by adding the same number of 1- and 2–handles and a 4-handle. So $\\pi_{1}(\\partial X)$ can be killed by adding the same number of generators and relators. However, a finitely presented group with a nontrivial representation to a compact connected Lie group cannot be trivialized by adding the same number of generators and relators by [GR62] and there are contractible X where $\\pi_{1}(\\partial X)$ is such a group. In fact, geometrization now implies all nontrivial 3-manifold groups admit nontrivial finite quotients by [Hem87], so any contractible 4-manifold with boundary other than $S^{3}$ requires 1-handles, such as the Mazur manifold with boundary $\\Sigma(2,3,13)$ [AK79b].\n\n(4) If a geometrically simply connected closed 4-manifold X has $b^{+}_{2}(X) >$ 1 and $b^{-}_{2}(X)$ =0, then all the stable cohomotopy Seiberg-Witten invariants of X vanish [Yas19]; so, e.g. it cannot admit a symplectic structure—see also [HL19]. In the same paper, Yasui shows that if X is geometrically simply connected, then every $\\alpha \\in H_{2}(X;\\mathbb{Z})$ has a neighborhood W diffeomorphic to a 2-handle attached to a 4-ball, where $\\alpha$ is the image of the generato r of $H_{2}(W;\\mathbb{Z}) \\cong \\mathbb{Z}$ under the inclusion induced homomorphism. Any simply connected X with some $\\alpha \\in H_{2}(X;\\mathbb{Z})$ that does not admit such a neighborhood would be a counter-example.\n\n(5) Admitting a handle decomposition without 1- and 3-handles has strong implications. For instance, if this is true for every homotopy $S^{4}$ or homotopy $\\mathbb{CP}^{2}$, then there are no exotic copies of these 4-manifolds. The conclusion for homotopy $\\mathbb{CP}^{2}s$ follows from [GL89].\n\nReferences cited:\n- [Sma62b] Stephen Smale. On the structure of 5-manifolds. Ann. of Math. (2), 75:38–46, 1962. doi:10.2307/1970417.\n- [Per02] Grisha Perelman. The entropy formula for the Ricci flow and its geometric applications, 2002. arXiv:math/0211159.\n- [Per03b] Grisha Perelman. Ricci flow with surgery on three-manifolds, 2003. arXiv:math/0303109.\n- [Per03a] Grisha Perelman. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, 2003. arXiv:math/0307245.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Yas08] Kouichi Yasui. Elliptic surfaces without 1-handles. J. Topol., 1(4):857–878, 2008. doi:10.1112/jtopol/jtn026.\n- [Akb12] Selman Akbulut. The Dolgachev surface. Disproving the Harer-Kas-Kirby conjecture. Comment. Math. Helv., 87(1):187–241, 2012. doi:10.4171/CMH/252.\n- [GR62] M. Gerstenhaber and O. S. Rothaus. The solution of sets of equations in groups. Proc. Nat. Acad. Sci. U.S.A., 48:1531–1533, 1962.\n- [Hem87] John Hempel. Residual finiteness for 3-manifolds. In Combinatorial group theory and topology (Alta, Utah, 1984), volume 111 of Ann. of Math. Stud., pages 379–396. Princeton Univ. Press, Princeton, NJ, 1987.\n- [AK79b] Selman Akbulut and Robion Kirby. Mazur manifolds. Michigan Math. J., 26(3):259–284, 1979. http://projecteuclid.org/euclid.mmj/1029002261.\n- [Yas19] Kouichi Yasui. Geometrically simply connected 4-manifolds and stable cohomotopy Seiberg-Witten invariants. Geom. Topol., 23(5):2685–2697, 2019. doi:10.2140/gt.2019.23.2685.\n- [HL19] Jennifer Hom and Tye Lidman. A note on positive-definite, symplectic fourmanifolds. J. Eur. Math. Soc. (JEMS), 21(1):257–270, 2019. doi:10.4171/JEMS/835.\n- [GL89] C. McA. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc., 2(2):371–415, 1989. doi:10.2307/1990979.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2995, "problem_number": "KP-4.119", "title": "Kirby Problem 4.119", "statement": "Is every topological 4-manifold homeomorphic to a CW complex?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.119.\n\nLiterature notes:\n(1) It follows from the work of Kirby and Siebenmann [KS77] that every topological manifold M has the homotopy type of a CW complex, and moreover there is a canonical simple homotopy type of CW complexes homotopy equivalent to M.\n\n(2) Every smooth manifold is triangulable and hence homeomorphic to a CW complex. See [Cai35], [Whi40]. In particular, non-compact 4-manifolds are smoothable [Qui82]; therefore, the question has a positive answer for those.\n\n(3) Every topological manifold of dimension $n \\ne$ 4 has a handlebody structure, and hence is homeomorphic to a CW complex. See [Moi77a] for n=3, [KS77, p.104] for $n \\geq$ 6 and [Qui86] for n=5.\n\n(4) Any 4-manifold with a handlebody structure is smooth. Indeed, when a handle is attached to a smooth 4-manifold, the attaching map is in dimension three, where everything is smoothable.\n\n(5) Topological, non-smoothable 4-manifolds (such as the $E_{8}$ manifold) are known to not be homeomorphic to simplicial complexes. See [AM90] and [Man13, Remark 4.2].\n\nReferences cited:\n- [KS77] Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations, volume 88 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J., 1977. With notes by John Milnor and Michael Atiyah.\n- [Cai35] S. S. Cairns. Triangulation of the manifold of class one. Bull. Amer. Math. Soc., 41(8):549–552, 1935. doi:10.1090/S0002-9904-1935-06140-3.\n- [Whi40] J. H. C. Whitehead. On C1-complexes. Ann. of Math. (2), 41:809–824, 1940. doi: 10.2307/1968861.\n- [Qui82] Frank Quinn. Ends of maps. III. Dimensions 4 and 5. J. Differential Geometry, 17(3):503–521, 1982. http://projecteuclid.org/euclid.jdg/1214437139.\n- [Moi77a] Edwin E. Moise. Geometric topology in dimensions 2 and 3, volume Vol. 47 of Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1977.\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n- [AM90] Selman Akbulut and John D. McCarthy. Casson’s invariant for oriented homology 3-spheres, volume 36 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1990. An exposition. doi:10.1515/9781400860623.\n- [Man13] Ciprian Manolescu. The Conley index, gauge theory, and triangulations. J. Fixed Point Theory Appl., 13(2):431–457, 2013. doi:10.1007/s11784-013-0134-3.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2996, "problem_number": "KP-4.120", "title": "Kirby Problem 4.120", "statement": "Which closed, smooth 4–manifolds admit achiral Lefschetz pencils? Does every simply connected 4–manifold have one?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.120.\n\nLiterature notes:\n(1) Achiral Lefschetz pencils are generalizations of Lefschetz pencils, where the local models for nodal singularities and base points are allowed to reverse orientations. So, they are also defined on nonorientable 4–manifolds. Here we allow achiral pencils to possibly have no critical points and/or no base points.\n\n(2) There are only a couple of known obstructions to the existence of an achiral Lefschetz pencil on a given closed, oriented 4–manifold X; see [GS99, Theorem 8.4.13] and [Sco03, Theorem 4.15]. These obstructions rule out definite 4–manifolds with $b_{2}$ +1 $< b_{1}$, such as $\\#_{m}(S^{1} \\times S^{3})$, for $m \\geq$ 2. A curious question is: Are there homotopy equivalent smooth 4–manifolds X and X1, where X admits an achiral pencil but X1 does not?\n\n(3) Any closed, orientable X admits an achiral Lefschetz fibration (without base points) after surgery along a curve, and specifically, $X\\#(S^{2} \\times S^{2})$ always admits one when X is simply connected [EF06].\n\n(4) Just like Lefschetz pencils can be equipped with certain symplectic forms making all the fibers symplectic, achiral Lefschetz pencils can be equipped with certain folded-symplectic forms. Furthermore, a variation of achiral Lefschetz pencils, defined in the complement of a 1–manifold, supsupport folded-Kähler forms on all closed, oriented, smooth 4–manifolds. See [Bay06, Hit16].\n\nReferences cited:\n- [GS99] Robert E. Gompf and András I. Stipsicz. 4-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1999. doi:10.1090/gsm/020.\n- [Sco03] Alexandru Scorpan. Existence of foliations on 4-manifolds. Algebr. Geom. Topol., 3:1225–1256, 2003. doi:10.2140/agt.2003.3.1225.\n- [EF06] John B. Etnyre and Terry Fuller. Realizing 4-manifolds as achiral Lefschetz fibrations. Int. Math. Res. Not., pages Art. ID 70272, 21, 2006. doi:10.1155/IMRN/2006/70272.\n- [Bay06] R. İnanç Baykur. Kähler decomposition of 4-manifolds. Algebr. Geom. Topol., 6:1239–1265, 2006. doi:10.2140/agt.2006.6.1239.\n- [Hit16] Nigel Hitchin. Higgs bundles and diffeomorphism groups. In Surveys in differential geometry 2016. Advances in geometry and mathematical physics, volume 21 of Surv. Differ. Geom., pages 139–163. Int. Press, Somerville, MA, 2016.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2997, "problem_number": "KP-4.121", "title": "Kirby Problem 4.121", "statement": "Which closed, smooth4–manifolds admit open book decompositions? In particular, does every closed, simply connected 4–manifold with signature zero admit one?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.121.\n\nLiterature notes:\n(1) An open book decomposition of an n-dimensional manifold X is given by a smooth fibration f: $X \\setminus L \\to S^{1}$, where the binding $L \\subset X$ is an (n−2)–dimensional embedded submanifold with a trivial normal bundle, and there is a neighborhood $N(L) \\cong L \\times D^{2}$ such that f conforms to the local model f(x,(r, $\\theta)) =\\theta$ for $r \\ne$ 0, where $x \\in L$ and (r, $\\theta) \\in D^{2}$ are the polar coordinates.\n\n(2) If a closed, oriented n–dimensional manifold X admits an open book, its signature vanishes. An extension of this necessary condition is the vanishing of the asymmetric signature; see [Ran98] for a definition and related discussion. Vanishing signature is also a sufficient condition for\n\n\\noindent$\\bullet$ any odd $n \\geq$ 3, by the works of Alexander, Lawson and Quinn [Law78, Qui79];\n\n\\noindent$\\bullet$ any even $n \\geq$ 6, when X is simply connected, by Winkelnkemper [Win73] for $n \\geq$ 8, and for n=6 by Quinn [Qui79].\n\n(3) Kastenholz [Kas25] claims that the simplicial volume vanishes for 4manifolds that admit an open book, and gives examples of non-simplyconnected 4-manifolds with vanishing asymmetric signature that do not admit open book decompositions.\n\n(4) Open books on 4–manifolds are related to Engel structures. If a closed, oriented 4–manifold X admits an open book with a binding that is a link of tori and a monodromy that preserves a framing on the fiber, then X admits an Engel structure [CPV18].\n\n(5) It would be interesting to see if there are smooth obstructions to admitting open books in dimension four. If $X_{1}$ and $X_{2}$ admit open book decompositions, then so does $X_{1}\\#X_{2}$. There are natural open books on any $\\Sigma_{g}–bundle$ over $S^{2}$, where the binding is a pair of fibers. Combining these general constructions, one gets an open book on every $\\#_{m}(S^{2} \\times S^{2})$ and $\\#_{n}(\\mathbb{CP}^{2}\\#\\mathbb{CP}^{2})$. Do their exotic copies always admit open books?\n\nReferences cited:\n- [Ran98] Andrew Ranicki. High-dimensional knot theory. Springer Monographs in Mathematics. Springer-Verlag, New York, 1998. Algebraic surgery in codimension 2, With an appendix by Elmar Winkelnkemper. doi:10.1007/978-3-662-12011-8.\n- [Law78] Terry Lawson. Open book decompositions for odd dimensional manifolds. Topology, 17(2):189–192, 1978. doi:10.1016/S0040-9383(78)90024-1.\n- [Qui79] Frank Quinn. Open book decompositions, and the bordism of automorphisms. Topology, 18(1):55–73, 1979. doi:10.1016/0040-9383(79)90014-4.\n- [Win73] H. E. Winkelnkemper. Manifolds as open books. Bull. Amer. Math. Soc., 79:45–51, 1973. doi:10.1090/S0002-9904-1973-13085-X.\n- [Kas25] Thorben Kastenholz. Simplicial volume of open books in dimension 4, 2025. arXiv: 2504.10975.\n- [CPV18] Vincent Colin, Francisco Presas, and Thomas Vogel. Notes on open book decompositions for Engel structures. Algebr. Geom. Topol., 18(7):4275–4303, 2018. doi:10.2140/agt.2018.18.4275.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2998, "problem_number": "KP-4.122", "title": "Kirby Problem 4.122", "statement": "Is there a universal branching surface $S \\subset S^{4}$ such that every closed, orientable 4-manifold W admits a branched covering $W \\to S^{4}$ with branchin g locus S?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.122.\n\nLiterature notes:\n(1) This question originated in the introduction of [PZ05] by Piergallini– Zuddas.\n\n(2) Using the signature, it can be shown that a universal branching surface in $S^{4}$ is necessarily disconnected [Vir84, IP02].\n\n(3) There exists an orientable ribbon surface $F \\subset B^{4}$, consisting of the disjoint union of one annulus and two discs, such that every compact orientable 4-manifold M constructed by adding only 1-handles and 2-handles to $B^{4}$ admits a branched covering $M \\to B^{4}$ with branching locus F [PZ05]. Question ([PZ05, Question 2]). Does there exist a connected universal branching surface in $B^{4}$?\n\n(4) A link $L \\subset S^{3}$ is said to bea universal branching link if every closed, orientable 3-manifold Y admits a branched covering $Y \\to S^{3}$ with branching locus L. The figure eight knot, the $9_{46}$ knot, the Whitehead link, the Borromean rings, and various other knots and links are known to be universal branching links [HLM83a, HLM83b, HLM85]. The first universal branching link was found by Thurston in unpublished work. Question ([PZ05, Question 4]). Which universal links in $S^{3}$ are boundaries of universal surfaces in $B^{4}$?\n\nReferences cited:\n- [PZ05] R. Piergallini and D. Zuddas. A universal ribbon surface in $B^{4}$. Proc. London Math. Soc. (3), 90(3):763–782, 2005. doi:10.1112/S0024611504015072.\n- [Vir84] O. Ya. Viro. The signature of a branched covering. Mat. Zametki, 36(4):549–557, 1984.\n- [IP02] Massimiliano Iori and Riccardo Piergallini. 4-manifolds as covers of the 4-sphere branched over non-singular surfaces. Geom. Topol., 6:393–401, 2002. doi:10.2140/gt.2002.6.393.\n- [HLM83a] Hugh M. Hilden, M. T. Lozano, and José Marı́a Montesinos. Universal knots. Bull. Amer. Math. Soc. (N.S.), 8(3):449–450, 1983. doi:10.1090/S0273-0979-1983-15114-5.\n- [HLM83b] Hugh M. Hilden, Marı́a Teresa Lozano, and José Marı́a Montesinos. The Whitehead link, the Borromean rings and the knot 946 are universal. Collect. Math., 34(1):19– 28, 1983.\n- [HLM85] Hugh M. Hilden, Marı́a Teresa Lozano, and José Marı́a Montesinos. On knots that are universal. Topology, 24(4):499–504, 1985. doi:10.1016/0040-9383(85)90019-9.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 2999, "problem_number": "KP-4.123", "title": "Kirby Problem 4.123", "statement": "Is every closed leaf of a two dimensional co-orientable smooth taut foliation of an oriented 4-manifold smoothly genus-minimizing in its homology class?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.123.\n\nLiterature notes:\n(1) This question is due to Kronheimer [Kro98, Question 7.12], and is the natural generalization of the corresponding three-dimensional result of Thurston [Thu86]. Here we say that the $foliation\\mathcal{F}$ is taut if there exists a 2-for $m \\omega$ such that:\n\n\\noindent$\\bullet$ for every leaf $L, \\omega∥_{L}$ is an area for m;\n\n\\noindent$\\bullet$ for every $v_{1}, v_{2} \\in T_{F}$ and $z \\in T_{M}, d \\omega(v_{1}, v_{2}$, z) =0. This directly generalizes the definition in dimension three (in which case the second condition simply says that $\\omega$ is closed). As Kronheimer points out, the question is also interesting when one allows foliations with singularities with a suitable local model, e.g. the foliations defined by the vanishing of a holomorphic 1-form.\n\n(2) If the foliation is calibrated to a symplectic form, then every closed leaf is a symplectic subsurface and hence genus-minimizing by the symplectic Thom conjecture [OS00].\n\n(3) If “taut,” is eliminated from the hypotheses, then the answer is “no.” Constructions of non-genus-minimizing compact leaves of coorientable foliations of 4-manifolds were constructed by Mitsumatsu–Vogt [MV08] and Bowden [Bow11].\n\n(4) The following specific subquestion would be an interesting first step: Must a compact leaf of a coorientable smooth taut foliation of $S^{2} \\times S^{2}$ be a 2sphere?\n\nReferences cited:\n- [Kro98] P. B. Kronheimer. Embedded surfaces and gauge theory in three and four dimensions. In Surveys in differential geometry, Vol. III (Cambridge, MA, 1996), pages 243–298. Int. Press, Boston, MA, 1998.\n- [Thu86] William P. Thurston. A norm for the homology of 3-manifolds. Mem. Amer. Math. Soc., 59(339):i–vi and 99–130, 1986.\n- [OS00] Peter Ozsváth and Zoltán Szabó. The symplectic Thom conjecture. Ann. of Math. (2), 151(1):93–124, 2000. doi:10.2307/121113.\n- [MV08] Yoshihiko Mitsumatsu and Elmar Vogt. Foliations and compact leaves on 4-manifolds. I. Realization and self-intersection of compact leaves. In Groups of diffeomorphisms, volume 52 of Adv. Stud. Pure Math., pages 415–442. Math. Soc. Japan, Tokyo, 2008. doi:10.2969/aspm/05210415.\n- [Bow11] Jonathan Bowden. On closed leaves of foliations, multisections and stable commutator lengths. J. Topol. Anal., 3(4):491–509, 2011. doi:10.1142/$S^{1}$793525311000696.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3000, "problem_number": "KP-4.124", "title": "Kirby Problem 4.124", "statement": "Does there exist a hyperbolic integer homology four-sphere? What about an arithmetic one? Homology four-spheres have Euler characteristic 2, so it makes sense to ask more generally if there exist any closed hyperbolic fourmanifold with Euler characteristic 2.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.124.\n\nLiterature notes:\n(1) If any closed, hyperbolic manifold with Euler characteristic 2 exists, there are only finitely many. This is because the Gauss-Bonnet Theorem says that $V ol(M) =^{4}_{3}\\pi^{2}\\chi(M)$ for a hyperbolic four-manifold M, and there are only finitely many manifolds with volume below any fixed value. Note that in general, the Euler characteristic of a closed, orientable hyperbolic four-manifold is always even, since such manifolds have zero signature (see [LR00]), so these would be the ones with smallest volume. The best known example seems to be the Conder-Maclachlan manifold with Euler characteristic 16 [CM05]. For comparison, there are one-cusped orientable hyperbolic four-manifolds with Euler characteristic one [RT00].\n\n(2) It is worth pointing out that there are various constructions of aspherical 4-manifolds with Euler characteristic 2. For example, Luo constructed an aspherical rational homology 4-sphere [Luo88] and Tschantz constructed aspherical integer homology 4-spheres [RT05] (answering [Kir97, Problem 4.17]). The latter examples even admit metrics of non-positive curvature.\n\n(3) If there exists an arithmetic hyperbolic integer homology sphere (or, more generally, an arithmetic, closed, hyperbolic manifold with Euler characteristic 2), then by Belolipetsky [Bel07, Theorem 5.5’] (see also [Bel04, Theorem 5.5]) it has to be an index-28800 cover of the 4-dimensional hyperbolic reflection group with the below Coxeter diagram.\n\nReferences cited:\n- [LR00] D. D. Long and A. W. Reid. On the geometric boundaries of hyperbolic 4-manifolds. Geom. Topol., 4:171–178, 2000. doi:10.2140/gt.2000.4.171.\n- [CM05] Marston Conder and Colin Maclachlan. Compact hyperbolic 4-manifolds of small volume. Proc. Amer. Math. Soc., 133(8):2469–2476, 2005. doi:10.1090/S0002-9939-05-07634-3.\n- [RT00] John G. Ratcliffe and Steven T. Tschantz. The volume spectrum of hyperbolic 4-manifolds. Experiment. Math., 9(1):101–125, 2000. http://projecteuclid.org/euclid.em/1046889595.\n- [Luo88] Feng Luo. The existence of $K(\\pi,1)$ 4-manifolds which are rational homology 4-spheres. Proc. Amer. Math. Soc., 104(4):1315–1321, 1988. doi:10.2307/2047635.\n- [RT05] John G. Ratcliffe and Steven T. Tschantz. Some examples of aspherical 4-manifolds that are homology 4-spheres. Topology, 44(2):341–350, 2005. doi:10.1016/j.top.2004.10.006.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Bel07] Mikhail Belolipetsky. Addendum to: “On volumes of arithmetic quotients of $\\mathrm{SO}(1,n)$” [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 4, 749–770; mr2124587]. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 6(2):263–268, 2007.\n- [Bel04] Mikhail Belolipetsky. On volumes of arithmetic quotients of $\\mathrm{SO}(1,n)$. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 3(4):749–770, 2004.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3001, "problem_number": "KP-4.125", "title": "Kirby Problem 4.125", "statement": "Is there a noncompact, finite volume, orientable hyperbolic four-manifold without a spin structure?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.125.\n\nLiterature notes:\nAll compact orientable manifolds with dimension at most three admit spin structures. Sullivan observed that every finite volume hyperbolic nmanifold has a finite cover that admits a spin structure [Sul79b, p.533]. Reid and Long showed in [LR20] that there are orientable, finite volume, non-compact hyperbolic n-manifolds with $n \\geq$ 5 that do not admit spin structures. MartelliRiolo-Slavich showed that there are closed orientable hyperbolic four-manifolds that do not admit spin structures [MRS20].\n\nReferences cited:\n- [Sul79b] Dennis Sullivan. Hyperbolic geometry and homeomorphisms. In Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pages 543–555. Academic Press, New York-London, 1979.\n- [LR20] D. D. Long and A. W. Reid. Virtually spinning hyperbolic manifolds. Proc. Edinb. Math. Soc. (2), 63(2):305–313, 2020. doi:10.1017/s0013091519000324.\n- [MRS20] Bruno Martelli, Stefano Riolo, and Leone Slavich. Compact hyperbolic manifolds without spin structures. Geom. Topol., 24(5):2647–2674, 2020. doi:10.2140/gt.2020.24.2647.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3002, "problem_number": "KP-4.126", "title": "Kirby Problem 4.126", "statement": "(a) If M is a closed, orientable hyperbolic 4-manifold then it always has signature 0, because its Pontryagin class vanishes [Che55]. This implies that M bounds a compact, orientable 5-manifold. Is M always a geometric boundary?\n\n(b) For general n, suppose that M is a closed hyperbolic n-manifold. Is M a geometric boundary?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.126.\n\nLiterature notes:\n(1) Following [LR00], we say that $M^{n}$ is a geometric boundary if it is the totally geodesic boundary of a compact hyperbolic manifold $W^{n}+^{1}$.\n\n(2) Perhaps a good example to start with for(a)is the Davis manifold [Dav85].\n\n(3) Every closed, orientable surface of genus at least 2 has a hyperbolic metric that is the totally geodesic boundary of a compact, orientable, hyperbolic 3-manifold by [Fuj90]. However, some closed, hyperbolic, orientable 3manifolds are not the geodesic boundary of any compact, orientable, hyperbolic 4-manifold [LR00]. It is still open if there is a hyperbolic rational homology 3-sphere which is the totally geodesic boundary of a compact, orientable, hyperbolic 4-manifold (see Problem 3.77).\n\n(4) We restrict to the setting of orientable 4-manifolds as a closed, nonorientable, hyperbolic 4-manifold may have odd Euler characteristic. In this case, the 4-manifold cannot bound any compact 5-manifold, without mention of geometry. Even in the setting of nonorientable 4-manifolds, it may be interesting to ask this question with the additional hypothesis that the nonorientable 4-manifold does bound some compact (nonorientable) 5-manifold.\n\n(5) For (b), the general problem can be posed either in the orientable or nonorientable setting; one might assume that M is null-bordant to start with. In the orientable case, Long and Reid [LR00] observe that when $n =$ 4k−1, the $\\eta$-invariant [APS75a] of M would have to be integral. They give examples of orientable hyperbolic 3-manifolds with non-integral $\\eta$-invariant, which are therefore not geometric boundaries in the oriented category. The oriented version in higher dimensions could similarly be answered by finding hyperbolic (4k −1)-manifolds with non-integral $\\eta invariant$ for $k >$ 1. Some constructions of higher-dimensional orientable hyperbolic manifolds that are geometric boundaries are given in [LR01].\n\n(6) In the nonorientable case, J. Chen [Che25] claims to construct, in all dimensions $n \\geq$ 4 not of the for m n =4k−1, examples of nonorientable closed hyperbolic n-manifolds that are not the boundary of any compact (n+1)-manifold (not assuming any geometric condition). The question of whether there are nonorientable hyperbolic manifolds that are boundaries but not geometric boundaries remains open.\n\nReferences cited:\n- [Che55] Shiing-shen Chern. On curvature and characteristic classes of a Riemann manifold. Abh. Math. Sem. Univ. Hamburg, 20:117–126, 1955. doi:10.1007/BF02960745.\n- [LR00] D. D. Long and A. W. Reid. On the geometric boundaries of hyperbolic 4-manifolds. Geom. Topol., 4:171–178, 2000. doi:10.2140/gt.2000.4.171.\n- [Dav85] Michael W. Davis. A hyperbolic 4-manifold. Proc. Amer. Math. Soc., 93(2):325– 328, 1985. doi:10.2307/2044771.\n- [Fuj90] Michihiko Fujii. Hyperbolic 3-manifolds with totally geodesic boundary. Osaka J. Math., 27(3):539–553, 1990. http://projecteuclid.org/euclid.ojm/1200782445.\n- [APS75a] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geometry. I. Math. Proc. Cambridge Philos. Soc., 77:43–69, 1975. doi:10.1017/S0305004100049410.\n- [LR01] D. D. Long and A. W. Reid. Constructing hyperbolic manifolds which bound geometrically. Math. Res. Lett., 8(4):443–455, 2001. doi:10.4310/MRL.2001.v8.n4.a5.\n- [Che25] Jacopo G. Chen. Non-cobordant hyperbolic manifolds, 2025. arXiv:2501.11610.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3003, "problem_number": "KP-4.127", "title": "Kirby Problem 4.127", "statement": "Given an aspherical closed (or compact and bounded by flat 3-manifolds) 4-manifold M and a self-diffeomorphism f of M, find necessary and sufficient conditions on f so that the resulting 5-dimensional mapping torus $M_{f}$ admits a hyperbolic structure.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.127.\n\nLiterature notes:\n(1) By hyperbolization, in dimension two the sufficient and necessary condition is that f is pseudo-Anosov (see Thurston [Thu82, Thu22] and Otal [Ota01] for a complete proof). There are some analogues in strictly higher dimensions, e.g. [Far72, Theorem 6.4] gives necessary and sufficient conditions for a manifold of dimension at least six to fiber over $S^{1}$. If such a theorem worked in dimension 5, then one could potentially check such a condition against hyperbolic 5-manifolds. However, similar higher-dimensional techniques have not yet been successfully applied in the context of 5-dimensional hyperbolic manifolds.\n\n(2) Italiano–Martelli–Migliorini recently found examples of (M, f) with M 4-dimensional that produce a hyperbolic 5-manifold $M_{f}$ [IMM23].\n\nReferences cited:\n- [Thu82] William P. Thurston. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.), 6(3):357–381, 1982. doi:10.1090/S0273-0979-1982-15003-0.\n- [Thu22] William P. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. In Collected works of William P. Thurston with commentary. Vol. I. Foliations, surfaces and differential geometry, pages 495–509. Amer. Math. Soc., Providence, RI, [2022] ©2022. Reprint of [ 0956596].\n- [Ota01] Jean-Pierre Otal. The hyperbolization theorem for fibered 3-manifolds, volume 7 of SMF/AMS Texts and Monographs. American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. Translated from the 1996 French original by Leslie D. Kay.\n- [Far72] F. T. Farrell. The obstruction to fibering a manifold over a circle. Indiana Univ. Math. J., 21:315–346, 1971/72. doi:10.1512/iumj.1971.21.21024.\n- [IMM23] Giovanni Italiano, Bruno Martelli, and Matteo Migliorini. Hyperbolic 5-manifolds that fiber over $S^{1}$. Invent. Math., 231(1):1–38, 2023. doi:10.1007/s00222-022-01141-w.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3004, "problem_number": "KP-4.128", "title": "Kirby Problem 4.128", "statement": "What is the structure of 4-manifolds that admit a Riemannian metric of positive scalar curvature? There are variations of this problem for different classes of manifolds.\n\n(a) Is every closed simply connected PSC 4-manifold diffeomorphic to a connected sum of copies of $\\mathbb{CP}^{2}, \\mathbb{CP}^{2}$, and $S^{2} \\times S^{2}$?\n\n(b) What is the structure of non-simply connected closed PSC 4-manifolds?\n\n(c) Which 4-manifolds with boundary have a PSC metric?\n\n(d) Which non-compact 4-manifolds have a complete PSC metric with uniformly positive scalar curvature?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.128.\n\nLiterature notes:\n(1) Let us say that a smooth manifold is a PSC manifold if it admits a Riemannian metric of positive scalar curvature.\n\n(2) The corresponding problem in dimension 2 is easy by the Gauss-Bonnet theorem, and is solved in dimension 3 as a consequence of Perelman’s work. In particular, every orientable closed PSC 3-manifold is a connected sum of spherical space forms and copies of $S^{1} \\times S^{2}$. In dimensions at least 5 the problem is completely solved for simply connected manifolds via index theory and surgery theory, and there is a well-developed obstruction theory in the non-simply connected case; see the survey articles [Ros07b, RS01]. The classical Lichnerowicz obstruction [Lic63] states that a spin PSC 4k-manifold has vanishing $A_{(}-genus$; in dimension 4 this is equivalent to having vanishing signature. Witten [Wit94] showed that PSC 4-manifolds with $b^{+}_{2}$ >1 have vanishing Seiberg-Witten invariants. This implies, for instance, that the existence of a PSC metric depends on the underlying smooth structure. In the non-simply connected case, there are further obstructions based on Rokhlin’s theorem [RS07] and gauge theory [Lin19, Kon19, KT20, KT23].\n\n(3) It is conjectured that the answer to(a)is positive. This seems wildly optimistic, but there are no known counterexamples. A weaker version would allow an exotic $S^{4}$ with a PSC metric as a summand in the connected sum decomposition. The conjecture was stated as a question in [Kir97,\n\nProblem 4.143].\n\n(4) Some standard examples of non-simply connected PSC 4-manifolds are $S^{1} \\times Y$ for Y a spherical space for m, as well as $S^{2} \\times \\Sigma$ and $\\mathbb{RP}^{2} \\times \\Sigma$ for any closed surface $\\Sigma$. Some other constructions are described in the survey [MT21]. The decomposition theorem for PSC 4-manifolds in [BLM23] reproduces some portion of the picture in dimension 3. Problem 4.129 has a discussion of a decomposition question for non-simply connected PSC 4-manifolds, which would reduce the general problem to the classification of PSC 4-dimensional orbifolds with finite orbifold fundamental group.\n\n(5) One standard setting for part (c) requires that the metric be a product near the boundary, in which case the boundary would have positive scalar curvature. There are obstructions to the existence of PSC metrics on bounding 4-manifolds coming from index theory [BG95] and SeibergWitten theory. Formulating a good conjecture here would be welcome. Rosenberg-Weinberger [RW23] discuss a different boundary condition, and conjecture that a manifold has a PSC metric whose boundary has positive mean curvature (with respect to the outward normal) if and only if its double has a PSC metric.\n\n(6) By [Ros07b, Theorem 0.1] every non-compact 4-manifold admits a PSC metric, typically not complete, so one needs to have additional constraints on the geometry at infinity. Using Gromov’s notion of $a \\mu-bubble$, ChodoshMaximo-Mukherjee [CMM24] show that there are exotic $\\mathbb{R}^{4}s$ that do not admit a complete PSC metric with uniformly positive scalar curvature. A particular case of interest, asked by A. Mukherjee, is whether the punctured K3 surface admits such a metric.\n\nReferences cited:\n- [Ros07b] Jonathan Rosenberg. Manifolds of positive scalar curvature: a progress report. In Surveys in differential geometry. Vol. XI, volume 11 of Surv. Differ. Geom., pages 259–294. Int. Press, Somerville, MA, 2007. doi:10.4310/SDG.2006.v11.n1.a9.\n- [RS01] Jonathan Rosenberg and Stephan Stolz. Metrics of positive scalar curvature and connections with surgery. In Surveys on surgery theory, Vol. 2, volume 149 of Ann. of Math. Stud., pages 353–386. Princeton Univ. Press, Princeton, NJ, 2001.\n- [Lic63] André Lichnerowicz. Spineurs harmoniques. C. R. Acad. Sci. Paris, 257:7–9, 1963.\n- [Wit94] Edward Witten. Monopoles and four-manifolds. Math. Res. Lett., 1(6):769–796, 1994. doi:10.4310/MRL.1994.v1.n6.a13.\n- [RS07] Daniel Ruberman and Nikolai Saveliev. Dirac operators on manifolds with periodic ends. J. Gökova Geom. Topol. GGT, 1:33–50, 2007.\n- [Lin19] Jianfeng Lin. The Seiberg-Witten equations on end-periodic manifolds and an obstruction to positive scalar curvature metrics. J. Topol., 12(2):328–371, 2019. doi:10.1112/topo.12090.\n- [Kon19] Hokuto Konno. Positive scalar curvature and higher-dimensional families of Seiberg-Witten equations. Journal of Topology, 12(4):1246–1265, 2019. https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/topo.12117. doi:10.1112/topo.12117.\n- [KT20] Hokuto Konno and Masaki Taniguchi. Positive scalar curvature and 10/8-type inequalities on 4-manifolds with periodic ends. Invent. Math., 222(3):833–880, 2020. doi:10.1007/s00222-020-00979-2.\n- [KT23] Hokuto Konno and Masaki Taniguchi. Positive scalar curvature and homology cobordism invariants. J. Topol., 16(2):679–719, 2023.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [MT21] Agnese Mantione and Rafael Torres. Geography of 4-manifolds with positive scalar curvature. Expo. Math., 39(4):566–582, 2021. doi:10.1016/j.exmath.2021.05.003.\n- [BLM23] Richard H. Bamler, Chao Li, and Christos Mantoulidis. Decomposing 4-manifolds with positive scalar curvature. Adv. Math., 430:Paper No. 109231, 17, 2023. doi: 10.1016/j.aim.2023.109231.\n- [BG95] Boris Botvinnik and Peter B. Gilkey. The eta invariant and metrics of positive scalar curvature. Math. Ann., 302(3):507–517, 1995. doi:10.1007/BF01444505.\n- [RW23] Jonathan Rosenberg and Shmuel Weinberger. Positive scalar curvature on manifolds with boundary and their doubles. Pure Appl. Math. Q., 19(6):2919–2950, 2023. doi:10.4310/pamq.2023.v19.n6.a12.\n- [CMM24] Otis Chodosh, Davi Maximo, and Anubhav Mukherjee. Complete riemannian 4-manifolds with uniformly positive scalar curvature, 2024. arXiv:2407.05574.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3005, "problem_number": "KP-4.129", "title": "Kirby Problem 4.129", "statement": "Given a closed, 4-dimensional PSC manifold M, is there a (possibly disconnected) 4-dimensional orbifold $M^{1}$ with isolated singularities such that the following hold. (I) The orbifold M1 also admits a metric of positive scalar curvature. (II) All components of $M^{1}$ have finite orbifold-fundamental group. (III) M can be obtained from $M^{1}$ by a series of 0 and 1 surgeries. Here we also allow 0-surgeries between two orbifold points of the same type, which amounts to a removal of two subsets of the for $m D^{4}/\\Gamma$ and an addition of a copy of $S^{3}/\\Gamma \\times$ [0,1].", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.129.\n\nLiterature notes:\n(1) Because the orbifold $M^{1}$ has isolated singularities, its orbifold fundamental group is just the fundamental group of its regular part.\n\n(2) The following converse statement is true due to the work of Gromov– Lawson [GL80]. If M1 satisfies Property (I) and M can be obtained from $M^{1}$ as in Property (III), then M admits a PSC metric. So if the answer to the problem is ‘yes’, then this would reduce the study of PSC 4-manifolds to the study of PSC 4-orbifolds with finite fundamental group. See problem4.128 for a conjectural picture in the simply connected case.\n\n(3) The examples of non-simply connected closed 4-manifolds admitting a PSC metric described in Question (b)of Problem 4.128 satisfy properties (I)-(III). For example, a bundle over $S^{1}$ with fiber a PSC 3-manifold can be obtained from the unreduced suspension of Y by 0-surgery at the two orbifold points. $Likewise,S^{2} \\times \\Sigma_{g}$ can be obtained from a connected sum of 2g copies of $S^{1} \\times S^{3}$ by surgery on a circle.\n\n(4) It seems likely that the problem can be approached using 4-dimensional Ricci flow once there is a reasonable construction of Ricci flow with surgery in this dimension. Here the 0 and 1 surgeries would correspond to geometric surgeries that excise cylindrical singularities. Other singularities, for example those modeled on non-cylindrical shrinking solitons, would contribute components to $M^{1}$ with finite fundamental group [Bam21a]. Partial progress to the problem was made via minimal surfaces in [BLM23], where Property (II) was proved with the weaker conclusion that every component of M1 has vanishing first Betti number.\n\n(5) Properties (I)–(III) impose nontrivial topological restrictions on M. For example, they imply that M cannot be aspherical; note, however, that non-asphericity was already shown in [CL24]. More generally, Properties (I)–(III) imply that any cover of M must have finite 2-dimensional Urysohn width [Gro88] (for the lift of one and thus any Riemannian metric on M). Here we say that a metric space (X, d) has k-dimensional Urysohn width of at most W if there is a continuous mapp: $X \\to$ Ato akdimensional simplicial complex such that the diameter of any $fiberp^{-1}(a)$ is at most W. The property that any cover of a has finite 2-dimensional Urysohn width imposes a restriction on homotopy type of the manifold. See also [CLL23, LM23a, Bol09] for related results.\n\nReferences cited:\n- [GL80] Mikhael Gromov and H. Blaine Lawson, Jr. The classification of simply connected manifolds of positive scalar curvature. Ann. of Math. (2), 111(3):423–434, 1980. doi:10.2307/1971103.\n- [Bam21a] Richard H. Bamler. On the fundamental group of non-collapsed ancient Ricci flows, 2021. arXiv:2110.02254.\n- [BLM23] Richard H. Bamler, Chao Li, and Christos Mantoulidis. Decomposing 4-manifolds with positive scalar curvature. Adv. Math., 430:Paper No. 109231, 17, 2023. doi: 10.1016/j.aim.2023.109231.\n- [CL24] Otis Chodosh and Chao Li. Generalized soap bubbles and the topology of manifolds with positive scalar curvature. Ann. of Math. (2), 199(2):707–740, 2024. doi:10.4007/annals.2024.199.2.3.\n- [Gro88] M. Gromov. Width and related invariants of Riemannian manifolds. In On the geometry of differentiable manifolds (Rome, 1986), number 163-164 in Astérisque, pages 93–109. Société mathématique de France, 1988.\n- [CLL23] Otis Chodosh, Chao Li, and Yevgeny Liokumovich. Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions. Geom. Topol., 27(4):1635–1655, 2023. doi:10.2140/gt.2023.27.1635.\n- [LM23a] Yevgeny Liokumovich and Davi Maximo. Waist inequality for 3-manifolds with positive scalar curvature. In Perspectives in scalar curvature. Vol. 2, pages 799– 831. World Sci. Publ., Hackensack, NJ, [2023] ©2023.\n- [Bol09] Dmitry Bolotov. About the macroscopic dimension of certain PSC-manifolds. Algebr. Geom. Topol., 9(1):21–27, 2009. doi:10.2140/agt.2009.9.21.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3006, "problem_number": "KP-4.130", "title": "Kirby Problem 4.130", "statement": "Does longitudinal knot surgery using a knot K, along a fiber in a K3 surface always yield a reducible 4–manifold? A completely decomposable 4–manifold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.130.\n\nLiterature notes:\n(1) Longitudinal knot surgeryis a variation of Fintushel-Stern knot surgery [FS98] with a different gluing map. The original Fintushel-Stern version is defined using a knot K and a square-0 torus T in a 4-manifold X. Then $X_{K}$ is defined as (X −T $\\times D^{2}) \\cup _{\\phi}(S^{1} \\times$ E(K)). Here $E(K)$ is the exterior of the knot, and the gluing map $\\phi$ sends the longitude of K to the boundary of a meridional disk of T. Under appropriate hypotheses, Fintushel and Stern show that this operation multiplies the Seiberg-Witten invariant of X by the Alexander polynomial of K. In particular, if the Seiberg-Witten invariant of X is nontrivial, the same holds for $X_{K}$.\n\n(2) In longitudinal knot surgery, the $S^{1}$ factor in $S^{1} \\times E(K)$ is identified with the boundary of the meridional disk of T. Denote the result by $X_{K}^{\\lambda}$. Taubes shows [Tau16] that in contrast to the standard surgery, the Seiberg-Witten invariant of $X_{K}^{\\lambda}$ vanishes, even if K is nontrivial. This raises the question of whether $X_{K}^{\\lambda}$ splits as a connected sum, or is completely decomposable, i.e. is diffeomorphic to a connected sum $\\#_{m}\\mathbb{CP}^{2}\\# \\#_{n}\\mathbb{CP}^{2}$. Taubes reports, based on communications with Akbulut, Baykur, and Fintushel, that when K is an unknot, then $(K3)^{\\lambda}_{K}$ completely decomposes.\n\n(3) One motivation comes from the search (starting with [Poo86], albeit with opposite orientation conventions) for 4-manifolds that admit a Riemannian metric whose self-dual Weyl curvature $W_{+}$ vanishes. Such metrics are called conformally anti-self-dual. Taubes [Tau16] shows that when K is hyperbolic, X is a K3 surface, and T is a fiber in an elliptic fibration coming from the Kummer construction, the manifold $X_{K}^{\\lambda}$ admits a Riemannian metric with $W_{2,+}$ arbitrarily small. After repeated blowing up by connected sum with $\\mathbb{CP}$, it will have [Tau92] a conformally anti-self-dual metric; for 2–bridge knots, three blowups suffice. Hence it would be of interest to identify the manifold $(K3)^{\\lambda}_{K}$. If K is hyperbolic and $(K3)^{\\lambda}_{K}$ is completely decomposable, then $\\#_{3}\\mathbb{CP}^{2}\\#_{19}\\overline{\\mathbb{CP}}{}^{2}$ would admit a conformally anti-self-dual metric. Such metrics are known on $\\#_{3}\\mathbb{CP}^{2}\\#_{N}\\overline{\\mathbb{CP}}{}^{2}$ when $N \\geq$ 30 by work of Le Brun [Le B95] and Rollin-Singer [RS09].\n\nReferences cited:\n- [FS98] Ronald Fintushel and Ronald J. Stern. Knots, links, and 4-manifolds. Invent. Math., 134(2):363–400, 1998. doi:10.1007/s002220050268.\n- [Tau16] Clifford Henry Taubes. Some 4-manifold geometry from hyperbolic knots in $S^{3}$, 2016. arXiv:1602.01687.\n- [Poo86] Y. Sun Poon. Compact self-dual manifolds with positive scalar curvature. J. Differential Geom., 24(1):97–132, 1986. http://projecteuclid.org/euclid.jdg/1214440260.\n- [Tau92] Clifford Henry Taubes. The existence of anti-self-dual conformal structures. J. Differential Geom., 36(1):163–253, 1992. http://projecteuclid.org/euclid.jdg/1214448445.\n- [LeB95] Claude LeBrun. Anti-self-dual Riemannian 4-manifolds. In Twistor theory (Plymouth), volume 169 of Lecture Notes in Pure and Appl. Math., pages 81–94. Dekker, New York, 1995.\n- [RS09] Yann Rollin and Michael Singer. Constant scalar curvature Kähler surfaces and parabolic polystability. J. Geom. Anal., 19(1):107–136, 2009. doi:10.1007/s12220-008-9053-8.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3007, "problem_number": "KP-4.131", "title": "Kirby Problem 4.131", "statement": "Does every Lipschitz 4-manifold admit a smooth structure? Is this smooth structure unique if so? Some more specific, related questions are as follows.\n\n(a) Is there a topological, spin, closed, indefinite 4-manifold X that admits a Lipschitz structure and violates the 10/8-inequality $b_{2}(X) \\geq 5|\\sigma(X)|$ +2 by Furuta [Fur01], or the “10/8+4” inequality $b_{2}(X) \\geq 5|\\sigma(X)|$ +4 by Hopkins–Lin–Shi–Xu [HLSX22] for $X \\ne S^{4}, S^{2} \\times S^{2}, K3$.\n\n(b) Let $X \\subset \\mathbb{R}^{4}$ be a Lipschitz embedded 4-manifold with boundary $\\partial X \\cong S^{3}$. Does X admit a unique smooth structure?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.131.\n\nLiterature notes:\n(1) In dimension $n \\ne$ 4, every topological n-manifold X admits a Lipschitz structure by a theorem of Sullivan [Sul79b]. On the other hand, Donaldson and Sullivan [DS89, Theorem 2] showed that there exist closed topological 4-manifolds X that admit more than one inequivalent Lipschitz structure. Donaldson and Sullivan established that the simplest numerical invariants of smooth 4-manifolds (due to Kotschick [Kot89]) are quasiconformal invariants of smooth 4-manifolds.\n\n(2) The inequalities in Problem (a) are proved for smooth 4-manifolds using the variation of Seiberg–Witten theory introduced in [Fur01]. Hence a Lipschitz manifold violating either of those inequalities would suggest that there is no extension of this version (or perhaps other versions) of Seiberg–Witten theory to the setting of Lipschitz 4-manifolds. One reason to suspect that there is no such extension is that the Seiberg–Witten equations on X are defined in terms of the Dirac operator associated to $a Spin^{c}$ structure on X. Sullivan has conjectured [Sul99, Sul95] that the existence of a Dirac operator on X (as part of a full ‘Dirac package’) implies that X is in fact smoothable.\n\n(3) The 4-dimensional Schoenflies conjecture (Problem 4.23) is known [Sul79b] to hold for Lipschitz embeddings $\\varphi: S^{3} \\to \\mathbb{R}^{4}$. Thus a positive solution to Problem (b) would imply the Schoenflies conjecture.\n\nReferences cited:\n- [Fur01] M. Furuta. Monopole equation and the 11 8 -conjecture. Math. Res. Lett., 8(3):279– 291, 2001. doi:10.4310/MRL.2001.v8.n3.a5.\n- [HLSX22] Michael J. Hopkins, Jianfeng Lin, XiaoLin Danny Shi, and Zhouli Xu. Intersection forms of spin 4-manifolds and the $\\mathrm{Pin}(2)$-equivariant Mahowald invariant. Comm. Amer. Math. Soc., 2:22–132, 2022. doi:10.1090/cams/4.\n- [Sul79b] Dennis Sullivan. Hyperbolic geometry and homeomorphisms. In Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pages 543–555. Academic Press, New York-London, 1979.\n- [DS89] SK Donaldson and DP Sullivan. Quasiconformal 4-manifolds. Acta Mathematica, 163:181–252, 1989.\n- [Kot89] Dieter Kotschick. On manifolds homeomorphic to $\\mathbb{CP}^{2}$\\#8$\\mathbb{CP}^{2}$. Invent. Math., 95(3):591–600, 1989. doi:10.1007/BF01393892.\n- [Sul99] Dennis Sullivan. On the foundation of geometry, analysis, and the differentiable structure for manifolds. In Topics in low-dimensional topology (University Park, PA, 1996), pages 89–92. World Sci. Publ., River Edge, NJ, 1999. doi:10.1142/4202.\n- [Sul95] Dennis Sullivan. Exterior d, the local degree, and smoothability. In Prospects in topology (Princeton, NJ, 1994), volume 138 of Ann. of Math. Stud., pages 328– 338. Princeton Univ. Press, Princeton, NJ, 1995.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3008, "problem_number": "KP-5.1", "title": "Kirby Problem 5.1", "statement": "Does every cellular set in the plane have the fixed point property?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.1.\n\nLiterature notes:\n- A space has the fixed point property (FPP) if every self-map has a fixed point. A set in an $n$-manifold is cellular if it is the intersection of a countable sequence of embedded closed $n$-cells, each contained in the interior of the preceding one. A subset of the plane is cellular if and only if it is non-empty, compact, connected and non-separating (meaning that its complement is connected). Since a (planar) cellular set is the intersection of cells, and since the FPP holds for cells, it is natural to ask if it also has the FPP.\n\n- The Mandelbrot set [BM81, Man80] in the plane is cellular, but it is unknown whether it has the FPP.\n\n- The answer in dimensions $\\geq 3$ is no; a counterexample is due to Kinoshita [Kin53]. See the expository article by Bing [Bin69].\n\nReferences cited:\n- [BM81] Robert Brooks and J. Peter Matelski. The dynamics of 2-generator subgroups of P$\\mathrm{SL}(2,\\mathbb{C})$. In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume No. 97 of Ann. of Math. Stud., pages 65–71. Princeton Univ. Press, Princeton, NJ, 1981.\n- [Man80] Benoit B. Mandelbrot. Fractal aspects of the iteration of z $\\to$ $\\lambda z(1-z)$ for complex λ and z. Annals of the New York Academy of Sciences, 357(1):249–259, 1980. doi:10.1111/j.1749-6632.1980.tb29690.x.\n- [Kin53] Shin’ichi Kinoshita. On some contractible continua without fixed point property. Fund. Math., 40:96–98, 1953. doi:10.4064/fm-40-1-96-98.\n- [Bin69] R. H. Bing. The elusive fixed point property. Amer. Math. Monthly, 76:119–132, 1969. doi:10.2307/2317258.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3009, "problem_number": "KP-5.2", "title": "Kirby Problem 5.2", "statement": "(Doubly-Small Morphisms of Manifolds).\n\n- Suppose that $h: \\R^{n} \\to \\R^{n}$ is a homeomorphism (or diffeomorphism) which satisfies two smallness hypotheses:\n\n- every orbit of $h$ (of any $x\\in \\R^{n}$, under all powers of $h$) is uniformly bounded in diameter (by 1 say), and\n\n- some subsequence of powers of $h$ converges to $\\mathrm{Id}_{\\R^{n}}$ in (say) the compact-open topology.\n\nThen must $h$ be the identity map?\n\n- A special case of this question is: let $h$ be a homeomorphism of $B^{n}$ that is the identity on $\\partial B^{n}$. If there is a subsequence of powers of $h$ which converge to the identity, must $h$ itself be the identity?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.2.\n\nLiterature notes:\n- The case $n=1$ is trivial, $n=2$ seems likely to be true, as a consequence of results of Brouwer and Cartwright-Littlewood (nicely and succinctly reproved in [Bro84] and [Bro77]), and $n\\geq 3$ is open.\n\n- The question is meant to be local in nature, i.e. the question can be adapted to any open subset of $\\R^{n}$. It also applies to any manifold, where in Condition (i) one would assume that every orbit of $h$ has diameter less than some $\\epsilon$ in the compact-open topology.\n\n- The answer is `yes' if $h$ is periodic. This is Newman's Theorem, with an excellent exposition in [Dre69].\n\n- For $h$ a diffeomorphism, and using $C^{\\infty}$ convergence, the answer seems likely to be yes.\n\n- Since a homeomorphism $h: \\R^{n}\\to \\R^{n}$ generates a homomorphism $\\varphi: \\Z \\to \\Homeo(\\R^{n})$ (and vice-versa) by $m\\mapsto h^{m}$, the Question can be rephrased in terms of such a $\\varphi$. Condition (i) becomes: Assume that $\\image(\\varphi)$ lies in a suitably small neighborhood of $\\mathrm{Id}_{\\R^{n}}$, and Condition (ii) becomes: Assume that $\\varphi$ accumulates at $\\mathrm{Id}_{\\R^{n}}$. And the Question becomes: Must $\\varphi$ be the trivial homomorphism?\n\n- The question is `stronger' than the Hilbert--Smith Conjecture, discussed in Problem 5.3 below. That is, an affirmative answer to it would imply the Hilbert--Smith Conjecture. The Hilbert--Smith Conjecture (in its Question form) is equivalent to the Question above if in addition one assumes that the closure of the union of the powers of $h$ in $\\Homeo(\\R^{n})$ is compact.\n\nReferences cited:\n- [Bro84] Morton Brown. A new proof of Brouwer’s lemma on translation arcs. Houston J. Math., 10(1):35–41, 1984.\n- [Bro77] Morton Brown. A short short proof of the Cartwright-Littlewood theorem. Proc. Amer. Math. Soc., 65(2):372, 1977. doi:10.2307/2041926.\n- [Dre69] Andreas Dress. Newman’s theorems on transformation groups. Topology, 8:203–207, 1969. doi:10.1016/0040-9383(69)90010-X.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3010, "problem_number": "KP-5.3", "title": "Kirby Problem 5.3", "statement": "(Hilbert--Smith Conjecture).\n\n- The Hilbert--Smith Conjecture [Smi41] asserts that a locally compact subgroup of the homeomorphism group of a connected manifold must be a Lie group.\n\n- Conjecture: The free-action subset of any Cantor group action on an ENR is a homology-$Z$-subset (of the ENR).", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.3.\n\nLiterature notes:\n- We first define the terms in Conjecture (b).\n\n- A Cantor group is a topological group $G$ whose underlying space is a Cantor set (space). That is, $G$ is a profinite group that is non-finite and 2nd countable (hence metrizable). A universal example of such a group is the direct product of all (of the countably many) finite groups (or, if you wish, the finite symmetry groups). Other important examples of Cantor groups are the $p$-adic integers.\n\n- An ENR is a Euclidean Neighborhood Retract, that is, a subset of some $\\R^{n}$ that has a neighborhood that retracts onto it. Such subsets are characterized as stably having manifold mapping cylinder neighborhoods (like tubular neighborhoods for manifolds, or regular neighborhoods for polyhedra).\n\n- A subset $A$ of $X$ is a homology-$Z$-set (in $X$) if for any $a\\in A$ and any open neighborhood $U$ of $a$ in $X$, the relative homology $H_{*}(U,U-A)$ is 0. If $X$ is a manifold, the only such $A$ are subsets of $\\partial X$.\n\n- Pardon [Par13a] proved the Hilbert--Smith Conjecture for dimension 3. The conjecture has been reduced to whether the locally compact subgroup in question can be the $p$-adic integers. See Pardon's papers [Par13a, Par19] for more discussion and additional references.\n\n- Conjecture (b) is known as the Free-Set (is a) Z-Set Conjecture (FSZSC) and is due to R. Edwards. It is a natural and stronger version of the Hilbert--Smith Conjecture. As a special case, the FSZSC asserts that a Cantor group cannot act freely on an ENR. Like the HSC, the FSZSC has been reduced to the case of proving it for the $p$-adic integers.\n\n- The Hilbert--Smith Conjecture is closely related to the preceding problem 5.2, as discussed above.\n\nReferences cited:\n- [Smi41] P. A. Smith. Periodic and nearly periodic transformations. In Lectures in Topology, pages 159–190. Univ. Michigan Press, Ann Arbor, MI, 1941.\n- [Par13a] John Pardon. The Hilbert-Smith conjecture for three-manifolds. J. Amer. Math. Soc., 26(3):879–899, 2013. doi:10.1090/S0894-0347-2013-00766-3.\n- [Par19] John Pardon. Totally disconnected groups (not) acting on two-manifolds. In Breadth in contemporary topology, volume 102 of Proc. Sympos. Pure Math., pages 187– 193. Amer. Math. Soc., Providence, RI, 2019. doi:10.1090/pspum/102/13.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3011, "problem_number": "KP-5.4", "title": "Kirby Problem 5.4", "statement": "Is the homeomorphism group of a manifold an absolute neighborhood retract (ANR)?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.4.\n\nLiterature notes:\n- This is [Kir97, Problem 5.27] and is also listed in [Wes90]. In dimension 2 it was shown to hold by Mason [Mas71] and Luke-Mason [LM72]; it is unknown for manifolds of dimension greater than 2.\n\n- It is not easy to distinguish between ANRs and more general locally contractible spaces, so it is worth recalling that the homeomorphism group of a manifold is locally contractible [EK71, Čer69].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Wes90] James E. West. Open problems in infinite-dimensional topology. In Open problems in topology, pages 523–597. North-Holland, Amsterdam, 1990.\n- [Mas71] W. K. Mason. The space of all self-homeomorphisms of a two-cell which fix the cell’s boundary is an absolute retract. Trans. Amer. Math. Soc., 161:185–205, 1971. doi:10.2307/1995936.\n- [LM72] R. Luke and W. K. Mason. The space of homeomorphisms on a compact two-manifold is an absolute neighborhood retract. Trans. Amer. Math. Soc., 164:275– 285, 1972. doi:10.2307/1995974.\n- [EK71] Robert D. Edwards and Robion C. Kirby. Deformations of spaces of imbeddings. Ann. of Math. (2), 93:63–88, 1971. doi:10.2307/1970753.\n- [Čer69] A. V. Černavskiĭ. Local contractibility of the group of homeomorphisms of a manifold. Mat. Sb. (N.S.), 79(121):307–356, 1969. English translation in Math. USSR, Sb. 8, 287–333 (1969).", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3012, "problem_number": "KP-5.5", "title": "Kirby Problem 5.5", "statement": "Is the connected sum of (locally flat) pairs\n\n$$\n(M^{n+k}_{1},N^{n}_{1})\\#(M^{n+k}_{2},N^{n}_{2})\n$$\n\nwell-defined in the topological category?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.5.\n\nLiterature notes:\nWhen $k=1$, this is easily true from Brown's paper showing that locally flat codimension one submanifolds are flat [Bro62]. According to [Liv24], it is well-defined in any dimension when $k=2$. This uses the uniqueness of normal bundles in codimension 2, which fails in higher codimensions; see [Liv24, Appendix C]. However this question appears to be open for codimension $>2$ and $n+k>4$. It would follow from a pairwise version of the `torus trick' [KS77] if a pairwise version of Wall's non-simply connected surgery [Wal99] were known and in print.\n\nReferences cited:\n- [Bro62] Morton Brown. Locally flat imbeddings of topological manifolds. Ann. of Math. (2), 75:331–341, 1962. doi:10.2307/1970177.\n- [Liv24] Charles Livingston. Connected sums of codimension two locally flat submanifolds. Proc. Roy. Soc. Edinburgh Sect. A, 154(6):1937–1944, 2024. doi:10.1017/prm.2022.87.\n- [KS77] Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations, volume 88 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J., 1977. With notes by John Milnor and Michael Atiyah.\n- [Wal99] C. T. C. Wall. Surgery on compact manifolds, volume 69 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition, 1999. Edited and with a foreword by A. A. Ranicki. doi:10.1090/surv/069.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3013, "problem_number": "KP-5.6", "title": "Kirby Problem 5.6", "statement": "- Does every closed, PL, orientable $n$-manifold admit an $n$-fold branched covering map over $S^{n}$?\n\nAssuming that the answer to this problem is \"yes\", the following follow-up questions would be natural to ask.\n\n- Does every $n$-manifold admit an $n$-fold branched covering over $S^{n}$ with branch locus a codimension-2 embedded submanifold?\n\n- If the answer to (b) is \"yes\", then one could naturally ask if the cover can additionally be taken to be simple, meaning the covering monodromy sends every meridian of the embedded submanifold to a transposition. For $n=2,3,4$, the above papers show the answer is, \"yes\".\n\n- If the answer to (b) is \"yes\", then in a separate direction one could ask when the branch locus can be taken to be orientable.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.6.\n\nLiterature notes:\n- A classical theorem of Alexander [Ale20] says that every $n$-manifold admits a branched covering map over $S^{n}$ with no restriction on degree of the covering. An easier version of this problem would be to show that for every $n$, there exists some natural number $m_{n}$ such that every $n$-manifold admits an $m_{n}$-fold branched covering over $S^{n}$. Certainly $m_{n}$ cannot be smaller than $n$: for example, because the $n$-torus $T^{n}$ has reduced cohomology ring of length $n$, any branched covering $f:T^{n}\\to S^{n}$ has degree at least $n$ [BE78].\n\n- It is well-known that every orientable surface is a 2-fold branched cover over $S^{2}$. Hilden [Hil74], Hirsch [Hir74], and Montesinos [Mon74] showed that every 3-manifold is a 3-fold branched cover over $S^{3}$. Piergallini [Pie95] showed that every PL 4-manifold is a 4-fold branched cover over $S^{4}$. The answer to this question is unknown in all higher dimensions.\n\n- Note by Berstein--Edmonds [BE78] the answer to Question (b) is \"no\" if we additionally require that this submanifold be locally flat. For $n=2,3$ the answer is known to be \"yes\", but for $n=4$, the best known result to date, by Piergallini in [Pie95], produces branched loci that are immersed surfaces. Iori--Piergallini [IP02] later showed that every 4-manifold admits a 5-fold simple branched covering over $S^{4}$ with branch locus an embedded surface. Whether every 4-manifold admits a 4-fold cover over $S^{4}$ with embedded branch locus remains open.\n\n- The answer to Question (d) is generally negative -- for example, when $n=4k$ the existence of such a covering implies the $n$-manifold has signature zero [Vir84]. For this question, it may be simpler to restrict to the case that the $n$-manifold is null-cobordant.\n\nReferences cited:\n- [Ale20] James W. Alexander. Note on Riemann spaces. Bull. Amer. Math. Soc., 26(8):370– 372, 1920. doi:10.1090/S0002-9904-1920-03319-7.\n- [BE78] Israel Berstein and Allan Edmonds. The degree and branch set of a branched covering. Inventiones Mathematicae, 45(3):213–220, 1978.\n- [Hil74] Hugh M. Hilden. Every closed orientable 3-manifold is a 3-fold branched covering space of $S^{3}$. Bull. Amer. Math. Soc., 80:1243–1244, 1974. doi:10.1090/S0002-9904-1974-13699-2.\n- [Hir74] Ulrich Hirsch. über offene Abbildungen auf die 3-Sphäre. Math. Z., 140:203–230, 1974. doi:10.1007/BF01214163.\n- [Mon74] José Marı́a Montesinos. A representation of closed orientable 3-manifolds as 3-fold branched coverings of $S^{3}$. Bulletin of the American Mathematical Society, 80(5):845–846, 1974.\n- [Pie95] R. Piergallini. Four-manifolds as 4-fold branched covers of $S^{4}$. Topology, 34(3):497– 508, 1995. doi:10.1016/0040-9383(94)00034-I.\n- [IP02] Massimiliano Iori and Riccardo Piergallini. 4-manifolds as covers of the 4-sphere branched over non-singular surfaces. Geom. Topol., 6:393–401, 2002. doi:10.2140/gt.2002.6.393.\n- [Vir84] O. Ya. Viro. The signature of a branched covering. Mat. Zametki, 36(4):549–557, 1984.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3014, "problem_number": "KP-5.7", "title": "Kirby Problem 5.7", "statement": "(Montgomery--Yang problem). Does there exist a pseudo-free, smooth, $S^{1}$ action on $S^{5}$ with more than three multiple orbits?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.7.\n\nLiterature notes:\n- This is Problem 4.123 in [Kir97]; see the surveys by Kollár [Kol08] and Şavk [Şav24]. Recall that an $S^{1}$ action is pseudo-free if there are no fixed points and the orbits of finite isotropy are isolated.\n\n- There is a good understanding of the situation for other odd dimensional spheres: $S^{1}$ actions on $S^{3}$ are linear by Seifert [Sei33], whereas every homotopy $2k-1$ sphere (for $k\\geq 4$) admits pseudo-free $S^{1}$ actions with arbitrarily many multiple orbits by Montgomery--Yang [MY72] (for $k=4$) and Petrie [Pet75] (for $k>4$).\n\n- The answer is positive if a Seifert fibered homology 3-sphere $\\Sigma$ with more than 3 multiple fibers bounds an acyclic 4-manifold $W$, where the induced homomorphism by the inclusion of boundary is surjective on the $\\pi_{1}$. Then $\\Sigma\\times D^{2}\\cup W\\times S^{1}=S^{5}$ and $S^{1}$ acts diagonally on $\\Sigma\\times B^{2}$ and trivially on $W$. See also Problem 4.58.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Kol08] János Kollár. Is there a topological Bogomolov-Miyaoka-Yau inequality? Pure Appl. Math. Q., 4(2, Special Issue: In honor of Fedor Bogomolov. Part 1):203– 236, 2008. doi:10.4310/PAMQ.2008.v4.n2.a1.\n- [Şav24] Oğuz Şavk. A survey of the homology cobordism group. Bull. Amer. Math. Soc. (N.S.), 61(1):119–157, 2024. doi:10.1090/bull/1806.\n- [Sei33] H. Seifert. Topologie Dreidimensionaler Gefaserter Räume. Acta Math., 60(1):147– 238, 1933. doi:10.1007/BF02398271.\n- [MY72] D. C. Montgomery and C. T. Yang. Differentiable pseudo-free circle actions on homotopy seven spheres. In H.T. Ku, L. N. Mann, J. L. Sicks, and J. C. Su, editors, Conference on Compact Transformation Groups, Lecture Notes in Math., pages 41–101. Springer-Verlag, 1972. Proc. of a Conference in Univ. of Massachusetts, Amherst,.\n- [Pet75] T. Petrie. Equivariant quasi-equivalence, transversality and normal cobordism. In R. D. James, editor, Proc. International Congress of Mathematicians, Vancouver, volume 1974, pages 537–541. Canadian Mathematical Congress, Montreal, QC, 1975.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3015, "problem_number": "KP-5.8", "title": "Kirby Problem 5.8", "statement": "Is there a closed aspherical 5-manifold that is not triangulable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.8.\n\nLiterature notes:\n- In this context, `triangulable' means homeomorphic to a simplicial complex. Davis and Januszkiewicz [DJ91] constructed aspherical non-triangulable 4-manifolds, by applying the process of `hyperbolization' to Freedman's $E_{8}$ manifold. Subsequent to Manolescu's (negative) solution to the triangulation conjecture [Man16b] in dimensions $\\geq 5$, Davis--Fowler--Lafont [DFL14] used a hyperbolization procedure to construct aspherical non-triangulable $n$-manifolds for all $n\\geq 6$. They explain why their procedure breaks down in dimension 5, and explicitly asked [DFL14, \\S3] about the 5-dimensional case.\n\n- It follows from [Sie70] (using the double suspension theorem [Edw06, Can79]; compare [GS80, Mat78]) that any orientable 5-manifold is triangulable. Hence any aspherical non-triangulable 5-manifold would have a triangulable cover. This suggests the following.\n\n\\medskip\\noindent\\textsc{Question.} \\emph{Are there examples of non-triangulable aspherical 4-manifolds in dimension 4 that are virtually triangulable, i.e., that admit a finite cover that can be triangulated?}\n\nThe examples of [DJ91, DFL14] have residually finite fundamental groups, so they would be a good place to start.\n\nReferences cited:\n- [DJ91] Michael W. Davis and Tadeusz Januszkiewicz. Hyperbolization of polyhedra. J. Differential Geom., 34(2):347–388, 1991. http://projecteuclid.org/euclid.jdg/1214447212.\n- [Man16b] Ciprian Manolescu. Pin(2)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture. J. Amer. Math. Soc., 29(1):147–176, 2016. doi:10.1090/jams829.\n- [DFL14] Michael W. Davis, Jim Fowler, and Jean-François Lafont. Aspherical manifolds that cannot be triangulated. Algebr. Geom. Topol., 14(2):795–803, 2014. doi:10.2140/agt.2014.14.795.\n- [Sie70] L. C. Siebenmann. Are nontriangulable manifolds triangulable? In Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), pages 77–84. Markham Publishing Co., Chicago, IL, 1970.\n- [Edw06] Robert D. Edwards. Suspensions of homology spheres, 2006. URL: https://arxiv.org/abs/math/0610573, arXiv:math/0610573.\n- [Can79] J. W. Cannon. Shrinking cell-like decompositions of manifolds. Codimension three. Ann. of Math. (2), 110(1):83–112, 1979. doi:10.2307/1971245.\n- [GS80] David E. Galewski and Ronald J. Stern. Classification of simplicial triangulations of topological manifolds. Ann. of Math. (2), 111(1):1–34, 1980. doi:10.2307/1971215.\n- [Mat78] Takao Matumoto. Triangulation of manifolds. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 3–6. Amer. Math. Soc., Providence, R.I., 1978.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3016, "problem_number": "KP-5.9", "title": "Kirby Problem 5.9", "statement": "Let $M_{1}$ and $M_{2}$ be smooth manifolds of dimension $n$. Suppose $M_{1}$ admits an $S$-map into $\\R^{p}$. If $M_{2}$ is homeomorphic to $M_{1}$, then does $M_{2}$ admit an $S$-map as well?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.9.\n\nLiterature notes:\n- For this problem, we consider $C^{\\infty}$ maps $f:M\\to \\R^{p}$ of smooth $n$-dimensional manifolds into $\\R^{p}$ with $n\\geq p\\geq 1$. Let $S$ be a set of (equivalence classes of) singularities of smooth map germs $(\\R^{n},0)\\to (\\R^{p},0)$. A smooth map $f:M\\to \\R^{p}$ is called an $S$-map if all its singularities are equivalent to a singularity belonging to $S$.\n\n- If $M_{2}$ is diffeomorphic to $M_{1}$, then by composing a diffeomorphism $M_{2}\\to M_{1}$ and an $S$-map $M_{1}\\to \\R^{p}$, we get an $S$-map on $M_{2}$. Therefore, if the answer to the above question is negative, then $M_{2}$ is not diffeomorphic to $M_{1}$; in other words, $(M_{1},M_{2})$ is an exotic pair of manifolds.\n\nIn this sense, for $n\\leq 3$, the answer is always affirmative.\n\n- Let $S$ be the singleton consisting of the definite fold singularity represented by the map germ\n\n$$\n(x_{1},x_{2},\\ldots,x_{p-1},x_{p}^{2}+x_{p+1}^{2}+\\cdots+x_{n}^{2}).\n$$\n\nIn this case, an $S$-map is called a \\emph{special generic map}. Then there are many examples of manifold pairs $(M_{1},M_{2})$ for which the answer is negative as follows.\n\n- For $n\\geq 7$ and $p=n-1,n-2$ and $n-3$, the pair of the standard $n$-sphere and an exotic $n$-sphere is such an example [Sae93b].\n\n- For $n=4$ and $p=1,2$ and 3, the pair of the standard $\\R^{4}$ and an exotic $\\R^{4}$ is such an example, provided that we consider proper special generic maps [Sae10].\n\n- For $n=4$ and $p=3$, there are quite a few such examples $(M_{1},M_{2})$, where $M_{1}$ is a connected sum of $S^{2}\\times S^{2}$ and $\\CP^{2}\\#\\overline{\\CP}{}^{2}$ [Sae93a, SS99].\n\n- According to the solution to the Poincaré Conjecture in high dimensions due to Smale [Sma61], for special generic functions with $p=1$, the answer is affirmative for $n\\geq 5$. In other words, only by the existence of a special generic function with $n\\geq 5$ and $p=1$ one cannot detect exotic differentiable structures. Note also that for $n=4$ and $p=1$, the problem for special generic functions is equivalent to the 4-dimensional smooth Poincaré Conjecture.\n\nOn the other hand, for $p=1$, let $S$ be the set of non-degenerate critical points of indices in the set $\\{0,2,3,\\ldots,n-2,n\\}$. Then, for $n\\geq 5$, by Smale's result [Sma62a], the answer to the problem is always affirmative, whereas for $n=4$, the problem seems to be still open.\n\n- Let us consider $C^{\\infty}$ stable maps of closed 4-manifolds into $\\R^{4}$. They have, in general, fold, cusp, swallowtail, butterfly and umbilic singularities. It is known that when the 4-manifold is oriented, each umbilic singularity can be given a sign $+1$ or $-1$ and the total number of umbilic points counted with signs is equal to 3 times the signature. Furthermore, if the signature vanishes, then all the umbilic singularities can be eliminated by homotopy [And82, Sti95]. Hence, for $S$ consisting of fold, cusp, swallowtail and butterfly singularities, the answer to the above question is affirmative in this case.\n\n- Let $F$ be a set of (equivalence classes) of singular fibers of $C^{\\infty}$ maps in the following sense [Sae04]. Let $f_{i}:M_{i}\\to N_{i}$ be smooth maps, $i=0,1$. For $y_{i}\\in N_{i}$, we say that the fibers over $y_{0}$ and $y_{1}$ are equivalent if for some open neighborhoods $U_{i}$ of $y_{i}$ there exist diffeomorphisms $\\widetilde{\\phi}:f_{0}^{-1}(U_{0})\\to f_{1}^{-1}(U_{1})$ and $\\phi:U_{0}\\to U_{1}$ with $\\phi(y_{0})=y_{1}$, which make the following diagram commutative:\n\n$$\n\\begin{CD}\n(f_{0}^{-1}(U_{0}),f_{0}^{-1}(y_{0})) @>{\\widetilde{\\phi}}>> (f_{1}^{-1}(U_{1}),f_{1}^{-1}(y_{1}))\\\\\n@V{f_{0}}VV @VV{f_{1}}V\\\\\n(U_{0},y_{0}) @>{\\phi}>> (U_{1},y_{1})\n\\end{CD}\n$$\n\nWhen the fibers over $y_{0}$ and $y_{1}$ are equivalent, we also say that for $i=0,1$, the map germs $f_{i}:(M_{i},f_{i}^{-1}(y_{i}))\\to (N_{i},y_{i})$ are right-left equivalent. A smooth map $f:M\\to \\R^{p}$ is called an $F$-map if all its singular fibers are equivalent to a singular fiber belonging to $F$. Then, we can ask the same question as Problem 5.9 for $F$-maps.\n\n- It is known that for maps of smooth closed oriented 4-manifolds into $\\R^{3}$, each singular fiber of type $\\mathrm{III}^{8}$ can be given a sign $+1$ or $-1$, and for a certain class of generic maps (so-called $C^{\\infty}$ stable maps), the number of $\\mathrm{III}^{8}$-fibers counted with signs coincides with the signature of the source 4-manifold [SY06]. Therefore, it is a natural question if singular fibers of type $\\mathrm{III}^{8}$ with opposite signs can be eliminated by homotopy. So far, we do not know if there exists a smooth closed oriented 4-manifold $M$ with zero signature such that an arbitrary $C^{\\infty}$ stable map $M\\to \\R^{3}$ necessarily has a singular fiber of type $\\mathrm{III}^{8}$. Though for connected sums of copies of $S^{2}\\times S^{2}$ and $\\CP^{2}\\#\\overline{\\CP}{}^{2}$, the answer is known to be affirmative.\n\n- In the problem, one can also consider the problem by replacing \"homeomorphic to\" by \"homotopy equivalent to\". For example, for $n\\geq p\\geq 1$, let $S$ be the set of fold singularities of all (absolute) indices. Then, the answer to the homotopy version of the problem is affirmative for closed orientable 4-manifolds and $1\\leq p\\leq 4$ [Sae03, Sad04].\n\nReferences cited:\n- [Sae93b] Osamu Saeki. Topology of special generic maps of manifolds into Euclidean spaces. Topology Appl., 49(3):265–293, 1993. doi:10.1016/0166-8641(93)90116-U.\n- [Sae10] Osamu Saeki. Special generic maps on open 4-manifolds. J. Singul., 1:1–12, 2010. doi:10.5427/jsing.2010.1a.\n- [Sae93a] Osamu Saeki. Topology of special generic maps into $\\mathbb{R}^{3}$. Mat. Contemp., 5:161– 186, 1993. Workshop on Real and Complex Singularities (São Carlos, 1992).\n- [SS99] Osamu Saeki and Kazuhiro Sakuma. Special generic maps of 4-manifolds and compact complex analytic surfaces. Math. Ann., 313(4):617–633, 1999. doi:10.1007/s002080050275.\n- [Sma61] Stephen Smale. Generalized Poincaré’s conjecture in dimensions greater than four. Ann. of Math. (2), 74:391–406, 1961. doi:10.2307/1970239.\n- [Sma62a] S. Smale. On the structure of manifolds. Amer. J. Math., 84:387–399, 1962. doi: 10.2307/2372978.\n- [And82] Yoshifumi Ando. Elimination of certain Thom-Boardman singularities of order two. J. Math. Soc. Japan, 34(2):241–267, 1982. doi:10.2969/jmsj/03420241.\n- [Sti95] Robert Paul Stingley. Singularities of maps between 4-manifolds. ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–State University of New York at Stony Brook. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt= info:ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri:pqdiss:9539899.\n- [Sae04] Osamu Saeki. Topology of singular fibers of differentiable maps, volume 1854 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2004. doi:10.1007/b100393.\n- [SY06] Osamu Saeki and Takahiro Yamamoto. Singular fibers of stable maps and signatures of 4-manifolds. Geom. Topol., 10:359–399, 2006. doi:10.2140/gt.2006.10.359.\n- [Sae03] Osamu Saeki. Fold maps on 4-manifolds. Comment. Math. Helv., 78(3):627–647, 2003. doi:10.1007/s00014-003-0758-9.\n- [Sad04] Rustam Sadykov. Elimination of singularities of smooth mappings of 4-manifolds into 3-manifolds. Topology Appl., 144(1-3):173–199, 2004. doi:10.1016/j.topol.2004.04.006.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3017, "problem_number": "KP-5.10", "title": "Kirby Problem 5.10", "statement": "The Andrews--Curtis Conjecture [AC65] for the trivial group: a presentation of the trivial group can be changed to the trivial presentation by Andrews--Curtis moves.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.10.\n\nLiterature notes:\n- This problem appears in Problem 5.2 in [Kir97].\n\n- Let $\\mathcal{P}$ be a finite presentation of a given group $\\pi$. The stabilized Andrews--Curtis moves (abbreviated AC moves) change the presentation $\\mathcal{P}=\\{x_{1},\\ldots,x_{n}:R_{1},\\ldots,R_{m}\\}$ as follows:\n\n- $R_{i}\\to R_{i}^{-1}$,\n\n- $R_{i}\\to R_{i}R_{j}$, $i\\neq j$,\n\n- $R_{i}\\to wR_{i}w^{-1}$, $w$ any word,\n\n- add generator $x_{n+1}$ and relation $wx_{n+1}$.\n\nThe AC Conjecture for the trivial group states that the presentation $\\mathcal{P}$ can be changed to the trivial presentation $(x_{1},\\ldots,x_{n}:x_{1},\\ldots,x_{n})$ by AC moves.\n\nNote that redundant relations cannot be added, so that $m-n$ is unchanged. Also note that the broader conjecture that any two presentations of an arbitrary finitely presented group are equivalent by AC moves is false for some nontrivial groups, e.g., the trefoil group [HAMS93]. (See also [AC66].)\n\n- Given a presentation $\\mathcal{P}$ of the trivial group, we can construct a 5-dimensional handlebody $Y$ from a 0-handle, 1-handles for each generator, and 2-handles for each relation; $Y$ is unique because the attaching maps are isotopic because they are homotopic.\n\nThen if the AC Conjecture is true, $Y$ is diffeomorphic to the 5-ball, because the AC moves correspond to handle moves (in particular the move $R_{i}\\to R_{i}R_{j}$, $i\\neq j$ corresponds to sliding the $i^{\\mathrm{th}}$ handle over the $j^{\\mathrm{th}}$ handle).\n\nReferences cited:\n- [AC65] J. J. Andrews and M. L. Curtis. Free groups and handlebodies. Proceedings of the American Mathematical Society, 16:192–195, 1965.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [HAMS93] C. Hog-Angeloni, W. Metzler, and A. J. Sieradski. Two-dimensional Homotopy and Combinatorial Group Theory, volume 197 of London Math. Soc. Lect. Note Ser. Cambridge Univ. Press, 1993.\n- [AC66] J. J. Andrews and M. L. Curtis. Extended Nielsen operations in free groups. Amer. Math. Monthly, 73:21–28, 1966. doi:10.2307/2313917.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3018, "problem_number": "KP-5.11", "title": "Kirby Problem 5.11", "statement": "(Whitehead's Asphericity Question). Is every subcomplex of an aspherical 2-complex aspherical?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.11.\n\nLiterature notes:\n- This was problem 5.4 in [Kir97]; a thorough discussion of older progress on this problem, also known as the Whitehead Conjecture, may be found in [HAMS93, Chapter X]. It was stated as a question on page 428 in [Whi41], with the implicit assumption that the complex is finite, but the question still makes sense without that assumption.\n\n- The fundamental group of a finite-dimensional aspherical complex is torsion free, so an \\emph{a priori} easier question is whether the fundamental group of a subcomplex of an aspherical two-complex is necessarily torsion-free. For partial results relating to fundamental group structure, see [Coc54, Ada55, How79].\n\n- Bestvina and Brady [BB97] showed that the Whitehead conjecture and the Eilenberg--Ganea conjecture cannot both be true. The Eilenberg--Ganea conjecture [EG57] is that a group with cohomological dimension 2 has a 2-dimensional Eilenberg-Mac Lane space.\n\nMore concretely, Bestvina and Brady showed that the (Bestvina--Brady) group $H_{L}$ associated with a flag triangulation $L$ of a spine of the Poincaré homology sphere is either a counterexample to the Eilenberg--Ganea conjecture, or there exists a contractible 2-complex $Y$ that contains a non-aspherical subcomplex [BB97, Theorem 8.7]. Note that the potential counterexample $Y$ to the Whitehead conjecture that they construct is infinite (since the group $H_{L}$ acts freely and cellularly on it).\n\n- In [How79], J. Howie reduced the problem to finding counterexamples $K\\subset L$ of two types: (i) $L$ is finite and contractible and $K=L-e$ for some 2-cell $e$; or (ii) $L$ is the union of an infinite chain of non-aspherical subcomplexes $K=K_{0}\\subset K_{1}\\subset \\cdots$ such that each inclusion is null-homotopic.\n\nAccording to Howie's results in [How79] and [How83], if the Andrews--Curtis conjecture (Problem 5.10) holds, then the standard 2-complexes associated with LOT presentations account for all test cases of type (i). Recall that a version of the Andrews--Curtis conjecture asserts that every finite contractible 2-complex can be 3-deformed to a point. A LOT presentation is a group presentation described by a (finite) labeled oriented tree, and the associated 2-complex has the homotopy type of a ribbon disc complement [How83].\n\nIt remains unknown whether all LOT complexes are aspherical, though significant progress has been made in the last decades, establishing the asphericity of various subfamilies. For instance, Harlander and Rosebrock [HR17] showed that alternating ribbon disk-complements, the ones that can be encoded by injective labeled oriented trees, are aspherical.\n\n- Costa and Farber [CF17] give a model for random simplicial complexes in which aspherical 2-complexes satisfy the Whitehead conjecture with probability 1.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [HAMS93] C. Hog-Angeloni, W. Metzler, and A. J. Sieradski. Two-dimensional Homotopy and Combinatorial Group Theory, volume 197 of London Math. Soc. Lect. Note Ser. Cambridge Univ. Press, 1993.\n- [Whi41] J. H. C. Whitehead. On adding relations to homotopy groups. Ann. of Math. (2), 42:409–428, 1941. doi:10.2307/1968907.\n- [Coc54] W. H. Cockroft. On 2-dimensional aspherical complexes. Proc. London Math. Soc., 4:375–384, 1954.\n- [Ada55] J. F. Adams. A new proof of a theorem of W. H. Cockcroft. J. London Math. Soc., 30:482–488, 1955.\n- [How79] James Howie. Aspherical and acyclic 2-complexes. J. London Math. Soc. (2), 20(3):549–558, 1979. doi:10.1112/jlms/s2-20.3.549.\n- [BB97] Mladen Bestvina and Noel Brady. Morse theory and finiteness properties of groups. Invent. Math., 129(3):445–470, 1997. doi:10.1007/s002220050168.\n- [EG57] Samuel Eilenberg and Tudor Ganea. On the Lusternik-Schnirelmann category of abstract groups. Ann. of Math. (2), 65:517–518, 1957. doi:10.2307/1970062.\n- [How83] James Howie. Some remarks on a problem of J. H. C. Whitehead. Topology, 22(4):475–485, 1983. doi:10.1016/0040-9383(83)90038-1.\n- [HR17] Jens Harlander and Stephan Rosebrock. Injective labeled oriented trees are aspherical. Math. Z., 287(1-2):199–214, 2017. doi:10.1007/s00209-016-1823-6.\n- [CF17] A. Costa and M. Farber. Large random simplicial complexes, II; the fundamental group. J. Topol. Anal., 9(3):441–483, 2017. doi:10.1142/$S^{1}$793525317500170.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3019, "problem_number": "KP-5.12", "title": "Kirby Problem 5.12", "statement": "(Zeeman Conjecture). If $K$ is a finite contractible 2-complex, then $K\\times I$ collapses to a point [Zee64, Conjecture (1)].", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.12.\n\nLiterature notes:\nThis problem appears as part of Problem 5.2 in [Kir97].\n\nThe Zeeman Conjecture for a special polyhedron that is the spine of a compact 3-manifold is equivalent [GR83] to the Poincaré Conjecture, and thus true by Perelman. The Zeeman conjecture for special polyhedra that do not embed in compact 3-manifolds is equivalent to the Andrews--Curtis Conjecture [Mat87]. A nice discussion of this general area may be found in Chapters I, XI and XII of [HAMS93] and the exposition in [Kup21].\n\nReferences cited:\n- [Zee64] E. C. Zeeman. On the dunce hat. Topology, 2:341–358, 1964. doi:10.1016/0040-9383(63)90014-4.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [GR83] D. Gillman and D. Rolfsen. The Zeeman conjecture for standard spines is equivalent to the Poincaré conjecture. Topology, 22(3):315–323, 1983. doi:10.1016/0040-9383(83)90017-4.\n- [Mat87] S. V. Matveev. The Zeeman conjecture for nonthickenable special polyhedra is equivalent to the Andrews-Curtis conjecture. Sibirsk. Mat. Zh., 28(6):66–80, 218, 1987.\n- [HAMS93] C. Hog-Angeloni, W. Metzler, and A. J. Sieradski. Two-dimensional Homotopy and Combinatorial Group Theory, volume 197 of London Math. Soc. Lect. Note Ser. Cambridge Univ. Press, 1993.\n- [Kup21] Alexander Kupers. Zeeman’s conjecture. Grad. J. Math., 6(1):35–42, 2021. https://gradmath.org/wp-content/uploads/2021/07/GJM2021-Kupers.pdf.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3020, "problem_number": "KP-5.13", "title": "Kirby Problem 5.13", "statement": "Let $M$ be a finite-volume hyperbolic $n$-manifold.\n\n- Does it always have a finite cover with $b_{1}>0$?\n\n- Does it always have a finite cover with fundamental group that surjects onto a non-abelian free group?\n\n- Does it have a finite cover with cubulated fundamental group?\n\n- When $n$ is odd, does it always have a finite cover that fibers over the circle?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.13.\n\nLiterature notes:\n- When $n=3$, these questions all have positive answers, by work of Agol [Ago13] and Wise [Wis21]. However, their methods break down when $n>3$. The crucial steps in their argument are as follows. By work of Kahn-Markovic [KM12], $\\pi_{1}(M^{3})$ contains lots of surface subgroups. These are `codimension 1' subgroups and hence $\\pi_{1}(M^{3})$ has a finite index subgroup that is the fundamental group of a compact non-positively curved cube complex, i.e., it is cubulated. Agol showed that such cubulated groups are virtually special, and Wise showed that virtually special groups have many excellent virtual properties. In particular, they have a finite index subgroup with $b_{1}>0$ and indeed a finite index subgroup that surjects onto a non-abelian free group. Using a 3-dimensional argument involving sutured manifolds, Agol [Ago08] was able to show that hyperbolic 3-manifolds virtually fiber. An alternative and more general argument using group rings was given by Kielak [Kie20].\n\n- This argument fails at the first step when $n>3$. The methods of Kahn-Markovic have been extended to all odd dimensions by Hamenstädt [Ham15], and hence $\\pi_{1}(M)$ is known to contain many surface subgroups. However, these groups are not codimension one, and therefore do not establish that the group is cubulated. However, it is known that some hyperbolic $n$-manifolds are cubulated when $n>3$. Indeed, any arithmetic hyperbolic $n$-manifold containing a totally geodesic $(n-1)$-dimensional (possibly immersed) submanifold is cubulated. When $n$ is even, this includes all arithmetic hyperbolic $n$-manifolds.\n\n- There are further results for arithmetic hyperbolic $n$-manifolds. There are 3 types of arithmetic lattices in $\\mathrm{SO}(n,1)$: those arising from quadratic forms (type I), those arising from quaternion algebras (type II), and those arising from octonion algebras, and type III - trialitarian lattices. The first type appears in all dimensions and are known to be cubulated after work of Bergeron--Wise [BW12]. The second type are not known to be cubulated, but only appear in even dimensions. The third type only occurs in dimension 7; for these it is known [BC17a] that the first Betti number of every congruence subgroup equals 0. Moreover, it is still open if the lattices have the congruence subgroup property.\n\n- If a manifold fibers over the circles, its Euler characteristic is zero. In even dimensions, the volume of a hyperbolic manifold is proportional to its Euler characteristic, and hence its Euler characteristic is necessarily non-zero. This explains the restriction to $n$ odd in the final question above. The first examples of hyperbolic 5-manifolds that fiber over the circle were given by Italiano--Martelli--Migliorini [IMM23].\n\n- When a hyperbolic 3-manifold fibers over the circle, the universal cover $\\widetilde{F}$ of the fiber has a circle at infinity, and the inclusion $\\widetilde{F}\\to \\mathbb{H}^{3}$ is known to extend continuously to a map from the circle at infinity of $\\widetilde{F}$ to the sphere at infinity of $\\mathbb{H}^{3}$. This forms a space-filling curve. This was established by Cannon and Thurston [CT07]. It would be interesting to know whether there is any analogue of this in higher dimensions. The fiber will not in general have word hyperbolic fundamental group, and so part of the challenge would be to define its space at infinity appropriately.\n\n- By work of Delzant and Gromov [DG05], complex hyperbolic manifolds do \\emph{not} have cubulated fundamental group.\n\nReferences cited:\n- [Ago13] Ian Agol. The virtual Haken conjecture. Doc. Math., 18:1045–1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning, https://elibm.org/article/10000267. doi:10.4171/DM/421.\n- [Wis21] Daniel T. Wise. The structure of groups with a quasiconvex hierarchy, volume 209 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, [2021] ©2021.\n- [KM12] Jeremy Kahn and Vladimir Markovic. Immersing almost geodesic surfaces in a closed hyperbolic three manifold. Ann. of Math. (2), 175(3):1127–1190, 2012. doi: 10.4007/annals.2012.175.3.4.\n- [Ago08] Ian Agol. Criteria for virtual fibering. J. Topol., 1(2):269–284, 2008. doi:10.1112/jtopol/jtn003.\n- [Kie20] Dawid Kielak. Residually finite rationally solvable groups and virtual fibring. J. Amer. Math. Soc., 33(2):451–486, 2020. doi:10.1090/jams/936.\n- [Ham15] Ursula Hamenstädt. Incompressible surfaces in rank one locally symmetric spaces. Geom. Funct. Anal., 25(3):815–859, 2015. doi:10.1007/s00039-015-0330-y.\n- [BW12] Nicolas Bergeron and Daniel T. Wise. A boundary criterion for cubulation. Amer. J. Math., 134(3):843–859, 2012. doi:10.1353/ajm.2012.0020.\n- [BC17a] Nicolas Bergeron and Laurent Clozel. Sur la cohomologie des variétés hyperboliques de dimension 7 trialitaires. Israel J. Math., 222(1):333–400, 2017. doi:10.1007/s11856-017-1593-9.\n- [IMM23] Giovanni Italiano, Bruno Martelli, and Matteo Migliorini. Hyperbolic 5-manifolds that fiber over $S^{1}$. Invent. Math., 231(1):1–38, 2023. doi:10.1007/s00222-022-01141-w.\n- [CT07] James W. Cannon and William P. Thurston. Group invariant Peano curves. Geom. Topol., 11:1315–1355, 2007. doi:10.2140/gt.2007.11.1315.\n- [DG05] Thomas Delzant and Misha Gromov. Cuts in Kähler groups. In Infinite groups: geometric, combinatorial and dynamical aspects, volume 248 of Progr. Math., pages 31–55. Birkhäuser, Basel, 2005. doi:10.1007/3-7643-7447-0\\\\_3.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3021, "problem_number": "KP-5.14", "title": "Kirby Problem 5.14", "statement": "Does there exist a 1-cusped finite-volume hyperbolic $n$-manifold for any $n\\geq 5$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.14.\n\nLiterature notes:\n- Hyperbolic $n$-manifolds with one cusp are abundant when $n=2$ or 3, and Kolpakov--Martelli [KM13a] constructed 1-cusped hyperbolic 4-manifolds. However, there are far fewer constructions of hyperbolic $n$-manifolds when $n\\geq 4$ and they tend to produce manifolds with a large number of cusps. There are no known examples in dimensions $\\geq 5$ with a single cusp; the question of their existence was raised in [LR02].\n\n- Stover [Sto13] proved that there are no 1-cusped arithmetic hyperbolic $n$-dimensional orbifolds when $n>30$. In fact, he showed that for each $m\\geq 1$, there is a $c_{m}\\geq 1$ such that there are no $m$-cusped arithmetic hyperbolic $n$-dimensional orbifolds when $n>c_{m}$.\n\nReferences cited:\n- [KM13a] Alexander Kolpakov and Bruno Martelli. Hyperbolic four-manifolds with one cusp. Geom. Funct. Anal., 23(6):1903–1933, 2013. doi:10.1007/s00039-013-0247-2.\n- [LR02] D. D. Long and A. W. Reid. All flat manifolds are cusps of hyperbolic orbifolds. Algebr. Geom. Topol., 2:285–296, 2002. doi:10.2140/agt.2002.2.285.\n- [Sto13] Matthew Stover. On the number of ends of rank one locally symmetric spaces. Geom. Topol., 17(2):905–924, 2013. doi:10.2140/gt.2013.17.905.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3022, "problem_number": "KP-5.15", "title": "Kirby Problem 5.15", "statement": "Suppose $M$ is a manifold with a complete Riemannian metric with nonnegative Ricci curvature. Is the fundamental group of $M$ finitely generated?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.15.\n\nLiterature notes:\n- This is a conjecture of Milnor, following on his paper [Mil68a]. The conjecture was known to hold in dimension 2 by work of Cohn-Vossen [CV35] and in dimension 3 by work of Liu [Liu13]. Bruè--Naber--Semola construct counterexamples in dimensions at least 7 [BNS25], and 6 [BNS23]. The issue is therefore to determine the status of the conjecture in dimensions 4 and 5. This question is explicitly posed in [BNS23, Question 1.1], along with some other interesting open problems in the area.\n\nPrevious work of Wilking [Wil00] shows that the fundamental group of any counterexample could be assumed to be abelian; the fundamental group of the example from [BNS25] in dimension 7 is $\\Q/\\Z$.\n\n- A crucial point in the Bruè--Naber--Semola paper [BNS25, Lemma 9.1] is that the orbit of the mapping class group of $S^{3}\\times S^{3}$ acting on the standard product metric lies in a single path component of the space of positive Ricci curvature metrics on $S^{3}\\times S^{3}$. This raises several questions.\n\n- What can be said about the action of the mapping class group of $S^{2}\\times S^{2}$ on the path components of the space of positive Ricci curvature metrics on $S^{2}\\times S^{2}$?\n\n- Is the space of positive Ricci curvature metrics on $S^{2}\\times S^{2}$ path connected?\n\n- One could ask the same questions for $S^{2}\\times S^{3}$; see [BNS23, \\S6] for a particular diffeomorphism of $S^{2}\\times S^{3}$ such that the pullback of the standard metric is connected to the standard metric through Ricci curvature metrics.\n\nReferences cited:\n- [Mil68a] J. Milnor. A note on curvature and fundamental group. J. Differential Geometry, 2:1–7, 1968. http://projecteuclid.org/euclid.jdg/1214501132.\n- [CV35] Stefan Cohn-Vossen. Kürzeste Wege und Totalkrümmung auf Flächen. Compositio Math., 2:69–133, 1935. URL: http://www.numdam.org/item?id=CM 1935 2 69 0.\n- [Liu13] Gang Liu. 3-manifolds with nonnegative Ricci curvature. Invent. Math., 193(2):367– 375, 2013. doi:10.1007/s00222-012-0428-x.\n- [BNS25] Elia Bruè, Aaron Naber, and Daniele Semola. Fundamental groups and the Milnor conjecture. Ann. of Math. (2), 201(1):225–289, 2025. doi:10.4007/annals.2025.201.1.4.\n- [BNS23] Elia Bruè, Aaron Naber, and Daniele Semola. Six dimensional counterexample to the Milnor Conjecture, 2023. arXiv:2311.12155.\n- [Wil00] Burkhard Wilking. On fundamental groups of manifolds of nonnegative curvature. Differential Geom. Appl., 13(2):129–165, 2000. doi:10.1016/S0926-2245(00) 00030-9.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3023, "problem_number": "KP-5.16", "title": "Kirby Problem 5.16", "statement": "Let $R$ be $\\Z$ or a field. Let $A$ and $B$ be differential graded algebras so that either:\n\\begin{itemize}\n\n- As a graded algebra, $A$ (respectively $B$) is isomorphic to the free, non-commutative $R$-algebra on finitely many homogeneous generators $x_{i}$, i.e., to the tensor algebra on the free $R$-module generated by $x_{1},\\ldots,x_{n}$.\n\n- As a graded algebra, $A$ (respectively $B$) is isomorphic to a free graded-commutative algebra on finitely many homogeneous generators $x_{i}$. (So, if the $x_{i}$ have even gradings, $A$ is a polynomial algebra.)\n\\end{itemize}\n(Different generators can have different gradings, $A$ and $B$ may have different numbers of generators, and gradings of generators may be negative.)\n\nAre the following questions decidable?\n\n- Is $A$ stable tame isomorphic to $B$?\n\n- Is $A$ quasi-isomorphic to $B$?\n\n- Is $A$ derived Morita equivalent to $B$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.16.\n\nLiterature notes:\n- This question is inspired by contact topology, where many invariants take the form of a finitely-generated differential graded algebras up to a notion of equivalence.\n\n- The kinds of dgas described in the problem are often called \\emph{semifree} in the literature. The information in a semifree dga $A$ consists of the grading of the generators $x_{1},\\ldots,x_{n}$ and the elements $d(x_{i})$, which are either polynomials in $x_{1},\\ldots,x_{n}$ (in the commutative case) or linear combinations of words in $x_{1},\\ldots,x_{n}$ (in the non-commutative case).\n\n- A dga homomorphism is a quasi-isomorphism if it induces an isomorphism on homology; two dgas are quasi-isomorphic if there is a dga $C$ and quasi-isomorphism $C\\to A$ and $C\\to B$. Stable tame isomorphism of semifree dgas was introduced in [Che02] (see also [ENS02, Section 3.3] and [EN22, Section 3.2]); it allows introducing a pair of canceling generators (stabilizing) and isomorphisms sending some $x_{i}$ to $x_{i}$ plus a word in the other variables. Two dgas are derived Morita equivalent if the derived categories of differential modules over them are equivalent (as triangulated categories). Stable tame isomorphism implies quasi-isomorphism implies derived Morita equivalence.\n\n- This question arises from contact homology and related invariants. For example, to a Legendrian knot $\\Lambda$ in $\\R^{3}$, one can associate a dga $\\mathcal{A}_{\\Lambda}$, the Legendrian contact dga, whose stable tame isomorphism class is an isotopy invariant of $\\Lambda$ (see the citations above). The Legendrian contact dga can also be defined for knots in other manifolds and higher-dimensional Legendrian knots. Contact homology can also be defined for (certain) closed contact manifolds; in this case, one is forced to work with a commutative dga. In practice, it seems to be hard to tell whether two dgas are stable tame isomorphic or quasi-isomorphic.\n\n- One strategy that has been developed to distinguish dgas is to study the set of augmentations of $A$, i.e., dga maps $A\\to R$ (where $R$ lies in grading 0 and has trivial differential); this set is called the augmentation variety of $A$. Given an augmentation of $A$, one can form the linearized homology with respect to that augmentation, which is a finitely generated $R$-module [Che02].\n\n- For Legendrian knots in $\\R^{3}$, the set of augmentations form the objects of a category, the augmentation category, which in the case of Legendrian contact homology is equivalent to a certain category of sheaves [NRS+20]. The set of augmentations and the linearized contact homology depend only on the abelianization of the dga, but the augmentation category needs the non-commutative version.\n\n- Contact homology can also be applied to study smooth objects in low-dimensional topology, for instance by considering the unit cotangent bundle or unit conormal bundle. An instance is Ng's knot contact homology [Ng05] (a variant of which is a complete knot invariant [ENS18]); another is given in Problem 4.106. It is possible that a result along these lines could also be applied to decision problems in symplectic or smooth topology.\n\n- In some of the applications, one actually works over $\\Z\\{t,t^{-1},x_{1},\\ldots,x_{n}\\}$, say, but this seems unlikely to affect the question.\n\n- In the special case that $A$ and $B$ are commutative with all of their generators in positive gradings, Sullivan's theory of minimal models gives an algorithm for answering the question; in particular, that case is well studied in rational homotopy theory. Note also that there is a unique augmentation in this case; in particular, the dgas arising from contact topology rarely have this property.\n\n- Given that the question has some similarity to Hilbert's tenth problem, the answer might be different for $R$ a finite field from $R=\\Q$ or $\\Z$, say.\n\nReferences cited:\n- [Che02] Yuri Chekanov. Differential algebra of Legendrian links. Invent. Math., 150(3):441– 483, 2002. doi:10.1007/s002220200212.\n- [ENS02] John B. Etnyre, Lenhard L. Ng, and Joshua M. Sabloff. Invariants of Legendrian knots and coherent orientations. J. Symplectic Geom., 1(2):321–367, 2002. http: //projecteuclid.org/euclid.jsg/1092316653.\n- [EN22] John B. Etnyre and Lenhard L. Ng. Legendrian contact homology in $\\mathbb{R}^{3}$. In Surveys in differential geometry 2020. Surveys in 3-manifold topology and geometry, volume 25 of Surv. Differ. Geom., pages 103–161. Int. Press, Boston, MA, [2022] ©2022.\n- [NRS+20] Lenhard Ng, Dan Rutherford, Vivek Shende, Steven Sivek, and Eric Zaslow. Augmentations are sheaves. Geom. Topol., 24(5):2149–2286, 2020. doi:10.2140/gt.2020.24.2149.\n- [Ng05] Lenhard Ng. Knot and braid invariants from contact homology. I. Geom. Topol., 9:247–297, 2005. doi:10.2140/gt.2005.9.247.\n- [ENS18] Tobias Ekholm, Lenhard Ng, and Vivek Shende. A complete knot invariant from contact homology. Invent. Math., 211(3):1149–1200, 2018. doi:10.1007/s00222-017-0761-1.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3024, "problem_number": "KP-5.17", "title": "Kirby Problem 5.17", "statement": "Let $(W,\\omega,V_{i},\\phi_{i})$ be two Weinstein structures on a fixed symplectic manifold $(W,\\omega)$ (or equivalently consider two Weinstein handle decompositions). Is there a Weinstein homotopy from $(W,\\omega,V_{1},\\phi_{1})$ to $(W,\\omega,V_{2},\\phi_{2})$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.17.\n\nLiterature notes:\n% The PDF prints V_2 in both terms of this convex combination; retained as printed.\n\n- The convex combination $V_{t}=(1-t)V_{2}+tV_{2}$ is a Liouville vector field positively transverse to the boundary, but it may fail to be gradient-like for some $t$. The question is whether we can find a family of convex Liouville vector fields $V_{t}$ that are all gradient-like.\n\n- There are various additional hypotheses one can add which will ensure a Weinstein homotopy exists. For example, by [CE12, Proposition 11.22], if we have a fixed complex Stein structure $J$ on $(W,\\omega)$, any two $J$-convex (plurisubharmonic) functions $\\phi_{1}$ and $\\phi_{2}$ will yield Weinstein homotopic Weinstein structures. There are numerous other specific Weinstein homotopies constructed in [CE12, Chapter 12], that assume some specific properties or estimates about the Liouville form and/or gradient-like function.\n\nFor Weinstein manifolds of dimension strictly greater than 4, there is a notion of \"flexible Weinstein manifolds.\" In this case any two flexible Weinstein structures are Weinstein homotopic [CE12, Theorem 14.5].\n\n- A special case, which may be easier, is the following. Suppose $(W,\\omega)$ admits a symplectomorphism that restricts to a contactomorphism on the boundary, but which is not isotopic to the identity in the class of symplectomorphisms that are contact on the boundary. Let $\\lambda$ be one Liouville form on $(W,\\omega)$, corresponding to a Liouville vector field $V_{1}$ which is gradient-like for some Morse function $\\phi_{1}$. Consider the Liouville form $f^{*}\\lambda$ and denote its corresponding Liouville vector field by $V_{2}$. Then $V_{2}$ is gradient-like for $\\phi_{2}=f^{*}\\phi_{1}$. We can ask if $(W,\\omega,V_{1},\\phi_{1})$ Weinstein homotopic to $(W,\\omega,V_{2},\\phi_{2})$ in this case.\n\n- It may be helpful to think about the Liouville form $\\lambda_{i}$ rather than the Liouville vector field $V_{i}$ (which are related by $\\iota_{V_{i}}\\omega=\\lambda_{i}$). While $\\lambda_{i}$ is not closed (since $d\\lambda_{i}=\\omega$ by the Liouville condition), $\\lambda_{1}-\\lambda_{2}$ is closed and thus represents an element of $H^{1}(W)$. It may be easier to construct a counterexample for manifolds where $H^{1}(W)$ is nontrivial, in which case, one should refine the question to consider manifolds where $H^{1}(W)=0$, or further that $W$ is simply connected.\n\n- This question is open in arbitrary dimensions $>2$, but dimension 4 could be a good place to start.\n\nReferences cited:\n- [CE12] Kai Cieliebak and Yakov Eliashberg. From Stein to Weinstein and back, volume 59 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2012. Symplectic geometry of affine complex manifolds. doi:10.1090/coll/059.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3025, "problem_number": "KP-5.18", "title": "Kirby Problem 5.18", "statement": "- In higher dimensions, find the `non-analytic' cohomology module $\\HH^{*}_{?}$ analogous to the analytic lattice cohomology $\\HH^{*}_{\\mathrm{an}}$, and a natural functor $\\HH^{*}_{\\mathrm{an}}\\to \\HH^{*}_{?}$ connecting them.\n\n- Define a version $\\ECH^{*}$ of Embedded Contact Homology for isolated complex singularity links (in any dimension) associated with the canonical contact structure of the link, together with a natural graded $\\Z[U]$-module morphism $\\HH^{*}_{\\mathrm{an}}\\to \\ECH^{*}$. More precisely,\n\n- Show that this morphism is injective.\n\n- Fix the diffeomorphism type of a link, and also a contact structure on it that can be realized as the canonical contact structure associated with some singularity analytic structure. Then characterize the family $\\{\\HH^{*}_{\\mathrm{an}}\\}$, indexed by all the possible analytic germs inducing this contact link, via those graded sub-$\\Z[U]$-modules of $\\ECH^{*}$ which satisfy certain specific properties.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.18.\n\nLiterature notes:\n- As background, consider a complex normal surface singularity with a rational homology sphere link. The (topological) lattice cohomology $\\HH^{*}_{\\mathrm{top}}$, associated with the link (which is a plumbed 3-manifold associated with a connected negative definite graph), was introduced by Némethi in [Ném05, Ném08]. It has an analytic analogue $\\HH^{*}_{\\mathrm{an}}$ constructed recently in [ÁN21a, ÁN21b]. Both theories are multigraded, $\\HH^{*}_{\\mathrm{top}}=\\bigoplus_{q\\geq 0}\\HH^{q}_{\\mathrm{top}}$, and $\\HH^{q}_{\\mathrm{top}}$ is a $2\\Z$-graded $\\Z[U]$-module, with an additional grading indexed by the $\\operatorname{spin}^{c}$ structures of the link. The same is valid for $\\HH^{*}_{\\mathrm{an}}$ as well. Even more, the existence of a graded $\\Z[U]$-module morphism $\\HH^{*}_{\\mathrm{an}}\\to \\HH^{*}_{\\mathrm{top}}$ was also established.\n\nFollowing on calculations in [OS03b], it was conjectured in [Ném08] that the topological version can be identified with Heegaard Floer homology. Further results from [Ném08, OSS14b] support the conjecture, whose full proof was announced in [Zem25].\n\nIn particular, it is also isomorphic with any other (co)homology theory of 3-manifolds that agree with Heegaard Floer homology, e.g. with Embedded Contact Homology (introduced in [Hut10, Hut14], for its equivalence with the Heegaard Floer homology see [CGH20]).\n\nOne of the $\\operatorname{spin}^{c}$ structures (determined from the analytic structure) is distinguished; it is called the `canonical $\\operatorname{spin}^{c}$ structure'. The analytic lattice cohomology associated with this canonical structure has an extension to any higher dimension, for complex isolated singularities (again, with certain restrictions).\n\n- Part (b) addresses the important issue that in higher dimension the link of the isolated singularity contains essentially less information than might be needed to construct such an invariant (e.g. the link can be even the usual sphere). In particular, in the above context, the `non-analytic' version shouldn't be `topological' or `smooth'. However, the smooth structure enhanced with its canonical contact structure (induced by the analytic structure of the germ) might produce the desired cohomology.\n\nA good candidate for this is some version of Embedded Contact Homology associated with the canonical contact structure of the link. For 3-dimensional singularity links the canonical contact structure can be uniquely determined from the link itself [CNPP06]. However, in higher dimensions, it is an essential enhancement of the smooth structure. For results regarding ECH in higher dimensions see e.g. [CHT24].\n\nReferences cited:\n- [Ném05] András Némethi. On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds. Geom. Topol., 9:991–1042, 2005. doi:10.2140/gt.2005.9.991.\n- [Ném08] András Némethi. Lattice cohomology of normal surface singularities. Publ. Res. Inst. Math. Sci., 44(2):507–543, 2008. doi:10.2977/prims/1210167336.\n- [ÁN21a] Tamás Ágoston and András Némethi. Analytic lattice cohomology of surface singularities, 2021. arXiv:2108.12294.\n- [ÁN21b] Tamás Ágoston and András Némethi. Analytic lattice cohomology of surface singularities, II (the equivariant case), 2021. arXiv:2108.12429.\n- [OS03b] Peter Ozsváth and Zoltán Szabó. On the Floer homology of plumbed threemanifolds. Geom. Topol., 7:185–224, 2003. doi:10.2140/gt.2003.7.185.\n- [OSS14b] Peter Ozsváth, András I. Stipsicz, and Zoltán Szabó. A spectral sequence on lattice homology. Quantum Topol., 5(4):487–521, 2014. doi:10.4171/QT/56.\n- [Zem25] Ian Zemke. The equivalence of lattice and Heegaard Floer homology. Duke Math. J., 174(5):857–910, 2025. doi:10.1215/00127094-2024-0044.\n- [Hut10] Michael Hutchings. Embedded contact homology and its applications. In Proceedings of the International Congress of Mathematicians. Volume II, pages 1022–1041. Hindustan Book Agency, New Delhi, 2010.\n- [Hut14] Michael Hutchings. Lecture notes on embedded contact homology. In Contact and symplectic topology, volume 26 of Bolyai Soc. Math. Stud., pages 389–484. János Bolyai Math. Soc., Budapest, 2014. doi:10.1007/978-3-319-02036-5\\\\_9.\n- [CGH20] Vincent Colin, Paolo Ghiggini, and Ko Honda. An exposition of the equivalence of Heegaard Floer homology and embedded contact homology. In Characters in low-dimensional topology, volume 760 of Contemp. Math., pages 45–101. Amer. Math. Soc., [Providence], RI, [2020] ©2020. doi:10.1090/conm/760/15286.\n- [CNPP06] Clément Caubel, András Némethi, and Patrick Popescu-Pampu. Milnor open books and Milnor fillable contact 3-manifolds. Topology, 45(3):673–689, 2006. doi:10.1016/j.top.2006.01.002.\n- [CHT24] Vincent Colin, Ko Honda, and Yin Tian. Applications of higher-dimensional Heegaard Floer homology to contact topology. J. Topol., 17(3):Paper No. e12349, 77, 2024. doi:10.1112/topo.12349.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3026, "problem_number": "OPG-3031", "title": "trace inequality", "statement": "Let $A,B$ be positive semidefinite, by Jensen's inequality, it is easy to see $[tr(A^s+B^s)]^{\\frac{1}{s}}\\leq [tr(A^r+B^r)]^{\\frac{1}{r}}$, whenever $s>r>0$.\n\nWhat about the $tr(A^s+B^s)^{\\frac{1}{s}}\\leq tr(A^r+B^r)^{\\frac{1}{r}}$, is it still valid?", "background": "Source: Open Problem Garden. Original node ID: 3031. URL: http://www.openproblemgarden.org/op/trace_inequality.\n\nSource subject path: Algebra.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/trace_inequality\n- Subject(s): Algebra\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 11th, 2008 by Miwa Lin\n\nComments:\n- August 28th, 2010 | ojs | clarification: I suppose that A and B are square hermitian matrixes and that s and r are real numbers. But what are the brackets supposed to represent? Sorry for my ignorance if it is obvious.\n- October 12th, 2008 | Miwa Lin | It is open.: It is open.\n- December 20th, 2010 | Anonymous | this inequality is not true.: this inequality is not true. It was denied some years ago. See X.Z Zhan's\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"trace inequality\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3027, "problem_number": "OPG-23298", "title": "Elementary symmetric of a sum of matrices", "statement": "Problem\n\nGiven a Matrix $A$, the $k$-th elementary symmetric function of $A$, namely $S_k(A)$, is defined as the sum of all $k$-by- $k$ principal minors.\n\nFind a closed expression for the $k$-th elementary symmetric function of a sum of N $n$-by- $n$ matrices, with $0\\le N\\le k\\le n$ by using partitions.", "background": "Source: Open Problem Garden. Original node ID: 23298. URL: http://www.openproblemgarden.org/op/elementary_symmetric_of_a_sum_of_matrices.\n\nSource subject path: Algebra.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/elementary_symmetric_of_a_sum_of_matrices\n- Subject(s): Algebra\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: December 8th, 2008 by rscosa\n\nProblem-page discussion:\nThe Newton-Girard formulas imply particular expressions for small values of $k$ and $N$, for example, $S_2(A+B)=S_2(A)+S_2(B)+S_1(A)S_1(B)-S_1(AB)$.\n\nSource links:\n- elementary symmetric function: http://en.wikipedia.org/wiki/elementary symmetric function\n\nDiscussion links:\n- Newton-Girard formulas: http://en.wikipedia.org/wiki/Newton-Girard formulas\n\nComments:\n- August 27th, 2010 | olivier | that's ok: that's ok\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Elementary symmetric of a sum of matrices\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3028, "problem_number": "OPG-37283", "title": "Finite Lattice Representation Problem", "statement": "Conjecture\n\nThere exists a finite lattice which is not the congruence lattice of a finite algebra.", "background": "Source: Open Problem Garden. Original node ID: 37283. URL: http://www.openproblemgarden.org/op/finite_congruence_lattice_problem.\n\nSource subject path: Algebra.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/finite_congruence_lattice_problem\n- Subject(s): Algebra\n- Keywords: congruence lattice; finite algebra\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: December 10th, 2010 by williamdemeo\n\nProblem-page discussion:\nA well-known result of universal algebra states: every algebraic lattice is isomorphic to the congruence lattice of an algebra. Thus there is essentially no restriction on the shape of a congruence lattice of a general algebra. It is natural to ask whether the same is true for finite lattices and finite algebras. That is, does every finite lattice occur as the congruence lattice of a finite algebra? This fundamental question, asked over 40 years ago, is among the most elusive problems of universal algebra.\n\nThis question is important because, until it is answered, we lack something very basic in our understanding of algebras -- namely, if we assume an algebra is finite, does this place any restriction on the shape of its congruence lattice? If so, then finite algebras are fundamentally different from infinite algebras in this sense.\n\nBibliography:\n[GS] Gratzer and Schmidt, Characterizations of congruence lattices of abstract algebras, Acta Sci. Math. (Szeged) 24 (1963), 34-59.\n\n[P5] Palfy and Pudlak. Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis, 11 (1980), 22–27.\n\n[PT] Pudlak and Tuma, Every finite lattice can be embedded in the lattice of all equivalences over a finite set. Algebra Universalis, 10 (1980), 74–95.\n\nRelated:\nRelated problems\nWhich lattices occur as intervals in subgroup lattices of finite groups?\n\nDiscussion links:\n- universal algebra: http://en.wikipedia.org/wiki/Universal_algebra\n- algebraic lattice: http://en.wikipedia.org/wiki/Algebraic_lattice\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Finite Lattice Representation Problem\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3029, "problem_number": "OPG-48715", "title": "Sub-atomic product of funcoids is a categorical product", "statement": "Conjecture In the category of continuous funcoids (defined similarly to the category of topological spaces) the following is a direct categorical product:\n\n- Product morphism is defined similarly to the category of topological spaces.\n- Product object is the sub-atomic product.\n- Projections are sub-atomic projections.\n\nSee details, exact definitions, and attempted proofs here.", "background": "Source: Open Problem Garden. Original node ID: 48715. URL: http://www.openproblemgarden.org/op/sub_atomic_product_of_funcoids_is_a_categorical_product.\n\nSource subject path: Algebra.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/sub_atomic_product_of_funcoids_is_a_categorical_product\n- Subject(s): Algebra\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 21st, 2013 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nRelated:\nRelated problems\nA construction of direct product in the category of continuous maps between endo-funcoids\n\nSource links:\n- here: http://planetmath.org/directproductsinacategoryoffuncoids\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Sub-atomic product of funcoids is a categorical product\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3030, "problem_number": "OPG-50149", "title": "inverse of an integer matrix", "statement": "Question I've been working on this for a long time and I'm getting nowhere. Could you help me or at least tell me where to look for help. Suppose D is an m-by-m diagonal matrix with integer elements all $\\ge 2$. Suppose X is an m-by-n integer matrix $(m \\le n)$. Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.", "background": "Source: Open Problem Garden. Original node ID: 50149. URL: http://www.openproblemgarden.org/op/inverse_of_an_integer_matrix.\n\nSource subject path: Algebra.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/inverse_of_an_integer_matrix\n- Author(s): Gregory, Steven A\n- Subject(s): Algebra\n- Keywords: invertable matrices, integer matrices\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 27th, 2013 by lvoyster\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"inverse of an integer matrix\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3031, "problem_number": "OPG-57824", "title": "Graphs of exact colorings", "statement": "Conjecture For $c \\geq m \\geq 1$, let $P(c,m)$ be the statement that given any exact $c$-coloring of the edges of a complete countably infinite graph (that is, a coloring with $c$ colors all of which must be used at least once), there exists an exactly $m$-colored countably infinite complete subgraph. Then $P(c,m)$ is true if and only if $m=1$, $m=2$, or $c=m$.", "background": "Source: Open Problem Garden. Original node ID: 57824. URL: http://www.openproblemgarden.org/op/graphs_of_exact_colorings.\n\nSource subject path: Algebra.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/graphs_of_exact_colorings\n- Subject(s): Algebra\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 3rd, 2013 by sabisood\n\nProblem-page discussion:\nStacey and Weidl have shown that given $m \\geq 3$, there is an integer $C(m)$ such that $P(c,m)$ is false for all $c \\geq C(m)$.\n\n* M. Erickson, \"A Conjecture Concerning Ramsey's Theorem,\" Discrete Mathematics 126, 395--398 (1994); MR 95b:05209\n\nA. Stacey and P. Weidl, \"The Existence of Exactly m-Coloured Complete Subgraphs,\" J. of Combinatorial Theory, Series B 75, 1-18 (1999)\n\n* indicates original appearance(s) of problem.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Graphs of exact colorings\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3032, "problem_number": "OPG-60031", "title": "Waring rank of determinant", "statement": "Question What is the Waring rank of the determinant of a $d \\times d$ generic matrix?\n\nFor simplicity say we work over the complex numbers. The $d \\times d$ generic matrix is the matrix with entries $x_{i,j}$ for $1 \\leq i,j \\leq d$. Its determinant is a homogeneous form of degree $d$, in $d^2$ variables. If $F$ is a homogeneous form of degree $d$, a power sum expression for $F$ is an expression of the form $F = \\ell_1^d+\\dotsb+\\ell_r^d$, the $\\ell_i$ (homogeneous) linear forms. The Waring rank of $F$ is the least number of terms $r$ in any power sum expression for $F$. For example, the expression $xy = \\frac{1}{4}(x+y)^2 - \\frac{1}{4}(x-y)^2$ means that $xy$ has Waring rank $2$ (it can't be less than $2$, as $xy \\neq \\ell_1^2$ ).\n\nThe $2 \\times 2$ generic determinant $x_{1,1}x_{2,2}-x_{1,2}x_{2,1}$ (or $ad-bc$ ) has Waring rank $4$. The Waring rank of the $3 \\times 3$ generic determinant is at least $14$ and no more than $20$, see for instance Lower bound for ranks of invariant forms, Example 4.1. The Waring rank of the permanent is also of interest. The comparison between the determinant and permanent is potentially relevant to Valiant's \"VP versus VNP\" problem.", "background": "Source: Open Problem Garden. Original node ID: 60031. URL: http://www.openproblemgarden.org/op/waring_rank_of_determinant.\n\nSource subject path: Algebra.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/waring_rank_of_determinant\n- Author(s): Teitler, Zach\n- Subject(s): Algebra\n- Keywords: Waring rank, determinant\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 15th, 2019 by Zach Teitler\n\nSource links:\n- Lower bound for ranks of invariant forms: https://arxiv.org/abs/1409.0061\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 20.\n\nAttempt notes:\nTarget:\nMake progress on \"Waring rank of determinant\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3033, "problem_number": "OPG-725", "title": "$C^r$ Stability Conjecture", "statement": "Conjecture Any $C^r$ structurally stable diffeomorphism is hyperbolic.", "background": "Source: Open Problem Garden. Original node ID: 725. URL: http://www.openproblemgarden.org/op/c_r_stability_conjecture.\n\nSource subject path: Analysis.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/c_r_stability_conjecture\n- Author(s): Palis, J.; Smale, S.\n- Subject(s): Analysis\n- Keywords: diffeomorphisms,; dynamical systems\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: December 20th, 2007 by m n\n\nProblem-page discussion:\nSee the definitions of: stractural stability and hyperbolicity.\n\nThe conjecture is due to J Palis and S Smale (1970's). In the case $r=1$ the conjecture was proved by R Mañé (Publ. IHES 1986). In higher regularity, $r>1$, the conjecture is one of the most important and difficult problems in dynamical systems.\n\nThere is a similar conjecture for the vector fields or flows, and in the $C^1$ topology has been proved by S Hayashi (Ann Math. 1997).\n\nDiscussion links:\n- stractural stability: http://en.wikipedia.org/wiki/Structural_stability\n- hyperbolicity: http://en.wikipedia.org/wiki/Hyperbolic_structure\n- R Mañé (Publ. IHES 1986): http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1987__66__161_0\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"$C^r$ Stability Conjecture\" in Analysis, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 8, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3034, "problem_number": "OPG-36697", "title": "Invariant subspace problem", "statement": "Problem Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?", "background": "Source: Open Problem Garden. Original node ID: 36697. URL: http://www.openproblemgarden.org/op/invariant_subspace_problem.\n\nSource subject path: Analysis.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/invariant_subspace_problem\n- Subject(s): Analysis\n- Keywords: subspace\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 9th, 2009 by tchow\n\nProblem-page discussion:\nLet $H$ be a Hilbert space. The subspaces $\\{0\\}$ and $H$ are trivially invariant under any linear operator on $H$, and so these are referred to as the trivial invariant subspaces. The problem is concerned with determining whether bounded operators necessarily have non-trivial invariant subspaces.\n\nThis is one of the most famous open problems in functional analysis. Enflo [1] constructed Banach spaces for which the corresponding question has a negative answer, and recently Argyros and Haydon constructed a Banach space for which the corresponding question has a positive answer [4].\n\nFor a nice overview to the problem see [2], [3] or [5].\n\nBibliography:\n[1] P. Enflo, On the invariant subspace problem for Banach spaces, Acta Math. 158 (1987), 213-313. MathSciNet\n\n[2] B. S. Yadav, The Present State and Heritages of the Invariant Subspace Problem, Milan J. Math. 73 (2005), 289-316. MathSciNet another link\n\n[3] H. Radjavi and P. Rosenthal, The Invariant Subspace Problem, The Mathematics Intelligencer 4 (1982), no. 1, 33-37. MathSciNet\n\n[4] S. A. Argyros and R. G. Haydon, A hereditarily indecomposable $L_\\infty$-space that solves the scalar-plus-compact problem, arXiv:0903.3921 (2009).\n\n[5] J. Noel. The Invariant Subspace Problem. Honours Thesis, Thompson Rivers University. Link to pdf.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0892591\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2175046\n- another link: http://www.math.leidenuniv.nl/%7Enaw/serie5/deel06/jun2005/pdf/yadav.pdf\n- H. Radjavi: http://www.math.uwaterloo.ca/PM_Dept/Homepages/Radjavi/radjavi.shtml\n- P. Rosenthal: http://www.math.toronto.edu/rosent/\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0678734\n- arXiv:0903.3921: http://www.arxiv.org/abs/0903.3921\n- J. Noel: http://www.math.mcgill.ca/jnoel\n- Link to pdf: http://www.math.mcgill.ca/jnoel/pdf/Honours.pdf\n\nComments:\n- April 11th, 2011 | Anonymous | closed: Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?\n- April 1st, 2011 | Anonymous | ستار اكاديمي 8: ستار اكاديمي 8\n\nthank you man, u'r good:)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Invariant subspace problem\" in Analysis, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 8, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3035, "problem_number": "OPG-36928", "title": "Criterion for boundedness of power series", "statement": "Question Give a necessary and sufficient criterion for the sequence $(a_n)$ so that the power series $\\sum_{n=0}^{\\infty} a_n x^n$ is bounded for all $x \\in \\mathbb{R}$.", "background": "Source: Open Problem Garden. Original node ID: 36928. URL: http://www.openproblemgarden.org/op/criterion_for_boundedness_of_power_series.\n\nSource subject path: Analysis.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/criterion_for_boundedness_of_power_series\n- Author(s): Rüdinger, Andreas\n- Subject(s): Analysis\n- Keywords: boundedness; power series; real analysis\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: May 9th, 2009 by andreasruedinger\n\nProblem-page discussion:\nConsider a power series $\\sum_{n=0}^{\\infty} a_n x^n$ that is convergent for all $x \\in {\\mathbb R}$, thus defining a function $f: {\\mathbb R} \\to {\\mathbb R}$. Are there criteria to decide whether $f$ is bounded (which e.g. is the case for the series with $a_n = (-1)^k/(2k)!$ for $n = 2k$ and $a_n = 0$ for n odd)? Some general remarks:\n\n- A necessary condition for $\\sum_n a_n x^n$ to be bounded is that $a_0$ is the only non-zero $a_n$ or there are infinitely many non-zero $a_n$ 's which change sign infinitely many times.\n- Changing a finite set of $a_n$ 's (except $a_0$ ) does leave the subspace of bounded power series.\n- The subspace of bounded power series is \"large\" in the sense that it is both a linear subspace (closed under sums and scalar multiples) and an algebra (closed under products). It includes all functions of the form $a \\cos( f(x))$, where $f$ is any entire function $\\mathbb{R} \\to \\mathbb{R}$. The question whether the subspace of bounded power series contains only these functions seems to be open.\n\nComments:\n- February 10th, 2011 | Anonymous | A necessary condition: It seems the sum would be bounded if there are only finitely many non-zero a sub n; it is not apparent to me that a sub 0 be the only non-zero a sub n.\n- June 21st, 2012 | Anonymous | What you have then is a: What you have then is a polynomial, and any nonconstant polynomial function is unbounded.\n- February 16th, 2011 | Comet | Re: A necessary condition: I posted the above comment anonymously, but now I have created an account. \"It seems the sum would be bounded if there are only finitely many non-zero a sub n; it is not apparent to me that a sub 0 be the only non-zero a sub n.\"\n- April 27th, 2012 | Anonymous | harder than that: Look at sin(x)=x-x^3/6+x^5/120-....\n\nJPB\n- June 21st, 2012 | Anonymous | sin x = cos(pi/2 - x): The sine function is in the class mentioned.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Criterion for boundedness of power series\" in Analysis, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 8, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3036, "problem_number": "OPG-37185", "title": "Something like Picard for 1-forms", "statement": "Conjecture Let $D$ be the open unit disk in the complex plane and let $U_1,\\dots,U_n$ be open sets such that $\\bigcup_{j=1}^nU_j=D\\setminus\\{0\\}$. Suppose there are injective holomorphic functions $f_j: U_j \\to \\mathbb{C},$ $j=1,\\ldots,n,$ such that for the differentials we have ${\\rm d}f_j={\\rm d}f_k$ on any intersection $U_j\\cap U_k$. Then those differentials glue together to a meromorphic 1-form on $D$.", "background": "Source: Open Problem Garden. Original node ID: 37185. URL: http://www.openproblemgarden.org/op/something_like_picard_for_1_forms.\n\nSource subject path: Analysis.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/something_like_picard_for_1_forms\n- Author(s): Elsner, B.\n- Subject(s): Analysis\n- Keywords: Essential singularity; Holomorphic functions; Picard's theorem; Residue of 1-form; Riemann surfaces\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 26th, 2010 by MathOMan\n\nProblem-page discussion:\nIt is an evidence that the 1-form is holomorphic on $D\\setminus\\{0\\}$. In the case that its residue at the origin vanishes we can use Picard's big theorem.\n\nBibliography:\n*B. Elsner: Hyperelliptic action integral, Annales de l'institut Fourier 49(1), p.303–331\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Something like Picard for 1-forms\" in Analysis, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 8, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3037, "problem_number": "OPG-41335", "title": "Inequality for square summable complex series", "statement": "Conjecture For all $\\alpha=(\\alpha_1,\\alpha_2,\\ldots)\\in l_2(\\cal{C})$ the following inequality holds $$\\sum_{n\\geq 1}|\\alpha_n|^2\\geq \\frac{6}{\\pi^2}\\sum_{k\\geq0}\\bigg| \\sum_{l\\geq0}\\frac{1}{l+1}\\alpha_{2^k(2l+1)}\\bigg|^2$$", "background": "Source: Open Problem Garden. Original node ID: 41335. URL: http://www.openproblemgarden.org/op/inequality_for_square_summable_complex_series.\n\nSource subject path: Analysis.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/inequality_for_square_summable_complex_series\n- Author(s): Retkes, Zoltan\n- Subject(s): Analysis\n- Keywords: Inequality\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: December 25th, 2012 by tigris35711\n\nRelated:\nRelated problems\nCriterion for boundedness of power series\n\nComments:\n- November 3rd, 2013 | Anonymous | Solution: It's a simple application of the Shwartz inequality:\n\n$$\\sum_{k}\\left|\\sum_{l} \\frac{1}{l+1}a_{2^k(2l+1)}\\right|^2 \\le$$$$\\le \\sum_{k}\\left|\\sum_{l} \\frac{1}{l+1}\\left|a_{2^k(2l+1)}\\right|\\right|^2 \\le$$Shwartz:$$\\le \\sum_{k} \\left(\\sum_{l}\\frac{1}{(l+1)^2}\\right)\\left(\\sum_{h}|a_{2^k(2l+1)}|^2\\right) =$$$$= \\sum_{k} \\frac{\\pi^2}{6}\\sum_{h}|a_{2^k(2l+1)}|^2 =$$$$= \\frac{\\pi^2}{6} \\sum_{k}\\sum_{h}|a_{2^k(2l+1)}|^2 =$$$$= \\frac{\\pi^2}{6} \\sum_{n}|a_n|^2$$because$A_k:=\\{ 2^k(2l+1)| l\\in \\mathbb N\\}$is a partition of$\\mathbb N^+$.\n- October 29th, 2014 | tigris35711 | Oh Yes.: Where shall I send the £10 prize?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Inequality for square summable complex series\" in Analysis, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 8, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3038, "problem_number": "OPG-426", "title": "Long rainbow arithmetic progressions", "statement": "For $k\\in \\mathbb{N}$ let $T_k$ denote the minimal number $t\\in \\mathbb{N}$ such that there is a rainbow $AP(k)$ in every equinumerous $t$-coloring of $\\{ 1,2,\\ldots,tn\\}$ for every $n\\in \\mathbb{N}$\n\nConjecture For all $k\\geq 3$, $T_k=\\Theta (k^2)$.", "background": "Source: Open Problem Garden. Original node ID: 426. URL: http://www.openproblemgarden.org/op/long_rainbow_arithemtic_progressions.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/long_rainbow_arithemtic_progressions\n- Author(s): Fox, Jacob; Jungic, Veselin; Mahdian, Mohammad; Nesetril, Jaroslav; Radoicic, Rados\n- Subject(s): Combinatorics\n- Keywords: arithmetic progression; rainbow\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 3rd, 2007 by vjungic\n\nProblem-page discussion:\nA $t$-coloring of $\\{ 1,2,\\ldots, tn\\}$ is equinumerous if each color is used $n$ times. An arithmetic progression is rainbow if it does not containt two terms of the same color.\n\nIn [JLMNR] it was proved that $\\lfloor \\frac{k^2}{4}\\rfloor 4$ ([CJR]). It is not hard to show that $T_k > k$ for all $k\\ge 5$ ([AF]).\n\nBibliography:\n[AF] Maria Axenovich, Dmitri Fon-Der-Flaass: On rainbow arithmetic progressions, Electronic Journal of Combinatorics, 11, (2004), R1.\n\n[CJR] David Conlon, Veselin Jungic, Rados Radoicic, On the existence of rainbow 4-term arithmetic progressions, Graphs and Combinatorics, 23 (2007), 249-254\n\n*[JLMNR] Veselin Jungic, Jacob Licht (Fox), Mohammad Mahdian, Jaroslav Nesetril, Rados Radoicic: Rainbow arithmetic progressions and anti-Ramsey results, Combinatorics, Probability, and Computing - Special Issue on Ramsey Theory, 12, (2003), 599--620.\n\n[JNR] Veselin Jungic, Jaroslav Nesetril, Rados Radoicic: Rainbow Ramsey theory, Integers, The Electronic Journal of Combinatorial Number Theory, Proceedings of the Integers Conference 2003 in Honor of Tom Brown, 5(2), (2005), A9.\n\n[JR] Veselin Jungic, Rados Radoicic: Rainbow 3-term arithmetic progressions, Integers, The Electronic Journal of Combinatorial Number Theory, 3, (2003), A18.\n\nRelated:\nRelated problems\nRainbow AP(4) in an almost equinumerous coloring\n\nBibliography links:\n- On the existence of rainbow 4-term arithmetic progressions: http://dx.doi.org/10.1007/s00373-007-0723-2\n\nComments:\n- May 12th, 2010 | Anonymous | Is this unsolved?: Looks like a nice problem, but is it still unsolved?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Long rainbow arithmetic progressions\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3039, "problem_number": "OPG-478", "title": "Rainbow AP(4) in an almost equinumerous coloring", "statement": "Problem Do 4-colorings of $\\mathbb{Z}_{p}$, for $p$ a large prime, always contain a rainbow $AP(4)$ if each of the color classes is of size of either $\\lfloor p/4\\rfloor$ or $\\lceil p/4\\rceil$?", "background": "Source: Open Problem Garden. Original node ID: 478. URL: http://www.openproblemgarden.org/op/rainbow_ap_4_in_an_almost_equinumerous_coloring.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/rainbow_ap_4_in_an_almost_equinumerous_coloring\n- Author(s): Conlon, David\n- Subject(s): Combinatorics\n- Keywords: arithmetic progression; rainbow\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 20th, 2007 by vjungic\n\nProblem-page discussion:\nIt is known that there are equinumerous colorings of $\\mathbb{Z}_{4m}$ (i.e. colorings of $\\mathbb{Z}_{4m}$ for some $m$ such that each color occurs $m$ times) within which we cannot find rainbow arithmetic progressions of length $4$. ([CJR])\n\nBibliography:\n*[C] David Conlon, Rainbow solutions of linear equations over $\\mathbb{Z}_p$, Discrete Mathematics, 306 (2006) 2056 - 2063.\n\n[CJR] David Conlon, Veselin Jungic, Rados Radoicic, On the existence of rainbow 4-term arithmetic progressions, Graphs and Combinatorics, 23 (2007), 249-254\n\nRelated:\nRelated problems\nLong rainbow arithmetic progressions\n\nBibliography links:\n- On the existence of rainbow 4-term arithmetic progressions: http://dx.doi.org/10.1007/s00373-007-0723-2\n\nComments:\n- September 1st, 2007 | Dino | Tight hypergraphs: It deservs to be mentioned that in any $3$-colouring of ${\\mathbb Z}_p^*/{\\mathbb Z}_3^*$, the equation $x+y=z$, have an heterochromatic (rainbow) solution; here, ${\\mathbb Z}_p^*$ denotes the multiplicative group of the field ${\\mathbb Z}_p$.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Rainbow AP(4) in an almost equinumerous coloring\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3040, "problem_number": "OPG-618", "title": "Monotone 4-term Arithmetic Progressions", "statement": "Question Is it true that every permutation of positive integers must contain monotone 4-term arithmetic progressions?", "background": "Source: Open Problem Garden. Original node ID: 618. URL: http://www.openproblemgarden.org/op/monotone_4_term_arithmetic_progressions.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/monotone_4_term_arithmetic_progressions\n- Author(s): Davis, James A.; Entringer, Roger C.; Graham, Ronald L.; Simmons, Gustavus J.\n- Subject(s): Combinatorics\n- Keywords: monotone arithmetic progression; permutation\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 3rd, 2007 by vjungic\n\nProblem-page discussion:\nIt is not difficult to see that any permutation of positive integers contains a monotone 3-term arithmetic progression, i.e., that for any permutation $\\pi:\\mathbb{N}\\to \\mathbb{N}$ there is a 3-term arithmetic progression $a, a+d, a+2d$ such that $\\pi (a)>\\pi (a+d)>\\pi (a+2d)$ or $\\pi (a)<\\pi (a+d)<\\pi (a+2d)$.\n\nIn [DEGS] an example of a permutation of $\\mathbb{N}$ that does not contain a monotone 5-term arithmetic progression is given.\n\nBibliography:\n*[DEGS] J. A. Davis, R. C. Entringer, R. L. Graham, and G. J. Simmons, On permutations containing no long arithmetic progression, Acta Arithmetica XXXIV.1 (1977), 81-90.\n\n[LR] Bruce M. Landman and Aaron Robertson, Ramsey Theory on the Integers, Stud. Math. Libr. 24, AMS Providence, RI, 2004.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Monotone 4-term Arithmetic Progressions\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3041, "problem_number": "OPG-636", "title": "Even vs. odd latin squares", "statement": "A latin square is even if the product of the signs of all of the row and column permutations is 1 and is odd otherwise.\n\nConjecture For every positive even integer $n$, the number of even latin squares of order $n$ and the number of odd latin squares of order $n$ are different.", "background": "Source: Open Problem Garden. Original node ID: 636. URL: http://www.openproblemgarden.org/op/even_vs_odd_latin_squares.\n\nSource subject path: Combinatorics.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/even_vs_odd_latin_squares\n- Author(s): Alon, Noga; Tarsi, Michael\n- Subject(s): Combinatorics\n- Keywords: latin square\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 7th, 2007 by mdevos\n\nProblem-page discussion:\nFor every positive integer $n$, let $ELS(n)$, ( $OLS(n)$ ) be the number of even (odd) latin squares of order $n$.\n\nThe inspiration for this conjecture comes from an attempt by Alon and Tarsi to use their polynomial technique to show that the complete bipartite graph $K_{n,n}$ is $n$-edge-choosable (a famous conjecture of Dinitz asserts that this is always true). They show (in [AT]) that whenever $ELS(n) \\neq OLS(n)$, the graph $K_{n,n}$ is $n$-edge-choosable. For odd integers $n>1$ it is easy to see that $ELS(n) = OLS(n)$, since interchanging the first two rows has no effect on the signs of the rows, but flips the signs of all of the columns. For even $n$, Alon and Tarsi checked that $ELS(n)$ and $OLS(n)$ were different for $n=2,4,6$ and conjectured that this pattern would continue. Although Dinitz' Conjecture has since been resolved, Alon and Tarsi's conjecture remains quite interesting. In particular, it has been shown by Huang and Rota [HR] that the truth of this conjecture would imply Rota's basis conjecture for even values of $n$ (see [O] for a nice proof of this).\n\nELS() and OLS() appear in the The Encyclopedia of Integer Sequences as A114628 and A114629. The following chart shows the first few values. Although the data here is quite limited, $ELS(n) > OLS(n)$ for every even $n$ in the chart, and as far as we know, this might hold in general.\n\nn\nELS(n)\nOLS(n)\n\n1\n1\n0\n\n2\n2\n0\n\n3\n6\n6\n\n4\n576\n0\n\n5\n80640\n80640\n\n6\n505958400\n306892800\n\n7\n30739709952000\n30739709952000\n\n8\n55019078005712486400\n53756954453370470400\n\nDrisko [D1] proved that whenever $p$ is prime, $ELS(p+1) - OLS(p+1) \\cong (-1)^{{(p+1)}/2} p^2$ (mod $p^3$ ), thus verifying the Alon-Tarsi conjecture for any even number which is one more than a prime. Shortly afterward, Zappa [Z] introduced a function $AT()$ which compares the number of even and odd latin squares which have all diagonal entries equal to one, and proved some interesting identities concerning $AT()$. By utilizing these identities, Drisko [D2] proved that $ELS(n) \\neq OLS(n)$ whenever $n$ is of the form $2^rp$ for a prime $p$.\n\nBibliography:\n*[AT] N. Alon, M. Tarsi, Coloring and Orientations of Graphs. Combinatorica 12, 125-143, 1992 MathSciNet\n\n[D1] A. Drisko, On the number of even and odd Latin squares of order $p+1$, Adv. Math. 128 (1997), no. 1, 20--35. MathSciNet\n\n[D2] A. Drisko, Proof of the Alon-Tarsi conjecture for $n=2\\sp rp$. Electron. J. Combin. 5 (1998) MathSciNet.\n\n[HR] R. Huang and G-C Rota, On the relations of various conjectures on Latin squares and straightening coefficients. Discrete Math. 128 (1994), no. 1-3, 225--236. MathSciNet.\n\n[O] S. Onn, A colorful determinantal identity, a conjecture of Rota, and Latin squares. Amer. Math. Monthly 104 (1997), no. 2, 156--159. MathSciNet.\n\n[Z] P. Zappa, The Cayley determinant of the determinant tensor and the Alon-Tarsi conjecture. Adv. in Appl. Math. 19 (1997), no. 1, 31--44. MathSciNet.\n\nRelated:\nRelated problems\nRota's basis conjecture\n\nSource links:\n- latin square: http://en.wikipedia.org/wiki/latin square\n\nDiscussion links:\n- Rota's basis conjecture: http://www.openproblemgarden.org/?q=node/631\n- The Encyclopedia of Integer Sequences: http://www.research.att.com/%7Enjas/sequences\n- A114628: http://www.research.att.com/%7Enjas/sequences/A114628\n- A114629: http://www.research.att.com/%7Enjas/sequences/A114629\n\nBibliography links:\n- Coloring and Orientations of Graphs: http://www.math.tau.ac.il/%7Enogaa/PDFS/chrom3.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1179249\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1451417\n- Proof of the Alon-Tarsi conjecture for $n=2\\sp rp$: http://www.combinatorics.org/Volume_5/PDF/v5i1r28.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1624999\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1271866\n- A colorful determinantal identity, a conjecture of Rota, and Latin squares: http://ie.technion.ac.il/%7Eonn/Preprints/AMM1.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1437419\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1453404\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Even vs. odd latin squares\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3042, "problem_number": "OPG-1797", "title": "2-accessibility of primes", "statement": "Question Is the set of prime numbers 2-accessible?", "background": "Source: Open Problem Garden. Original node ID: 1797. URL: http://www.openproblemgarden.org/op/2_accessibility_of_primes.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/2_accessibility_of_primes\n- Author(s): Landman, Bruce M.; Robertson, Aaron\n- Subject(s): Combinatorics\n- Keywords: monochromatic diffsequences; primes\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 9th, 2008 by vjungic\n\nProblem-page discussion:\nA set $S\\subseteq \\mathbb{N}$ is $r$-accessible if for any $r$-coloring of $\\mathbb{N}$, $r\\in \\mathbb{N}$, there exist long monochromatic $S$-diffsequences, i.e., for any $k\\in \\mathbb{N}\\backslash \\{ 1\\}$ there is a monochromatic sequence $\\{ x_1,x_2,\\ldots,x_k\\}$ such that $x_{i+1}-x_i\\in S$, for all $i\\in \\{ 1,\\ldots,k-1\\}$.\n\nThe set of primes $P$ is not 3-accessible. [LR2]\n\nLandman and Robertson proved [LR1] that for any odd $t$, the set $t+P$ is 2-accessible.\n\nIt is known that a 2-coloring of any 33 consecutive positive integers yields a monochromatic 7-term $P$-diffsequence.\n\nBibliography:\n[J] Jungi\\'c, Veselin, {\\it On a conjecture of Brown concerning accessible sets}, J. Combin. Theory Ser. A 110 (2005), MathSciNet\n\n[KL] Abdollah Khodkar and Bruce M. Landman, {\\it Recent progress in Ramsey theory on the integers}, Combinatorial number theory, 305--313, de Gruyter, Berlin, 2007. MathSciNet\n\n[LR1] Bruce M. Landman and Aaron Robertson, {\\it Avoiding Monochromatic Sequences With special Gaps}, SIAM J. Discrete Math Vol. 21 (2007), no. 3, 794--801. MathSciNet\n\n*[LR2] Bruce M. Landman and Aaron Robertson, Ramsey Theory on the Integers, Stud. Math. Libr. 24, AMS Providence, RI, 2004.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2128973\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2337054\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2354006\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"2-accessibility of primes\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3043, "problem_number": "OPG-1825", "title": "3-accessibility of Fibonacci numbers", "statement": "Question Is the set of Fibonacci numbers 3-accessible?", "background": "Source: Open Problem Garden. Original node ID: 1825. URL: http://www.openproblemgarden.org/op/3_accessibility_of_fibonacci_numbers.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/3_accessibility_of_fibonacci_numbers\n- Author(s): Landman, Bruce M.; Robertson, Aaron\n- Subject(s): Combinatorics\n- Keywords: Fibonacci numbers; monochromatic diffsequences\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 24th, 2008 by vjungic\n\nProblem-page discussion:\nA set $S$ is $r$-accessible if for any $r$-coloring of $\\mathbb{N}$, $r\\in \\mathbb{N}$, there exist long monochromatic $S$-diffsequences, i.e., for any $k\\in \\mathbb{N}\\backslash \\{ 1\\}$ there is a monochromatic sequence $\\{ x_1,x_2,\\ldots,x_k\\}$ such that $x_{i+1}-x_i \\in S$, for all $i\\in \\{ 1,2,\\ldots,k-1\\}$.\n\nThe set of Fibonacci numbers $F$ is 2-accessible. [LR1]\n\n$F$ is not 6-accessible. [AGJL]\n\nIt is known that a 3-coloring of any 27 consecutive positive integers yields a monochromatic 4-term $F$-diffsequence.\n\nBibliography:\n[AGJL] Hayri Ardal, David Gunderson, Veselin Jungi\\'c, and Bruce Landman, {\\it On Accessibility of the Set of Fibonacci Numbers}, In Preparation\n\n*[LR1] Bruce M. Landman and Aaron Robertson, {\\it Avoiding Monochromatic Sequences With special Gaps}, SIAM J. Discrete Math Vol. 21 (2007), no. 3, 794--801.\n\n[LR2] Bruce M. Landman and Aaron Robertson, Ramsey Theory on the Integers, Stud. Math. Libr. 24, AMS Providence, RI, 2004\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"3-accessibility of Fibonacci numbers\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3044, "problem_number": "OPG-2063", "title": "Wide partition conjecture", "statement": "Conjecture An integer partition is wide if and only if it is Latin.", "background": "Source: Open Problem Garden. Original node ID: 2063. URL: http://www.openproblemgarden.org/op/wide_partition_conjecture.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/wide_partition_conjecture\n- Author(s): Chow, Timothy Y.; Taylor, Brian D.\n- Subject(s): Combinatorics\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 24th, 2008 by tchow\n\nProblem-page discussion:\nAn integer partition $\\lambda$ is wide if $\\mu \\ge \\mu'$ for every subpartition $\\mu$ of $\\lambda$. (Here $\\mu'$ denotes the conjugate of $\\mu$, $\\ge$ denotes dominance or majorization order, and a subpartition of $\\lambda$ is a submultiset of the parts of $\\lambda$.) An integer partition $\\lambda$ is Latin if there exists a tableau $T$ of shape $\\lambda$ such that for every $i$, the $i$ th row of $T$ contains a permutation of $\\{1,2,\\ldots,\\lambda_i\\}$, and such that every column of $T$ contains distinct integers. It is easy to show that if $\\lambda$ is Latin then $\\lambda$ is wide, but the converse remains open.\n\nBibliography:\n*[CFGV] Timothy Y. Chow, C. Kenneth Fan, Michel X. Goemans, Jan Vondrak, Wide partitions, Latin tableaux, and Rota's basis conjecture, Advances Appl. Math. 21 (2003), 334-358.\n\nRelated:\nRelated problems\nRota's basis conjecture\n\nBibliography links:\n- Wide partitions, Latin tableaux, and Rota's basis conjecture: http://alum.mit.edu/www/tchow/wide.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Wide partition conjecture\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3045, "problem_number": "OPG-37167", "title": "Shuffle-Exchange Conjecture", "statement": "Given integers $k,n\\ge2$, let $d(k,n)$ be the smallest integer $d\\ge2$ such that the symmetric group $\\frak S$ on the set of all words of length $n$ over a $k$-letter alphabet can be generated as $\\frak S = (\\sigma \\frak G)^d:=\\sigma\\frak G \\sigma\\frak G \\dots \\sigma\\frak G$ ( $d$ times), where $\\sigma\\in \\frak S$ is the shuffle permutation defined by $\\sigma(x_1 x_2 \\dots x_{n}) = x_2 \\dots x_{n} x_1$, and $\\frak G$ is the exchange group consisting of all permutations in $\\frak S$ preserving the first $n-1$ letters in the words.\n\nProblem (SE) Evaluate $d(k,n)$.\n\nConjecture (SE) $d(k,n)=2n-1$, for all $k,n\\ge2$.", "background": "Source: Open Problem Garden. Original node ID: 37167. URL: http://www.openproblemgarden.org/op/shuffle_exchange_conjecture.\n\nSource subject path: Combinatorics.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/shuffle_exchange_conjecture\n- Author(s): Beneš, Václav E.; Folklore; Stone, Harold S.\n- Subject(s): Combinatorics\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 27th, 2009 by Vadim Lioubimov\n\nProblem-page discussion:\nThis beautiful and difficult problem arises in switching networks theory and has important applications in parallel processing, sorting networks, card shuffling, etc. In this area it is perhaps the most famous open question which is at the center of the quest to understand the phenonemon of network rearrangeability. Both the problem and conjecture are referred to as Shuffle-Exchange (SE) ones. The case $k=2$ of SE problem (but not the conjecture) can be traced back to the work of Stone [S71], where he showed that $d(2,n)\\le n^2$. The upper bound $d(k,n)\\le 2n-1$ is the central case of Beneš conjecture [B75], while the lower bound $d(k,n)\\ge 2n-1$ can be easily seen (it is also a special case of the stronger version [B75] of Beneš conjecture, which turned out to be generally false). Since 1975 SE conjecture, especially its case $k=2$, has received a lot of attention, mostly in the context of switching networks, with rather modest results.\n\nNote that $d(k,n)\\le m$ is equivalent to $\\frak S = (\\sigma \\frak G)^m$, for any integer $m\\ge2$. Furthermore, it is easy to see that the latter decomposition is equivalent to $\\frak S = \\frak G_1 \\frak G_2\\dots \\frak G_{m}$, where $\\frak G_i:=\\sigma^{i} \\frak G \\sigma^{-i}$ is the subgroup of $\\frak S$ consisting of all permutations which may only change the letters on the position $i-1\\ (\\text{mod } n) + 1$ in the words.\n\nAlso, the case $n=2$ of SE conjecture can be reformulated as the following\n\nTheorem ( $\\star$ ) Every permutation of entries of a square matrix can be obtained in 3 steps as follows: first by permuting entries in the columns, then - in the rows, and then - in the columns again. Moreover, some permutations cannot be obtained in less than 3 of such steps.\n\n(The parameter $k$ in the case $n=2$ of SE conjecture corresponds to the size $k\\times k$ of a matrix in the theorem.) Moreover, the general case of SE conjecture can be reformulated as a straightforward generalization of Theorem ( $\\star$ ) to the $n$-dimensional cubic matrices (of size $k\\times\\dots\\times k$ ) stating that every permutation of entries of such a matrix can be obtained in $2n-1$ steps in a similar way, and this number is generally a precise lower bound.\n\nTheorem ( $\\star$ ) holds as being easily equivalent to the special case of the following classical result when each part of the multigraph has size $k$:\n\nTheorem (König) A $k$-regular bipartite multigraph is $k$-edge-colorable.\n\nThe function $d(k,n)$ admits 3 main interpretations (that are not immediately equivalent), \"group-theoretic\" (presented in the beginning), \"combinatorial\" (below), and \"graph-theoretic\", each of which provides its own framework for SE problem and suggests its own interesting natural generalizations and extensions. Accordingly, there are 3 equivalent forms of SE problem/conjecture. Although the group-theoretic interpretation of $d(k,n)$ is the shortest and most elegant among the three, it seems the least natural when it comes to proving the known results and studying SE problem more deeply. I believe that SE problem is very deep and combinatorial by nature. I also strongly believe in the validity of SE conjecture.\n\n2. Combinatorial form of SE problem/conjecture\n\nGiven a pure abstract simplicial complex $\\Delta$ of rank $n\\ge2$ and a positive integer $\\ell$, an $\\ell$-transition is a map that assigns to evey pair of ordered facets, $x_1,\\dots,x_{n}$ and $y_1,\\dots,y_{n}$, a sequence of vertices $z_1,\\dots,z_{\\ell}$ such that every $n$-segment of the sequence $x_1,\\dots,x_{n},z_1,\\dots,z_{\\ell}, y_1,\\dots,y_{n}$ forms a facet. Let $\\text{tr}(\\Delta)$ be the smallest $\\ell$, or $\\infty$ if none exists, for which there exists an $\\ell$-transition. Note that $\\text{tr}(\\Delta)\\le \\ell$ is equivalent to the existence of $\\ell$-transition for $\\Delta$, for any $\\ell\\ge 1$.\n\nGiven integers $k,n\\ge2$, let $\\Delta_{k,n}$ be the pure abstract simplicial complex of rank $n$ whose vertex set is the set $V_{k,n}$ of all uniform $k$-partitions (i.e., ones consisting of $k$ equal-sized blocks) of a $k^n$-set, and whose facets are all $n$-subsets of $V_{k,n}$ with zero infinum. Using normal reasoning, it is not hard to show [L04] the following\n\nTheorem $d(k,n) = \\text{tr}(\\Delta_{k,n})+n$.\n\nThus, the combinatorial forms of SE problem and conjecture can be formulated as to find $\\text{tr}(\\Delta_{k,n})$ and that $\\text{tr}(\\Delta_{k,n})=n-1$, respectively.\n\nThe infinum (or meet) of two partitions $\\mathbf{a}$ and $\\mathbf{b}$ of a set $E$ is the partition of $E$ defined by $$\\mathbf{a\\wedge b}:= \\big\\{\\, a\\cap b\\ne\\varnothing \\ | \\ a\\in\\mathbf{a} \\ \\&\\ b\\in\\mathbf{b} \\,\\big\\}.$$Note that together with the operation$\\wedge$, the collection of all partitions of$E$forms a semilattice (i.e., a commutative and idempotent semigroup) with the identity and zero being the partitions$\\mathbf{1}_E:=\\{E\\}$and$\\mathbf{0}_E:=\\big\\{\\{x\\} \\ | \\ x\\in E \\big\\}$, respectively.\n\nObserve that the complex $\\Delta_{k,n}$ is non-matroidal for all $(k,n)\\ne (2,2)$.\n\n3. Constructive version of SE problem/conjecture\n\nApplication-wise it is important not only to establish a certain decomposition $\\frak S = (\\sigma \\frak G)^m$ or, equivalently, rearrangeability of the graph $(\\text{SE}(k,n))^{m-1}$ or, equivalently, the existence of an $(m-n)$-transition for the complex $\\Delta_{k,n}$, but also to find a corresponding efficient factorization/routing/transition algorithm.\n\nGiven an identity $A = A_1A_2\\dots A_m$, where all $A_i$ are subsets of a multiplicative group, a factorization algorithm finds for every $a\\in A$ an $m$-tuple $(a_1,\\dots,a_m)\\in A_1\\times \\dots \\times A_m$ such that $a = a_1a_2\\dots a_m$. Given a rearrangeable graph $(\\text{SE}(k,n))^{m-1}$, a routing algorithm takes a mask of the graph as input and returns a corresponding routing. Given a pure simplicial complex $\\Delta$ with $\\text{tr}(\\Delta)\\le r$, an $r$-transition algorithm realizes an $r$-transition for $\\Delta$. It is not hard to prove\n\nTheorem Any factorization algorithm for $\\frak S = (\\sigma \\frak G)^m$ translates into a routing algorithm for $(\\text{SE}(k,n))^{m-1}$ and into an $(m-n)$-transition algorithm for $\\Delta_{k,n}$ of the same complexity, and vise versa. Consequently, $D^{*}(k,n) = R^{*}(k,n) = T^{*}(k,n)$.\n\nHere $D^*(k,n)$, $R^*(k,n)$, and $T^{*}(k,n)$ are the sets of all $m\\ge 2$, respectively, for which there exists an efficient polynomial-time (in $k^n$ ) factorization/routing/transition algorithm mentioned in the above theorem (we will also write $A \\buildrel{*}\\over= A_1A_2\\dots A_m$ to indicate the existence of such a factorization algorithm for $A = A_1A_2\\dots A_m$, where each $A_i\\subseteq \\frak S$ ). Clearly, $$d^*(k,n)\\ge d(k,n)= r(k,n)=\\text{tr}(\\Delta_{k,n})+n,$$where$d^*(k,n):= \\min D^*(k,n)$with the usual convention$\\min\\varnothing:=\\infty$, and$r(k,n)$ is defined here.\n\nIt is easy to see that $d\\in D^*(k,n)$ implies $[d,\\infty)\\subseteq D^*(k,n)$ (equivalently, the same is true for $R^*(k,n)$ and $T^{*}(k,n)$ ). Consequently, $d^*(k,n)\\le m$ is equivalent to $\\frak S \\buildrel{*}\\over= (\\sigma \\frak G)^m$.\n\nProblem (CSE) Evaluate $d^*(k,n)$ and specify the corresponding factorization/routing/transition algorithm for the upper bound.\n\nConjecture (CSE) $d^*(k,n)=2n-1$.\n\nBoth the problem and conjecture are referred here to as Constructive Shuffle-Exchange (CSE) ones. The conjecture was proposed in [L04]. Clearly, CSE conjecture implies SE one as $2n-1\\le d(k,n)\\le d^*(k,n)$.\n\n4. Main results\n\nSo far SE/CSE conjecture has been only settled in the following 3 cases: $n=2$, $(k,n)=(2,3)$, and $(k,n)=(2,4)$. That is, the following 3 identities holds:\n\n$$(1)\\ d(k,2) = d^*(k,2) = 3,\\ \\ (2)\\ d(2,3) = d^*(2,3) = 5,\\ \\ (3)\\ d(2,4) = d^*(2,4) = 7.$$\n\nAlso, there are 2 the following major results on SE/CSE problem:\n\n(4) $d(k,n)\\ge 2n-1$.\n\n(5) $d^{(*)}(k,n)\\le d^{(*)}(k,r)+3(n-r)$, for all $n > r\\ge2$.\n\nThe lower bound (4) follows immediately from the obvious observation that $\\text{tr}(\\Delta) \\ge \\dim(\\Delta)$, for any pure complex $\\Delta$.\n\nNote that (4) reduces SE (CSE) conjecture to $d^{(*)}(k,n)\\le 2n-1$ which is equivalent to $\\frak S \\buildrel{(*)}\\over= (\\sigma \\frak G)^{2n-1}$. In fact, the main reason why SE/CSE conjecture is widely believable, apart from results (1-4), is a close similarity between the latter decomposition and the following well known result [B65, L04] (that is not hard to derive from the constructive version of the König's theorem):\n\nTheorem (Beneš) $\\frak S \\buildrel{*}\\over= (\\frak G\\sigma^{-1})^{n-1}\\frak G(\\sigma \\frak G)^{n-1}$.\n\nCombining (1) and (3) with (5) yields respectively the following 2 best known upper bounds (in addition to (2)) for both $d(k,n)$ and $d^*(k,n)$:\n\n$(6)\\quad d(k,n)\\le d^*(k,n)\\le 3n-3$, for all $k\\ge3$ and $n\\ge2$\n\n$(7)\\quad d(2,n)\\le d^*(2,n)\\le 3n-5$, for all $n\\ge4$.\n\nAs it was mentioned earlier, the case $n=2$ of SE conjecture is easily equivalent to the following case of the Konig's theorem: a $k$-regular bipartite multigraph $B$ with $k$-vertex parts is $k$-edge-colorable. Moreover, any $k$-edge-coloring algorithm for the graph $B$ easily translates into a factorization/routing/1-transition algorithm of the same complexity for $\\frak S = (\\sigma \\frak G)^3$ (at $n=2$ ) or the graph $(\\text{SE}(k,2))^{2}$ or the complex $\\Delta_{k,2}$, respectively, and vise versa. Consequently, as there are many efficient polynomial-time (in $k^2$ ) $k$-edge-coloring algorithms well known for the graph $B$, the case $n=2$ of CSE conjecture also holds.\n\nThere are at least 6 alternative proofs proposed for the case $(k,n)=(2,3)$ of CSE conjecture. Although they may look quite different, each proof is essentially based on either of 3 similar short and elegant algorithms which we refer to as A1 [RV87, LT89, L04], A2 [ND00, L04] and A3 [KR91]. Each algorithm is based on a 2-edge-coloring algorithm for a 2-regular bipartite multigraph with 4-vertex parts. Namely, A1 uses 2, A2 uses at most 2, and A3 uses 1 application(s) of such an algorithm. Each algorithm Ai deals with 2 cases in which the procedure is especially simple. The algorithms A1 and A2 are very efficient (with A2 being slightly faster than A1), while A3 is not so (contrary to what is claimed in [KR91]) as it relies on an exhausting search to determine the case for each input permutation. However, A3 has some theoretical advantage over A1 and A2 as its 2 cases partition the symmetric group $S_8$ into 2 classes that do not depend on a realization of the algorithm. In [L04], both algorithms A1 and A2 are explicitly described as 2-transition algorithms for the complex $\\Delta_{2,3}$, and the corresponding 2 proofs for the statement $\\text{\\rm tr}(\\Delta_{2,3})=2$ are particularly transparent. Moreover, the latter statement, the algorithms and the proofs are straightforwardly generalized [L05] to a wide class of 2-dimensional pure abstract simplicial complexes.\n\nA brute force verification for the case $(k,n)=(2,4)$ of SE conjecture was first reported in [R95]. The first theoretical proof for such case of CSE conjecture was proposed (in graph-theoretic terms) in [ND00]. Although the ideas behind the underlying algorithm for this proof are simple, the algorithm deals with a huge and intricate tree of cases and is substantially more complicated (and not so elegant) than that of the case $(k,n)=(2,3)$. As a result, the proof is very tedious, hard to verify, and leaves little hope for using a similar approach to prove the next case $(k,n)=(2,5)$ of CSE conjecture. An essentially similar but slightly better organized algorithm and proof for (3) were proposed in [DS08] (with no reference to [ND00]).\n\nThe upper bound (5) was first obtained in [VR88] for the case $k=2$ and (i) $d^{(*)}(k,r)=2r-1$. In other words, it was shown that (i) at $k=2$ implies $d^{(*)}(2,n)\\le 3n-r-1$, if $n > r$. A much simpler proof of (5) for the case $r=2,3$ and (i) appeared in [LT89]. The latter proof was easily extended [ND00] to an arbitrary $r\\ge2$. A transparent combinatorial proof in terms of the complex $\\Delta_{k,n}$ for the general case of (5) was proposed in [L04]. This proof (together with its underlying transition algorithm) was generalized [L05] to a wide class of pure abstract simplicial complexes of arbitrary dimensions. Namely, it was shown that, given a complex $\\Delta$ in this class and an integer $1\\le m<\\dim(\\Delta)$,\n\n$$:\\qquad\\qquad \\text{tr}(\\Delta) \\le 2m + \\max \\big\\{ \\text{tr}(\\Delta/F) \\mid F\\in\\Delta,\\ |F|=m \\big\\}$$\n\nand, moreover, that any $\\ell$-transition algorithm for the complexes $\\Delta/F$ can be efficiently used to make a $(2m+\\ell)$-transition algorithm for $\\Delta$. Note that (5) can be easily obtained as an instance of the latter result. Here $\\Delta/ F$ is the link of a face $F$ in $\\Delta$, i.e., a subcomplex of $\\Delta$ defined by\n\n$$:\\qquad\\qquad \\Delta/ F:= \\{ A\\in \\Delta \\ | \\ A\\cap F = \\varnothing \\ \\&\\ A\\cup F\\in\\Delta \\big\\}.$$\n\nIt is worth noting that there are many flawed proofs for SE conjecture in the literature. Most notably, in [Ba01] (the general case) and [C03] (the case $k=2$ ). The latter proof was first refuted in [BHL06], while the former remains unrefuted in the literature.\n\nBibliography:\n[B65] V.E. Benes, Mathematical theory of connecting networks and telephone traffic, Academic Press, New York, 1965.\n\n*[S71] H.S. Stone, Parallel processing with the perfect shuffle, IEEE Trans. on Computers C-20 (1971), 153-161.\n\n*[B75] V.E. Beneš, Proving the rearrangeability of connecting networks by group calculation, Bell Syst. Tech. J. 54 (1975), 421-434.\n\n[RV87] C.S. Raghavendra, A. Varma, Rearrangeability of 5-stage shuffle/exchange network for N=8, IEEE Trans. on Commun. COM-35 (1987), 808-812.\n\n[VR88] A. Varma, C.S. Raghavendra, Rearrangeability of multistage shuffle/exchange networks, IEEE Trans. on Commun. 36 (1988), 1138-1147.\n\n[LT89] N. Linial, M. Tarsi, Interpolation between bases and the shuffle-exchange networks, European J. of Combinatorics, 10(1) (1989), 29-39.\n\n[KR91] K. Kim, C.S. Raghavendra, A Simple Algorithm to Route Arbitrary Permutations on 8-input 5-stage Shuffle/Exchange Network, Proc. 5th International Parallel Processing Symposium (1991), 398-403.\n\n[R95] C.S. Raghavendra, On the rearrangeability conjecture of $(2\\log_2 N -1)$-stage shuffle/exchange network, IEEE Computer Society, Tech. Committee on Comp. Arch. Newsletter, Position paper (Winter 1995), 10-12.\n\n[ND00] H.Q. Ngo, D.Z. Du, On the rearrangeability of shuffle-exchange networks, Tech. Report TR00-045, Dept. of Computer Science, Univ. of Minnesota (2000)\n\n[Ba01] R.E. Bashirov, On the rearrangeability of 2s-1 stage networks employing uniform interconnection pattern Calcolo, Springer Verlag, 38(2) (2001), 85-97.\n\n[C03] H. Cam, Rearrangeability of (2n-1)-stage shuffle-exchange networks, SIAM J. on Computing 32(3) (2003), 557-585.\n\n[L04] V. Lioubimov, Decomposition of symmetric group into product of stabilizers and Shuffle-Exchange problem, manuscript (2004).\n\n[L05] V. Lioubimov, Facet transitions in abstract simplicial complexes, manuscript (2005).\n\n[BHL06] X. Bao, F.K. Hwang, Q. Li, Rearrangeability of bit permutation networks, Theoretical Computer Science, 352(1) (2006), 197-214.\n\n[DS08] H. Dai, X. Shen, Rearrangeability of 7-stage 16x16 shuffle-exchange networks, Frontiers of Electrical and Electronic Engineering in China, 3(4) (2008), 440-458.\n\nRelated:\nRelated problems\nShuffle-Exchange Conjecture (graph-theoretic form)\nBeneš Conjecture\nBeneš Conjecture (graph-theoretic form)\n\nDiscussion links:\n- Beneš conjecture: http://www.openproblemgarden.org/?q=node/37181\n- stronger version: http://www.openproblemgarden.org/?q=node/37181\n- \"graph-theoretic\": http://www.openproblemgarden.org/?q=node/37089\n- rearrangeability: http://www.openproblemgarden.org/?q=node/37089\n- graph $(\\text{SE}(k,n))^{m-1}$: http://www.openproblemgarden.org/?q=node/37089\n- mask: http://www.openproblemgarden.org/?q=node/37089\n- routing: http://www.openproblemgarden.org/?q=node/37089\n- here: http://www.openproblemgarden.org/?q=node/37089\n\nBibliography links:\n- On the rearrangeability of shuffle-exchange networks: http://www.cs.umn.edu/research/technical_reports.php?page=report&report_id=00-045\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 122.\n\nAttempt notes:\nTarget:\nMake progress on \"Shuffle-Exchange Conjecture\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3046, "problem_number": "OPG-37181", "title": "Beneš Conjecture", "statement": "Let $E$ be a non-empty finite set. Given a partition $\\bf h$ of $E$, the stabilizer of $\\bf h$, denoted $S(\\bf h)$, is the group formed by all permutations of $E$ preserving each block of $\\mathbf h$.\n\nProblem ( $\\star$ ) Find a sufficient condition for a sequence of partitions ${\\bf h}_1, \\dots, {\\bf h}_\\ell$ of $E$ to be complete, i.e. such that the product of their stabilizers $S({\\bf h}_1) S({\\bf h}_2) \\dots S({\\bf h}_\\ell)$ is equal to the whole symmetric group $\\frak S(E)$ on $E$. In particular, what about completeness of the sequence $\\bf h,\\delta(\\bf h),\\dots,\\delta^{\\ell-1}(\\bf h)$, given a partition $\\bf h$ of $E$ and a permutation $\\delta$ of $E$?\n\nConjecture (Beneš) Let $\\bf u$ be a uniform partition of $E$ and $\\varphi$ be a permutation of $E$ such that $\\bf u\\wedge\\varphi(\\bf u)=\\bf 0$. Suppose that the set $\\big(\\varphi S({\\bf u})\\big)^{n}$ is transitive, for some integer $n\\ge2$. Then $$\\frak S(E) = \\big(\\varphi S({\\bf u})\\big)^{2n-1}.$$", "background": "Source: Open Problem Garden. Original node ID: 37181. URL: http://www.openproblemgarden.org/op/bene_conjecture.\n\nSource subject path: Combinatorics.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/bene_conjecture\n- Author(s): Beneš, Václav E.\n- Subject(s): Combinatorics\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: January 3rd, 2010 by Vadim Lioubimov\n\nProblem-page discussion:\nThis conjecture was essentially proposed by Václav E. Beneš in 1975 [B75] and bears his name. It remains open for all $n\\ge3$.\n\nA partition of a set is uniform if all its blocks have the same size. Given a subset $P$ of a multiplicative group and a positive integer $m$, by $P^m$ we mean the product $PP\\dots P$ ( $m$ times). A set $T\\subseteq\\frak S(E)$ is transitive if for every $x,y\\in E$ there exists a permutation $\\tau\\in T$ such that $\\tau(x)=y$. The infinum of two partitions $\\bf a$ and $\\bf b$ of $E$ is the partition of $E$ defined by\n\n$$: \\qquad\\qquad\\qquad {\\bf a\\wedge b}:= \\big\\{\\, a\\cap b\\ne\\varnothing \\ | \\ a\\in{\\bf a} \\ \\&\\ b\\in{\\bf b} \\,\\big\\}.$$\n\nThe partition $\\bf 0$ of $E$ is defined by ${\\bf 0}:={\\bf 0}_E:=\\big\\{\\{x\\} \\ | \\ x\\in E \\big\\}$. So the condition ${\\bf h}\\wedge\\delta({\\bf h})={\\bf 0}$ is equivalent to saying that for every pair of blocks $a,b\\in{\\bf h}$, the intersection $a\\cap\\delta(b)$ consists of at most one element.\n\nObserve that the decomposition $\\frak S(E) = \\big(\\delta S({\\bf h})\\big)^{\\ell}$ is equivalent to completeness of the sequence ${\\bf h},\\delta({\\bf h}),\\dots,\\delta^{\\ell-1}({\\bf h})$ due to the obvious identity $\\delta S({\\bf h}) \\delta^{-1} = S(\\delta{\\bf h})$. Thus Problem ( $\\star$ ) is indeed underlying for Beneš conjecture.\n\nProblem ( $\\star$ ) is a special case of a broader fundamental problem of description of product of stabilizers on a finite set. The latter problem, which I believe is combinatorial by nature, is of great interest in switching network study. However, despite many years of extensive research on its various cases in the context of switching networks, this fascinating problem remains unsolved in all but a very few interesting instances. Very little is understood about such products beyond what is obvious. In particular, it is unclear how to efficiently compute their cardinalities. Even for some rather simple sequences of partitions, the product of their stabilizers is surprisingly difficult to describe. Beneš conjecture, if proven (even under some additional assumptions on $E,{\\bf u},\\varphi$ ), would provide a very useful and easy-to-check sufficient condition for completeness of the sequences ${\\bf u},\\varphi({\\bf u}),\\dots,\\varphi^{\\ell-1}({\\bf u})$ that are of particular interest.\n\nAnother important and interesting problem related to ( $\\star$ ) is to find an efficient polynomial-time (in $|E|$ ) factorization algorithm for the identity $\\frak S(E) = S({\\bf h}_1) S({\\bf h}_2) \\dots S({\\bf h}_\\ell)$. Given an identity $A = A_1A_2\\dots A_\\ell$, where all $A_i$ are subsets of a multiplicative group, a factorization algorithm finds for every $a\\in A$ an $\\ell$-tuple $(a_1,\\dots,a_\\ell)\\in A_1\\times \\dots \\times A_\\ell$ such that $a = a_1a_2\\dots a_\\ell$.\n\nBeneš conjecture is mainly famous for its central case, Shuffle-Exchange (SE) conjecture, stating essentially that $\\frak S(\\tilde X) = \\big(\\sigma S({\\bf g})\\big)^{2n-1}$, where $(\\tilde X,{\\bf g},\\sigma)$ is an instance of $(E,{\\bf u},\\varphi)$ defined, given arbitrary integer parameters $k,n\\ge2$, as follows:\n\n$\\bullet$ $\\tilde X$ is the set of all words of length $n$ over a $k$-letter alphabet $X$.\n\n$\\bullet$ $\\bf g$ is the $k^{n-1}$-partition of $\\tilde X$ formed by the equivalence relation $\\sim$ on $\\tilde X$ defined by\n\n$:\\qquad\\qquad x_1\\dots x_{n} \\sim y_1\\dots y_{n}: \\Leftrightarrow x_1\\dots x_{n-1}=y_1\\dots y_{n-1}$.\n\n$\\bullet$ $\\sigma$ is the shuffle permutation of $\\tilde X$ defined by $\\sigma(x_1 x_2 \\dots x_{n}):= x_2 \\dots x_{n} x_1$.\n\nWhereas SE conjecture, especially its case $k=2$, has received enormous attention in the study of switching networks with relatively little progress, the general case of Beneš conjecture, despite importance of Problem ( $\\star$ ) in that area, has virtually generated no literature and had no progress. While I strongly believe in the validity of SE conjecture, I am not so sure about the general case of Beneš conjecture and even do not rule out that it could be disproved by a low-scale counterexample. On the other hand, I cannot rule out that Beneš conjecture (possibly under some mild additional assumptions on $E,{\\bf u},\\varphi$ ) may be reduced to SE conjecture.\n\nIt is easy to see that the case $n=2$ of Beneš conjecture coincides with that of SE conjecture. The latter case is well known to be valid (discussed here).\n\nUnlike completeness of a sequence of partitions of $E$, the condition of transitivity of the product of their stabilizers is very easy to check. In particular, transitivity of the set $\\big(\\delta S({\\bf h})\\big)^{n}$ with $n\\ge2$ is equivalent to the following assertion:\n\n$$:\\qquad\\qquad \\forall\\,h_1,h_n\\in{\\bf h} \\ \\exists\\,h_2,\\dots,h_{n-1}\\in{\\bf h} \\ \\forall\\, i\\in[n-1]: h_i\\cap \\delta(h_{i+1}) \\ne \\varnothing.$$\n\nBeneš conjecture (as well as its underlying Problem ( $\\star$ ) and a broader problem of description of product of stabilizers on a finite set) admits a nice equivalent graph-theoretic form.\n\nCounterexamples\n\nIn what follows we present 3 counterexamples showing that certain stronger versions of Beneš conjecture are false.\n\nCounterexample 1. The condition ${\\bf u}\\wedge\\varphi({\\bf u})={\\bf 0}$ is necessary for Beneš conjecture. This can be shown by the following simple counterexample:\n\n$$:\\qquad\\qquad E:= \\{1,2,3,4,5,6\\}, \\ {\\bf u}:= \\big\\{\\{1,2,3\\}, \\{4,5,6\\}\\big\\}, \\text{ and } \\varphi:= (3,4).$$\n\nIndeed, ${\\bf u}\\wedge\\varphi({\\bf u}) \\ne {\\bf 0}$ as $\\{1,2,3\\}\\cap\\varphi\\{1,2,3\\} = \\{1,2\\}$. Also, the set $\\big(\\varphi S({\\bf u})\\big)^{2}$ is obviously transitive. However, it can be easily seen that any permutation $\\alpha$ of $E$ satisfying $\\alpha \\{1,2,3\\} = \\{4,5,6\\}$ does not belong to $S({\\bf u})\\varphi S({\\bf u})\\varphi S({\\bf u})$. Thus, $\\frak S(E) \\ne \\big(\\varphi S({\\bf u})\\big)^{3}$. In fact, the condition ${\\bf u}\\wedge\\varphi({\\bf u})={\\bf 0}$ is missing in the original statement [B75] of Beneš conjecture (however, such condition is commonly (but not always) assumed in the context of switching networks).\n\nCounterexample 2. Beneš conjecture is not directly generalizable to the products of stibilizers of the form $P:=S({\\bf u})\\varphi_1 S({\\bf u})\\dots\\varphi_{n-1} S({\\bf u})$. More precisely, transitivity of $P$ does not always imply $\\frak S(E) = P^2$, where ${\\bf u}$ is a uniform partition of $E$ and all $\\varphi_i$ are permutations of $E$ such that ${\\bf u}\\wedge\\varphi_i({\\bf u})={\\bf 0}$ (while Beneš conjecture states that this implication is always true as long as $\\varphi_1=\\dots=\\varphi_{n-1}$ ). For that I constructed the following counterexample:\n\n$$:\\qquad\\qquad E:= \\{1,2,\\dots,12\\}, \\ {\\bf u}:= \\big\\{\\{1,2\\}, \\{3,4\\},\\dots,\\{11,12\\}\\big\\}, \\ \\varphi_1:= (2,3)(6,7)(10,11) \\text{ and } \\varphi_2:= (2,7)(4,9)(6,11).$$\n\nIndeed, it is obvious that both permutations $\\varphi_1, \\varphi_2$ are satisfying ${\\bf u}\\wedge\\varphi_i({\\bf u})={\\bf 0}$ and the set $Q:=S({\\bf u})\\varphi_1 S({\\bf u})\\varphi_2 S({\\bf u})\\varphi_1S({\\bf u})$ is transitive. However, $\\frak S(E) \\ne Q^2$ as, in particular, it can be easily seen that any permutation $\\alpha$ of $E$ satisfying $\\alpha \\{1,2,3,4\\} = \\{5,6,7,8\\}$ does not belong to $Q^2$.\n\nCounterexample 3. In the same paper [B75], Beneš also proposed the following\n\nConjecture ( $\\diamond$ ) Let $\\bf u$ be a uniform partition of $E$ and $\\varphi$ be a permutation of $E$ such that $\\bf u\\wedge\\varphi(\\bf u)=\\bf 0$. Suppose that $n\\ge2$ is the smallest integer such that the set $\\big(\\varphi S({\\bf u})\\big)^{n}$ is transitive. Then $\\frak S(E) \\ne \\big(\\varphi S({\\bf u})\\big)^{2n-2}$.\n\nIn other words, this conjecture together with Beneš one, asserts that if $n\\ge2$ is the smallest integer such that $\\big(\\varphi S({\\bf u})\\big)^{n}$ is transitive, then $2n-1$ is the the smallest integer $\\ell$ such that $\\frak S(E) = \\big(\\varphi S({\\bf u})\\big)^{\\ell}$. However, Conjecture ( $\\diamond$ ) turned out to be generally false as I found the following counterexample for it:\n\n$$:\\qquad\\qquad E:= \\{1,2,\\dots,8\\}, \\ {\\bf u}:= \\big\\{\\{1,2\\}, \\{3,4\\}, \\{5,6\\}, \\{7,8\\}\\big\\}, \\text{ and } \\varphi:= (2,3)(4,5,6,7).$$\n\nIndeed, it is easy to verify that ${\\bf u}\\wedge\\varphi({\\bf u})={\\bf 0}$ and the set $\\big(\\varphi S({\\bf u})\\big)^{4}$ is transitive while $\\big(\\varphi S({\\bf u})\\big)^{3}$ is not as, in particular, $\\{4,7\\} \\cap \\big(\\varphi S({\\bf u})\\big)^{3}\\{1,2\\} =\\varnothing$. However, a brute force verification confirmed that $\\frak S(E) = \\big(\\varphi S({\\bf u})\\big)^{6}$.\n\nBibliography:\n*[B75] V.E. Beneš, Proving the rearrangeability of connecting networks by group calculation, Bell Syst. Tech. J. 54 (1975), 421-434.\n\nRelated:\nRelated problems\nShuffle-Exchange Conjecture\nBeneš Conjecture (graph-theoretic form)\nShuffle-Exchange Conjecture (graph-theoretic form)\n\nDiscussion links:\n- Václav E. Beneš: http://en.wikipedia.org/wiki/Václav E. Beneš\n- Shuffle-Exchange (SE) conjecture: http://www.openproblemgarden.org/?q=node/37167\n- SE conjecture: http://www.openproblemgarden.org/?q=node/37167\n- here: http://www.openproblemgarden.org/?q=node/37167\n- graph-theoretic form: http://www.openproblemgarden.org/?q=node/37210\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 101.\n\nAttempt notes:\nTarget:\nMake progress on \"Beneš Conjecture\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3047, "problem_number": "OPG-37222", "title": "Dividing up the unrestricted partitions", "statement": "Begin with the generating function for unrestricted partitions:\n\n(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...\n\nNow change some of the plus signs to minus signs. The resulting series will have coefficients congruent, mod 2, to the coefficients of the generating series for unrestricted partitions. I conjecture that the signs may be chosen such that all the coefficients of the series are either 1, -1, or zero.", "background": "Source: Open Problem Garden. Original node ID: 37222. URL: http://www.openproblemgarden.org/op/dividing_up_the_unrestricted_partitions.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/dividing_up_the_unrestricted_partitions\n- Author(s): David S.; Newman\n- Subject(s): Combinatorics\n- Keywords: congruence properties; partition\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 11th, 2010 by DavidSNewman\n\nProblem-page discussion:\nI've been thinking about this problem since about 1970. Emory Starke thought that it was a good problem, but not suitable for the Problems section of the AMM, because it was unsolved. George Andrews and Freeman Dyson also thought that it is a good problem, but neither had any ideas how to solve it.\n\nI've found choices of sign which yield series with coefficients 1, -1, or 0 for all exponents about as high as 110 using computer searches. One thing which mitigates against finding a meaningful solution is that there is no known pattern for the number of unrestricted partitions modulo 2.\n\nBibliography:\nAndrews, George E., The Theory of Partitions, Cambridge University Press (1984)\n\nComments:\n- May 18th, 2010 | Benjamin Young | a question on the numerical work: I'm also curious to know a little more about the experimental work that was done -- roughly how many ways were there to choose the signs to make things work up to degree 110? were there any choices which gave you lots of zeros?\n- May 18th, 2010 | Benjamin Young | pentagonal number theorem: Euler's famous pentagonal number theorem is somewhat like this problem, except it deals with the generating function for partitions into distinct parts:\n\n(1+x)(1+x^2)(1+x^3)...\n\nIf you change *all* of the + signs in the above into minus signs, then the statement of your conjecture holds; indeed there is an explicit formula for the terms of the generating function involving the pentagonal numbers, hence the name of the theorem. This theorem has several pretty and well-publicized proofs (see Chapter 1 of the introduction to \"The Theory of Partitions\" by George Andrews, or Chapter 14.5 of \"Introduction to Analytic Number Theory\" by Tom Apostol, or \"Proofs from the book\" by Aigner-Ziegler, or Wikipedia).\n\nI would wager that this observation isn't terribly helpful, but still. Was this was the motivation of the problem?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Dividing up the unrestricted partitions\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3048, "problem_number": "OPG-37226", "title": "Sequence defined on multisets", "statement": "Conjecture Define a $2 \\times n$ array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array $[1; 1]$, the sequence is: $[1; 1]$-> $[1; 2]$-> $[1, 2; 1, 1]$-> $[1, 2; 3, 1]$-> $[1, 2, 3; 2, 1, 1]$-> $[1, 2, 3; 3, 2, 1]$-> $[1, 2, 3; 2, 2, 2]$-> $[1, 2, 3; 1, 4, 1]$-> $[1, 2, 3, 4; 3, 1, 1, 1]$-> $[1, 2, 3, 4; 4, 1, 2, 1]$-> $[1, 2, 3, 4; 3, 2, 1, 2]$-> $[1, 2, 3, 4; 2, 3, 2, 1]$, and we now have a fixed point (loop of one array).\n\nThe process always results in a loop of 1, 2, or 3 arrays.", "background": "Source: Open Problem Garden. Original node ID: 37226. URL: http://www.openproblemgarden.org/op/a_sequence_defined_on_multisets.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_sequence_defined_on_multisets\n- Author(s): Erickson, Martin\n- Subject(s): Combinatorics\n- Keywords: multiset; sequence\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: June 29th, 2010 by Martin Erickson\n\nBibliography:\n* Erickson, Martin J., \"Introduction to Combinatorics,\" Wiley, 1996.\n\nComments:\n- July 18th, 2011 | Anonymous | Solution: This problem has recently been solved by the author.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Sequence defined on multisets\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3049, "problem_number": "OPG-37228", "title": "Square achievement game on an n x n grid", "statement": "Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $n \\times n$ grid. The first player (if any) to occupy four cells at the vertices of a square with horizontal and vertical sides is the winner. What is the outcome of the game given optimal play? Note: Roland Bacher and Shalom Eliahou proved that every 15 x 15 binary matrix contains four equal entries (all 0's or all 1's) at the vertices of a square with horizontal and vertical sides. So the game must result in a winner (the first player) when n=15.", "background": "Source: Open Problem Garden. Original node ID: 37228. URL: http://www.openproblemgarden.org/op/a_game_on_an_n_x_n_grid.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_game_on_an_n_x_n_grid\n- Author(s): Erickson, Martin\n- Subject(s): Combinatorics\n- Keywords: game\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: June 29th, 2010 by Martin Erickson\n\nBibliography:\nR. Bacher and S. Eliahou, \"Extremal binary matrices without constant 2-squares,\" J. of Combinatorics, Volume 1, Number 1, 77-100, 2010.\n\n* Erickson, Martin, \"Pearls of Discrete Mathematics,\" CRC Press, 2010\n\nComments:\n- September 13th, 2011 | Carolus | A new reference: My new article \"Guaranteed successful strategies for a square achievement game on an n by n grid\" concerning that problem is now accessible via http://arxiv.org/abs/1109.2341.\n- August 13th, 2011 | porton | This is not mathematics: I suspect this problem can be solved only with brute force for every particular n and is not a nice mathematical conjecture (just like the question who wins in chess, white or black, playing both with the optimal strategy).\n\n-- Victor Porton - http://www.mathematics21.org\n- August 14th, 2011 | Carolus | On drawn games for n up to 14: I should have read the referenced article by Bacher and Eliahou before trying to find drawn games. But I've read it now.\n\nAmong many other things, the authors describe parametric families of square-free configurations for n=14.\n\nNot all of those configurations do provide the required relation of the numbers of symbols of both sorts needed to be an end-configuration of the game.\n\nBut one can, for example, take the family A1, give the variables x1 up to x8 the value 0 (or O, resp.) and the variables x9 up to x16 the value 1 (or X, resp.).\n\nThen the numbers of symbols of both sorts are equal and that square-free configuration is also a correct end-configuration of the game for n=14.\n- August 13th, 2011 | Anonymous | An URL of the technical report version of the referenced article: http://www-lmpa.univ-littoral.fr/publications/articles/lmpa404.pdf\n- August 13th, 2011 | Carolus | Drawn game for n=12: I've had not much hope to find one in an acceptable time but after running another couple of hours my program delivered this square-free end-configuration for n=12:\n\noooooooxxxox\n\nxoxoxoxoxoxx\n\nxoxxooxooxxo\n\noooxxxooxxoo\n\noxooxoxxxoxo\n\nxxxooxoxoxox\n\nxoxxxxooooxx\n\noxoxoxxoxoox\n\nooxoooxxoxox\n\nxxooxooxoxxo\n\noooxoxoxooxx\n\nxoxxxoxxxoox\n- August 11th, 2011 | Carolus | Drawn games for grid sizes from 3 to 11: Running a self-written program I've found the square-less end-configurations to be listed (because of the 1000 characters limit) in a comment of this comment for the grid sizes (n) from 3 to 11.\n\nIf I understand the note in the problem text right, this proofs that there is no sure winning strategy for those sizes.\n\nOn my 1 GHz Celeron/PIII the search for n=11 took 65 seconds.\n\nBut I gave up the search for n=12 after circa 12 hours of calculation estimating the 130-fold duration for the complete search. On the other side: In the first 8 hours there was a square-less configuration just before the occupation of the last field.\n- August 13th, 2011 | Carolus | I withdraw the conclusion: The note in the problem text let me assume that it is proven that if the game for some grid size n must have a winner there is a sure winning strategy for the first player.\n\nTaking in account that implication only, one cannot, as I did, conclude that there is no sure winning strategy if a game could be drawn.\n- August 12th, 2011 | Carolus | Text corrections: It should stand 'proves' instead of 'proofs' and 'square-free' instead of 'square-less'.\n- August 13th, 2011 | Carolus | A list of square-free end-configurations for n from 3 to 11: 3\n\nooo\n\noxo\n\nxxx\n\n4\n\noooo\n\noxox\n\nxoxx\n\nxxxo\n\n5\n\nooooo\n\noxoxo\n\nooxxo\n\nxoxox\n\nxxxxx\n\n6\n\noooooo\n\noxoxox\n\nooxxoo\n\nxoxoxx\n\nxxxxox\n\nxooxxx\n\n7\n\nooooooo\n\noxoxoxo\n\nooxxoox\n\nxxooxxx\n\noxoxxox\n\nxxxooox\n\nxoxxxxo\n\n8\n\noooooooo\n\noxoxoxox\n\nooxxooxx\n\nxxooxxxo\n\noxoxxoxx\n\nxxxoxoox\n\nxoxoxxox\n\nxooooxxx\n\n9\n\nooooooooo\n\noxoxoxoxo\n\nooxxooxxo\n\nxoooxxxoo\n\noxoxxoxxx\n\nxxxoxooox\n\nxoxoxxoxx\n\nxxoxoxoox\n\noxxxoxxox\n\n10\n\nooooooooox\n\noxoxoxoxoo\n\noxxooxxxxo\n\nooxxxoxoxo\n\noxooxoxxoo\n\nxxxooooxxx\n\nxoxoxxoxox\n\nxooxoxxoxx\n\nxxoxxoxoox\n\nxoxxooxxox\n\n11\n\noooooooooxx\n\noxoxoxxoxox\n\noxxoxxooxox\n\nooxxxooxxoo\n\noxooxoxxoox\n\nxxxxoooooxx\n\nxoxooxoxxxo\n\nxooxoxxoxoo\n\nxxooxxoooxo\n\nxoxxxooxoxx\n\nxoxoxoxxxxo\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Square achievement game on an n x n grid\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3050, "problem_number": "OPG-37230", "title": "Transversal achievement game on a square grid", "statement": "Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $n \\times n$ grid. The first player (if any) to occupy a set of $n$ cells having no two cells in the same row or column is the winner. What is the outcome of the game given optimal play?", "background": "Source: Open Problem Garden. Original node ID: 37230. URL: http://www.openproblemgarden.org/op/a_transversal_achievement_game_on_a_square_grid.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_transversal_achievement_game_on_a_square_grid\n- Author(s): Erickson, Martin\n- Subject(s): Combinatorics\n- Keywords: game\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: June 29th, 2010 by Martin Erickson\n\nComments:\n- January 6th, 2021 | Anonymous | Solution: The problem has been solved. Link:- https://arxiv.org/abs/2101.00770\n- February 13th, 2013 | Anonymous | Are there a simple solution?: I suspect, there are no simple answer and it can be solved only by heavy calculations, that is essentally there is no solution to this problem.\n- February 22nd, 2013 | mshj | history and application: i'm not sure but i think to solve this problem, i was wondering if any body gives me some information about the history and application of this problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Transversal achievement game on a square grid\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3051, "problem_number": "OPG-37416", "title": "Length of surreal product", "statement": "Conjecture Every surreal number has a unique sign expansion, i.e. function $f: o\\rightarrow \\{-, +\\}$, where $o$ is some ordinal. This $o$ is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of $s$ as $\\ell(s)$.\n\nIt is easy to prove that\n\n$$\\ell(s+t) \\leq \\ell(s)+\\ell(t)$$\n\nWhat about\n\n$$\\ell(s\\times t) \\leq \\ell(s)\\times\\ell(t)$$?", "background": "Source: Open Problem Garden. Original node ID: 37416. URL: http://www.openproblemgarden.org/op/length_of_surreal_product.\n\nSource subject path: Combinatorics.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/length_of_surreal_product\n- Author(s): Gonshor, Harry\n- Subject(s): Combinatorics\n- Keywords: surreal numbers\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: April 7th, 2012 by Lukáš Lánský\n\nProblem-page discussion:\nThis is strongly conjectured to be true by Gonshor in [Gon86]. There is an easy way to prove that\n\n$$\\ell(s\\times t) \\leq 3^{\\ell(s)+\\ell(t)}$$\n\nBibliography:\n*[Gon86] Harry Gonshor, An Introduction to the Theory of Surreal Numbers, Cambridge University Press, Cambridge, 1986.\n\nSource links:\n- surreal number: http://en.wikipedia.org/wiki/surreal number\n\nComments:\n- May 30th, 2012 | vprusso | Proof Already Exists?: I believe the proof for the conjectured statement was proven in the affirmative in the paper \"Fields of Surreal Numbers and Exponentiation\" by Dries and Ehrlich. Specifically, Lemma 3.3 on page 6: http://www.ohio.edu/people/ehrlich/EhrlichvandenDries.pdf\n\nIf this satisfies the conjecture adequately great, if not, let me know if you would like to work toward a solution together on something similar or related.\n\nThanks.\n\n-Vincent Russo\n- June 5th, 2012 | Lukáš Lánský | Maybe!: Thank you! I wasn't aware of this paper. At first sight I think that the part you refer to establish the required result just for surreals in the form $r\\cdot\\omega^x$, but I'll find time to go through it thoroughly as it is most relevant for the matter.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Length of surreal product\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3052, "problem_number": "OPG-58213", "title": "Roller Coaster permutations", "statement": "Let $S_n$ denote the set of all permutations of $[n]=\\set{1,2,\\ldots,n}$. Let $i(\\pi)$ and $d(\\pi)$ denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in $\\pi$. Let $X(\\pi)$ denote the set of subsequences of $\\pi$ with length at least three. Let $t(\\pi)$ denote $\\sum_{\\tau\\in X(\\pi)}(i(\\tau)+d(\\tau))$.\n\nA permutation $\\pi\\in S_n$ is called a Roller Coaster permutation if $t(\\pi)=\\max_{\\tau\\in S_n}t(\\tau)$. Let $RC(n)$ be the set of all Roller Coaster permutations in $S_n$.\n\nConjecture For $n\\geq 3$,\n\n- If $n=2k$, then $|RC(n)|=4$.\n- If $n=2k+1$, then $|RC(n)|=2^j$ with $j\\leq k+1$.\n\nConjecture (Odd Sum conjecture) Given $\\pi\\in RC(n)$,\n\n- If $n=2k+1$, then $\\pi_j+\\pi_{n-j+1}$ is odd for $1\\leq j\\leq k$.\n- If $n=2k$, then $\\pi_j + \\pi_{n-j+1} = 2k+1$ for all $1\\leq j\\leq k$.", "background": "Source: Open Problem Garden. Original node ID: 58213. URL: http://www.openproblemgarden.org/op/roller_coaster_permutations.\n\nSource subject path: Combinatorics.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/roller_coaster_permutations\n- Author(s): Ahmed, Tanbir; Snevily, Hunter S.\n- Subject(s): Combinatorics\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 14th, 2013 by Tanbir Ahmed\n\nBibliography:\n*[AS] Tanbir Ahmed, Hunter Snevily, Some properties of Roller Coaster permutations. To appear in Bull. Institute of Combinatorics and its Applications, 2013.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"Roller Coaster permutations\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3053, "problem_number": "OPG-60000", "title": "The Double Cap Conjecture", "statement": "Conjecture The largest measure of a Lebesgue measurable subset of the unit sphere of $\\mathbb{R}^n$ containing no pair of orthogonal vectors is attained by two open caps of geodesic radius $\\pi/4$ around the north and south poles.", "background": "Source: Open Problem Garden. Original node ID: 60000. URL: http://www.openproblemgarden.org/op/the_double_cap_conjecture.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_double_cap_conjecture\n- Author(s): Kalai, Gil\n- Subject(s): Combinatorics\n- Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 15th, 2015 by Jon Noel\n\nProblem-page discussion:\nThe problem of determining the maximum was first considered by Witsenhausen [Wit] who proved that the measure of such a set is at most $\\frac{1}{n}$ times the surface measure of the sphere. In $\\mathbb{R}^3$, DeCorte and Pikhurko [DP] improved the multiplicative constant to $0.313< 1/3$. The conjecture above would imply that the measure is at most $1-1/\\sqrt{2} \\approx 0.2928$.\n\nBibliography:\n[DP] E. DeCorte and O. Pikhurko, Spherical sets avoiding a prescribed set of angles, arXiv:1502.05030v2.\n\n[Kalai] G. Kalai, How Large can a Spherical Set Without Two Orthogonal Vectors Be? https://gilkalai.wordpress.com/2009/05/22/how-large-can-a-spherical-set-without-two-orthogonal-vectors-be/\n\n[Wit] H. S. Witsenhausen. Spherical sets without orthogonal point pairs. American Mathematical Monthly, pages 1101–1102, 1974.\n\nRelated:\nRelated problems\nCircular colouring the orthogonality graph\nPartitioning the Projective Plane\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"The Double Cap Conjecture\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3054, "problem_number": "OPG-60002", "title": "Saturation in the Hypercube", "statement": "Question What is the saturation number of cycles of length $2\\ell$ in the $d$-dimensional hypercube?", "background": "Source: Open Problem Garden. Original node ID: 60002. URL: http://www.openproblemgarden.org/op/saturation_in_the_hypercube.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/saturation_in_the_hypercube\n- Author(s): Morrison, Natasha; Noel, Jonathan A.; Scott, Alex\n- Subject(s): Combinatorics\n- Keywords: cycles; hypercube; minimum saturation; saturation\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 20th, 2015 by Jon Noel\n\nProblem-page discussion:\nLet $G$ and $H$ be graphs. Say that a spanning subgraph $F$ of $G$ is $(G,H)$-saturated if $F$ contains no copy of $H$ but $F+e$ contains a copy of $H$ for every edge $e\\in E(G)\\setminus E(F)$. Let $\\text{sat}(G,H)$ denote the minimum number of edges in a $(G,H)$-saturated graph. Saturation was introduced by Erdős, Hajnal and Moon [EHM] who proved the following:\n\nTheorem (Erdős, Hajnal and Moon) For $n\\geq k\\geq2$ we have $\\text{sat}(K_n,K_k) = \\binom{n}{2} = \\binom{n-k+2}{2}$.\n\nLet $Q_d$ denote the $d$-dimensional hypercube. Saturation of $4$-cycles in the hypercube has been studied by Choi and Guan [CG] who proved that $\\text{sat}(Q_d,C_4)\\leq \\left(\\frac{1}{4} + o(1)\\right)|E(Q_d)|$. This was drastically improved by Johnson and Pinto [JP] to $\\text{sat}(Q_d,C_4) < 10\\cdot 2^d$. The saturation number for longer cycles in the hypercube is not known, though. The question above addresses this.\n\nAnother open problem is to determine the saturation number of sub-hypercubes in $Q_d$. This was first considered by Johnson and Pinto [JP] who proved that $\\text{sat}(Q_d,Q_m) = o\\left(|E(Q_d)|\\right)$ for fixed $m$ and $d\\to \\infty$. This upper bound was improved to $(1+o(1))72m^2 2^d$ by Morrison, Noel and Scott [MNS]. The best known lower bound on $\\text{sat}(Q_d,Q_m)$ for fixed $m$ and large $d$, also due to [MNS], is $(m-1-o(1))2^d$.\n\nProblem Improve the upper and lower bounds on $\\text{sat}(Q_d,Q_m)$ for fixed $m$ and large $d$.\n\nThe results of [MNS] show that $\\text{sat}(Q_d,Q_m) = \\Theta(2^d)$ for fixed $m$. Howver, the precise asymptotic behaviour of this quantity is unknown.\n\nQuestion (Morrison, Noel and Scott) For fixed $m\\geq 2$, is it true that $\\frac{\\text{sat}(Q_d,Q_m)}{2^d}$ converges as $d\\to \\infty$?\n\nBibliography:\n[CG] S. Choi and P. Guan, Minimum critical squarefree subgraph of a hypercube, Proceedings of the Thirty-Ninth Southeastern International Conference on Combinatorics, Graph Theory and Computing, vol. 189, 2008, pp. 57–64.\n\n[EHM] P. Erdős, A. Hajnal, and J. W. Moon, A problem in graph theory, Amer. Math. Monthly 71 (1964), 1107–1110.\n\n[JP] J. R. Johnson and T. Pinto, Saturated subgraphs of the hypercube, arXiv:1406.1766v1, preprint, June 2014.\n\n[MNS] N. Morrison, J. A. Noel and A. Scott, Saturation in the Hypercube and Bootstrap Percolation, arXiv:1408.5488v2, June 2015.\n\nRelated:\nRelated problems\nSaturated $k$-Sperner Systems of Minimum Size\nTurán Problem for $10$-Cycles in the Hypercube\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 26.\n\nAttempt notes:\nTarget:\nMake progress on \"Saturation in the Hypercube\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3055, "problem_number": "OPG-60003", "title": "Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube", "statement": "Problem Determine the smallest percolating set for the $4$-neighbour bootstrap process in the hypercube.", "background": "Source: Open Problem Garden. Original node ID: 60003. URL: http://www.openproblemgarden.org/op/extremal_4_neighbour_bootstrap_percolation_in_the_hypercube.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/extremal_4_neighbour_bootstrap_percolation_in_the_hypercube\n- Author(s): Morrison, Natasha; Noel, Jonathan A.\n- Subject(s): Combinatorics\n- Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 20th, 2015 by Jon Noel\n\nProblem-page discussion:\nThe $r$-neighbour bootstrap process starts with an initial set of \"infected\" vertices in a graph and, at each step, a healthy vertex becomes infected if it has at least $r$ infected neighbours. Say that the initial set of infected vertices percolates if every vertex of $G$ is eventually infected. Let $m(G,r)$ denote the smallest percolating set in $G$ under the $r$-neighbour process.\n\nLet $Q_d$ denote the hypercube of dimension $d$. Balogh and Bollobás [BB] proved the following.\n\nTheorem (Balogh and Bollobás) $m(Q_d,2) = \\left\\lceil \\frac{d}{2}\\right\\rceil +1$ for all $d\\geq 2$.\n\nThey also conjectured that $m(Q_d,r) = \\frac{1+o(1)}{r}\\binom{d}{r-1}$ for fixed $r$ and $d\\to\\infty$. This conjecture was proved by Morrison and Noel [MN], who also showed the following.\n\nTheorem (Morrison and Noel) $m(Q_d,3) = \\left\\lceil \\frac{d(d+3)}{6} \\right\\rceil +1$ for all $d\\geq 3$.\n\nIt seems possible that one could obtain a general formula for $m(Q_d,r)$ for all $r$ and $d\\geq r$. However, the precise formula for $m(Q_d,r)$ (in terms of $d$ ) is not known for any fixed $r\\geq4$. A solution to this problem may have applications in proving probabilistic results for bootstrap percolation in the hypercube; see [BBM].\n\nBibliography:\n[BB] J. Balogh and B. Bollobás, Bootstrap percolation on the hypercube, Probab. Theory Related Fields 134 (2006), no. 4, 624–648.\n\n[BBM] J. Balogh, B. Bollobás and R. Morris, Bootstrap percolation in high dimensions, Combin. Probab. Comput. 19 (2010), no. 5-6, 643–692.\n\n[MN] N. Morrison and J. A. Noel, Extremal Bounds for Bootstrap Percolation in the Hypercube, preprint, arXiv:1506.04686v1.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3056, "problem_number": "OPG-60006", "title": "Turán Problem for $10$-Cycles in the Hypercube", "statement": "Problem Bound the extremal number of $C_{10}$ in the hypercube.", "background": "Source: Open Problem Garden. Original node ID: 60006. URL: http://www.openproblemgarden.org/op/turan_problem_for_10_cycles_in_the_hypercube.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/turan_problem_for_10_cycles_in_the_hypercube\n- Author(s): Erdos, Paul\n- Subject(s): Combinatorics\n- Keywords: cycles; extremal combinatorics; hypercube\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 20th, 2015 by Jon Noel\n\nProblem-page discussion:\nThe problem of bounding the extremal number for cycles in the hypercube was first considered by Erdős [Erd1,Erd2] who conjectured that $\\text{ex}(Q_d,C_4) = (1/2 +o(1)) |E(Q_d)|$ and that $\\text{ex}(Q_d,C_{2t}) = o(|E(Q_d)|)$ for all $t\\geq3$. The first conjecture is still open, and the second is known to be false in the case $t=3$ (see [BDT, Chu, Cond]).\n\nChung [Chu] proved that $\\text{ex}(Q_d,C_{2t}) = o(|E(Q_d)|)$ for even $t\\geq 4$ and Füredi and Özkahya [FO1,FO2] proved the same for odd $t\\geq 7$. Conlon [Conl] gave a unified proof of these results, which also applies to more general subgraphs of the hypercube. However, the case of $C_{10}$ remains unsolved.\n\nBibliography:\n[BDT] A. E. Brouwer, I. J. Dejter and C. Thomassen, Highly symmetric subgraphs of hypercubes, J. Algebraic Combin. 2 (1993), 25–29.\n\n[Chu] F. Chung, Subgraphs of a hypercube containing no small even cycles, J. Graph Theory 16 (1992), 273–286.\n\n[Cond] M. Conder, Hexagon-free subgraphs of hypercubes, J. Graph Theory 17 (1993), 477–479.\n\n[Conl] D. Conlon, An extremal theorem in the hypercube, Electron. J. Combin. 17 (2010), no. 1, Research Paper 111, 7 pages\n\n[Erd1] P. Erdős, On some problems in graph theory, combinatorial analysis and combinatorial number theory, in: Graph Theory and Combinatorics (Cambridge, 1983), Academic Press, London, 1984, 1–17.\n\n[Erd2] P. Erdős, Some of my favourite unsolved problems, in: A tribute to Paul Erdős, Cambridge University Press, 1990, 467–478.\n\n[FO1] Z. Füredi and L. Özkahya, On 14-cycle-free subgraphs of the hypercube, Combin. Probab. Comput. 18 (2009), 725–729.\n\n[FO2] Z. Füredi and L. Özkahya, On even-cycle-free subgraphs of the hypercube, Electronic Notes in Discrete Mathematics 34 (2009), 515–517.\n\nRelated:\nRelated problems\nSaturation in the Hypercube\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Turán Problem for $10$-Cycles in the Hypercube\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3057, "problem_number": "OPG-37196", "title": "Perfect 2-error-correcting codes over arbitrary finite alphabets.", "statement": "Conjecture Does there exist a nontrivial perfect 2-error-correcting code over any finite alphabet, other than the ternary Golay code?", "background": "Source: Open Problem Garden. Original node ID: 37196. URL: http://www.openproblemgarden.org/op/perfect_2_error_correcting_codes_over_arbitrary_finite_alphabets.\n\nSource subject path: Combinatorics > Codes.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/perfect_2_error_correcting_codes_over_arbitrary_finite_alphabets\n- Subject(s): Combinatorics; Codes\n- Keywords: 2-error-correcting; code; existence; perfect; perfect code\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 3rd, 2010 by davidcullen\n\nProblem-page discussion:\nVery few perfect codes are known to exist over any alphabet. The trivial examples are codes with 1 or 2 codewords, or q-ary (n, M, d) codes with all of the q^n vectors being codewords. Other than this, we have an infinite family of perfect 1-error-correcting Hamming codes, and two unique Golay codes, the binary one which corrects 1 error, the ternary one which corrects 2 errors. Recent research activity has discovered a large number of previously unknown perfect 1-error correcting codes which are not isomorphic to the Hamming codes.\n\nIt is well known (see Van Lint) that the answer is negative for codes over alphabets of size equal to a power of a prime number. Further results (see Hong, Best) establish that there are no perfect t-error-correcting codes for any t > 2 over any finite alphabet, which establishes the fact that 2 is the largest number of errors which any new perfect code could possibly correct. Lloyd's theorem plays a key role in ruling out t > 2, but provides less information than needed in the t = 2 case. Establishing the result in the negative would likely require an ad-hoc combinatorial argument, while establishing it in the positive could be done by any clever construction.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Perfect 2-error-correcting codes over arbitrary finite alphabets.\" in Combinatorics; Codes, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3058, "problem_number": "OPG-762", "title": "Combinatorial covering designs", "statement": "A $(v, k, t)$ covering design, or covering, is a family of $k$-subsets, called blocks, chosen from a $v$-set, such that each $t$-subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by $C(v, k, t)$.\n\nProblem Find a closed form, recurrence, or better bounds for $C(v,k,t)$. Find a procedure for constructing minimal coverings.", "background": "Source: Open Problem Garden. Original node ID: 762. URL: http://www.openproblemgarden.org/op/covering_designs.\n\nSource subject path: Combinatorics > Designs.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/covering_designs\n- Author(s): Gordon, D.M.; Mills, W.H.; Rödl, V.; Schönheim, J.\n- Subject(s): Combinatorics; Designs\n- Keywords: recreational mathematics\n- Importance: Low ✭\n- Recommended for undergraduates: no\n- Posted: April 11th, 2008 by Pseudonym\n\nProblem-page discussion:\nThe problem has applications in file design, but is also known at the \"lottery cover problem\", for its strategic application in playing lotteries.\n\nCurrent \"best\" covers have been collected by Dan Gordon.\n\nThe trivial lower bound is $C(v,k,t) \\geq \\dfrac{\\binom{v}{t}}{\\binom{k}{t}}$. When equality holds, the resulting design is called a Steiner system, and often denoted $S(t,k,v)$. If $S(t,k,v)$ exists, so does $S(t-1,k-1,v-1)$: just remove all occurrences of a point from the blocks containing it, and discard the blocks that didn't contain it before the deletion.\n\nBibliography:\nJ. Schönheim, On coverings, Pacific Journal of Mathematics, 14:1405–1411, 1964.\n\nDaniel M. Gordon, Oren Patashnik, Greg Kuperberg (1995) New constructions for covering designs, J. Combinatorial Designs 3(4), 269-284.\n\nD. T. Todorov. Combinatorial Coverings. PhD thesis, University of Sofia, 1985.\n\nDiscussion links:\n- Dan Gordon: http://www.ccrwest.org/cover.html\n\nBibliography links:\n- New constructions for covering designs: http://www.ccrwest.org/cover/cover.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Combinatorial covering designs\" in Combinatorics; Designs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3059, "problem_number": "OPG-148", "title": "A nowhere-zero point in a linear mapping", "statement": "Conjecture If ${\\mathbb F}$ is a finite field with at least 4 elements and $A$ is an invertible $n \\times n$ matrix with entries in ${\\mathbb F}$, then there are column vectors $x,y \\in {\\mathbb F}^n$ which have no coordinates equal to zero such that $Ax=y$.", "background": "Source: Open Problem Garden. Original node ID: 148. URL: http://www.openproblemgarden.org/op/a_nowhere_zero_point_in_a_linear_mapping.\n\nSource subject path: Combinatorics > Matrices.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_nowhere_zero_point_in_a_linear_mapping\n- Author(s): Jaeger, Francois\n- Subject(s): Combinatorics; Matrices\n- Keywords: invertible; nowhere-zero flow\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 8th, 2007 by mdevos\n\nProblem-page discussion:\nThe motivation for this problem comes from the study of nowhere-zero flows on graphs. If $A$ is the directed incidence matrix of a graph $G$, then a nowhere-zero ${\\mathbb F}$-flow on $G$ is precisely a vector $x$ so that $x$ has all entries nonzero, and $Ax=0$. The above conjecture is similar, but is for general (invertible) matrices. Alon and Tarsi have resolved this conjecture for all fields not of prime order using their polynomial technique.\n\nDefinition: Say that a $m \\times n$ matrix $A$ is $(a,b)$-choosable if for all $X_1,X_2,\\ldots,X_m \\subseteq {\\mathbb F}$ with $|X_i|=a$ and for all $Y_1,Y_2,\\ldots,Y_n \\subseteq {\\mathbb F}$ with $|Y_j|=b$, there exists a vector $x \\in X_1 \\times X_2 \\ldots \\times X_m$ and a vector $y \\in Y_1 \\times Y_2 \\ldots \\times Y_n$ so that $Ax=y$. Note that every matrix is $(1,|{\\mathbb F}|)$-choosable, but that an $n \\times n$ matrix is $(|{\\mathbb F}|,1)$-choosable if and only if it is invertible.\n\nAlon and Tarsi actually prove a stronger property than Jaeger conjectured for fields not of prime order. They prove that if ${\\mathbb F}$ has characteristic $p$, then every invertible matrix over ${\\mathbb F}$ is $(p,|{\\mathbb F}|-1)$-choosable. This result has been extended by DeVos [D] who showed that every such matrix is $(p,|{\\mathbb F}|-p+1)$-choosable. Yang Yu [Y] has verified that the conjecture holds for $n \\times n$ matrices with entries in ${\\mathbb Z}_p$ when $n < 2^{p-2}$.\n\nJaeger's conjecture is true in a very strong sense for fields of characteristic 2. DeVos [D] proved that every invertible matrix over such a field is $(k+1,|{\\mathbb F}|-k)$-choosable for every $k$. The following conjecture asserts that invertible matrices over fields of prime order have choosability properties nearly as strong.\n\nConjecture [The choosability in ${\\mathbb Z}_p$ conjecture (DeVos)] Every invertible matrix with entries in ${\\mathbb Z}_p$ for a prime $p$ is $(k+2,p-k)$-choosable for every $k$.\n\nThis is essentially the strongest choosability conjecture one might hope to be true over fields of prime order. I (M. DeVos) don't have any experimental evidence for this at all, so it could be false already for some small examples. However, I suspect that if The permanent conjecture is true, that this conjecture should also be true. In any case, I (M. DeVos) am offering a bottle of wine for this conjecture.\n\nBibliography:\n[A] N. Alon, Combinatorial Nullstellensatz, Combinatorics Probability and Computing 8 (1999) no. 1-2, 7-29. MathSciNet\n\n[AT] N. Alon, M. Tarsi, A Nowhere-Zero Point in Linear Mappings, Combinatorica 9 (1989), 393-395. MathSciNet\n\n[BBLS] R. Baker, J. Bonin, F. Lazebnik, and E. Shustin, On the number of nowhere-zero points in linear mappings, Combinatorica 14 (2) (1994), 149-157. MathSciNet\n\n[D] M. DeVos, Matrix Choosability, J. Combinatorial Theory, Ser. A 90 (2000), 197-209. MathSciNet\n\n[Y] Y. Yu, The Permanent Rank of a Matrix, J. Combinatorial Theory Ser. A 85 (1999), 237-242. MathSciNet\n\nBibliography links:\n- Combinatorial Nullstellensatz: http://www.math.tau.ac.il/%7Enogaa/PDFS/null2.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1684621\n- A Nowhere-Zero Point in Linear Mappings: http://www.springerlink.com/content/y9766rt81243188l/\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1054015\n- On the number of nowhere-zero points in linear mappings: http://www.springerlink.com/content/jx477824728366p0/\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1289069\n- Matrix Choosability: http://www.ams.org/leavingmsn?url=http://www.sciencedirect.com/science/journal/00973165\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1749430\n- The Permanent Rank of a Matrix: http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.1006/jcta.1998.2904\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1673948\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 26.\n\nAttempt notes:\nTarget:\nMake progress on \"A nowhere-zero point in a linear mapping\" in Combinatorics; Matrices, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3060, "problem_number": "OPG-150", "title": "The additive basis conjecture", "statement": "Conjecture For every prime $p$, there is a constant $c(p)$ (possibly $c(p)=p$ ) so that the union (as multisets) of any $c(p)$ bases of the vector space $({\\mathbb Z}_p)^n$ contains an additive basis.", "background": "Source: Open Problem Garden. Original node ID: 150. URL: http://www.openproblemgarden.org/op/the_additive_basis_conjecture.\n\nSource subject path: Combinatorics > Matrices.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_additive_basis_conjecture\n- Author(s): Jaeger, Francois; Linial, Nathan; Payan, Charles; Tarsi, Michael\n- Subject(s): Combinatorics; Matrices\n- Keywords: additive basis; matrix\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 8th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: Let $V$ be a finite dimensional vector space over the field ${\\mathbb Z}_p$. We call a multiset $B$ with elements in $V$ an additive basis if for every $v \\in V$, there is a subset of $B$ which sums to $v$.\n\nIt is worth noting that this conjecture would also imply that every $2c(3)$-edge-connected graph has a nowhere-zero 3-flow, thus resolving The weak 3-flow conjecture.\n\nDiscussion links:\n- The weak 3-flow conjecture: http://www.openproblemgarden.org/?q=op/3_flow_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"The additive basis conjecture\" in Combinatorics; Matrices, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3061, "problem_number": "OPG-151", "title": "The permanent conjecture", "statement": "Conjecture If $A$ is an invertible $n \\times n$ matrix, then there is an $n \\times n$ submatrix $B$ of $[A A]$ so that $perm(B)$ is nonzero.", "background": "Source: Open Problem Garden. Original node ID: 151. URL: http://www.openproblemgarden.org/op/the_permanent_conjecture.\n\nSource subject path: Combinatorics > Matrices.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_permanent_conjecture\n- Author(s): Kahn, Jeff\n- Subject(s): Combinatorics; Matrices\n- Keywords: invertible; matrix; permanent\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 8th, 2007 by mdevos\n\nProblem-page discussion:\nIf true, this conjecture would imply the nowhere-zero point in a linear mapping conjecture via the Alon-Tarsi polynomial technique. I believe Yang Yu was the first to suggest the following generalization of the permanent conjecture.\n\nConjecture (Yu) If $A,B$ are invertible $n \\times n$ matrices over the same field, then there is an $n \\times n$ submatrix $C$ of $[A B]$ so that $perm(C)$ is nonzero.\n\nThis conjecture when restricted to the field ${\\mathbb Z}_3$ is a consequence of the Alon-Tarsi basis conjecture. In addition to implying the above conjecture, the truth of this conjecture for matrices over the field ${\\mathbb Z}_3$ would imply that every 6-edge-connected graph has a nowhere-zero 3-flow, thus resolving The weak 3-flow conjecture.\n\nDiscussion links:\n- nowhere-zero point in a linear mapping conjecture: http://www.openproblemgarden.org/?q=op/a_nowhere_zero_point_in_a_linear_mapping\n- The weak 3-flow conjecture: http://www.openproblemgarden.org/?q=op/3_flow_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"The permanent conjecture\" in Combinatorics; Matrices, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3062, "problem_number": "OPG-152", "title": "The Alon-Tarsi basis conjecture", "statement": "Conjecture If $B_1,B_2,\\ldots B_p$ are invertible $n \\times n$ matrices with entries in ${\\mathbb Z}_p$ for a prime $p$, then there is a $n \\times (p-1)n$ submatrix $A$ of $[B_1 B_2 \\ldots B_p]$ so that $A$ is an AT-base.", "background": "Source: Open Problem Garden. Original node ID: 152. URL: http://www.openproblemgarden.org/op/the_alon_tarsi_basis_conjecture.\n\nSource subject path: Combinatorics > Matrices.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_alon_tarsi_basis_conjecture\n- Author(s): Alon, Noga; Linial, Nathan; Meshulam, Roy\n- Subject(s): Combinatorics; Matrices\n- Keywords: additive basis; matrix\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 8th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: If $A$ is an $n \\times (p-1)n$ matrix over a field of characteristic $p$, then we say that $A$ is an Alon-Tarsi basis (or AT-basis) if the permanent of the $(p-1)n \\times (p-1)n$ matrix obtained by stacking $p-1$ copies of $A$ is nonzero.\n\nIt follows from the Alon-Tarsi polynomial technique that if $A$ is an AT-base then for every $X_1,X_2,\\ldots,X_{(p-1)n} \\subseteq {\\mathbb Z}_p$ of size 2 and for every $y \\in {\\mathbb Z}_p^n$, there exists a vector $x \\in X_1 \\times X_2 \\ldots \\times X_{(p-1)n}$ so that $Ax=y$ (using the notation from A nowhere-zero point in a linear mapping, $A$ is (2,1)-choosable). It follows from this that every Alon-Tarsi base over ${\\mathbb Z}_p$ is also an additive basis. Thus, the above conjecture, if true, would imply The additive basis conjecture. The following strengthening of this conjecture was suggested in [D]\n\nConjecture (The strong Alon-Tarsi basis conjecture (DeVos)) If $B_1,B_2,\\ldots,B_p$ are invertible $n \\times n$ matrices with entries in a field of characteristic $p$, then we may partition the columns of $[B_1 B_2 \\ldots B_p]$ into an $n \\times (p-1)n$ matrix $A$ and an $n \\times n$ matrix $C$ so that $A$ is an AT-base and $C$ is invertible.\n\nIn addition to implying the conjecture, above, if true, this conjecture would imply both The permanent conjecture and The choosability in ${\\mathbb Z}_p$ conjecture.\n\nDiscussion links:\n- A nowhere-zero point in a linear mapping: http://www.openproblemgarden.org/?q=op/a_nowhere_zero_point_in_a_linear_mapping\n- The additive basis conjecture: http://www.openproblemgarden.org/?q=op/the_additive_basis_conjecture\n- The permanent conjecture: http://www.openproblemgarden.org/?q=op/the_permanent_conjecture\n- The choosability in ${\\mathbb Z}_p$ conjecture: http://www.openproblemgarden.org/?q=op/a_nowhere_zero_point_in_a_linear_mapping\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"The Alon-Tarsi basis conjecture\" in Combinatorics; Matrices, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3063, "problem_number": "OPG-361", "title": "Rota's unimodal conjecture", "statement": "Let $M$ be a matroid of rank $r$, and for $0 \\le i \\le r$ let $w_i$ be the number of closed sets of rank $i$.\n\nConjecture $w_0,w_1,\\ldots,w_r$ is unimodal.\n\nConjecture $w_0,w_1,\\ldots,w_r$ is log-concave.", "background": "Source: Open Problem Garden. Original node ID: 361. URL: http://www.openproblemgarden.org/op/rotas_unimodal_conjecture.\n\nSource subject path: Combinatorics > Matroid Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/rotas_unimodal_conjecture\n- Author(s): Rota, Gian-Carlo\n- Subject(s): Combinatorics; Matroid Theory\n- Keywords: flat; log-concave; matroid\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 8th, 2007 by mdevos\n\nProblem-page discussion:\nA sequence $a_0,a_1,\\ldots a_n$ is log-concave if $a_i^2 \\ge a_{i-1} a_{i+1}$ for all $1 \\le i \\le n-1$.\n\nThe first of these conjectures is due to Rota [R], the second is folklore as far as I (M. DeVos) know. The special case of proving the second conjecture for $w_1,w_2,w_3$ amounts to showing that $(\\#lines)^2 \\ge (\\#points)(\\#planes)$ and has been called the points-lines-planes conjecture. Seymour [S] proved this conjecture in the special case where every line contains at most four points, but it is still open in general.\n\nBibliography:\n*[R] Rota, Gian-Carlo, Combinatorial theory, old and new. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3, pp. 229--233. Gauthier-Villars, Paris, 1971. MathSciNet\n\n[S] Seymour, P. D. On the points-lines-planes conjecture, J. Combin. Theory Ser. B 33 (1982), no. 1, 17--26. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0505646\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0678168\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Rota's unimodal conjecture\" in Combinatorics; Matroid Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3064, "problem_number": "OPG-369", "title": "Bases of many weights", "statement": "Let $G$ be an (additive) abelian group, and for every $S \\subseteq G$ let ${\\mathit stab}(S) = \\{ g \\in G: g + S = S \\}$.\n\nConjecture Let $M$ be a matroid on $E$, let $w: E \\rightarrow G$ be a map, put $S = \\{ \\sum_{b \\in B} w(b): B \\mbox{ is a base} \\}$ and $H = {\\mathit stab}(S)$. Then $$|S| \\ge |H| \\left( 1 - rk(M) + \\sum_{Q \\in G/H} rk(w^{-1}(Q)) \\right).$$", "background": "Source: Open Problem Garden. Original node ID: 369. URL: http://www.openproblemgarden.org/op/bases_of_many_weights.\n\nSource subject path: Combinatorics > Matroid Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/bases_of_many_weights\n- Author(s): Schrijver, Alexander; Seymour, Paul D.\n- Subject(s): Combinatorics; Matroid Theory\n- Keywords: matroid; sumset; zero sum\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 10th, 2007 by mdevos\n\nProblem-page discussion:\nAlthough this conjecture may look a bit technical, it is in fact very natural, and important.\n\nThere is an interesting branch of combinatorial number theory which begins with the Cauchy-Davenport theorem, and M. Kneser's generelization of this theorem. We highlight these two theorems below. For a positive integer $n$, we let ${\\mathbb Z}_n = {\\mathbb Z} / n {\\mathbb Z}$.\n\nTheorem (Cauchy-Davenport) If $p$ is prime and $A,B \\subseteq {\\mathbb Z}_p$ are nonempty, then $|A+B| \\ge \\min\\{p, |A| + |B| - 1 \\}$.\n\nTheorem (Kneser) Let $A,B \\subseteq G$ be finite and nonempty, and let $H = {\\mathit stab}(A+B)$. Then $|A+B| \\ge |A+H| + |B+H| - |H|$.\n\nIn a somewhat underappreciated paper of Schrijver and Seymour, they find a generalization of the Cauchy-Davenport theorem to matroids. Namely, they prove the following.\n\nTheorem (Schrijver, Seymour) Let $M$ be a matroid on $E$, let $p$ be prime, and let $w: E \\rightarrow {\\mathbb Z}_p$ be a map. Then $\\#\\{ \\sum_{b \\in B} w(b): B \\mbox{ is a base} \\} \\ge \\min \\{p, \\sum_{g \\in {\\mathbb Z}_p} rk(w^{-1}(g)) \\}$.\n\nThe special case of this theorem when the underlying matroid is obtained from the free matroid on two elements by adding parallel edges is exactly the Cauchy-Davenport theorem. Further, their conjecture is precisely the common generalization of their theorem and Kneser's theorem.\n\nDeVos, Goddyn, and Mohar have proved this conjecture in the special case when the underlying matroid is obtained from a uniform matroid by adding parallel elements, but apart from that, little is known.\n\nBibliography:\n[C] A.L. Cauchy, Recherches sur les nombers, J. Ecole Polytechniques, 9 (1813), 99-123.\n\n[D] H. Davenport, On the addition of residue classes, J. London Math. Soc., 10 (1935), 30-32.\n\n[K] M. Kneser, Abschätzung der aymptotischen dichte von summenmengen, Math. Z. (1953) 459-484.\n\n[N] M.B. Nathanson, Additive Number Theory, GTM 165, Springer, 1996.\n\n*[SS] A. Schrijver and P.D. Seymour, Spanning trees of different weights. Polyhedral combinatorics, DIMACS Ser. Discrete Math. Theoret. Comp. Sci., 1, 281-288.\n\nBibliography links:\n- Spanning trees of different weights: http://repository.cwi.nl/search/fullrecord.php?publnr=1609\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 20.\n\nAttempt notes:\nTarget:\nMake progress on \"Bases of many weights\" in Combinatorics; Matroid Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3065, "problem_number": "OPG-382", "title": "Aharoni-Berger conjecture", "statement": "Conjecture If $M_1,\\ldots,M_k$ are matroids on $E$ and $\\sum_{i=1}^k rk_{M_i}(X_i) \\ge \\ell (k-1)$ for every partition $\\{X_1,\\ldots,X_k\\}$ of $E$, then there exists $X \\subseteq E$ with $|X| = \\ell$ which is independent in every $M_i$.", "background": "Source: Open Problem Garden. Original node ID: 382. URL: http://www.openproblemgarden.org/op/aharoni_berger_conjecture.\n\nSource subject path: Combinatorics > Matroid Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/aharoni_berger_conjecture\n- Author(s): Aharoni, Ron; Berger, Eli\n- Subject(s): Combinatorics; Matroid Theory\n- Keywords: independent set; matroid; partition\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 11th, 2007 by mdevos\n\nProblem-page discussion:\nLet us begin by stating two classic results. For a graph (or hypergraph) we let $\\tau$ denote the size of the smallest (vertex) cover and we let $\\nu$ denote the size of the largest matching.\n\nTheorem (König) $\\nu = \\tau$ for every bipartite graph.\n\nTheorem (Matroid Intersection) If $M_1,M_2$ are matroids on $E$ and $rk_{M_1}(X_1) + rk_{M_2}(X_2) \\ge \\ell$ for every partition $\\{X_1,X_2\\}$ of $E$, then there exists $X \\subseteq E$ with $|X| = \\ell$ which is independent in both $M_1$ and $M_2$.\n\nThe matroid intersection theorem is exactly the $k=2$ case of the above conjecture, but it may also be viewed as a generalization of König's theorem. To see this, let $G$ be a bipartite graph with edge set $E$ and bipartition $\\{A_1,A_2\\}$ and for $i=1,2$ let $M_i$ be the (uniform) matroid on $E$ where a subset $S \\subseteq E$ is independent if no two edges in $S$ share an endpoint in $A_i$. Then $rk_{M_i}(S)$ is the number of vertices in $A_i$ which are incident with an edge in $S$, so $rk_{M_1}(X_1) + rk_{M_2}(X_2)$ has minimum value $\\tau$, and a set of edges is independent in both $M_1$ and $M_2$ if and only if it is a matching, so the size of the largest such set is $\\nu$.\n\nA famous conjecture of Ryser suggests a generalization of König's theorem to hypergraphs. It claims that every $k$-partite $k$-uniform hypergraph satisfies $\\tau \\le (k-1) \\nu$. The above conjecture is the common generalization of this conjecture of Ryser and the matroid intersection theorem. Aharoni [A] proved the 3-partite 3-uniform case of Ryser's conjecture, and this was extended by Aharoni-Berger [AB] to the $k=3$ case of the above conjecture. The conjecture remains open for $k \\ge 4$.\n\nBibliography:\n[A] R. Aharoni, Ryser's conjecture for tripartite 3-graphs, Combinatorica 21 (2001), 1-4. MathSciNet\n\n*[AB] R. Aharoni, E. Berger, The intersection of a matroid with a simplicial complex. Trans. Amer. Math. Soc. 358 (2006), no. 11 MathSciNet\n\nRelated:\nRelated problems\nRyser's conjecture\n\nDiscussion links:\n- cover: http://en.wikipedia.org/wiki/covering (graph theory)\n- conjecture of Ryser: http://www.openproblemgarden.org/?q=node/165\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1805710\n- The intersection of a matroid with a simplicial complex: http://www.math.princeton.edu/%7Eeberger/matcom.ps\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2231877\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 28.\n\nAttempt notes:\nTarget:\nMake progress on \"Aharoni-Berger conjecture\" in Combinatorics; Matroid Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3066, "problem_number": "OPG-692", "title": "Equality in a matroidal circumference bound", "statement": "Question Is the binary affine cube $AG(3,2)$ the only 3-connected matroid for which equality holds in the bound $$E(M) \\leq c(M) c(M^*) / 2$$where$c(M)$is the circumference (i.e. largest circuit size) of$M$?", "background": "Source: Open Problem Garden. Original node ID: 692. URL: http://www.openproblemgarden.org/op/equality_in_a_matroidal_circumference_bound.\n\nSource subject path: Combinatorics > Matroid Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/equality_in_a_matroidal_circumference_bound\n- Author(s): Oxley, James; Royle, Gordon\n- Subject(s): Combinatorics; Matroid Theory\n- Keywords: circumference\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 2nd, 2007 by Gordon Royle\n\nProblem-page discussion:\nIf $M$ is a 2-connected matroid with at least two elements then it was proved in [LO] that $$E(M) \\leq c(M) c(M^*) / 2$$where$c(M)$is the size of the largest circuit in$M$.\n\nEquality can hold in this bound -- in particular the binary affine cube $AG(3,2)$ is an 8-element self-dual matroid with circumference 4. There are various graphic matroids for which equality holds, and these have been classified in [W] where it is shown that they are all series-parallel networks and hence not 3-connected.\n\nThis question is therefore asking whether $AG(3,2)$ is the sole $3$-connected example where equality holds; this is known to be true for all matroids on up to 9 elements.\n\n(A variant of this question would be to ask if $AG(3,2)$ is the only non-graphic example other than trivial modifications like replacing every element with an equally sized parallel class.)\n\nBibliography:\n[LO] Lemos, Manoel; Oxley, James A sharp bound on the size of a connected matroid. Trans. Amer. Math. Soc. 353 (2001), no. 10, 4039--4056 MathSciNet\n\n[W] Wu, Pou-Lin Extremal graphs with prescribed circumference and cocircumference. Discrete Math. 223 (2000), no. 1-3, 299--308 MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1837219\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1782055\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Equality in a matroidal circumference bound\" in Combinatorics; Matroid Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3067, "problem_number": "OPG-696", "title": "Ding's tau_r vs. tau conjecture", "statement": "Conjecture Let $r \\ge 2$ be an integer and let $H$ be a minor minimal clutter with $\\frac{1}{r}\\tau_r(H) < \\tau(H)$. Then either $H$ has a $J_k$ minor for some $k \\ge 2$ or $H$ has Lehman's property.", "background": "Source: Open Problem Garden. Original node ID: 696. URL: http://www.openproblemgarden.org/op/dings_tau_r_vs_tau_conjecture.\n\nSource subject path: Combinatorics > Optimization.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/dings_tau_r_vs_tau_conjecture\n- Author(s): Ding, Guoli\n- Subject(s): Combinatorics; Optimization\n- Keywords: clutter; covering; MFMC property; packing\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 11th, 2007 by mdevos\n\nProblem-page discussion:\nSee Wikipedia's Clutter for definitions of clutter and clutter minors. The clutter $J_k$ is the degenerate projective plane with vertex set $\\{0,1,\\ldots,k\\}$ and edge set $\\{ \\{0,1\\}, \\{1,2\\},\\ldots,\\{0,k\\},\\{0,1,\\ldots,k\\} \\}$. If $H=(V,E)$ is a clutter, then for every positive integer $r$ we let $\\tau_r(H)$ denote the largest multiset of vertices of $H$ which hit every edge at least $r$ times. Note that $\\tau(H) = \\tau_1(H)$ and that $\\tau_r(H) \\le r \\tau(H)$.\n\nWe say that a clutter $H$ with $|V(H)| = n$, $\\tau(H) = s$ and $\\tau(b(H)) = r$ has Lehman's property if $rs > n$, $E(H) = \\{A_1,\\ldots,A_n\\}$, $E(b(H)) = \\{B_1,\\ldots,B_n\\}$, and the following properties are satisfied.\n\n- $|A_i| = r$ for every $1 \\le i \\le n$.\n- $|B_i| = s$ for every $1 \\le i \\le n$.\n- $|A_i \\cap B_i| = rs - n +1$ for $1 \\le i \\le n$\n- $|A_i \\cap B_j| = 1$ if $1 \\le i,j \\le n$ and $i \\neq j$.\n- every $v \\in V(H)$ lies in exactly $r$ edges of $H$, $s$ edges of $b(H)$, and $rs-n+1$ members of $\\{A_1 \\cap B_1, \\ldots,A_n \\cap B_n\\}$.\n\nAlthough the conditions in Lehman's condition are extremely stringent, Lehman [L] showed that every minor minimal clutter with the MFMC property satisfies these properties. Since the MFMC property for $H$ implies $\\frac{1}{r}\\tau_r(H) = \\tau(H)$ (and the degenerate projective planes are minor minimal without MFMC), if true, the above conjecture would be a nice extension of Lehman's theorem.\n\nDing [D] proved this conjecture for $r=2$, but it is open for all other cases.\n\nBibliography:\n*[D] G. Ding, Clutters with tau_2=2 tau, Discrete Math. 115 (1993), no. 1-3, 141--152. MathSciNet.\n\n[L] A. Lehman, On the width-length inequality, mimeographic notes, published 1979. Math. Program. 17, 403--417 MathSciNet.\n\nSource links:\n- clutter: http://en.wikipedia.org/wiki/clutter\n\nDiscussion links:\n- Wikipedia's Clutter: http://en.wikipedia.org/wiki/clutter (mathematics)\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1217624\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0550854\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 32.\n\nAttempt notes:\nTarget:\nMake progress on \"Ding's tau_r vs. tau conjecture\" in Combinatorics; Optimization, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3068, "problem_number": "OPG-59928", "title": "Saturated $k$-Sperner Systems of Minimum Size", "statement": "Question Does there exist a constant $c>1/2$ and a function $n_0(k)$ such that if $|X|\\geq n_0(k)$, then every saturated $k$-Sperner system $\\mathcal{F}\\subseteq \\mathcal{P}(X)$ has cardinality at least $2^{(1+o(1))ck}$?", "background": "Source: Open Problem Garden. Original node ID: 59928. URL: http://www.openproblemgarden.org/op/saturated_k_sperner_systems_of_minimum_size.\n\nSource subject path: Combinatorics > Posets.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/saturated_k_sperner_systems_of_minimum_size\n- Author(s): Morrison, Natasha; Noel, Jonathan A.; Scott, Alex\n- Subject(s): Combinatorics; Posets\n- Keywords: antichain; extremal combinatorics; minimum saturation; saturation; Sperner system\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: April 12th, 2014 by Jon Noel\n\nProblem-page discussion:\nThe power set of a set $X$, denoted $\\mathcal{P}(X)$, is the collection of all subsets of $X$. A collection $\\mathcal{F}\\subseteq\\mathcal{P}(X)$ is said to be a $k$-Sperner system if there does not exist a subcollection $\\{A_1,\\dots,A_{k+1}\\}\\subseteq \\mathcal{F}$ such that $A_1\\subsetneq \\dots\\subsetneq A_{k+1}$; such a subcollection is called a $(k+1)$-chain. A $k$-Sperner system $\\mathcal{F}\\subseteq\\mathcal{P}(X)$ is said to be saturated if for every subset $S$ of $X$ not contained in $\\mathcal{F}$, the collection $\\mathcal{F}\\cup\\{S\\}$ contains a $(k+1)$-chain.\n\nGerbner et al. [1] proved that if $|X|\\geq k$, then every saturated $k$-Sperner System in $\\mathcal{P}(X)$ has cardinality at least $2^{k/2-1}$. Moreover, they conjectured that there exists a function $n_0(k)$ such that if $|X|\\geq n_0(k)$, then the minimum size of a saturated $k$-Sperner System in $\\mathcal{P}(X)$ has size $2^{k-1}$. This was disproved by Morrison, Noel and Scott in [2], who showed the following:\n\nTheorem (Morrison, Noel and Scott (2014)) There exists a constant $\\varepsilon>0$ and a function $n_0(k)$ such that for every $k$ and every set $X$ such that $|X|\\geq n_0(k)$ there exists a saturated $k$-Sperner system in $\\mathcal{P}(X)$ of cardinality at most $2^{(1-\\varepsilon)k}$.\n\nThe value of $\\varepsilon$ which can be deduced from their proof is approximately $\\left(1-\\frac{\\log_2(15)}{4}\\right)\\approx 0.023277$. Moreover, in [2] it was shown that there exists a function $n_0(k)$ and a constant $c\\in [1/2,1-\\varepsilon]$ such that if $|X|\\geq n_0(k)$, then the size of the smallest $k$-Sperner System in $\\mathcal{P}(X)$ is asymptotically $2^{(1+o(1))ck}$. The problem stated here is to determine whether $c>1/2$.\n\nA $1$-Sperner system is called an antichain. As was observed in [2], a positive answer to the above question would follow from a positive answer to the following question:\n\nQuestion (Morrison, Noel, Scott (2014)) Does there exist a constant $c>1/2$ and a function $n_0(k)$ such that if $|X|\\geq n_0(k)$ and $\\mathcal{A}\\subseteq\\mathcal{P}(X)$ is a saturated antichain in which every element of $\\mathcal{A}$ has cardinality between $\\left\\lfloor\\frac{k}{2}\\right\\rfloor$ and $|X|-\\left\\lfloor\\frac{k}{2}\\right\\rfloor +1$, then $|\\mathcal{A}|\\geq 2^{(1+o(1))ck}$?\n\nA more general problem is the following:\n\nQuestion Given integers $a,b$ and a set $X$, what is the minimum size of a saturated antichain $\\mathcal{A}$ in $\\mathcal{P}(X)$ in which every set of $\\mathcal{A}$ has cardinality between $a$ and $|X|-b$?\n\nBibliography:\n[1] D. Gerbner, B. Keszegh, N. Lemons, C. Palmer, D. Palvolgyi, and B. Patkos, Saturating Sperner Families, Graphs Combin. 29 (2013), no. 5, 1355–1364. arXiv:1105.4453\n\n*[2] N. Morrison, J. A. Noel, A. Scott. On Saturated k-Sperner Systems. arXiv:1402.5646 (2014). arXiv:1402.5646\n\nRelated:\nRelated problems\nSaturation in the Hypercube\n\nBibliography links:\n- arXiv:1105.4453: http://www.arxiv.org/abs/1105.4453\n- arXiv:1402.5646: http://www.arxiv.org/abs/1402.5646\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 32.\n\nAttempt notes:\nTarget:\nMake progress on \"Saturated $k$-Sperner Systems of Minimum Size\" in Combinatorics; Posets, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3069, "problem_number": "OPG-351", "title": "Diagonal Ramsey numbers", "statement": "Let $R(k,k)$ denote the $k^{th}$ diagonal Ramsey number.\n\nConjecture $\\lim_{k \\rightarrow \\infty} R(k,k) ^{\\frac{1}{k}}$ exists.\n\nProblem Determine the limit in the above conjecture (assuming it exists).", "background": "Source: Open Problem Garden. Original node ID: 351. URL: http://www.openproblemgarden.org/op/diagonal_ramsey_numbers.\n\nSource subject path: Combinatorics > Ramsey Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/diagonal_ramsey_numbers\n- Author(s): Erdos, Paul\n- Subject(s): Combinatorics; Ramsey Theory\n- Keywords: Ramsey number\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 4th, 2007 by mdevos\n\nProblem-page discussion:\nErdos offered $100 for a solution to the highlighted conjecture and$250 for a solution to the associated problem (these prizes are now provided by Graham).\n\nClassic results of Erdos [E] and Erdos-Szekeres [ESz] give bounds on $R(k,k)$ which show that if $\\lim_{k \\rightarrow \\infty} R(k,k)^{\\frac{1}{k}}$ exists, then it is in the interval $[\\sqrt{2},4]$. Although these arguments are quite basic, little progress has been made in improving these bounds. The best known lower bound on $R(k,k)$ is due to Spencer [S] and the best known upper bound is due to Thomason [T]. They are as follows: $$(1 + o(1)) \\frac{ \\sqrt 2 }{e} k 2 ^{k/2} < R(k,k) < k^{-1/2 + c / \\sqrt{ \\log k}} {2k-2 \\choose k-1}.$$\n\nGowers [G] has suggested that resolving these problems might require a rough structure theorem.\n\nBibliography:\n[E] P. Erdos, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294. MathSciNet\n\n[ESz] P. Erdos and G. Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935), 463–470.\n\n[G] W. T. Gowers, Rough structure and classification, GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal. 2000, Special Volume, Part I, 79--117. MathSciNet\n\n[S] J. Spencer, Ramsey’s theorem—a new lower bound, J. Comb. Theory Ser. A 18 (1975), 108–115. MathSciNet\n\n[T] A. Thomason, An upper bound for some Ramsey numbers, J. Graph Theory 12 (1988), 509–517. MathSciNet\n\nSource links:\n- Ramsey number: http://en.wikipedia.org/wiki/ramsey number\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0019911\n- Rough structure and classification: http://www.dpmms.cam.ac.uk/%7Ewtg10/gafavisions.ps\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1826250\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0366726\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0968746\n\nComments:\n- September 13th, 2010 | Anonymous | diagonal Ramsey numbers: Hi, My name is steve waterman.\n\nre - diagonal Ramsey numbers\n\nI have no proof of my conjectured formulas. However, the results are within the limits established in ALL cases. There is also a logic to these numbers as you will see.\n\nhttp://www.watermanpolyhedron.com/RAMSEY.html\n\nIt is my belief that these values are indeed exact...that is, no bounds required, and thus an answer to this riddle - and as I also see it....only to be proven later. It is a big claim no doubt. I doubt that I will ever see a counter-example nor a single proof of say R(5,5) as long as I live. Lastly, knowing that these MAY INDEED BE the correct values...give us a chance to zero in upon these numbers specifically.\n\nsteve\n- July 9th, 2007 | Anonymous | Upper bound: The upper bound has been improved by David Conlon (to appear in Annals)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Diagonal Ramsey numbers\" in Combinatorics; Ramsey Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3070, "problem_number": "OPG-373", "title": "The large sets conjecture", "statement": "Conjecture If $A$ is 2-large, then $A$ is large.", "background": "Source: Open Problem Garden. Original node ID: 373. URL: http://www.openproblemgarden.org/op/the_large_sets_conjecture.\n\nSource subject path: Combinatorics > Ramsey Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_large_sets_conjecture\n- Author(s): Brown, Tom C.; Graham, Ronald L.; Landman, Bruce M.\n- Subject(s): Combinatorics; Ramsey Theory\n- Keywords: 2-large sets; large sets\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 10th, 2007 by vjungic\n\nProblem-page discussion:\nFor $r\\in \\mathbb{N}$, a set of positive integers $L$ is said to be $r$-large if for any $r$-coloring $f$ of positive integers there are arbitrarily long $f$- monochromatic arithmetic progressions whose common differences belong to $L$. Then $L$ is large if and only if it is $r$-large for all $r$. From Bergelson-Leibman's Polynomial van der Waerden's Theorem [BL] it follows that $\\{ |p(n)|: n \\in \\mathbb{N} \\} \\cap \\mathbb{N}$ is large for any polynomal $p$ with rational coefficients and such that $p(0)=0$.\n\nThe conjecture was stated in 1995 and published in 1999 [BGL].\n\nBibliography:\n[BL] V. Bergelson and A. Leibman, Polynomial extension of van der Waerden’s and Szemer\\'{e}di’s theorems, J. Amer. Math. Soc. 9 (1996) 725-753.\n\n*[BGL] T.C. Brown, R. L. Graham, and B. M. Landman, On the set of common differences in van der Waerden’s theorem on arithmetic progressions, Canadian Math. Bull. 42 (1999) 25-36.\n\n[J] V. Jungic, On Brown’s conjecture on Accessible Sets, J. Comb. Theory, Ser. A 110(1) (2005), 175-178\n\n[LR] B. M. Landman and A. Robertson, Ramsey Theory on the Integers, American Mathematical Society, 2004.\n\nBibliography links:\n- On the set of common differences in van der Waerden’s theorem on arithmetic progressions: http://www.math.ucsd.edu/%7Efan/ron/papers/99_02_common_differences.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"The large sets conjecture\" in Combinatorics; Ramsey Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3071, "problem_number": "OPG-404", "title": "Concavity of van der Waerden numbers", "statement": "For $k$ and $\\ell$ positive integers, the (mixed) van der Waerden number $w(k,\\ell)$ is the least positive integer $n$ such that every (red-blue)-coloring of $[1,n]$ admits either a $k$-term red arithmetic progression or an $\\ell$-term blue arithmetic progression.\n\nConjecture For all $k$ and $\\ell$ with $k \\geq \\ell$, $w(k,\\ell) \\geq w(k+1,\\ell-1)$.", "background": "Source: Open Problem Garden. Original node ID: 404. URL: http://www.openproblemgarden.org/op/concavity_of_van_der_waerden_numbers.\n\nSource subject path: Combinatorics > Ramsey Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/concavity_of_van_der_waerden_numbers\n- Author(s): Landman, Bruce M.\n- Subject(s): Combinatorics; Ramsey Theory\n- Keywords: arithmetic progression; van der Waerden\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 21st, 2007 by Bruce Landman\n\nProblem-page discussion:\nThe conjecture was stated in 2000 and published 2003 [LR] and 2007 [KL].\n\nBibliography:\n*[BL] Bruce Landman and Aaron Robertson, Ramsey Theory on the Integers, American Mathematical Society, Providence, Rhode Island, 2003.\n\n[KL] Abdollah Khodkar and Bruce Landman, Recent progress in Ramsey theory on the integers, in Combinatorial Number Theory, 305-313, de Gruyter, Berlin, 2007.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Concavity of van der Waerden numbers\" in Combinatorics; Ramsey Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3072, "problem_number": "OPG-2359", "title": "Edge-antipodal colorings of cubes", "statement": "We let $Q_d$ denote the $d$-dimensional cube graph. A map $\\phi: E(Q_d) \\rightarrow \\{0,1\\}$ is called edge-antipodal if $\\phi(e) \\neq \\phi(e')$ whenever $e,e'$ are antipodal edges.\n\nConjecture If $d \\ge 2$ and $\\phi: E(Q_d) \\rightarrow \\{0,1\\}$ is edge-antipodal, then there exist a pair of antipodal vertices $v,v' \\in V(Q_d)$ which are joined by a monochromatic path.", "background": "Source: Open Problem Garden. Original node ID: 2359. URL: http://www.openproblemgarden.org/op/edge_antipodal_colorings_of_cubes.\n\nSource subject path: Combinatorics > Ramsey Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/edge_antipodal_colorings_of_cubes\n- Author(s): Norine, Serguei\n- Subject(s): Combinatorics; Ramsey Theory\n- Keywords: antipodal; cube; edge-coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 6th, 2008 by mdevos\n\nProblem-page discussion:\nThis conjecture has been verified by hand for $d \\le 5$.\n\nComments:\n- August 20th, 2010 | Anonymous | 2-colorings of edges of the cube: Q_n denotes the n-dimensional cube. For any x in Q_n, x_bar denotes the antipodal of x in Q_n.\n\nWe conjecture the following: Conj1 Let c:E_n --> {0, 1} be a coloring of the edges of Q_n. Then, there exists a pair of antipodal points x, x_bar and a path p from x to x_bar that it is either monochromatic or it changes colors exactly once.\n\nIt is easy to see that this conjecture implies an affirmative answer to the \"antipodal\" coloring open problem. We have verified that Conj1 holds for dimensions n=2, 3, and 4. We have also found that if the coloring is simple, that is, it does not contain squares colored 0101, then Conj1 holds (in fact, we find a monochromatic path joining a pair of antipodals).\n- May 18th, 2009 | leshabirukov | proof?: Let's suppose G is minimal counterexample. We are denote vertices as \"x1 x2 x3...\" xi={0|1} so, for example, \"110...01\" and \"001...10\" are antipodes. Consider G' is subgraph induced by x1=1. G' is lesser hypercube, so where exists connected pair of G'-antipodes, for clarification, \"100001111\" and \"111110000\". but (\"100001111\",\"000001111\") and (\"111110000\",\"011110000\" ) are edge-antipodes in G, so either (\"100001111\", \"011110000\") or (\"000001111\", \"111110000\") are connected! (And path length is equal to G dimension.)\n\nLooks too simple, am I misunderstood something? Or where is my medal?:))\n- May 18th, 2009 | md | not quite!: the edge-coloring of the subcube consisting of those vertices with x1=1 need not be edge-antipodal.\n- May 19th, 2009 | leshabirukov | Yes, indeed: Got it. edge-antipodes in G' are not antipodes in G, so can have same color. Thanks.\n- May 14th, 2009 | Anonymous | special case proven: We prove the conjecture in the special case where there is no square xyzt in the cube such that xy and zt get value 0, while yz and xt get value 1. The paper by Tomas Feder and Carlos Subi (submitted) can be found at\n\nhttp://theory.stanford.edu/~tomas/antipod.ps\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Edge-antipodal colorings of cubes\" in Combinatorics; Ramsey Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3073, "problem_number": "OPG-357", "title": "A conjecture on iterated circumcentres", "statement": "Conjecture Let $p_1,p_2,p_3,\\ldots$ be a sequence of points in ${\\mathbb R}^d$ with the property that for every $i \\ge d+2$, the points $p_{i-1}, p_{i-2}, \\ldots p_{i-d-1}$ are distinct, lie on a unique sphere, and further, $p_i$ is the center of this sphere. If this sequence is periodic, must its period be $2d+4$?", "background": "Source: Open Problem Garden. Original node ID: 357. URL: http://www.openproblemgarden.org/op/a_conjecture_on_iterated_circumcentres.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_conjecture_on_iterated_circumcentres\n- Author(s): Goddyn, Luis A.\n- Subject(s): Geometry\n- Keywords: periodic; plane geometry; sequence\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: June 8th, 2007 by mdevos\n\nProblem-page discussion:\nLuis Goddyn discovered this curiosity, and proved the above conjecture for $d \\le 5$. He also studied related sequences, for instance, the sequence in ${\\mathbb R}^2$ where the $i^{th}$ point is the circumcentre of the points with index $i-2$, $i-3$, and $i-4$. See Iterated Circumcenters for a delightful and interactive discussion of this problem.\n\nBibliography:\n*[G] Luis Goddyn, Iterated Circumcenters\n\nDiscussion links:\n- Iterated Circumcenters: http://www.math.sfu.ca/%7Egoddyn/Circles\n\nBibliography links:\n- Iterated Circumcenters: http://www.math.sfu.ca/%7Egoddyn/Circles\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"A conjecture on iterated circumcentres\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3074, "problem_number": "OPG-588", "title": "Big Line or Big Clique in Planar Point Sets", "statement": "Let $S$ be a set of points in the plane. Two points $v$ and $w$ in $S$ are visible with respect to $S$ if the line segment between $v$ and $w$ contains no other point in $S$.\n\nConjecture For all integers $k,\\ell\\geq2$ there is an integer $n$ such that every set of at least $n$ points in the plane contains at least $\\ell$ collinear points or $k$ pairwise visible points.", "background": "Source: Open Problem Garden. Original node ID: 588. URL: http://www.openproblemgarden.org/op/big_line_or_big_clique_in_planar_point_sets.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/big_line_or_big_clique_in_planar_point_sets\n- Author(s): Kara, Jan; Por, Attila; Wood, David R.\n- Subject(s): Geometry\n- Keywords: Discrete Geometry; Geometric Ramsey Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: September 25th, 2007 by David Wood\n\nProblem-page discussion:\nThe conjecture is trivial for $\\ell \\leq 3$.\n\nKára et al. [KPW] proved the conjecture for $k \\leq 4$ and all $\\ell$.\n\nAddario-Berry et al. [AFKCW] proved the conjecture for $k=5$ and $\\ell=4$.\n\nAbel et al. [ABBCDHKLPW] proved the conjecture for $k=5$ and all $\\ell$.\n\nThe conjecture is open for $k=6$ or $\\ell=4$.\n\nNote that it is easily proved that for all $k,\\ell\\geq2$, every set of at least $\\Omega(\\ell k^2)$ points in the plane contains $\\ell$ collinear points or $k$ points with no three collinear [Brass].\n\nSee [Matousek] for related results and questions.\n\nBibliography:\n[ABBCDHKLPW] Zachary Abel, Brad Ballinger, Prosenjit Bose, Sébastien Collette, Vida Dujmović, Ferran Hurtado, Scott D. Kominers, Stefan Langerman, Attila Pór, David R. Wood. Every Large Point Set contains Many Collinear Points or an Empty Pentagon, Graphs and Combinatorics 27(1): 47-60, 2011.\n\n[AFKCW] Louigi Addario-Berry, Cristina Fernandes, Yoshiharu Kohayakawa, Jos Coelho de Pina, and Yoshiko Wakabayashi. On a geometric Ramsey-style problem, 2007.\n\n[Brass] Peter Brass. On point sets without k collinear points. In Discrete Geometry, vol. 253 of Monographs and Textbooks in Pure and Applied Mathematics, pp. 185–192. Dekker, New York, 2003.\n\n*[KPW] Jan Kára, Attila Pór, David R. Wood. On the chromatic number of the visibility graph of a set of points in the plane, Discrete and Computational Geometry 34(3):497-506, 2005.\n\n[Matousek] Jiří Matoušek. Blocking visibility for points in general position, Discrete and Computational Geometry 42(2): 219-223, 2009.\n\nBibliography links:\n- Every Large Point Set contains Many Collinear Points or an Empty Pentagon: http://dx.doi.org/10.1007/s00373-010-0957-2\n- On a geometric Ramsey-style problem: http://crm.umontreal.ca/cal/en/mois200708.html\n- On the chromatic number of the visibility graph of a set of points in the plane: http://dx.doi.org/10.1007/s00454-005-1177-z\n- Blocking visibility for points in general position: http://dx.doi.org/10.1007/s00454-009-9185-z\n\nComments:\n- September 25th, 2012 | cibulka | Improved bound in the proof for k=5 and arbitrary l: In their proof of the case $k=5$ and arbitrary $\\ell$, Abel et al. [ABBCDHKLPW] proved a doubly exponential upper bound on the number $p(\\ell)$ of points that guarantees the occurrence of an $\\ell$-tuple of collinear points or a $5$-tuple of points with no other point in their convex hull (an empty pentagon). The upper bound on $p(\\ell)$ was improved by Barát et al. [BDJPSSVW] to $p(\\ell) \\le 328\\ell^2$.\n\n[BDJPSSVW] János Barát, Vida Dujmović, Gwenaël Joret, Michael S. Payne, Ludmila Scharf, Daria Schymura, Pavel Valtr, and David R. Wood. Empty pentagons in point sets with collinearities, arxiv:1207.3633, 2012.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Big Line or Big Clique in Planar Point Sets\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3075, "problem_number": "OPG-605", "title": "Average diameter of a bounded cell of a simple arrangement", "statement": "Conjecture The average diameter of a bounded cell of a simple arrangement defined by $n$ hyperplanes in dimension $d$ is not greater than $d$.", "background": "Source: Open Problem Garden. Original node ID: 605. URL: http://www.openproblemgarden.org/op/average_diameter_of_a_bounded_cell_of_a_simple_arrangement.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/average_diameter_of_a_bounded_cell_of_a_simple_arrangement\n- Author(s): Deza, Antoine; Terlaky, Tamas; Zinchenko, Yuriy\n- Subject(s): Geometry\n- Keywords: arrangement; diameter; polytope\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 30th, 2007 by deza\n\nProblem-page discussion:\nLet $\\mathcal{A}$ be a simple arrangement formed by $n$ hyperplanes in dimension $d$. The number of bounded cells of $\\mathcal{A}$ is $I={n-1\\choose d}$. Let $\\delta(\\mathcal{A})$ denote the average diameter of a bounded cell $P_i$ of $\\mathcal{A}$; that is, $$\\delta(\\mathcal{A})=\\frac{\\sum_{i=1}^{i=I}\\delta(P_i)}{I}.$$Let$\\Delta_{\\mathcal{A}}({d,n})$denote the largest possible average diameter of a bounded cell of a simple arrangement defined by$n$inequalities in dimension$d$.\n\nWe have [DTZ,DX]:\n\nIf the conjecture of Hirsch holds, then $\\Delta_{\\mathcal{A}}(d,n)\\leq d+\\frac{2d}{n-1}$.\n\n$\\Delta_{\\mathcal{A}}({2,n})=2-\\frac{2\\lceil\\frac{n}{2}\\rceil}{(n-1)(n-2)}$ for $n\\geq 4$.\n\n$3-\\frac{6}{n-1}+\\frac{6(\\lfloor\\frac{n}{2}\\rfloor-2)}{(n-1)(n-2)(n-3)}\\leq \\Delta_{{\\mathcal A}}(3,n)\\leq 3 + \\frac{4(2n^2-16n+21)}{3(n-1)(n-2)(n-3)}$ for $n\\geq 5$.\n\n$\\Delta_{{\\mathcal A}}(d,n)\\geq d{n-d \\choose d}/{n-1 \\choose d}$ for $n\\geq 2d$.\n\nBibliography:\n*[DTZ] A. Deza, T. Terlaky and Y. Zinchenko: Polytopes and arrangements: diameter and curvature. Operations Research Letters (to appear).\n\n[DX] A. Deza and F. Xie: Hyperplane arrangements with large average diameter. Centre de Recherches Mathematiques and American Mathematical Society series (to appear).\n\nRelated:\nRelated problems\nHirsch Conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Average diameter of a bounded cell of a simple arrangement\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3076, "problem_number": "OPG-720", "title": "Convex 'Fair' Partitions Of Convex Polygons", "statement": "Basic Question: Given any positive integer n, can any convex polygon be partitioned into n convex pieces so that all pieces have the same area and same perimeter?\n\nDefinitions: Define a Fair Partition of a polygon as a partition of it into a finite number of pieces so that every piece has both the same area and the same perimeter. Further, if all the resulting pieces are convex, call it a Convex Fair Partition.\n\nQuestions: 1. (Rephrasing the above 'basic' question) Given any positive integer n, can any convex polygon be convex fair partitioned into n pieces?\n\n2. If the answer to the above is \"Not always\", how does one decide the possibility of such a partition for a given convex polygon and a given n? And if fair convex partition is allowed by a specific convex polygon for a give n, how does one find the optimal convex fair partition that minimizes the total length of the cut segments?\n\n3. Finally, what could one say about higher dimensional analogs of this question?\n\nConjecture: The authors tend to believe that the answer to the above 'basic' question is \"yes\". In other words they guess: Every convex polygon allows a convex fair partition into n pieces for any n", "background": "Source: Open Problem Garden. Original node ID: 720. URL: http://www.openproblemgarden.org/op/textbf_convex_fair_partitions_of_convex_polygons.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/textbf_convex_fair_partitions_of_convex_polygons\n- Author(s): Nandakumar, R.; Ramana, Rao N.\n- Subject(s): Geometry\n- Keywords: Convex Polygons; Partitioning\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: December 12th, 2007 by Nandakumar\n\nProblem-page discussion:\n1. The above conjecture is easily seen to hold for n=2. for n=3 and above, it is not clear.\n\n2. The n = 2 case does not appear to allow a recursive generalization for values of n equal to powers of 2.\n\n3. It can be shown that any polygon (not necessarily convex) allows a fair partitioning into n pieces for any n, provided the pieces need not be convex (this is not a convex fair partition). See (4) in references below.\n\n4. It appears that the fair parition of a convex polygon which minimizes the total length of cuts (or equivalently, the sum of the perimeters of the pieces) need not be a convex fair partition.\n\n5. There is no known work in this specific area. The problem of partitioning convex polygons into equal area convex pieces so that every piece equally shares the boundary of the input polygon has been studied (references below)\n\nBibliography:\n(*)1. The original 'mainstream' statement of this problem: http://maven.smith.edu/~orourke/TOPP/P67.html#Problem.67\n\n2. Jin Akiyama, A. Kaneko, M. Kano, Gisaku Nakamura, Eduardo Rivera-Campo, S. Tokunaga, and Jorge Urrutia. Radial perfect partitions of convex sets in the plane. In Japan Conf. Discrete Comput. Geom., pages 1-13, 1998.\n\n3. Jin Akiyama, Gisaku Nakamura, Eduardo Rivera-Campo, and Jorge Urrutia. Perfect divisions of a cake. In Proc. Canad. Conf. Comput. Geom., pages 114-115, 1998.\n\n4. This blog maintained by the authors has tentative thoughts, examples, etc on 'Fair Partitions': http://nandacumar.blogspot.com\n\nBibliography links:\n- http://maven.smith.edu/~orourke/TOPP/P67.html#Problem.67: http://maven.smith.edu/%7Eorourke/TOPP/P67.html#Problem.67\n- http://nandacumar.blogspot.com: http://nandacumar.blogspot.com\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Convex 'Fair' Partitions Of Convex Polygons\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3077, "problem_number": "OPG-1761", "title": "Dense rational distance sets in the plane", "statement": "Problem Does there exist a dense set $S \\subseteq {\\mathbb R}^2$ so that all pairwise distances between points in $S$ are rational?", "background": "Source: Open Problem Garden. Original node ID: 1761. URL: http://www.openproblemgarden.org/op/dense_rational_distance_sets_in_the_plane.\n\nSource subject path: Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/dense_rational_distance_sets_in_the_plane\n- Author(s): Ulam, Stanislaw M.\n- Subject(s): Geometry\n- Keywords: integral distance; rational distance\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 4th, 2008 by mdevos\n\nProblem-page discussion:\nThis famous problem was asked by Ulam, who guessed the answer would be negative.\n\nA cute theorem of Erdos shows that if $S \\subseteq {\\mathbb R}^2$ is non-collinear and all pairwise distances between points in $S$ are integral, then $S$ is finite. For the proof, first note that if $x,y \\in {\\mathbb R}^2$ have distance $k \\in {\\mathbb Z}$, then every point which has integer distance to both $x$ and $y$ must lie on one of the $k+1$ hyperbolas consisting of those $z \\in {\\mathbb R}^2$ with $|{\\mathit dist}(x,z) - {\\mathit dist}(y,z)| = j$ for some $0 \\le j \\le k$. So, if all pairwise distances between points in $S$ are integral, and $x,y,z \\in S$ are non-collinear, then every other point in $S$ must lie on an intersection between one of finitely many hyperbola with foci $x,y$ and one of finitely many with foci $x,z$. This set is necessarily finite, thus completing the proof.\n\nOf course, the above argument gives no upper bound on the size of a non-collinear set of points in ${\\mathbb R}^2$ with pairwise integral distances. Indeed, if Ulam's conjecture is true, then there exist such sets of arbitrary size. Surprisingly, it is very difficult to construct such sets $S$ of even rather small size. Recently Kreisel and Kurz [KK] found such a set of size 7, but it is unknown if there exists one of size 8.\n\nIt is trivial to find infinitely many points on a line with all pairwise distances rational. Less trivially, there exist infinite subsets of a circle with all pairwise distances rational. Very recently, Solymosi and De Zeeuw [SZ] proved that these are the only two irreducible algebraic curves with this property. This suggests that, if the answer to Ulam's problem is affirmative, such a set $S$ must be extremely special.\n\nBibliography:\n[KK] T. Kreisel and S. Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete & Computational Geometry, Online first: DOI 10.1007/s00454-007-9038-6\n\n[SZ] J. Solymosi and F. de Zeeuw, On a question of Erdos and Ulam.\n\nBibliography links:\n- On a question of Erdos and Ulam: http://arxiv.org/PS_cache/arxiv/pdf/0806/0806.3095v1.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Dense rational distance sets in the plane\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3078, "problem_number": "OPG-1820", "title": "Simplexity of the n-cube", "statement": "Question What is the minimum cardinality of a decomposition of the $n$-cube into $n$-simplices?", "background": "Source: Open Problem Garden. Original node ID: 1820. URL: http://www.openproblemgarden.org/op/simplexity_of_the_cube.\n\nSource subject path: Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/simplexity_of_the_cube\n- Subject(s): Geometry\n- Keywords: cube; decomposition; simplex\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: August 6th, 2008 by mdevos\n\nProblem-page discussion:\nA decomposition of a polytope $P$ into $n$-simplices is a set of $n$-simplices which have pairwise disjoint interiors and have union equal to $P$. This is also known as a (generalized) triangulation.\n\nLet $T(n)$ be the minimum cardinality of a decomposition of the $n$-cube into $n$-simplices (the answer to our question). It is trivial that $T(1) = 1$ and easy to see that $T(2) = 2$. A 3-dimensional cube may be decomposed into five simplices by cutting off every other corner as shown in the figure (from [JW]). This division is optimal, so $T(3) = 5$.\n\nChopping off every other corner of a 4-cube leaves a 16-cell (the 4-dimensional cross-polytope) which can then be decomposed into eight simplices (fix a vertex $x$ and then take each of the eight 4-simplices formed as the convex hull of $x$ and a facet which is not incident with $x$ ). This is also optimal, so $T(4) = 16$. Computer assisted searches have yielded other good decompositions in low dimensions (see [S]).\n\nThe decompositions of the 3 and 4-dimensional cubes described here do not generalize to higher dimensions. However, there is a naive decomposition of an $n$-cube into $n!$ simplices. Take the cube to be $[0,1]^n$ and let $S$ be the set of all points $(x_1,\\ldots,x_n)$ for which $0 \\le x_1 \\le x_2 \\ldots \\le x_n \\le 1$. Then $S$ is a simplex contained in our cube which contains the main diagonal from the origin to $(1,1,\\ldots,1)$. Further, by permuting the terms $x_1,\\ldots,x_n$ in the chain of inequlities, we get a total of $n!$ simplices which form a decomposition of the cube.\n\nThis naive decomposition is not optimal in dimensions 3 and 4 since our constructions show $T(3) \\le 5 < 3!$ and $T(4) \\le 16 < 4!$. Haiman [H] found a clever way to lift efficient lower dimensional decompositions to high dimensions thus achieving a significant improvement on our $n!$ upper bound. To state his result precisely, we require another parameter. Let $T^*(n)$ be the minimum cardinality of a decomposition of an $n$-cube into $n$-simplices with the following additional constraints:\n\n- Every vertex of a simplex is a vertex of the cube.\n- The intersection of any two simplices is a face of both of them.\n\nIt is immediate that $T(n) \\le T^*(n)$, but to the best of our knowledge these parameters may always be identical. Indeed, this is a separate interesting question. Anyway, back to Haiman's bound. He proved that $T^*(kn) \\le (T^*(n)/n!)^k (kn)!$. Using this inequality with either the 3 or 4-dimensional example from above would give an improvement on the $n!$ upper bound. However, best known is to plug in $T^*(7) \\le 1493$, which gives a general upper bound of $T(7n) \\le T^*(7n) <.840463^{7n}(7n)!$.\n\nA natural lower bound on $T(n)$ can be obtained by a volume argument. Clearly, $T(n)$ must be at least the volume of an $n$-dimensional cube divided by the volume of the largest simplex it contains. Smith [S] improved upon this by moving the argument to hyperbolic space (where the volume of a cube is comparatively much larger than that of a simplex). His volume estimate here yields $T(n) \\ge \\frac{1}{2} \\cdot 6^{n/2}(n+1)^{- \\frac{n+1}{2} } n!$.\n\nBibliography:\n[H] M. Haiman, A simple and relatively efficient triangulation of the n-cube, Discr. Comp. Geom. 6, 4 (1991) 287-289.\n\n[JW] Jackson, Frank and Weisstein, Eric W. \"Tetrahedron.\" From MathWorld--A Wolfram Web Resource.\n\n[S] W. Smith, A lower bound for the simplexity of the n-cube via hyperbolic volume.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 23.\n\nAttempt notes:\nTarget:\nMake progress on \"Simplexity of the n-cube\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3079, "problem_number": "OPG-2089", "title": "Kneser–Poulsen conjecture", "statement": "Conjecture If a finite set of unit balls in $\\mathbb{R}^n$ is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.", "background": "Source: Open Problem Garden. Original node ID: 2089. URL: http://www.openproblemgarden.org/op/kneser_poulsen_conjecture.\n\nSource subject path: Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/kneser_poulsen_conjecture\n- Author(s): Kneser, M.; Poulsen, E. T.\n- Subject(s): Geometry\n- Keywords: pushing disks\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 25th, 2008 by tchow\n\nProblem-page discussion:\nThis problem dates from the mid-1950's. The planar case was solved by Bezdek and Connelly in 2003, who also showed that the area of the intersection does not increase, and that the result holds even if the disks have unequal radii. In higher dimensions the problem remains open.\n\nThe conjecture is known to hold if the rearrangement can be executed by a continuous motion such that the distance between every pair of centers monotonically increases throughout the motion.\n\nBibliography:\n*[BC] K. Bezdek and R. Connelly, Pushing disks apart: The Kneser-Poulsen conjecture in the plane, J. Reine Angew. Math. 553 (2002), 221--236.\n\n*[K] M. Kneser, Einige Bemerkungen über das Minkowskische Flächenmass, Arch. Math. 6 (1955), 382--390.\n\n*[P] E. T. Poulsen, Problem 10, Math. Scand. 2 (1954), 346.\n\nBibliography links:\n- Pushing disks apart: The Kneser-Poulsen conjecture in the plane: http://arxiv.org/abs/math.MG/0108098\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Kneser–Poulsen conjecture\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3080, "problem_number": "OPG-2400", "title": "Erdös-Szekeres conjecture", "statement": "Conjecture Every set of $2^{n-2} + 1$ points in the plane in general position contains a subset of $n$ points which form a convex $n$-gon.", "background": "Source: Open Problem Garden. Original node ID: 2400. URL: http://www.openproblemgarden.org/op/erdos_szekeres_conjecture.\n\nSource subject path: Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/erdos_szekeres_conjecture\n- Author(s): Erdos, Paul; Szekeres, George\n- Subject(s): Geometry\n- Keywords: combinatorial geometry; Convex Polygons; ramsey theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 8th, 2008 by mdevos\n\nProblem-page discussion:\nThis is one of the most famous unsolved problems in combinatorial geometry, perhaps due in part to its lovely history. The problem of showing that every sufficiently large set of points in general position determine a convex $n$-gon was the original inspiration of Esther Klein. Erdös called this the Happy end problem since it led to the marriage of Esther Klein and George Szekeres. This problem was also one of the original sources of Ramsey Theory.\n\nLet $f(n)$ denote the smallest integer so that every set of $f(n)$ points in the plane in general position contains $n$ points which form a convex $n$-gon. The fact that $f(n)$ exists for every $n$ was first established in a seminal paper of Erdös and Szekeres who proved the following bounds on $f(n)$.\n$$\n2^{n-2} + 1 \\le f(n) \\le { 2n-4 \\choose n-2 } + 1\n$$\n\nThe lower bound is conjectured to be the truth, and this is known to hold for $n \\le 5$. A handful of recent papers on this problem have improved the upper bound to\n$$\nf(n) \\le {2n-5 \\choose n-3} + 1.\n$$\n\nDiscussion links:\n- Happy end problem: http://en.wikipedia.org/wiki/Happy end problem\n\nComments:\n- April 14th, 2009 | Anonymous | Now true for n = 6 as well: Now true for n = 6 as well by computer-aided solution of Szekeres and Peters (2006).\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Erdös-Szekeres conjecture\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3081, "problem_number": "OPG-2435", "title": "Monochromatic empty triangles", "statement": "If $X \\subseteq {\\mathbb R}^2$ is a finite set of points which is 2-colored, an empty triangle is a set $T \\subseteq X$ with $|T|=3$ so that the convex hull of $T$ is disjoint from $X \\setminus T$. We say that $T$ is monochromatic if all points in $T$ are the same color.\n\nConjecture There exists a fixed constant $c$ with the following property. If $X \\subseteq {\\mathbb R}^2$ is a set of $n$ points in general position which is 2-colored, then it has $\\ge cn^2$ monochromatic empty triangles.", "background": "Source: Open Problem Garden. Original node ID: 2435. URL: http://www.openproblemgarden.org/op/monochromatic_empty_triangles.\n\nSource subject path: Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/monochromatic_empty_triangles\n- Subject(s): Geometry\n- Keywords: empty triangle; general position; ramsey theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 8th, 2008 by mdevos\n\nProblem-page discussion:\nIt is known that any set of $n$ points in the plane in general position contains $\\ge cn^{5/4}$ monochromatic empty triangles.\n\nComments:\n- September 27th, 2009 | Anonymous | This has a trivial: This has a trivial counterexample for c > 0.\n\nConsider X = {(0,0), (0,1), (1,0)}, colored {red, blue, blue} respectively. There is only one empty triangle in X, and it is not monochromatic. So it has 0 monochromatic empty triangles, and 0 is not > c*(3^2) for c > 0.\n- August 27th, 2010 | Anonymous | Yes indeed. However in this: Yes indeed. However in this types of problems it is generally implied that the statement is for a sufficently large n.\n- May 12th, 2009 | Anonymous | Original source and one improvement.: The conjecture appeared first in \"Oswin Aichholzer, Ruy Fabila-Monroy, David Flores-Peñaloza, Thomas Hackl, Clemens Huemer, and Jorge Urrutia. Empty monochromatic triangles. In Proceedings of the 20th Canadian Conference on Computational Geometry (CCCG2008), pages 75-78, 2008.\"\n\nIn this paper the authors show that any set of n points in general position has $cn^{5/4}$ empty monochromatic triangles. You can get this paper from http://cccg.ca/proceedings/2008/paper18.pdf\n\nThere is one improvement showing that any set of n points in general position has $cn^{4/3}$ empty monochromatic triangles in: \"J. Pach, G. Toth. Monochromatic empty triangles in two-colored point sets. In: Geometry, Games, Graphs and Education: the Joe Malkevitch Festschrift (S. Garfunkel, R. Nath, eds.), COMAP, Bedford, MA, 2008, 195--198.\" Get it from: http://www.math.nyu.edu/~pach/publications/emptytriangle102408.pdf\n- May 8th, 2009 | Anonymous | The lower bound has been improved: The lower bound has been improved to cn4/3.\n\nJ. Pach and G. Toth: Monochromatic empty triangles in two-colored point sets, in: Geometry, Games, Graphs and Education: the Joe Malkevitch Festschrift (S. Garfunkel, R. Nath, eds.), COMAP, Bedford, MA, 2008, 195--198.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Monochromatic empty triangles\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3082, "problem_number": "OPG-36901", "title": "Inequality of the means", "statement": "Question Is is possible to pack $n^n$ rectangular $n$-dimensional boxes each of which has side lengths $a_1,a_2,\\ldots,a_n$ inside an $n$-dimensional cube with side length $a_1 + a_2 + \\ldots a_n$?", "background": "Source: Open Problem Garden. Original node ID: 36901. URL: http://www.openproblemgarden.org/op/inequality_of_the_means.\n\nSource subject path: Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/inequality_of_the_means\n- Subject(s): Geometry\n- Keywords: arithmetic mean; geometric mean; Inequality; packing\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 6th, 2009 by mdevos\n\nProblem-page discussion:\nTaking the arithmetic/geometric mean inequality\n$$\n(a_1 a_2 \\ldots a_n)^{1/n} \\le \\frac{a_1 + a_2 + \\ldots a_n}{n}\n$$\n multiplying both sides by $n$ and then raising both sides to the $n^{th}$ power yields:\n$$\nn^n \\cdot a_1 a_2 \\ldots a_n \\le (a_1 + a_2 + \\ldots a_n)^{n}.\n$$\n So, in the above question, the volume of the cube is at least the sum of the volumes of the rectangular boxes. Furthermore, a positive solution to this question would yield a strengthening of the arithmetic/geometric mean inequality.\n\nFor $n=1$ the problem is trivial, for $n=2$ it is immediate, and for $n=3$ it is tricky, but possible. It is also known that a solution for dimensions $n$ and $m$ can be combined to yield a solution for dimension $nm$. Thus, the question has a positive answer whenever $n$ has the form $2^a 3^b$. It is open for all other values.\n\nSee Bar-Natan's page for more.\n\nBibliography:\n[BCG] E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways for Your Mathematical Plays, Academic Press, New York 1983.\n\nDiscussion links:\n- Bar-Natan's page: http://www.math.toronto.edu/%7Edrorbn/projects/ArithGeom/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Inequality of the means\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3083, "problem_number": "OPG-37084", "title": "Edge-Colouring Geometric Complete Graphs", "statement": "Question What is the minimum number of colours such that every complete geometric graph on $n$ vertices has an edge colouring such that:\n\n\\item[Variant A] crossing edges get distinct colours, \\item[Variant B] disjoint edges get distinct colours, \\item[Variant C] non-disjoint edges get distinct colours, \\item[Variant D] non-crossing edges get distinct colours.", "background": "Source: Open Problem Garden. Original node ID: 37084. URL: http://www.openproblemgarden.org/op/edge_colouring_geometric_complete_graphs.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/edge_colouring_geometric_complete_graphs\n- Author(s): Hurtado, Ferran\n- Subject(s): Geometry\n- Keywords: geometric complete graph, colouring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: October 19th, 2009 by David Wood\n\nProblem-page discussion:\nLet $P$ be a set of $n$ points in the plane with no three collinear. Draw a straight line-segment between each pair of points in $P$. We obtain the complete geometric graph with vertex set $P$, denoted by $K_P$.\n\nTwo edges in $K_P$ are either:\n\n- adjacent if they have a vertex in common,\n- crossing if they intersect at a point in the interior of both edges.\n- disjoint if they do not intersect.\n\nLet $A(n)$, $B(n)$, $C(n)$ and $D(n)$ be the minimum number of colours for the four variants.\n\nVariant A: Here each colour class is a plane subgraph. Since there are point sets for which $\\frac{n}{2}$ edges are pairwise crossing, $A(n)\\geq\\frac{n}{2}$. For an upper bound, say $P=\\{v_1,\\dots,v_n\\}$. Colour each edge $v_iv_j$ with $i0$.\n\nVariant B: Here edges receiving the same colour must intersect. So each colour class is a geometric thrackle. Since there are point sets for which $\\frac{n}{2}$ edges are pairwise disjoint, $B(n)\\geq \\frac{n}{2}$. The $(n-1)$-colouring given in Variant A also works here. So $B(n)\\leq n-1$.\n\nConjecture. $B(n)\\leq (1-\\epsilon)n$ for some $\\epsilon>0$.\n\nVariant C: Here each colour class is a plane matching. So each colour class has at most $\\frac{n}{2}$ edges, and thus at least $n-1$ colours are always needed. Thus $C(n)\\geq n-1$. Araujo [ADHNU] proved an upper bound of $C(n)\\in O(n^{3/2})$.\n\nConjecture. $C(n)\\in O(n\\log n)$.\n\nStrong Conjecture. $C(n)\\in O(n)$.\n\nVariant D: (This variant was recently mentioned in [Mat].) Here edges receiving the same colour must cross. Each colour class is called a crossing family [ADHNU]. Every edge in any triangulation of $P$ requires its own colour. So if the convex hull of $P$ has only three points, then at least $3n-6$ colours are needed. Thus $D(n)\\geq 3n-6$.\n\nConjecture. A super-linear number of colours are always needed; i.e., $\\frac{D(n)}{n}\\rightarrow\\infty$ as $n\\rightarrow\\infty$.\n\nA better lower bound is obtained by taking $P$ in convex position. Then $\\Theta(n\\log n)$ is the minimum number of colours [KK]. I am not aware of any non-trivial upper bound for arbitrary point sets $P$.\n\nBibliography:\n[ADHNU] G. Araujo, A. Dumitrescu, F. Hurtado, M. Noy, J. Urrutia, On the chromatic number of some geometric type Kneser graphs, Computational Geometry: Theory & Applications 32(1):59–69, 2005 MathSciNet\n\n[BHRW] Prosenjit Bose, Ferran Hurtado, Eduardo Rivera-Campo, David R. Wood. Partitions of complete geometric graphs into plane trees, Computational Geometry: Theory & Applications 34(2):116-125, 2006. MathSciNet\n\n[AEGKKPS] B. Aronov, P. Erdos, W. Goddard, D.J. Kleitman, M. Klugerman, J. Pach, L.J. Schulman, Crossing families, Combinatorica 14(2):127–134, 1994. MathSciNet\n\n[KK] Alexandr Kostochka and Jan Kratochvil. Covering and coloring polygon-circle graphs, Discrete Math. 163(1--3):299--305, 1997. MathSciNet\n\n[Mat] Jiří Matoušek. Blocking visibility for points in general position. Discrete Comput. Geom. 42(2):219--223, 2009. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2155418\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2222887\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1289067\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1428585\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2519877\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 31.\n\nAttempt notes:\nTarget:\nMake progress on \"Edge-Colouring Geometric Complete Graphs\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3084, "problem_number": "OPG-37086", "title": "Partition of Complete Geometric Graph into Plane Trees", "statement": "Conjecture Every complete geometric graph with an even number of vertices has a partition of its edge set into plane (i.e. non-crossing) spanning trees.", "background": "Source: Open Problem Garden. Original node ID: 37086. URL: http://www.openproblemgarden.org/op/partition_of_complete_geometric_graph_into_plane_trees.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partition_of_complete_geometric_graph_into_plane_trees\n- Subject(s): Geometry\n- Keywords: complete geometric graph, edge colouring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 19th, 2009 by David Wood\n\nProblem-page discussion:\nFor a set $P$ of $n$ points in the plane with no three collinear, the complete geometric graph $K_P$ has vertex set $P$ and edge set consisting of the $\\binom{n}{2}$ straight line-segments between each pair of points in $P$.\n\nSince each subtree of $K_P$ has at most $n-1$ edges, every partition of $E(K_P)$ into subtrees has at least $\\frac{n}{2}$ parts. The conjecture asks for such a partition into exactly $\\frac{n}{2}$ subtrees, such that in addition, no two edges in each subtree cross.\n\nIt is folklore that the conjecture is true if $P$ is in convex partition. In fact, the edge set of the complete convex graph can be partitioned into plane Hamiltonian paths. Bose et al. [BHRW] characterised all possible partitions of the complete convex graph into plane spanning trees. Bose et al. [BHRW] also proved that every complete geometric graph on $n$ vertices can be partitioned into at most $n-\\sqrt{\\frac{n}{12}}$ plane subtrees.\n\nI heard about this conjecture from Ferran Hurtado in 2003, but the problem is much older than that.\n\nBibliography:\n[BHRW] Prosenjit Bose, Ferran Hurtado, Eduardo Rivera-Campo, David R. Wood. Partitions of complete geometric graphs into plane trees, Computational Geometry: Theory & Applications 34(2):116-125, 2006. MathSciNet\n\nRelated:\nRelated problems\nEdge-Colouring Geometric Complete Graphs\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2222887\n\nComments:\n- January 6th, 2022 | Anonymous | This conjecture is false: This conjecture has recently been disproved, see arXiv:2108.05159 and arXiv:2112.08456.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Partition of Complete Geometric Graph into Plane Trees\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3085, "problem_number": "OPG-37286", "title": "Point sets with no empty pentagon", "statement": "Problem Classify the point sets with no empty pentagon.", "background": "Source: Open Problem Garden. Original node ID: 37286. URL: http://www.openproblemgarden.org/op/point_sets_with_no_empty_pentagon.\n\nSource subject path: Geometry.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/point_sets_with_no_empty_pentagon\n- Author(s): Wood, David R.\n- Subject(s): Geometry\n- Keywords: combinatorial geometry; visibility graph\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: December 14th, 2010 by David Wood\n\nProblem-page discussion:\nLet $P$ be a finite set of points in the plane (not necessarily in general position). Two points $x,y\\in P$ are visible if the line segment $xy$ contains no other point in $P$. The visibility graph of $P$ has vertex set $P$, where two vertices are adjacent if and only if they are visible. An empty pentagon in $P$ consists of 5 points in $P$ that are the vertices of a strictly convex pentagon whose interior contains none of the points in $P$.\n\nConsider the following three closely related classes of point sets:\n\n$A:=$ point sets with no empty pentagon (called a 5-hole),\n$B:=$ point sets with no 5 pairwise visible points,\n$C:=$ point sets whose visibility graph is 4-colourable.\n\nBy definition, $C \\subseteq B \\subseteq A$, and it is easy to show that $A \\neq B$ and $B \\neq C$.\n\nA key example of a point set in $C$ is the planar grid (intersected with a convex set so that it is finite): colour each grid point $(x,y)$ by $(x \\bmod 2, y \\bmod 2)$. If $(x,y)$ and $(v,w)$ receive the same colour then $|x-v|$ and $|y-w|$ are both even, and thus the midpoint of $(x,y)$ and $(v,w)$ is a blocker. Hence the visibility graph of the grid is 4-colourable. [This result and proof is folklore.] Many other examples of point sets in these classes can be found in the references.\n\nConsider the following open problems:\n\n- Classify the point sets in $A$, $B$, or $C$\n(i.e. list all examples; this is easy for point sets with no empty quadrilateral, or no 4 pairwise visible points).\n\n- Does the visibility graph of every point set in $A$ have bounded chromatic number?\n\n- Does the visibility graph of every point set in $A$ have bounded clique number?\n\n- Does the visibility graph of every point set in $B$ have bounded chromatic number?\n\nKára-Pór-Wood gave an example of a point set in $B$ with chromatic number $5$.\n\nBibliography:\nZ. Abel, B. Ballinger, P. Bose, S. Collette, V. Dujmovic, F. Hurtado, S. D. Kominers, S. Langerman, A. Pór, D. R. Wood. Every large point set contains many collinear points or an empty pentagon, Graphs and Combinatorics 27(1):47-60, 2011.\n\nEppstein, David. Happy endings for flip graphs. J. Computational Geometry 1(1):3-28, 2010. MathSciNet\n\nKára, Jan; Pór, Attila; Wood, David R. On the chromatic number of the visibility graph of a set of points in the plane. Discrete Comput. Geom. 34(3):497-506, 2005. MathSciNet\n\nPfender, Florian. Visibility graphs of point sets in the plane. Discrete Comput. Geom. 39 (2008), no. 1-3, 455–459. MathSciNet\n\nRabinowitz, Stanley. Consequences of the pentagon property. Geombinatorics 14:208-220, 2005.\n\nRelated:\nRelated problems\nBig Line or Big Clique in Planar Point Sets\n\nDiscussion links:\n- empty pentagon: http://en.wikipedia.org/wiki/http://en.wikipedia.org/wiki/Happy_Ending_problem\n- colourable: http://en.wikipedia.org/wiki/http://en.wikipedia.org/wiki/Graph_coloring\n\nBibliography links:\n- Every large point set contains many collinear points or an empty pentagon: http://dx.doi.org/10.1007/s00373-010-0957-2\n- Happy endings for flip graphs: http://jocg.org/index.php/jocg/article/view/21\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2469149\n- On the chromatic number of the visibility graph of a set of points in the plane: http://dx.doi.org/10.1007/s00454-005-1177-z\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2160051\n- Visibility graphs of point sets in the plane: http://dx.doi.org/10.1007/s00454-008-9056-z\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2383770\n- Consequences of the pentagon property: http://www.mathpropress.com/stan/bibliography/consequences.pdf\n\nComments:\n- February 1st, 2014 | David Wood | Problem 3 solved: Cibulka, Kyncl and Valtr have solved problem 3. They constructed point sets with no empty pentagon, and whose visibility graph has arbitrarily large clique number.\n\nJosef Cibulka, Jan Kyncl and Pavel Valtr: On planar point sets with the pentagon property, Proc. SoCG 2013, pp. 81-90. http://dl.acm.org/citation.cfm?id=2462406\n- March 25th, 2019 | Leonard Nguyen... | Problem 2 solved: Since the chromatic number is always equal or greater than the clique number, the chromatic number of the visibility graph of sets in A is also unbounded.\n\nCibulka, Kyncl and Valtr have also indirectly solved problem 2.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Point sets with no empty pentagon\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3086, "problem_number": "OPG-37327", "title": "Covering a square with unit squares", "statement": "Conjecture For any integer $n \\geq 1$, it is impossible to cover a square of side greater than $n$ with $n^2+1$ unit squares.", "background": "Source: Open Problem Garden. Original node ID: 37327. URL: http://www.openproblemgarden.org/op/covering_a_square_with_unit_squares.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/covering_a_square_with_unit_squares\n- Subject(s): Geometry\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 18th, 2011 by Martin Erickson\n\nProblem-page discussion:\nAlexander Soifer in [S] raises the question of the smallest number $\\Pi (n)$ of unit squares that can cover a square of side $n+\\epsilon$. He shows the asymptotic upper bound $n^2+o(1)n+O(1)$, and the small values $\\Pi (1)=3$, $5 \\leq \\Pi (2) \\leq 7$, and $10 \\leq \\Pi (3) \\leq 14$. He conjectures the asymptotic lower bound $n^2+O(1)$.\n\nBibliography:\n[S] Soifer, Alexander, \"Covering a square of side n+epsilon with unit squares,\" J. of Combinatorial Theory, Series A 113 (2006):380-383.\n\nComments:\n- August 1st, 2011 | Carolus | A lower bound of the upper bound from polyomino-covering in [S]: (Using $e$ instead of $\\epsilon$ )\n\nIn [S], Soifer derives $\\Pi(n) < (n-k)^2+2(k+1)[[\\frac{k^2-1}{k^2+k-\\sqrt{2k+2}} n]]$.\n\nAs he mentioned, one can improve the covering construction. Holding the square of side length $n-k$ in the lower left corner, putting a square of side length $k$ in the upper right corner, covering the remaining uncovered area by 2 polyomino-coverings of rectangles of sides $n-k+e$ by $k+e$, removing useless unit squares in polyominos, we get a lower bound for the rhs of that inequality:\n\n$(n-k)^2+k^2+2[[(k+1)(n-k)\\frac{k^2-1}{k^2+k-\\sqrt{2k+2}} ]]$\n\nDenote by $U(n)$ the minimal value of this expression when varying $k$ from 2 to $n-2$.\n\nResults of computer calculations:\n\n$U(n) Algebraic Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/jacobian_conjecture\n- Author(s): Keller, Otto-Heinrich\n- Subject(s): Geometry; Algebraic Geometry\n- Keywords: Affine Geometry; Automorphisms; Polynomials\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 6th, 2008 by Charles\n\nProblem-page discussion:\nThe Jacobian determinant is the determinant of the matrix $A$ with $a_{ij}=\\frac{\\partial f_i}{\\partial x_j}$. It is elementary to show that if the map $F:k^n\\to k^n$ is an automorphism, then the Jacobian determinant is a nonzero constant, by using the inverse map. The other direction has turned out to be rather difficult.\n\nIt is known that the Conjecture holds for polynomials of degree 2, and that the general case follows from a special case in degree 3.\n\nComments:\n- March 4th, 2021 | Anonymous | Problem statement: iff the determinant of the Jacobian matrix is a nonzero constant.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Jacobian Conjecture\" in Geometry; Algebraic Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3096, "problem_number": "OPG-1803", "title": "The Hodge Conjecture", "statement": "Conjecture Let $X$ be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of $X$.", "background": "Source: Open Problem Garden. Original node ID: 1803. URL: http://www.openproblemgarden.org/op/the_hodge_conjecture.\n\nSource subject path: Geometry > Algebraic Geometry.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_hodge_conjecture\n- Author(s): Hodge, W. V. D.\n- Subject(s): Geometry; Algebraic Geometry\n- Keywords: Hodge Theory; Millenium Problems\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 13th, 2008 by Charles\n\nProblem-page discussion:\nA complex projective variety is the set of zeros of a finite collection of homogeneous polynomials on projective space, and we are concerned with the singular cohomology ring. There is a well known Hodge Decomposition of the cohomology into groups $H^{p,q}(X.\\mathbb{C})$ which hare holomorphic in $p$ variables and antiholomorphic in $q$ variables with the property that $\\oplus_{p+q=k}H^{p,q}=H^k$.\n\nSo we define the Hodge classes to be those in the intersection $H^{k,k}(X,\\mathbb{C})\\cap H^{2k}(X,\\mathbb{Q})$. It is fairly easy to show that the cohomology class of a subvariety is Hodge. We say that a cycle is algebraic if it is a rational linear combination of the classes of subvarieties. So every algebraic cycle is Hodge. In dimension one, we have the following result:\n\nTheorem (Lefshetz (1,1) Theorem) Any element of $H^2(X,\\mathbb{Q})\\cap H^{1,1}$ is the cohomology class of a divisor, and so is algebraic.\n\nIt's also true that if the Hodge Conjecture holds for cycles of degree $p2n-p$. So this and the (1,1) Theorem show that the Hodge Conjecture is true for complex curves, surfaces and threefolds.\n\nBibliography:\n*[Hod] Hodge, W. V. D. \"The topological invariants of algebraic varieties\". Proceedings of the International Congress of Mathematicians, Cambridge, MA, 1950, vol. 1, pp. 181–192.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"The Hodge Conjecture\" in Geometry; Algebraic Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3097, "problem_number": "OPG-316", "title": "Fat 4-polytopes", "statement": "The fatness of a 4-polytope $P$ is defined to be $(f_1 + f_2)/(f_0 + f_3)$ where $f_i$ is the number of faces of $P$ of dimension $i$.\n\nQuestion Does there exist a fixed constant $c$ so that every convex 4-polytope has fatness at most $c$?", "background": "Source: Open Problem Garden. Original node ID: 316. URL: http://www.openproblemgarden.org/op/fat_4_polytopes.\n\nSource subject path: Geometry > Polytopes.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/fat_4_polytopes\n- Author(s): Eppstein, David; Kuperberg, Greg; Ziegler, Gunter M.\n- Subject(s): Geometry; Polytopes\n- Keywords: f-vector; polytope\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 12th, 2007 by mdevos\n\nProblem-page discussion:\nThe $f$-vector of a $d$-dimensional polytope $P$ is the vector $(f_0,f_1,\\ldots,f_{d-1})$ where $f_i$ is the number of faces of dimension $i$. Let us denote by ${\\mathcal F}_d$ the collection of all $f$-vectors of convex $d$-dimensional polytopes. Steinitz proved that the set ${\\mathcal F}_3$ is completely characterized by the following three conditions:\n\n- $f_0 - f_1 + f_2 = 2$,\n- $f_2 \\le 2f_0 - 4$,\n- $f_0 \\le 2f_2 - 4$.\n\nThe first of these conditions is Euler's formula. The second and third are easy inequalities which are tight for simplicial (all faces triangles) and simple (all vertices of degree 3) polytopes, respectively.\n\nIn sharp contrast to this, the situation for ${\\mathcal F}_4$ seems to be quite complicated. For instance, it has been shown that ${\\mathcal F}_4$ does not contain all elements of ${\\mathbb Z}^4$ which lie in the convex hull of ${\\mathcal F}_4$; i.e., ${\\mathcal F}_4$ has \"holes\" in it. For the extreme examples of simple and simplicial polytopes, the $g$-theorem of Billera-Lee and Stanley gives a complete description of all possible $f$-vectors, but in general very little is known.\n\nSource links:\n- polytope: http://en.wikipedia.org/wiki/polytope\n\nDiscussion links:\n- Euler's formula: http://en.wikipedia.org/wiki/Euler's formula\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Fat 4-polytopes\" in Geometry; Polytopes, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3098, "problem_number": "OPG-610", "title": "Continous analogue of Hirsch conjecture", "statement": "Conjecture The order of the largest total curvature of the primal central path over all polytopes defined by $n$ inequalities in dimension $d$ is $n$.", "background": "Source: Open Problem Garden. Original node ID: 610. URL: http://www.openproblemgarden.org/op/continous_analogue_of_hirsch_conjecture.\n\nSource subject path: Geometry > Polytopes.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/continous_analogue_of_hirsch_conjecture\n- Author(s): Deza, Antoine; Terlaky, Tamas; Zinchenko, Yuriy\n- Subject(s): Geometry; Polytopes\n- Keywords: curvature; polytope\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 30th, 2007 by deza\n\nProblem-page discussion:\nLet $\\lambda^c(P)$ denote the total curvature of the central path corresponding to the linear optimization problem $\\min \\{ c^Tx: x\\in P\\}$. The quantity $\\lambda^c(P)$ can be regarded as the continuous analogue of the edge-length of the shortest path between a pair of vertices. Considering the largest $\\lambda^c(P)$ over all possible $c$ we obtain the quantity $\\lambda(P)$, referred to as the curvature of a polytope. Following the analogy with the diameter, let $\\Lambda(d,n)$ be the largest total curvature $\\lambda(P)$ of the primal central path over all polytopes $P$ defined by $n$ inequalities in dimension $d$.\n\nHolt and Klee~[HK] showed that, for $n> d\\geq 13$, the conjecture of Hirsch is tight. We have the following continuous analogue of the result of Holt and Klee:\n\n[DTZa] $\\liminf_{n\\rightarrow\\infty}\\frac{\\Lambda(d,n)}{n}\\geq \\pi$, that is, $\\Lambda(d,n)$ is bounded below by a constant times $n$.\n\nThe special case of $n=2d$ of the conjecture of Hirsch is known as the $d$-step conjecture, and it has been shown by Klee and Walkup~[KW] that the $d$-step conjecture is equivalent to the Hirsch conjecture. We have the following continuous analogue of the result of Klee and Walkup:\n\n[DTZb] If the order of the curvature is less than the dimension $d$ for all polytope defined by $2d$ inequalities and for all $d$, then the order of the curvature is less that the number of inequalities for all polytopes; that is, if $\\Lambda(d,2d)=\\mathcal{O}(d)$ for all $d$, then $\\Lambda(d,n)=\\mathcal{O}(n)$.\n\nBibliography:\n[DMS] J.-P. Dedieu, G. Malajovich and M. Shub: On the curvature of the central path of linear programming\n\n*[DTZa] A. Deza, T. Terlaky and Y. Zinchenko: Polytopes and arrangements: diameter and curvature. Operations Research Letters (to appear).\n\n[DTZb] A. Deza, T. Terlaky and Y. Zinchenko: The continuous d-step conjecture for polytopes. AdvOL-Report 2007/16, McMaster University (2007).\n\n[HK] F. Holt and V. Klee: Many polytopes meeting the conjectured Hirsch bound. Discrete and Computational Geometry 20 (1998) 1--17.\n\n[KW] V. Klee and D. Walkup: The $d$-step conjecture for polyhedra of dimension $d<6$. Acta Mathematica 133 (1967) 53--78.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Continous analogue of Hirsch conjecture\" in Geometry; Polytopes, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3099, "problem_number": "OPG-778", "title": "Cube-Simplex conjecture", "statement": "Conjecture For every positive integer $k$, there exists an integer $d$ so that every polytope of dimension $\\ge d$ has a $k$-dimensional face which is either a simplex or is combinatorially isomorphic to a $k$-dimensional cube.", "background": "Source: Open Problem Garden. Original node ID: 778. URL: http://www.openproblemgarden.org/op/cube_simplex_conjecture.\n\nSource subject path: Geometry > Polytopes.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cube_simplex_conjecture\n- Author(s): Kalai, Gil\n- Subject(s): Geometry; Polytopes\n- Keywords: cube; facet; polytope; simplex\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 9th, 2008 by mdevos\n\nProblem-page discussion:\nIt is an easy consequence of Euler's formula that every 3-polytope has a face which is either a triangle, a quadrilateral, or a pentagon. The 120-cell is a 4-polytope in which every 2-face is a pentagon (in fact every 3-face is a regular dodecahedron). Perles and Shephard asked whether there exist higher dimensional polytopes in which all 2-faces have at least 5 vertices. This question was answered in the negative by Kalai [K] who showed that every 5-polytope has a 2-face with at most 4 vertices. So, if we define $f(k)$ to be the smallest integer $d$ satisfying the above conjecture for $k$, or $\\infty$ if none exists, then $f(2) = 5$.\n\nThis conjecture is still open for simple polytopes. However, it is known that for every positive integer $k$, there exists an integer $d$ so that every simple polytope of dimension $\\ge d$ either has a 2-dimensional face which is a triangle, or a $k$-dimensional face which is combinatorially isomorphic to a cube. This was proved by Kalai [K] using some earlier results of Nikulin and of Blind and Blind. Actually, something much stronger holds here: simple polytopes of sufficiently high dimension without 2-faces which are triangles must have most $k$-dimensional faces combinatorially isomorphic to the $k$-cube.\n\nThe following is an interesting weakening of the above conjecture.\n\nConjecture For every positive integer $k$, there exists an integer $d$ and a finite list $L$ of $k$-dimensional polytopes, so that every polytope of dimension $\\ge d$ has a $k$-dimensional face which appears in $L$.\n\nDefining $h(k)$ to be the smallest integer $d$ satisfying this conjecture for $k$, or $\\infty$ if none exists, we find that $h(2) = 3$ (by the consequence of Euler's formula in the first paragraph). Meisinger, Kleinschmidt, and Kalai [MKK] proved that $h(3) \\le 9$ with the help of FLAGTOOL, a computer program which can compute linear relations for $f$-vectors. This weaker conjecture is known to be true for simple polytopes.\n\nBibliography:\n[MKK] G. Meisinger, P. Kleinschmidt, and G. Kalai, Three theorems, with computer-aided proofs, on three-dimensional faces and quotients of polytopes. The Branko Grünbaum birthday issue. Discrete Comput. Geom. 24 (2000), no. 2-3, 413--420. MathSciNet\n\n*[K] G. Kalai, On low-dimensional faces that high-dimensional polytopes must have. Combinatorica 10 (1990), no. 3, 271--280. MathSciNet\n\nDiscussion links:\n- 120-cell: http://en.wikipedia.org/wiki/120-cell\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1758060\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1092544\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Cube-Simplex conjecture\" in Geometry; Polytopes, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3100, "problem_number": "OPG-37341", "title": "Extension complexity of (convex) polygons", "statement": "The extension complexity of a polytope $P$ is the minimum number $q$ for which there exists a polytope $Q$ with $q$ facets and an affine mapping $\\pi$ with $\\pi(Q) = P$.\n\nQuestion Does there exists, for infinitely many integers $n$, a convex polygon on $n$ vertices whose extension complexity is $\\Omega(n)$?", "background": "Source: Open Problem Garden. Original node ID: 37341. URL: http://www.openproblemgarden.org/op/extension_complexity_of_convex_polygons.\n\nSource subject path: Geometry > Polytopes.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/extension_complexity_of_convex_polygons\n- Subject(s): Geometry; Polytopes\n- Keywords: polytope, projection, extension complexity, convex polygon\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 13th, 2011 by DOT\n\nProblem-page discussion:\nThe extension complexity of a polytope is bounded from above by its number of vertices. Thus, a convex polygon with $n$ vertices has extension complexity $O(n)$.\n\nSome regular convex polygons have extension complexity $O(\\log n)$ [BTN].\n\nA convex polygon whose points are drawn randomly on a circle has extension complexity $\\Omega(\\sqrt n)$ with probability one (follows from [FRT]).\n\nThe question asks for the maximal extension complexity of a convex polygon.\n\nA strongly related question is the following.\n\nQuestion What is the extension complexity of an $n$-vertex convex polygon whose vertices are drawn randomly on a circle?\n\nBibliography:\n*[BTN] Ben-Tal, A and Nemirovski, A. On polyhedral approximations of the second-order cone. Math. Oper. Res. 26:2 193-205 (2001)\n\n[FRT] Fiorini, S. and Rothvoss, T. and Tiwary, H.R. Extended formulations of polygons. arXiv:1107.0371\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Extension complexity of (convex) polygons\" in Geometry; Polytopes, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3101, "problem_number": "OPG-37459", "title": "Durer's Conjecture", "statement": "Conjecture Every convex polytope has a non-overlapping edge unfolding.", "background": "Source: Open Problem Garden. Original node ID: 37459. URL: http://www.openproblemgarden.org/op/d_urers_conjecture.\n\nSource subject path: Geometry > Polytopes.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/d_urers_conjecture\n- Author(s): Durer, A.; Shephard, G.C.\n- Subject(s): Geometry; Polytopes\n- Keywords: folding; polytope\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: June 4th, 2012 by dmoskovich\n\nProblem-page discussion:\nIn 1525, Albrecht Dürer represented polytopes by cutting them open along edges and then flattening the surface onto the plane, without overlaps and without distorting the individual faces. Self-intersections are allowed during the unfolding process, but the final flattened surface must be free of overlaps. Whether a non-overlapping edge unfolding, as defined above, is possible for any convex polytopes was formulated by Shephard as a conjecture in 1975.\n\nBibliography:\n[D] A. Dürer, Unterweysung der Messung mit dem Zyrkel und Rychtscheyd. Nürnberg (1525). English translation with commentary by Walter L. Strauss The Painter's Manual, New York (1977).\n\n[O] J. O'Rourke, How to fold it, Cambridge University Press, 2011, Book website\n\n[P] K. Polthier Imagining maths- unfolding polyhedra\n\n*[S] G.C. Shephard, Convex Polytopes with Convex Nets. Math. Proc. Camb. Phil. Soc., 78:389-403 (1975).\n\nBibliography links:\n- Book website: http://howtofoldit.org/\n- Imagining maths- unfolding polyhedra: http://plus.maths.org/content/os/issue27/features/mathart/index/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Durer's Conjecture\" in Geometry; Polytopes, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3102, "problem_number": "OPG-586", "title": "Pebbling a cartesian product", "statement": "We let $p(G)$ denote the pebbling number of a graph $G$.\n\nConjecture $p(G_1 \\Box G_2) \\le p(G_1) p(G_2)$.", "background": "Source: Open Problem Garden. Original node ID: 586. URL: http://www.openproblemgarden.org/op/pebbling_a_cartesian_product.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/pebbling_a_cartesian_product\n- Author(s): Graham, Ronald L.\n- Subject(s): Graph Theory\n- Keywords: pebbling; zero sum\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 24th, 2007 by mdevos\n\nProblem-page discussion:\nThe pebbling number of a graph $G$, denoted $p(G)$, is the smallest integer $k$ so that however $k$ pebbles are distributed onto the vertices of $G$, it is possible to move a pebble to any vertex by a sequence of steps, where in each step we remove two pebbles from one vertex, and then place one on an adjacent vertex. The cartesian product of two graphs $G_1$ and $G_2$, denoted $G_1 \\Box G_2$, is the graph with vertex set $V(G_1) \\times V(G_2)$ and an edge from $(v,w)$ to $(v',w')$ if either $v=v'$ and $w \\sim w'$ (in $G_2$ ) or $w=w'$ and $v \\sim v'$ (in $G_1$ ).\n\nGraph Pebbling arose out of the study of zero-sum subsequences in groups, but has proved to be a rich and interesting topic in graph theory. See Glenn Hurlbert's wonderful graph pebbling page for much more on this topic (and this problem in particular). Graham's conjecture was stated in one of the first papers on this subject by Fan Chung [C], and has since generated considerable interest.\n\nThere are a number of partial results toward this conjecture. Chung [C] proved that $p(P_{d_1+1} \\Box P_{d_2+1} \\ldots \\Box P_{d_{\\ell}+1}) = 2^{d_1 + d_2 \\ldots + d_{\\ell}}$, thus settling the conjecture for products of paths (here $P_n$ denotes a path with $n$ vertices). It is also known when $G_1,G_2$ are both trees, both cycles, and for graphs with high minimum degree. Again, we encourage the interested reader to see the graph pebbling page for more details.\n\nBibliography:\n*[C] F. Chung, Pebbling in hypercubes SIAM J. Disc. Math. 2 (1989), 467--472.\n\nDiscussion links:\n- graph pebbling page: http://mingus.la.asu.edu/%7Ehurlbert/pebbling/pebb.html\n- the graph pebbling page: http://mingus.la.asu.edu/%7Ehurlbert/pebbling/pebb.html\n\nBibliography links:\n- Pebbling in hypercubes: http://mingus.la.asu.edu/%7Ehurlbert/pebbling/papers/Chun_PIH.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Pebbling a cartesian product\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3103, "problem_number": "OPG-658", "title": "Reconstruction conjecture", "statement": "The deck of a graph $G$ is the multiset consisting of all unlabelled subgraphs obtained from $G$ by deleting a vertex in all possible ways (counted according to multiplicity).\n\nConjecture If two graphs on $\\ge 3$ vertices have the same deck, then they are isomorphic.", "background": "Source: Open Problem Garden. Original node ID: 658. URL: http://www.openproblemgarden.org/op/reconstruction_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/reconstruction_conjecture\n- Author(s): Kelly, Paul J.; Ulam, Stanislaw M.\n- Subject(s): Graph Theory\n- Keywords: reconstruction\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 18th, 2007 by zitterbewegung\n\nProblem-page discussion:\nSee Wikipedia's Reconstruction Conjecture for more on this problem.\n\nBibliography:\n*[K] P. J. Kelly, A congruence theorem for trees, Pacific J. Math., 7 (1957), 961–968.\n\n*[U] S. M. Ulam, A collection of mathematical problems, Wiley, New York, 1960.\n\nDiscussion links:\n- Reconstruction Conjecture: http://en.wikipedia.org/wiki/Reconstruction_conjecture\n\nComments:\n- May 4th, 2010 | Anonymous | A strategy for simple graphs?: How about 1) Prove for a 4-node, or tetrahedral graph. 2) Demonstrate that all graphs with > 4 nodes are composed of multiple overlapping tetrahedrons 3) Figure out how coupled tetrahedra function when nodes are deleted. 4) Induct on number of tetrahedra in graph.?\n\nObviously (3) is the tough part, but why should it be impossible?\n- May 23rd, 2008 | melch | correction and partial results: this should be on 3 or more vertices. False for digraphs, hypergraphs, and infinite graphs. It is open for simple graphs and multigraphs\n- November 28th, 2007 | Anonymous | Question.: Does G have to be simple, or can it be a multigraph?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Reconstruction conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3104, "problem_number": "OPG-804", "title": "Edge Reconstruction Conjecture", "statement": "Conjecture\n\nEvery simple graph with at least 4 edges is reconstructible from it's edge deleted subgraphs", "background": "Source: Open Problem Garden. Original node ID: 804. URL: http://www.openproblemgarden.org/op/edge_reconstruction_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/edge_reconstruction_conjecture\n- Author(s): Harary, Frank\n- Subject(s): Graph Theory\n- Keywords: reconstruction\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 23rd, 2008 by melch\n\nProblem-page discussion:\nIt is known that if a graph is vertex reconstructible then it is edge reconstructible.\n\nBibliography:\nJ.A.Bondy, A graph reconstruction manual, Surveys in Combinatorics, LMS-Lecture Note Series 166(1991)\n\nRelated:\nRelated problems\nReconstruction conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Edge Reconstruction Conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3105, "problem_number": "OPG-34908", "title": "Book Thickness of Subdivisions", "statement": "Let $G$ be a finite undirected simple graph.\n\nA $k$-page book embedding of $G$ consists of a linear order $\\preceq$ of $V(G)$ and a (non-proper) $k$-colouring of $E(G)$ such that edges with the same colour do not cross with respect to $\\preceq$. That is, if $v\\prec x\\prec w\\prec y$ for some edges $vw,xy\\in E(G)$, then $vw$ and $xy$ receive distinct colours.\n\nOne can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.\n\nThe book thickness of $G$, denoted by bt $(G)$ is the minimum integer $k$ for which there is a $k$-page book embedding of $G$.\n\nLet $G'$ be the graph obtained by subdividing each edge of $G$ exactly once.\n\nConjecture There is a function $f$ such that for every graph $G$, $$\\text{bt}(G) \\leq f( \\text{bt}(G') )\\enspace.$$", "background": "Source: Open Problem Garden. Original node ID: 34908. URL: http://www.openproblemgarden.org/op/book_thickness_of_subdivisions.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/book_thickness_of_subdivisions\n- Author(s): Blankenship, Robin; Oporowski, Bogdan\n- Subject(s): Graph Theory\n- Keywords: book embedding; book thickness\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: January 19th, 2009 by David Wood\n\nProblem-page discussion:\nThe conjecture is due to [B099]. The conjecture is true for complete graphs [BO99,EM99,E02]. The conjecture is discussed in depth in [DW05].\n\nBibliography:\n*[BO99] Robin Blankenship and Bogdan Oporowski. Drawing Subdivisions Of Complete And Complete Bipartite Graphs On Books, Technical Report 1999-4, Department of Mathematics, Louisiana State University, 1999.\n\n[DW05] Vida Dujmovic and David Wood. Stacks, queues and tracks: Layouts of graph subdivisions. Discrete Mathematics & Theoretical Computer Science 7:155-202, 2005.\n\n[EM99] Hikoe Enomoto and Miki Shimabara Miyauchi. Embedding graphs into a three page book with $O(M \\log N)$ crossings of edges over the spine. SIAM J. Discrete Math., 12(3):337–341, 1999.\n\n[E02] David Eppstein. Separating thickness from geometric thickness. In Proc. 10th International Symp. on Graph Drawing (GD ’02), pp. 150–161. vol. 2528 of Lecture Notes in Comput. Sci. Springer, 2002.\n\nBibliography links:\n- Stacks, queues and tracks: Layouts of graph subdivisions: http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/67\n- Embedding graphs into a three page book with $O(M \\log N)$ crossings of edges over the spine: http://dx.doi.org/10.1137/S0895480195280319\n\nComments:\n- July 10th, 2021 | Anonymous | This problem is solved: This problem is solved in:\n\nV. Dujmović, D. Eppstein, R. Hickingbotham, P. Morin, D. R. Wood. Stack-number is not bounded by queue-number. Combinatorica, accepted in 2021 https://arxiv.org/abs/2011.04195\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Book Thickness of Subdivisions\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3106, "problem_number": "OPG-36879", "title": "Shannon capacity of the seven-cycle", "statement": "Problem What is the Shannon capacity of $C_7$?", "background": "Source: Open Problem Garden. Original node ID: 36879. URL: http://www.openproblemgarden.org/op/shannon_capacity_of_the_seven_cycle.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/shannon_capacity_of_the_seven_cycle\n- Subject(s): Graph Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 19th, 2009 by tchow\n\nProblem-page discussion:\nLet $\\alpha(G)$ denote the independence number of the graph $G$, and let $G*H$ denote the strong graph product of $G$ and $H$ (in which $(g,h)$ is adjacent to $(g',h')$ if $g=g'$ and $h$ is adjacent to $h'$, or if $h=h'$ and $g$ is adjacent to $g'$, or if $g$ is adjacent to $g'$ and $h$ is adjacent to $h'$ ). Then the Shannon capacity of $G$ is defined by $$\\theta(G) = \\lim_{k\\to\\infty} \\biggl({\\alpha(G*G*\\cdots*G) \\over k}\\biggr)^{1/k},$$where the strong graph product is over$k$copies of$G$. The Shannon capacity is important because it represents the effective size of an alphabet in a communication model represented by$G$, but it is notoriously difficult to compute. Lovász [L] famously proved that the Shannon capacity of the five-cycle$C_5$is$\\sqrt{5}$, but even the Shannon capacity of$C_7$remains unknown. However, Bohman [B] has shown that$$\\lim_{k\\to\\infty}(k+(1/2)-\\theta(C_{2k+1}))=0.$$\n\nBibliography:\n[B] Tom Bohman, A limit theorem for the Shannon capacity of odd cycles II, Proc. Amer. Math. Soc. 133 (2005), no. 2, 537-543.\n\n[L] László Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Th. IT-25 (1979), 1-7.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 24.\n\nAttempt notes:\nTarget:\nMake progress on \"Shannon capacity of the seven-cycle\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3107, "problem_number": "OPG-37081", "title": "Number of Cliques in Minor-Closed Classes", "statement": "Question Is there a constant $c$ such that every $n$-vertex $K_t$-minor-free graph has at most $c^tn$ cliques?", "background": "Source: Open Problem Garden. Original node ID: 37081. URL: http://www.openproblemgarden.org/op/number_of_cliques_in_minor_closed_classes.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/number_of_cliques_in_minor_closed_classes\n- Author(s): Wood, David R.\n- Subject(s): Graph Theory\n- Keywords: clique; graph; minor\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 12th, 2009 by David Wood\n\nProblem-page discussion:\nHere a clique is a (not neccessarily maximal) set of pairwise adjacent vertices in a graph.\n\nSee [RW, NSTW] for early bounds on the number of cliques. Wood [W] proved that the number of cliques in an $n$-vertex $K_t$-minor-free graph is at most $c^{t\\sqrt{\\log t}}n\\enspace.$ Fomin et al. [FOT] improved this bound to $c^{t\\log\\log t}n\\enspace.$\n\nThese results are based on the fact that every $n$-vertex $K_t$-minor-free graph has at most $ct\\sqrt{\\log t}n$ edges. This bound is tight for certain random graphs. So it is reasonable to expect that random graphs might also provide good lower bounds on the number of cliques.\n\nUpdate 2014: Choongbum Lee and Sang-il Oum [LO] recently answered this question in the affirmative, and even proved it for excluded subdivisions. In particular, they proved that every $n$-vertex graph with no $K_t$-subdivision has at most $2^{474t}n$ cliques and also at most $2^{14t+o(t)}n$ cliques.\n\nThe question now is to determine the minimum constant. Wood [W] proved a lower bound of $3^{2t/3-o(t)}n$ using an appropriate sized complete graph minus a perfect matching. The same graph gives a lower bound of $3^{s-o(s)}n$ on the number of cliques in a graph with no $K_s$ subdivision.\n\nUpdate (2019): Fox and Wei [FW] have proved that every graph on $n$ vertices with no $K_t$-minor has at most $3^{2t/3+o(t)}n$ cliques. This bound is tight for $n \\geq 4t/3$.\n\nBibliography:\n[FOT] Fedor V. Fomin, Sang il Oum, and Dimitrios M. Thilikos. Rank-width and tree-width of $H$-minor-free graphs, European J. Combin. 31 (7), 1617–1628, 2010.\n\n[NSTW] Serguei Norine, Paul Seymour, Robin Thomas, Paul Wollan. Proper minor-closed families are small. J. Combin. Theory Ser. B, 96(5):754--757, 2006.\n\n[RW] Bruce Reed and David R. Wood. Fast separation in a graph with an excluded minor. In 2005 European Conf. on Combinatorics, Graph Theory and Applications (EuroComb '05), vol. AE of Discrete Math. Theor. Comput. Sci. Proceedings, pp. 45--50. 2005.\n\n* [W] David R. Wood. On the maximum number of cliques in a graph. Graphs Combin., 23(3):337--352, 2007.\n\n[LO] Choongbum Lee and Sang-il Oum. Number of cliques in graphs with forbidden minor, 2014.\n\n[FW] Jacob Fox, Fan Wei. On the number of cliques in graphs with a forbidden minor\n\nBibliography links:\n- Rank-width and tree-width of $H$-minor-free graphs: http://www.arxiv.org/abs/math.CO/0910.0079\n- On the maximum number of cliques in a graph: http://www.arxiv.org/abs/math.CO/0602191\n- Number of cliques in graphs with forbidden minor: http://www.arxiv.org/abs/1407.7707\n- On the number of cliques in graphs with a forbidden minor: http://www.arxiv.org/abs/1603.07056\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Number of Cliques in Minor-Closed Classes\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3108, "problem_number": "OPG-37089", "title": "Shuffle-Exchange Conjecture (graph-theoretic form)", "statement": "Given integers $k,n \\ge 2$, the 2-stage Shuffle-Exchange graph/network, denoted $\\text{SE}(k,n)$, is the simple $k$-regular bipartite graph with the ordered pair $(U,V)$ of linearly labeled parts $U:=\\{u_0,\\dots,u_{t-1}\\}$ and $V:=\\{v_0,\\dots,v_{t-1}\\}$, where $t:=k^{n-1}$, such that vertices $u_i$ and $v_j$ are adjacent if and only if $(j - ki) \\text{ mod } t < k$ (see Fig.1).\n\nGiven integers $k,n,r \\ge 2$, the $r$-stage Shuffle-Exchange graph/network, denoted $(\\text{SE}(k,n))^{r-1}$, is the proper (i.e., respecting all the orders) concatenation of $r-1$ identical copies of $\\text{SE}(k,n)$ (see Fig.1).\n\nLet $r(k,n)$ be the smallest integer $r\\ge 2$ such that the graph $(\\text{SE}(k,n))^{r-1}$ is rearrangeable.\n\nProblem Find $r(k,n)$.\n\nConjecture $r(k,n)=2n-1$.", "background": "Source: Open Problem Garden. Original node ID: 37089. URL: http://www.openproblemgarden.org/op/shuffle_exchange_conjecture_graph_theoretic_form.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/shuffle_exchange_conjecture_graph_theoretic_form\n- Author(s): Beneš, Václav E.; Folklore; Stone, Harold S.\n- Subject(s): Graph Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 30th, 2009 by Vadim Lioubimov\n\nProblem-page discussion:\nA mask for the graph $G:=(\\text{SE}(k,n))^{r-1}$ is a $k$-regular bipartite multigraph with the bipartition $\\{U,V\\}$. The graph $G$ is said to be rearrangeable if for every its mask there exists a collection, called routing, of corresponding mutually edge-disjoint paths in $G$ connecting its end parts. (For simplicity, we do not provide here a more general definition for rearrangeability of graphs.)\n\nNote that $G$ is a simple $r$-partite graph with $r k^{n-1}$ vertices and $(r-1)k^{n}$ edges, and any route for it consists exactly of $k^{n}$ paths. Also, $r(k,n)\\le r$ is equivalent to rearrangeability of $G$.\n\nFigure 1. Examples of multistage Shuffle-Exchange graphs.\n\nFor example, according to the conjecture, the graph $(\\text{SE}(2,3))^{4}$ (see Fig. 1) is rearrangeable, which is a well known result.\n\nThe problem and conjecture are equivalent \"graph-theoretic\" forms of remarkable Shuffle-Exchange (SE) problem and conjecture due to the following identity (that is not hard to show by normal reasoning):\n\nTheorem $r(k,n)=d(k,n)$.\n\nThe definition of $d(k,n)$ and more on SE problem/conjecture including the other 2 main forms of them, combinatorial and group-theoretic, and a survey of results can be found here.\n\nBibliography:\n*[S71] H.S. Stone, Parallel processing with the perfect shuffle, IEEE Trans. on Computers C-20 (1971), 153-161.\n\n*[B75] V.E. Beneš, Proving the rearrangeability of connecting networks by group calculation, Bell Syst. Tech. J. 54 (1975), 421-434.\n\nRelated:\nRelated problems\nShuffle-Exchange Conjecture\nBeneš Conjecture (graph-theoretic form)\nBeneš Conjecture\n\nDiscussion links:\n- Shuffle-Exchange (SE) problem and conjecture: http://www.openproblemgarden.org/?q=node/37167\n- here: http://www.openproblemgarden.org/?q=node/37167\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 27.\n\nAttempt notes:\nTarget:\nMake progress on \"Shuffle-Exchange Conjecture (graph-theoretic form)\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3109, "problem_number": "OPG-37182", "title": "Odd cycles and low oddness", "statement": "Conjecture If in a bridgeless cubic graph $G$ the cycles of any $2$-factor are odd, then $\\omega(G)\\leq 2$, where $\\omega(G)$ denotes the oddness of the graph $G$, that is, the minimum number of odd cycles in a $2$-factor of $G$.", "background": "Source: Open Problem Garden. Original node ID: 37182. URL: http://www.openproblemgarden.org/op/odd_cycles_and_low_oddness.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/odd_cycles_and_low_oddness\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: January 15th, 2010 by Gagik\n\nComments:\n- June 29th, 2010 | Anonymous | This conjecture is false.: For odd $n$ and $i\\in[n]$, let $H_i$ be the graph obtained by deleting an edge, say $x_iy_i$, from the Petersen graph. Define $G_n$ to be the graph obtained by joining vertex $y_i$ and vertex $x_{i+1}$ with an edge (with subscripts reduced modulo $n$ ). For each $i\\in[n]$, the set $\\{y_{i-1}x_{i},y_ix_{i+1}\\}$ is an edge cut. Hence, in any 2-factor of $G_n$, either none of the edges of the form $y_ix_{i+1}$ are contained in a cycle, or all of them are contained in the same cycle.\n\nCase 1: If none of the edges described above are contained in a cycle of a 2-factor of $G_n$, then this 2-factor contains a 2-factor of $H_i$ for each $i$. These 2-factors are also 2-factors of the graphs $H_i+x_iy_i$, that is, each is a 2-factor of the Petersen graph. Each 2-factor of the Petersen graph consists of two cycles of 5 vertices, hence, any such 2-factor of $G_n$ contains $2n$ cycles of odd length.\n\nCase 2: See the comment below.\n- July 2nd, 2010 | Anonymous | This conecture is false.: Case 2: If all of the edges of the form $y_ix_{i+1}$ are contained in the same cycle in a 2-factor of $G_n$, then replacing the edges $y_ix_{i+1}$ with the edges $x_iy_i$ converts this 2-factor of $G_n$ into a 2-factor of $n$ disjoint copies of the Petersen graph. Hence, when restricted to each $H_i$, the 2-factor of $G_n$ consists of a cycle with 5 vertices and a $x_iy_i$-path containing a total of 5 vertices. These paths must be joined together through the edges of the form $y_ix_{i+1}$ creating a cycle of length 5n. Hence, in this case the 2-factor of $G_n$ contains $n$ cycles of length 5 and one cycle of length 5n (which is odd).\n\nNow, $G_n$ is a bridgeless cubic graph whose 2-factors contain only odd cycles, but no 2-factor of $G_n$ contains fewer than $n+1$ cycles.\n- January 16th, 2010 | Anonymous | what is oddness: The notion of oddness of a graph requires explanation.\n- December 13th, 2011 | Anonymous | Let be a bridgeless cubic: Let $G$ be a bridgeless cubic graph. The oddness of a 2-factor $F$ is the number of odd circuits of $F$. The oddness of $G$ is the smallest oddness over all 2-factors. For example, a 3-edge-colorable cubic graph has oddness zero and the Petersen graph has oddness two.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Odd cycles and low oddness\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3110, "problem_number": "OPG-37210", "title": "Beneš Conjecture (graph-theoretic form)", "statement": "Problem ( $\\dag$ ) Find a sufficient condition for a straight $\\ell$-stage graph to be rearrangeable. In particular, what about a straight uniform graph?\n\nConjecture ( $\\diamond$ ) Let $L$ be a simple regular ordered $2$-stage graph. Suppose that the graph $L^m$ is externally connected, for some $m\\ge1$. Then the graph $L^{2m}$ is rearrangeable.", "background": "Source: Open Problem Garden. Original node ID: 37210. URL: http://www.openproblemgarden.org/op/bene_conjecture_graph_theoretic_form_0.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/bene_conjecture_graph_theoretic_form_0\n- Author(s): Beneš, Václav E.\n- Subject(s): Graph Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 17th, 2010 by Vadim Lioubimov\n\nProblem-page discussion:\nGiven an integer $\\ell\\ge2$, an $\\ell$-stage graph is an $\\ell$-partite graph $G$ with a list of its parts $V_1,\\dots,V_{\\ell}$ such that every edge of $G$ has endpoints in both $V_i$ and $V_{i+1}$, for some $i\\in[\\ell-1]$. A vertex in $V_1$ ( $V_\\ell$ ) is a source (target) of $G$. A path in $G$ is plain if it goes from a source to a target through each part of $G$ exactly once. The graph $G$ is externally connected if for every source $s$ and target $t$ there exists a plain path from $s$ to $t$. A mask for $G$ is a $2$-stage multigraph $M$ whose sources and targets are exactly those of $G$ and such that every vertex of $M$ has the same degree in $G$. The graph $G$ is rearrangeable if for every its mask there exists a collection, called routing, of corresponding mutually edge-disjoint plain paths in $G$.\n\nThe graph $G$ is ordered if each of its parts is linearly ordered. The graph $G$ is uniform and denoted $B^{\\ell-1}$ if there is an ordered 2-stage graph $B$ with equal-sized parts such that $G$ is the proper (i.e., respecting all the orders in $B$ ) concatenation of $\\ell-1$ identical copies of $B$. The graph $G$ is straight if for any $2\\le i\\le\\ell-1$ and any $v\\in V_i$, the number of edges joining $v$ with $V_{i-1}$ equals that of $V_{i+1}$.\n\nConjecture ( $\\diamond$ ) can be reformulated as $R(L) \\le 2F(L)$, where $R(B)$ ( $F(B)$ ) denotes the smallest positive integer $n$, or $\\infty$ if none exists, such that the graph $B^n$ is rearrangeable (externally connected).\n\nExamples\n\nConsider the simple 2-regular $2$-stage ordered graphs $A, C, D$ shown in Fig.1. It is easy to see that $F(A) = 2$ and $F(C) = F(D) = 3$ (the corresponding externally connected graphs $A^2, C^3, D^3$ are depicted in blue). Therefore, according to Conjecture ( $\\diamond$ ), the graphs $A^4, C^6, D^6$ should be rearrangeable, which is indeed the case. The graph $A$ is the 2-stage Shuffle-exchange graph $\\text{SE}(2,3)$, and there are several nice proofs known for $R(A)=4$. Although I am not aware of any theoretical proof for rearrangeability of $C^5$ or $D^6$, I have verified by brute force without difficulty that $R(C)=5$ and $R(D)=6$.\n\nFigure 1. Examples for Conjecture ( $\\diamond$ ).\n\nLink to Beneš Conjecture\n\nProblem ( $\\dag$ ) and Conjecture ( $\\diamond$ ) are equivalent \"graph-theoretic\" forms of Problem ( $\\star$ ) and Beneš conjecture [B75], respectively.\n\nThe equivalence is based on the natural bijection between the $\\ell$-systems of partitions and the straight $\\ell$-stage graphs, given any $\\ell\\ge2$. Here an $\\ell$-system of partitions is an $\\ell$-tuple ${\\bf H}:=({\\bf h}_1,\\dots,{\\bf h}_\\ell)$ of partitions of some finite set $E$. The image of ${\\bf H}$ under this bijection is the straight $\\ell$-stage graph denoted $G({\\bf H})$ and defined as follows. The edge set of $G({\\bf H})$ is $[\\ell-1]\\times E$, the $i$ th vertex part is $U_i:=\\{i\\}\\times {\\bf h}_i$, for all $i\\in[\\ell]$, and the edge-vertex incidence is such that every edge $(j,e)$ has endpoints $(j,a)\\in U_j$ and $(j+1,b)\\in U_{j+1}$ uniquely determined by $e\\in a\\cap b$.\n\nThe bijection ${\\bf H} \\mapsto G({\\bf H})$ provides a convenient two-way link between the frameworks for Problems ( $\\star$ ) and ( $\\dag$ ) via numerous easily seen equivalences. Here is some basic ones:\n\n$\\bullet$ Simplicity of $G({\\bf H})$ is equivalent to the condition ${\\bf h}_i\\wedge{\\bf h}_{i+1}={\\bf 0}$, for all $i\\in[\\ell-1]$.\n\n$\\bullet$ Uniformity of $G({\\bf H})$ is equivalent to the existence of a permutation $\\delta$ of $E$ such that ${\\bf h}_{i+1}=\\delta ({\\bf h}_i)$, for all $i\\in[\\ell-1]$.\n\n$\\bullet$ $k$-quasi-regularity of $G({\\bf H})$ is equivalent to every block of ${\\bf h}_i$ being of size $k$, for all $i\\in[\\ell]$. Here the graph $G$ is $k$-quasi-regular if the induced bipartite subgraph on $V_i\\cup V_{i+1}$ is $k$-regular, for all $i\\in[\\ell-1]$. Note that a quasi-regular multistage graph is straight. Also, $k$-quasi-regularity of $B^n$ is equivalent to $k$-regularity of $B$.\n\n$\\bullet$ External connectivity of $G({\\bf H})$ is equivalent to transitivity of $S({\\bf h}_\\ell)\\dots S({\\bf h}_2)S({\\bf h}_1)$.\n\n$\\bullet$ Given a permutation $\\xi$ of $E$, the membership $\\xi \\in S({\\bf h}_1)S({\\bf h}_2) \\dots S({\\bf h}_\\ell)$ is equivalent to routability of the mask $M(\\xi)$ for $G({\\bf H})$ defined as follows. The edge set of $M(\\xi)$ is $E$ and the edge-vertex incidence is such that every edge $e\\in E$ has endpoints $(1,a)\\in U_1$ and $(\\ell,b)\\in U_{\\ell}$ uniquely determined by $e\\in \\xi^{-1}(a)\\cap b$. Note that given ${\\bf H}$, the map $\\xi \\mapsto M(\\xi)$ is surjective (but generally not injective).\n\n$\\bullet$ Consequently, rearrangeability of $G({\\bf H})$ is equivalent to completeness of ${\\bf H}$. Here ${\\bf H}$ is complete if it satisfies $\\frak S(E) = S({\\bf h}_1)S({\\bf h}_2) \\dots S({\\bf h}_\\ell)$.\n\n$\\bullet$ If $G({\\bf H})$ is rearrangeable, then any routing algorithm for $G({\\bf H})$ easily translates to a factorization algorithm of the same complexity for the latter identity, and vise versa. Here, given a rearrangeable multistage graph, a routing algorithm is one that takes a mask of the graph as input and returns a corresponding routing.\n\n$\\bullet$ Contracting all edges between $U_i$ and $U_{i+1}$ in $G({\\bf H})$ is equivalent to replacing the partitions ${\\bf h}_i$ and ${\\bf h}_{i+1}$ in ${\\bf H}$ with their supremum ${\\bf h}_i\\vee{\\bf h}_{i+1}$, given any fixed $i\\in[\\ell-1]$. In other words, $G_i=G({\\bf H}_i)$, where $G_i$ is the contracted graph and ${\\bf H}_i:=({\\bf h}_1,\\dots,{\\bf h}_i\\vee{\\bf h}_{i+1},\\dots,{\\bf h}_\\ell)$. In fact, the procedure ${\\bf H} \\mapsto {\\bf H}_i$ preserves completeness of ${\\bf H}$, as $S({\\bf h}_i)S({\\bf h}_{i+1})\\subseteq S({\\bf h}_i\\vee{\\bf h}_{i+1})$. Equivalently, the procedure $G({\\bf H}) \\mapsto G_i$ preserves rearrangeability of $G({\\bf H})$.\n\nCounterexamples\n\nAlthough the presented graph-theoretic statement ( $\\dag$ ) of Problem ( $\\star$ ) may look more complex, it provides somewhat more intuitive framework to study the problem and, in particular, Beneš conjecture. To illustrate this, let us now reconsider in terms of this framework and in more detail the 3 counterexamples for some extensions of Beneš conjecture discussed here.\n\nCounterexample 1. The condition of simplicity of the graph $L$ (essentially missing in the original statement [B75] of Beneš conjecture) is necessary for Conjecture ( $\\diamond$ ). To see this, consider the following 2-stage 3-regular non-simple ordered graph $Q$:\n\nWhereas $Q$ is obviously externally connected, the graph $Q^2$ is not rearrageable. This is because it is evidently impossible to connect the two red vertices in $Q^2$ (a source and a target) with 3 mutually edge-disjoint plain paths.\n\nCounterexample 2. Conjecture ( $\\diamond$ ) is not directly generalizable to non-uniform graphs. More precisely, the condition of uniformity of $X$ is necessary for the following reformulation of ( $\\diamond$ ):\n\nConjecture Let $X$ be a simple quasi-regular ordered multistage graph. Suppose that $X$ is uniform and externally connected. Then the graph $X^{2}$ is rearrangeable.\n\nHere $X^{2}$ denotes the proper concatenation of 2 identical copies of $X$. To see the necessity, consider the following simple 4-stage 2-quasi-regular non-uniform ordered graph $Y$:\n\nWhereas $Y$ is obviously externally connected, the graph $Y^2$ is not rearrangeable. To see this, recall that contracting all edges between two consecutive parts in a straight multistage graph preserves its rearrangeability. Therefore, if $Y^2$ were rearrangeable then so would be the 3-stage graph $W$ obtained from $Y^2$ by contracting all edges in the shadowed areas. However, this is not true as it is evidently impossible to connect the two red vertices in $W$ (a source and a target) with 4 mutually edge-disjoint plain paths.\n\nCounterexample 3. The stronger version of Conjecture ( $\\diamond$ ) (proposed essentially in the same paper [B75]), claiming that $R(L) = 2F(L)$, is false. The graph $C$ shown in Fig.1 is a counterexample as $R(C) = 2F(C)-1$.\n\nMore information on Problem ( $\\dag$ ) and Conjecture ( $\\diamond$ ) can be found here (via Problem ( $\\star$ ) and Beneš conjecture).\n\nBibliography:\n*[B75] V.E. Beneš, Proving the rearrangeability of connecting networks by group calculation, Bell Syst. Tech. J. 54 (1975), 421-434.\n\nRelated:\nRelated problems\nBeneš Conjecture\nShuffle-Exchange Conjecture\nShuffle-Exchange Conjecture (graph-theoretic form)\n\nDiscussion links:\n- 2-stage Shuffle-exchange graph $\\text{SE}(2,3)$: http://www.openproblemgarden.org/?q=node/37089\n- proofs known for $R(A)=4$: http://www.openproblemgarden.org/?q=node/37167\n- Problem ( $\\star$ ) and Beneš conjecture: http://www.openproblemgarden.org/?q=node/37181\n- ( $\\star$ ): http://www.openproblemgarden.org/?q=node/37181\n- complete: http://www.openproblemgarden.org/?q=node/37181\n- factorization algorithm: http://www.openproblemgarden.org/?q=node/37181\n- Problem ( $\\star$ ): http://www.openproblemgarden.org/?q=node/37181\n- Beneš conjecture: http://www.openproblemgarden.org/?q=node/37181\n- here: http://www.openproblemgarden.org/?q=node/37181\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 95.\n\nAttempt notes:\nTarget:\nMake progress on \"Beneš Conjecture (graph-theoretic form)\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3111, "problem_number": "OPG-37211", "title": "Approximation Ratio for Maximum Edge Disjoint Paths problem", "statement": "Conjecture Can the approximation ratio $O(\\sqrt{n})$ be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than $\\mathcal{APX}$-hardness?", "background": "Source: Open Problem Garden. Original node ID: 37211. URL: http://www.openproblemgarden.org/op/approximation_ratio_for_maximum_edge_disjoint_paths_problem.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/approximation_ratio_for_maximum_edge_disjoint_paths_problem\n- Author(s): Bentz, Cedric\n- Subject(s): Graph Theory\n- Keywords: approximation algorithms; Disjoint paths; planar graph; polynomial algorithm\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 18th, 2010 by jcmeyer\n\nProblem-page discussion:\nAssume a flow graph $G = (V, E)$ with $n$ vertices and $m$ edges. Each edge has a capacity function $c: E \\rightarrow \\mathbb{Z}^+$ (Flow network). The graph contains a list $\\mathcal{N}$ of terminal vertices called sources ( $s_i$ ) and sinks ( $s_i'$ ). Each pair ( $s_i, s_i'$ ) defines a net or commodity.\n\nA Multiflow is a way of routing commodities from their sources to the respective sinks while ensuring that the flow of each commodity is conserved at each non-terminal vertex and that the sum of the flows of all commodities through an edge does not exceed the capacity of the edge.\n\nThe Maximum Integer Multiflow problem (MaxIMF) seeks to maximize the number of flow units routed between the nets in the graph. The Maximum Edge Disjoint Paths (MaxEDP) problem seeks to find the maximum number of disjoint paths between the sources and sinks. When the capacities for all edges are set to one, MaxIMF simplifies into the MaxEDP problem.\n\nBentz provides an algorithm to find the MaxEDP with a proven approximation ratio (Approximation and integrality gap) of $O(\\sqrt{n})$. Can the approximation ratio be improved for MaxEDP in planar graphs, or can an inapproximability result stronger than $\\mathcal{APX}$-hardness be proved for this problem? And what about the general graphs?\n\nBibliography:\nCédric Bentz, Edge disjoint paths and max integral muliflow/min multicut theorems in planar graphs, Electronic Notes in Discrete Mathematics 22 (2005), 55–60\n\nDiscussion links:\n- Flow network: http://en.wikipedia.org/wiki/Flow network\n- Approximation and integrality gap: http://en.wikipedia.org/wiki/Linear_programming_relaxation#Approximation_and_integrality_gap\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Approximation Ratio for Maximum Edge Disjoint Paths problem\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3112, "problem_number": "OPG-37217", "title": "Approximation ratio for k-outerplanar graphs", "statement": "Conjecture Is the approximation ratio for the Maximum Edge Disjoint Paths (MaxEDP) or the Maximum Integer Multiflow problem (MaxIMF) bounded by a constant in $k$-outerplanar graphs or tree-width graphs?", "background": "Source: Open Problem Garden. Original node ID: 37217. URL: http://www.openproblemgarden.org/op/approximation_ratio_for_k_outerplanar_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/approximation_ratio_for_k_outerplanar_graphs\n- Author(s): Bentz, Cedric\n- Subject(s): Graph Theory\n- Keywords: approximation algorithms; planar graph; polynomial algorithm\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 18th, 2010 by jcmeyer\n\nProblem-page discussion:\nAssume a flow graph $G = (V, E)$ with $n$ vertices and $m$ edges (Flow network). Each edge has a capacity function $c: E \\rightarrow \\mathbb{Z}^+$. The graph contains a list $\\mathcal{N}$ of terminal vertices called sources ( $s_i$ ) and sinks ( $s_i'$ ). Each pair ( $s_i, s_i'$ ) defines a net or commodity.\n\nA Multiflow is a way of routing commodities from their sources to the respective sinks while ensuring that the flow of each commodity is conserved at each non-terminal vertex and that the sum of the flows of all commodities through an edge does not exceed the capacity of the edge.\n\nThe Maximum Integer Multiflow problem (MaxIMF) seeks to maximize the number of flow units routed between the nets in the graph. The Maximum Edge Disjoint Paths (MaxEDP) problem seeks to find the maximum number of disjoint paths between the sources and sinks. When the capacities for all edges are set to one, MaxIMF simplifies into the MaxEDP problem.\n\nIs the approximation ratio (Approximation and integrality gap) for MaxEDP or MaxIMF bounded by a constant in $k$-outerplanar graphs (Outerplanar graphs) or tree-width graphs?\n\nBibliography:\nC. Bentz, “Disjoint paths in sparse graphs,” Discrete Appl. Math., vol. 157, no. 17, pp. 3558–3568, 2009\n\nDiscussion links:\n- Flow network: http://en.wikipedia.org/wiki/Flow network\n- Approximation and integrality gap: http://en.wikipedia.org/wiki/Linear_programming_relaxation#Approximation_and_integrality_gap\n- Outerplanar graphs: http://en.wikipedia.org/wiki/Planar_graph#Outerplanar_graphs\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Approximation ratio for k-outerplanar graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3113, "problem_number": "OPG-37218", "title": "Finding k-edge-outerplanar graph embeddings", "statement": "Conjecture It has been shown that a $k$-outerplanar embedding for which $k$ is minimal can be found in polynomial time. Does a similar result hold for $k$-edge-outerplanar graphs?", "background": "Source: Open Problem Garden. Original node ID: 37218. URL: http://www.openproblemgarden.org/op/finding_k_edge_outerplanar_graph_embeddings.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/finding_k_edge_outerplanar_graph_embeddings\n- Author(s): Bentz, Cedric\n- Subject(s): Graph Theory\n- Keywords: planar graph; polynomial algorithm\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 18th, 2010 by jcmeyer\n\nProblem-page discussion:\nA $k$-outerplanar graph [Baker] with $k > 0$ is a planar graph having an embedding with at most $k$ layers of vertices such that after removing iteratively the vertices (and their adjacent edges) lying on the outer face $k$ times, we obtain the empty graph.\n\nA $k$-edge-outerplanar graph [Bentz] is defined to be a planar graph having an embedding with at most $k$ layers of edges such that after removing iteratively the edges lying on the outer face $k$ times, we obtain a graph with no edge. All $k$-edge-outerplanar graphs are $k$-outerplanar graphs.\n\nGiven a planar graph, Bienstock and Monma have shown that a $k$-outerplanar embedding for which $k$ is minimal can be found in polynomial time. Does a similar result hold for $k$-edge-outerplanar graphs?\n\nBibliography:\nC. Bentz, “Disjoint paths in sparse graphs,” Discrete Appl. Math., vol. 157, no. 17, pp. 3558–3568, 2009\n\nD. Bienstock and C. L. Monma, “On the complexity of embedding planar graphs to minimize certain distance measures,” Algorithmica, vol. 5, no. 1–4, pp. 93–109, 1990\n\nB. S. Baker, “Approximation algorithms for np-complete problems on planar graphs,” J. ACM, vol. 41, no. 1, pp. 153–180, 1994\n\nDiscussion links:\n- $k$-outerplanar graph: http://en.wikipedia.org/wiki/Planar_graph#Outerplanar_graphs\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Finding k-edge-outerplanar graph embeddings\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3114, "problem_number": "OPG-37229", "title": "Exact colorings of graphs", "statement": "Conjecture For $c \\geq m \\geq 1$, let $P(c,m)$ be the statement that given any exact $c$-coloring of the edges of a complete countably infinite graph (that is, a coloring with $c$ colors all of which must be used at least once), there exists an exactly $m$-colored countably infinite complete subgraph. Then $P(c,m)$ is true if and only if $m=1$, $m=2$, or $c=m$.", "background": "Source: Open Problem Garden. Original node ID: 37229. URL: http://www.openproblemgarden.org/op/exact_colorings_of_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/exact_colorings_of_graphs\n- Author(s): Erickson, Martin\n- Subject(s): Graph Theory\n- Keywords: graph coloring; ramsey theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 29th, 2010 by Martin Erickson\n\nProblem-page discussion:\nStacey and Weidl have shown that given $m \\geq 3$, there is an integer $C(m)$ such that $P(c,m)$ is false for all $c \\geq C(m)$.\n\nBibliography:\n* M. Erickson, \"A Conjecture Concerning Ramsey's Theorem,\" Discrete Mathematics 126, 395--398 (1994); MR 95b:05209\n\nA. Stacey and P. Weidl, \"The Existence of Exactly m-Coloured Complete Subgraphs,\" J. of Combinatorial Theory, Series B 75, 1-18 (1999)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Exact colorings of graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3115, "problem_number": "OPG-37271", "title": "Star chromatic index of cubic graphs", "statement": "The star chromatic index $\\chi_s'(G)$ of a graph $G$ is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored.\n\nQuestion Is it true that for every (sub)cubic graph $G$, we have $\\chi_s'(G) \\le 6$?", "background": "Source: Open Problem Garden. Original node ID: 37271. URL: http://www.openproblemgarden.org/op/star_chromatic_index_of_cubic_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/star_chromatic_index_of_cubic_graphs\n- Author(s): Dvorak, Zdenek; Mohar, Bojan; Samal, Robert\n- Subject(s): Graph Theory\n- Keywords: edge coloring; star coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 16th, 2010 by Robert Samal\n\nProblem-page discussion:\nThe star chromatic number is the more usual concept [ACKKR,FRR]. Star chromatic index of a graph $G$ is simply the star chromatic number of the line graph $L(G)$; the definition given above is easily seen to be equivalent.\n\nDvořák, Mohar, and Šámal [DMS] show that every (sub)cubic graph $G$, satisfies $\\chi_s'(G) \\le 7$? On the other hand, it is simple to check that $\\chi_s'(K_{3,3])=6$, so the conjecture, if true, is tight.\n\nBibliography:\n[ACKKR] Albertson, Michael O.; Chappell, Glenn G.; Kierstead, Hal A.; Kündgen, André; Ramamurthi, Radhika: Coloring with no 2-Colored P4's, The Electronic Journal of Combinatorics 11 (1).\n\n[FRR] Fertin, Guillaume; Raspaud, André; Reed, Bruce, Star coloring of graphs, Journal of Graph Theory 47 (3): 163-182, doi:10.1002/jgt.20029.\n\n*[DMS] Dvořák, Zdeněk; Mohar, Bojan; Šámal, Robert: Star chromatic index, arXiv:1011.3376.\n\nDiscussion links:\n- star chromatic number: http://en.wikipedia.org/wiki/Star coloring\n- line graph: http://en.wikipedia.org/wiki/line graph\n\nBibliography links:\n- doi:10.1002/jgt.20029: http://dx.doi.org/10.1002/jgt.20029\n- arXiv:1011.3376: http://www.arxiv.org/abs/1011.3376\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Star chromatic index of cubic graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3116, "problem_number": "OPG-37275", "title": "Star chromatic index of complete graphs", "statement": "Conjecture Is it possible to color edges of the complete graph $K_n$ using $O(n)$ colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?\n\nEquivalently: is the star chromatic index of $K_n$ linear in $n$?", "background": "Source: Open Problem Garden. Original node ID: 37275. URL: http://www.openproblemgarden.org/op/star_chromatic_index_of_complete_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/star_chromatic_index_of_complete_graphs\n- Author(s): Dvorak, Zdenek; Mohar, Bojan; Samal, Robert\n- Subject(s): Graph Theory\n- Keywords: complete graph; edge coloring; star coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: November 16th, 2010 by Robert Samal\n\nProblem-page discussion:\nThe star chromatic index $\\chi_s'(G)$ of a graph $G$ is the minimum number of colors needed to properly color the edges of $G$ so that no path or cycle of length four is bi-colored. An equivalent definition is that $\\chi_s'(G)$ is the star chromatic number of the line graph $L(G)$.\n\nDvořák, Mohar, and Šámal [DMS] show that $\\chi_s'(G) \\ge (2+o(1))n$. On the other hand, the best known upper bound (also in \\cite{DMS]) is superlinear: $$\\chi_s'(K_n) \\le n \\cdot \\frac{ 2^{ 2\\sqrt2(1+o(1)) \\sqrt{\\log n} } }{(\\log n)^{1/4}} \\,.$$\n\nIt may be possible to decrease the upper bound by elementary methods.\n\nBibliography:\n*[DMS] Dvořák, Zdeněk; Mohar, Bojan; Šámal, Robert: Star chromatic index, arXiv:1011.3376.\n\nRelated:\nRelated problems\nStar chromatic index of cubic graphs\n\nDiscussion links:\n- line graph: http://en.wikipedia.org/wiki/line graph\n\nBibliography links:\n- arXiv:1011.3376: http://www.arxiv.org/abs/1011.3376\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Star chromatic index of complete graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3117, "problem_number": "OPG-37316", "title": "Vertex Coloring of graph fractional powers", "statement": "Conjecture Let $G$ be a graph and $k$ be a positive integer. The $k-$ power of $G$, denoted by $G^k$, is defined on the vertex set $V(G)$, by connecting any two distinct vertices $x$ and $y$ with distance at most $k$. In other words, $E(G^k)=\\{xy:1\\leq d_G(x,y)\\leq k\\}$. Also $k-$ subdivision of $G$, denoted by $G^\\frac{1}{k}$, is constructed by replacing each edge $ij$ of $G$ with a path of length $k$. Note that for $k=1$, we have $G^\\frac{1}{1}=G^1=G$.\nNow we can define the fractional power of a graph as follows:\nLet $G$ be a graph and $m,n\\in \\mathbb{N}$. The graph $G^{\\frac{m}{n}}$ is defined by the $m-$ power of the $n-$ subdivision of $G$. In other words $G^{\\frac{m}{n}}\\isdef (G^{\\frac{1}{n}})^m$.\nConjecture. Let $G$ be a connected graph with $\\Delta(G)\\geq3$ and $m$ be a positive integer greater than 1. Then for any positive integer $n>m$, we have $\\chi(G^{\\frac{m}{n}})=\\omega(G^\\frac{m}{n})$.\nIn [1], it was shown that this conjecture is true in some special cases.", "background": "Source: Open Problem Garden. Original node ID: 37316. URL: http://www.openproblemgarden.org/op/vertex_coloring_of_graph_fractional_powers.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/vertex_coloring_of_graph_fractional_powers\n- Author(s): Iradmusa, Moharram\n- Subject(s): Graph Theory\n- Keywords: chromatic number, fractional power of graph, clique number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: April 23rd, 2011 by Iradmusa\n\nBibliography:\n[1] Iradmusa, Moharram N., On colorings of graph fractional powers. Discrete Math. 310 (2010), no. 10-11, 1551–1556.\n\nComments:\n- July 13th, 2011 | Anonymous | Needs revision: Note that if K_t is the complete graph on t vertices with t even, then the 2-power of the 2-subdivision of K_t is isomorphic to the total graph of K_t. That is the graph T(K_t) whose vertex set is V(K_t) union E(K_t) and two vertices are adjacent in T(K_t) if their either adjacent or incident in K_t.\n\nclique number of T(K_t) is t + 1 and the chromatic number of T(K_t) is >= t+2.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"Vertex Coloring of graph fractional powers\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3118, "problem_number": "OPG-37325", "title": "Covering powers of cycles with equivalence subgraphs", "statement": "Conjecture Given $k$ and $n$, the graph $C_{n}^k$ has equivalence covering number $\\Omega(k)$.", "background": "Source: Open Problem Garden. Original node ID: 37325. URL: http://www.openproblemgarden.org/op/covering_powers_of_cycles_with_equivalence_subgraphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/covering_powers_of_cycles_with_equivalence_subgraphs\n- Subject(s): Graph Theory\n- Importance: Low ✭\n- Recommended for undergraduates: no\n- Posted: July 7th, 2011 by Andrew King\n\nProblem-page discussion:\nGiven a graph $G$, a subgraph $H$ of $G$ is an equivalence subgraph of $G$ if $H$ a disjoint union of cliques. The quivalence covering number of $G$, denoted $eq(G)$, is the least number of equivalence subgraphs needed to cover the edges of $G$.\n\nThis problem has been studied by various people since the 80s [A]. For line graphs, the equivalence covering number is known to within a constant factor [EGK]. It is therefore tempting to examine the situation for quasi-line graphs and claw-free graphs. Powers of cycles are perhaps the simplest interesting class of claw-free graphs that are not necessarily line graphs. However, even for $n$ very large compared to $k$, no upper bound is known beyond trivial linear bounds of order $\\Theta(k)$. Furthermore, it is not even certain that a nontrivial lower bound (i.e. going to infinity as $k$ goes to infinity) is known. It is possible that this can be related somehow to a known result, but for now it seems at least superficially that this problem is wide open.\n\nBibliography:\n[A] N. Alon, Covering graphs with the minimum number of equivalence relations, Combinatorica 6 (1986) 201–206.\n\n[EGK] L. Esperet, J. Gimbel, A. King, Covering line graphs with equivalence relations, Discrete Applied Mathematics Volume 158, Issue 17, 28 October 2010, Pages 1902-1907.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Covering powers of cycles with equivalence subgraphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3119, "problem_number": "OPG-37357", "title": "Obstacle number of planar graphs", "statement": "Does there exist a planar graph with obstacle number greater than 1? Is there some $k$ such that every planar graph has obstacle number at most $k$?", "background": "Source: Open Problem Garden. Original node ID: 37357. URL: http://www.openproblemgarden.org/op/obstacle_number_of_planar_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/obstacle_number_of_planar_graphs\n- Author(s): Alpert, Hanna; Koch, Christina; Laison, Joshua D.\n- Subject(s): Graph Theory\n- Keywords: graph drawing; obstacle number; planar graph; visibility graph\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: November 23rd, 2011 by Andrew King\n\nProblem-page discussion:\nA $k$-obstacle drawing of a graph $G$ is a mapping of the vertices of $G$ to points in the plane, along with a set of polygonal obstacles $P_1,\\ldots, P_k$, such that two vertices are adjacent precisely if the line segment connecting their corresponding points in $\\mathbb R^2$ does not intersect any obstacle. The {\\em obstacle number} of a graph $G$ is the minimum $k$ such that $G$ has a $k$-obstacle drawing.\n\nThis invariant was recently introduced by Alpert, Koch, and Laison [AKL], who proved that every outerplanar graph has obstacle number 1. The next question, then, follows naturally: what is the obstacle number of a planar graph? So far no planar graph has been proved to have obstacle number greater than 1. Alpert, Koch, and Laison specifically ask what the obstacle numbers of the icosahedron and dodecahedron are [AKL].\n\nBibliography:\n[AKL] Hannah Alpert, Christina Koch, and Joshua D. Laison: Obstacle numbers of graphs. Discrete Comput. Geom. (2010) 44:223-244.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Obstacle number of planar graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3120, "problem_number": "OPG-37364", "title": "Matching cut and girth", "statement": "Question For every $d$ does there exists a $g$ such that every graph with average degree smaller than $d$ and girth at least $g$ has a matching-cut?", "background": "Source: Open Problem Garden. Original node ID: 37364. URL: http://www.openproblemgarden.org/op/matching_cut_and_girth.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/matching_cut_and_girth\n- Subject(s): Graph Theory\n- Keywords: matching cut, matching, cut\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 30th, 2011 by w\n\nProblem-page discussion:\nLet $G=(V,E)$ be a graph. A matching $M$ is a matching-cut if there exists a set $S\\subset V$ such that $M = E(S:V\\setminus S)$. Graphs having a matching-cut are called decomposable.\n\nIt is known that every graph with $|E| < 3(|V|-1)/2$ is decomposable [BFP11].\n\nBibliography:\n[C84] V. Chvátal, Recognizing decomposable graphs, J Graph Theory 8 (1984), 51–53\n\n[BFP11] P. Bonsma, A. Farley, A. Proskurowski, Extremal graphs having no matching cuts, J Graph Theory (2011)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Matching cut and girth\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3121, "problem_number": "OPG-37420", "title": "Minimal graphs with a prescribed number of spanning trees", "statement": "Conjecture Let $n \\geq 3$ be an integer and let $\\alpha(n)$ denote the least integer $k$ such that there exists a simple graph on $k$ vertices having precisely $n$ spanning trees. Then $\\alpha(n) = o(\\log{n}).$", "background": "Source: Open Problem Garden. Original node ID: 37420. URL: http://www.openproblemgarden.org/op/minimal_graphs_with_a_prescribed_number_of_spanning_trees.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/minimal_graphs_with_a_prescribed_number_of_spanning_trees\n- Author(s): Azarija, Jernej; Skrekovski, Riste\n- Subject(s): Graph Theory\n- Keywords: number of spanning trees, asymptotics\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: April 22nd, 2012 by azi\n\nProblem-page discussion:\nObserve that $\\alpha(n)$ is well defined for $n \\geq 3$ since $C_n$ has $n$ spanning trees.\n\nThe function was introduced by Sedlacek [S] who has shown that for large enough $n$ $\\alpha(n) \\leq \\frac{n+6}{3} \\mbox{if } n \\equiv 0 \\pmod{3}$ and $\\alpha(n) \\leq \\frac{n+4}{3} \\mbox{if } n \\equiv 2 \\pmod{3}.$\n\nUsing the fact that almost all positive integers $n$ are expressible as $n = ab+ac+bc$ for integers $0 < a < b < c$ it can be shown [A] that for large enough $n$\n\n$\\alpha(n) \\leq \\frac{n+4}{3} \\mbox{if } n \\equiv 2 \\pmod{3}$ and $\\alpha(n) \\leq \\frac{n+9}{4}$ otherwise.\n\nMoreover, the only fixed points of $\\alpha$ are 3, 4, 5, 6, 7, 10, 13 and 22.\n\nThe conjecture is motivated by the following graph (ploted for a very small sample of vertices)\n\nThe conjecture [C] is justifiable for highly composite numbers $n$ since in this case one can construct the graph obtained after taking cycles $C_{p_1}, \\ldots,C_{p_k}$ for every odd prime factor $p_i$ of $n$.\n\nBibliography:\n[S] J. Sedlacek, On the minimal graph with a given number of spanning trees, Canad. Math. Bull. 13 (1970) 515-517.\n\n[A] J. Azarija, R. Skrekovski, Euler's idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees, IMFM preprints 49 (2011) Link to paper\n\n* [C] Minimal graphs with a prescribed number of spanning trees\n\nBibliography links:\n- Link to paper: http://www.imfm.si/preprinti/PDF/01157.pdf\n- Minimal graphs with a prescribed number of spanning trees: http://mathoverflow.net/questions/93656/minimal-graphs-with-a-prescribed-number-of-spanning-trees\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Minimal graphs with a prescribed number of spanning trees\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3122, "problem_number": "OPG-37670", "title": "The Borodin-Kostochka Conjecture", "statement": "Conjecture Every graph with maximum degree $\\Delta \\geq 9$ has chromatic number at most $\\max\\{\\Delta-1, \\omega\\}$.", "background": "Source: Open Problem Garden. Original node ID: 37670. URL: http://www.openproblemgarden.org/op/the_borodin_kostochka_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_borodin_kostochka_conjecture\n- Author(s): Borodin, Oleg V.; Kostochka, Alexandr V.\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 10th, 2012 by Andrew King\n\nProblem-page discussion:\nThe Borodin-Kostochka conjecture proposes that for any graph $G$ with maximum degree $\\Delta$ and clique number $\\omega < \\Delta$, $G$ is $\\Delta-1$ colourable so long as $\\Delta$ is sufficiently large (specifically, $\\Delta\\geq 9$ ). The requirement that $\\Delta \\geq 9$ is necessary, as one can see by looking at the strong product of $C_5$ and $K_3$.\n\nReed [R] proved that there exists a $\\Delta_0$ for which the conjecture holds whenever $\\Delta \\geq \\Delta_0$. Specifically he proved that $\\Delta_0 \\leq 10^{14}$, but claims that more careful analysis could reduce $\\Delta_0$ to 1000.\n\nThe conjecture was recently proven by Cranston and Rabern for claw-free graphs [CR]. In their paper they mention an unpublished strengthening proposed by Borodin and Kostochka, namely that one can replace the chromatic number in the statement of the conjecture with the list chromatic number.\n\nBibliography:\n[BK] O. V. Borodin and A. V. Kostochka. On an upper bound of a graph's chromatic number, depending on the graph's degree and density. JCTB 23 (1977), 247--250.\n\n[CR] D. W. Cranston and L. Rabern. Coloring claw-free graphs with $\\Delta-1$ colors, arXiv 1206.1269, 2012.\n\n[R] B. A. Reed. A strengthening of Brooks’ Theorem. J. Comb. Theory Ser. B, 76:136–149, 1999.\n\nBibliography links:\n- Coloring claw-free graphs with $\\Delta-1$ colors: http://www.arxiv.org/abs/1206.1269\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"The Borodin-Kostochka Conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3123, "problem_number": "OPG-46443", "title": "Stable set meeting all longest directed paths.", "statement": "Conjecture Every digraph has a stable set meeting all longest directed paths", "background": "Source: Open Problem Garden. Original node ID: 46443. URL: http://www.openproblemgarden.org/op/stable_set_meeting_all_longest_directed_paths.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/stable_set_meeting_all_longest_directed_paths\n- Author(s): Laborde, Jean-Marie; Payan, Charles; Xuong N.H.\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 1st, 2013 by fhavet\n\nProblem-page discussion:\nIf the stability number is 1, that is if the digraph is a tournament, it follows Redei's Theorem stating that every tournament has a directed hamiltonian path. The conjecture has been proved by Havet [H] for digraphs having stability number 2.\n\nThe conjecture would give an easy inductive proof of Gallai-Roy Theorem: every digraph with chromatic number $k$ contains a directed path on $k$ vertices.\n\nHahn and Jackson [HJ] conjectured that in contrast there is no directed path meeting every maximum stable set. In fact, they conjectured the following: For each positive integer $k$, there is a digraph $D$ with stability number $k$ such that deleting the vertices of any $k-1$ directed paths in $D$ leaves a digraph with stability number $k$. This was proved by Fox and Sudakov [FS] by a probabilistic argument.\n\nBibliography:\n[FS] J. Fox and B. Sudakov, Paths and stability number in digraphs, Electronic Journal of Combiantorics, 16 (2009), no.1, N23.\n\n[HJ] G. Hahn and B. Jackson, A note concerning paths and independence number in digraphs, Discrete Math. 82 (1990), 327–329.\n\n[H] F. Havet. Stable set meeting every longest path. Discrete Mathematics, 289 (2004), no. 1-3, 169-173.\n\n*[LPX] J.M. Laborde, C. Payan, and N.H. Xuong, Independent sets and longest directed paths in digraphs. In Graphs and other Combinatorial Topics (Prague, 1982)}, Teubner-Texte Math., Vol. 59 (1983), 173-177, Teubner, Leipzig.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Stable set meeting all longest directed paths.\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3124, "problem_number": "OPG-46496", "title": "Arc-disjoint strongly connected spanning subdigraphs", "statement": "Conjecture There exists an ineteger $k$ so that every $k$-arc-connected digraph contains a pair of arc-disjoint strongly connected spanning subdigraphs?", "background": "Source: Open Problem Garden. Original node ID: 46496. URL: http://www.openproblemgarden.org/op/arc_disjoint_strongly_connected_spanning_subdigraphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/arc_disjoint_strongly_connected_spanning_subdigraphs\n- Author(s): Bang-Jensen, Joergen; Yeo, Anders\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 2nd, 2013 by fhavet\n\nProblem-page discussion:\nBang-Jensen and Yeo [BY] proved the conjecture for several classes like tournaments. There is stronger conjecture for tournaments. Yeo (See [BG, Theorem 13.10.1]) showed that it is NP-complete to decide whether a 2-regular digraph has two arc-disjoint strongly connected spanning subdigraphs.\n\nA similar question can be asked about arc-disjoint out-branching and in-branching. Several related problems are mentioned in the survey of Bang-Jensen and Kriesell [BK].\n\nBibliography:\n[BG] J. Bang-Jensen, G. Gutin, Digraphs: Theory, Algorithms and Applications, 2nd. ed., Springer Verlag (2009).\n\n[BK] J. Bang-Jensen, M. Kriesell, Disjoint sub(di)graphs in digraphs, Electronic Notes in Discrete Mathematics 34 (2009), 179-183.\n\n*[BY] J. Bang-Jensen, A. Yeo, Decomposing k-arc-strong tournaments into strong spanning subdigraphs, Combinatorica 24 (2004), 331-349.\n\nRelated:\nRelated problems\nArc-disjoint out-branching and in-branching\nDecomposing k-arc-strong tournament into k spanning strong digraphs\n\nDiscussion links:\n- stronger conjecture for tournaments: http://www.openproblemgarden.org/?q=node/4765\n- arc-disjoint out-branching and in-branching: http://www.openproblemgarden.org/?q=node/46495\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Arc-disjoint strongly connected spanning subdigraphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3125, "problem_number": "OPG-46538", "title": "Do any three longest paths in a connected graph have a vertex in common?", "statement": "Conjecture Do any three longest paths in a connected graph have a vertex in common?", "background": "Source: Open Problem Garden. Original node ID: 46538. URL: http://www.openproblemgarden.org/op/do_any_three_longest_paths_in_a_connected_graph_have_a_vertex_in_common.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/do_any_three_longest_paths_in_a_connected_graph_have_a_vertex_in_common\n- Author(s): Gallai, Tibor\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 3rd, 2013 by fhavet\n\nProblem-page discussion:\nIt is a well-known exercise that every two longest paths in a connected graph have a common vertex. Skupien [S] showed connected graphs where 7 longest paths do not share a common vertex.\n\nBibliography:\n*[G] T. Gallai, Problem 6. In Theory of Graphs (Proc. Colloq., Tihany, 1966), 362 Academic Press, New York, 1968.\n\nZ. Skupień, Smallest sets of longest paths with empty intersection. Combin. Probab. Comput. 5 (1996), no. 4, 429–436.\n\nComments:\n- October 14th, 2023 | Robert Samal | Possible solution: Possible solution to this appears in https://arxiv.org/abs/2006.16245\n\n(I am not sure whether it is being in a referee process or whether somebody may have found a gap in the presented proof.)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Do any three longest paths in a connected graph have a vertex in common?\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3126, "problem_number": "OPG-46629", "title": "Lovász Path Removal Conjecture", "statement": "Conjecture There is an integer-valued function $f(k)$ such that if $G$ is any $f(k)$-connected graph and $x$ and $y$ are any two vertices of $G$, then there exists an induced path $P$ with ends $x$ and $y$ such that $G-V(P)$ is $k$-connected.", "background": "Source: Open Problem Garden. Original node ID: 46629. URL: http://www.openproblemgarden.org/op/lovasz_path_removal_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/lovasz_path_removal_conjecture\n- Author(s): Lovasz, Laszlo\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nIt follows from a theorem of Tutte that any 3-connected graph contains a non-separating path connecting any two vertices, and consequently, $f(1)=3$. When $k=2$, it was independently shown by Chen, Gould, and Yu [CGY] and Kriesell [K] that $f(2) = 5$.\n\nAnwering a conjecture of Kriesell, Kawarabayashi et al. [KLRW] proved the following weaker statement, in which one only removes the edges of the path.\n\nTheorem There exists a function $f(k)$ such that for every $f(k)$-connected graph $G$ and any two vertices $x$ and $y$ of $G$, there exists an induced path $P$ with ends $x$ and $y$ such that $G\\setminus E(P)$ is $k$-connected.\n\nBibliography:\n[CGY] G. Chen, R. Gould, X. Yu, Graph connectivity after path removal, Combinatorica 23 (2003) 185--203.\n\n[KLRW] K. Kawarabayashi, O. Lee, B. Reed, and P. Wollan, A weaker version of Lovasz's path removal conjecture, Journal of Combinatorial Theory, Series B 98 (2008) 972--979.\n\n[K] M. Kriesell, Induced paths in 5-connected graphs, J. of Graph Theory, 36 (2001), 52--58.\n\n*[T] C. Thomassen, Graph decompositions with applications to subdivisions and path systems modulo k, J. of Graph Theory, 7 (1983), 261--271.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Lovász Path Removal Conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3127, "problem_number": "OPG-46706", "title": "Turán number of a finite family.", "statement": "Given a finite family ${\\cal F}$ of graphs and an integer $n$, the Turán number $ex(n,{\\cal F})$ of ${\\cal F}$ is the largest integer $m$ such that there exists a graph on $n$ vertices with $m$ edges which contains no member of ${\\cal F}$ as a subgraph.\n\nConjecture For every finite family ${\\cal F}$ of graphs there exists an $F\\in {\\cal F}$ such that $ex(n, F ) = O(ex(n, {\\cal F}))$.", "background": "Source: Open Problem Garden. Original node ID: 46706. URL: http://www.openproblemgarden.org/op/turan_number_of_a_finite_family.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/turan_number_of_a_finite_family\n- Author(s): Erdos, Paul; Simonovits, Miklos\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 5th, 2013 by fhavet\n\nProblem-page discussion:\nFor the case when ${\\cal F}$ consists of even cycles, this would mean that (up to constants) the Turán number of ${\\cal F}$ is given by that of the longest cycle in ${\\cal F}$. Verstraëte (see [KO]) conjectured something stronger:\n\nConjecture For all integers $k < \\ell$ there exists a positive c = c(\\ell) such that every $C_{2\\ell}$-free graph $G$ has a $C_{2k}$-free subgraph $H$ with $e(H) ≥ e(G)/c$.\n\nThis conjecture was motivated by a result of Györi [G] who showed that every bipartite $C_6$-free graph $G$ has a $C_4$-free subgraph which contains at least half of the edges of $G$. The case $k=2$ was proved in [KO].\n\nBibliography:\n*[ES] P.Erdös and M. Simonovits, Compactness results in extremal graph theory, Combinatorica 2 (1982), 275–288.\n\n[KO] D. Kühn and D. Osthus, 4-cycles in graphs without a given even cycle, J. Graph Theory 48 (2005), 147-156.\n\n[G] E. Györi, $C_6$-free bipartite graphs and product representation of squares, Discrete Math. 165/166 (1997), 371-375.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Turán number of a finite family.\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3128, "problem_number": "OPG-46951", "title": "Switching reconstruction conjecture", "statement": "Conjecture Every simple graph on five or more vertices is switching-reconstructible.", "background": "Source: Open Problem Garden. Original node ID: 46951. URL: http://www.openproblemgarden.org/op/switching_reconstruction_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/switching_reconstruction_conjecture\n- Author(s): Stanley, Richard P.\n- Subject(s): Graph Theory\n- Keywords: reconstruction\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2013 by fhavet\n\nProblem-page discussion:\nTo switch a vertex of a simple graph is to exchange its sets of neighbours and non-neighbours. The graph so obtained is called a switching of the graph. The collection of switchings of a graph G is called the switching deck of $G$. A graph is switching-reconstructible if every graph with the same deck as $G$ is isomorphic to $G$.\n\nThere are four pairs of non-isomorphic graphs of order $4$ with the same switching deck. One of them consists of the empty graph and the $4$-cycle.\n\nStanley [S] proved that a graph on $n$ vertices is switching-reconstructible if $n \\not\\equiv 0 (\\mod 4)$.\n\nAn analogous problem was posed for digraphs. Instead of complementing the edges at a vertex, one reverses each of its incident arc.\n\nBibliography:\n*[S] R. P. Stanley Reconstruction from vertex-switching. J. Combin. Theory Ser. B, 38 (1985), 132--138.\n\nRelated:\nRelated problems\nSwitching reconstruction of digraphs\n\nDiscussion links:\n- analogous problem: http://www.openproblemgarden.org/?q=node/46952\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Switching reconstruction conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3129, "problem_number": "OPG-46952", "title": "Switching reconstruction of digraphs", "statement": "Question Are there any switching-nonreconstructible digraphs on twelve or more vertices?", "background": "Source: Open Problem Garden. Original node ID: 46952. URL: http://www.openproblemgarden.org/op/switching_reconstruction_of_digraphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/switching_reconstruction_of_digraphs\n- Author(s): Bondy, J. Adrian; Mercier, Fabien\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2013 by fhavet\n\nProblem-page discussion:\nTo switch a vertex of a digraph is to reverse all the arcs incident to it. The digraph so obtained is called a switching of the digraph. The collection of switchings of a digraph $D$ is called the switching deck of $D$. A digraph is switching-reconstructible if every digraph with the same deck as $D$ is isomorphic to $D$.\n\nThe problem is a directed analogue of switching reconstruction of graphs in which one complements the edges at a vertex, instead of reversing each of its incident arcs.\n\nBondy and Mercier proved an analogue to Stanley's result for switching reconstruction of graphs. They proved that a digraph on $n$ vertices is switching-reconstructible if $n \\not\\equiv 0 (\\mod 4)$. They also proved many other common results for both switching reconstructions.\n\nOne significant difference between the directed and undirected problems is that there exist switching-nonreconstructible directed graphs on eight vertices, while Stanley's conjecture that every simple graph on five or more vertices is switching-reconstructible.\n\nBibliography:\n*[BM] J. A. Bondy and F. Mercier. Switching reconstruction of digraphs. J. Graph Theory 67(2011), no. 4, 332-348.\n\nRelated:\nRelated problems\nSwitching reconstruction conjecture\n\nDiscussion links:\n- switching reconstruction of graphs: http://www.openproblemgarden.org/?q=node/46951\n- Stanley's conjecture: http://www.openproblemgarden.org/?q=node/46951\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Switching reconstruction of digraphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3130, "problem_number": "OPG-48264", "title": "Signing a graph to have small magnitude eigenvalues", "statement": "Conjecture If $A$ is the adjacency matrix of a $d$-regular graph, then there is a symmetric signing of $A$ (i.e. replace some $+1$ entries by $-1$ ) so that the resulting matrix has all eigenvalues of magnitude at most $2 \\sqrt{d-1}$.", "background": "Source: Open Problem Garden. Original node ID: 48264. URL: http://www.openproblemgarden.org/op/signing_a_graph_to_have_small_magnitude_eigenvalues.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/signing_a_graph_to_have_small_magnitude_eigenvalues\n- Author(s): Bilu, Yonatan; Linial, Nathan\n- Subject(s): Graph Theory\n- Keywords: eigenvalue; expander; Ramanujan graph; signed graph; signing\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 24th, 2013 by mdevos\n\nProblem-page discussion:\nA graph $H$ is a $k$-lift of a graph $G$ if there is a $k$-to- $1$ map $f: V(H) \\rightarrow V(G)$ which is locally injective in the sense that the restriction of $f$ to the neighbourhood of every vertex is an injection. We can construct a random $k$-lift of $G$ with vertex set $V(G) \\times \\{1,\\ldots,k\\}$ by adding a (uniformly chosen) random matching between $\\{v\\} \\times \\{1,\\ldots,k\\}$ and $\\{w\\} \\times \\{1,\\ldots,k\\}$ whenever $vw \\in E(G)$. If $H$ is a $k$-lift of $G$, then every eigenvalue of $G$ will also be an eigenvalue of $H$, but in addition $H$ will have $(k-1) |V(G)|$ new eigenvalues. There has been considerable interest and investigation into the behaviour of these new eigenvalues for a random $k$-lift, since it is expected that they should generally be small in magnitude. In particular, if $G$ is a Ramanujan graph (a $d$-regular graph for which all nontrivial eigenvalues are at most $2 \\sqrt{d-1}$ ) it may be possible to construct a new Ramanujan graph by taking a suitable $k$-lift of $G$. A series of increasingly strong results have shown that a random $k$-lift of a $d$-regular Ramanujan graph will have all new eigenvalues at most $O(d^{3/4})$ (Friedman [F]), $O(d^{2/3})$ (Linial and Pruder [LP]) and $O(\\sqrt{d} \\log d)$ (Lubetzky, Sudakov, and Vu [LSV]).\n\nAn interesting paper of Bilu and Linial [BL] investigates 2-lifts of graphs. Let $G$ be a graph and let $H$ be a 2-lift of $G$ with vertex set $V(G) \\times \\{1,2\\}$ as above. Every eigenvector of $G$ extends naturally to an eigenvector of $H$ which is constant on each fiber (set of the form $\\{u\\} \\times \\{1,2\\}$ ). Thus, we may assume that all of the new eigenvalues are associated with eigenvectors which sum to zero on each fiber. So, each of these new eigenvectors is completely determined by its behaviour on $V(G) \\times \\{1\\}$. Now we assign a signature $\\pm 1$ to each edge of $G$ to form a signed graph $G^*$ by assigning each edge $uv \\in E(G)$ for which $(u,1)(v,1) \\in E(H)$ a sign of $1$ and every other edge of $G$ sign $-1$. It is straightforward to verify that the restriction of any new eigenvector of $H$ to $V(G) \\times \\{1\\}$ will then be an eigenvector of $G^*$. Thus, the above conjecture is equivalent to the conjecture that every $d$-regular graph has a $2$-lift so that all new eigenvalues have magnitude at most $2 \\sqrt{d-1}$. Furthermore, a positive solution to this conjecture for $d$-regular Ramanujan graphs would yield families of $d$-regular expanders.\n\nBibliography:\n*[BL] Y. Bilu, N. Linial, Lifts, discrepancy and nearly optimal spectral gap, Combinatorica 26 (5) (2006) 495–519. MathSciNet\n\n[F] J. Friedman, Relative expanders or weakly relatively Ramanujan graphs, Duke Math. J. 118 (1) (2003) 19–35. MathSciNet\n\n[LP] N. Linial, D. Puder, Word maps and spectra of random graph lifts, Random Structures Algorithms 37 (1) (2010) 100–135. MathSciNet\n\n[LSV] E. Lubetzky, B. Sudakov, V Vu, Spectra of lifted Ramanujan graphs. Adv. Math. 227 (2011), no. 4, 1612–1645. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2279667\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1978881\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2674623\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2799807\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 27.\n\nAttempt notes:\nTarget:\nMake progress on \"Signing a graph to have small magnitude eigenvalues\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3131, "problem_number": "OPG-48368", "title": "Are almost all graphs determined by their spectrum?", "statement": "Problem Are almost all graphs uniquely determined by the spectrum of their adjacency matrix?", "background": "Source: Open Problem Garden. Original node ID: 48368. URL: http://www.openproblemgarden.org/op/are_almost_all_graphs_determined_by_their_spectrum.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_almost_all_graphs_determined_by_their_spectrum\n- Subject(s): Graph Theory\n- Keywords: cospectral; graph invariant; spectrum\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 26th, 2013 by mdevos\n\nProblem-page discussion:\nWe say that two non-isomorphic graphs are cospectral if their adjacency matrices have the same spectrum (counted with multiplicity). A graph is spectrally determined if no other graphs are cospectral to it. It is unclear to me (M. DeVos) how to attribute this problem, but it was considered already in the 1950's and resonates with the famous problem \"Can you hear the shape of a drum?\" ([vDH]).\n\nA priori, it might seem plausible for all graphs to be spectrally determined.. but this is false. The smallest counterexample is the cospectral pair given by $K_{1,4}$ and the graph obtained from $C_4$ by adding an isolated vertex. Some rich families of cospectral graphs are provided by strongly regular graphs, since any two strongly regular graphs with the same parameters will be cospectral.\n\nFor the special case of trees, Schwenk proved almost all trees are not spectrally determined. This was sharpened by Godsil and Mckay who showed that almost every tree $T$ has a cospectral graph $T'$ so that in addition the complements of $T$ and $T'$ are cospectral. Furthermore, an operation called Godsil-Mckay Switching defined by these authors gives a powerful tool to produce general graphs which are cospectral.\n\nOn the flip side, we seem to have a lack of good tools to prove that a given graph is spectrally determined.\n\nBibliography:\n[vDH] E. R. van Dam and W. H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra and its Applications 373 (2003) 241–272.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Are almost all graphs determined by their spectrum?\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3132, "problem_number": "OPG-49795", "title": "Minimum number of arc-disjoint transitive subtournaments of order 3 in a tournament", "statement": "Conjecture If $T$ is a tournament of order $n$, then it contains $\\left \\lceil n(n-1)/6 - n/3\\right\\rceil$ arc-disjoint transitive subtournaments of order 3.", "background": "Source: Open Problem Garden. Original node ID: 49795. URL: http://www.openproblemgarden.org/op/minimum_number_of_transitive_subtournaments_of_order_3_in_a_tournament.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/minimum_number_of_transitive_subtournaments_of_order_3_in_a_tournament\n- Author(s): Yuster, Raphael\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 21st, 2013 by fhavet\n\nProblem-page discussion:\nIf true the conjecture would be tight as shown by any tournament whose vertex set can be decomposed into $3$ sets $V_1, V_2, V_3$ of size $\\lceil n/3 \\rceil$ or $\\lfloor n/3\\rfloor$ and such that $V_1\\rightarrow V_2$, $V_2\\rightarrow V_3$ and $V_3\\rightarrow V_1$.\n\nLet $TT_3$ denote the transitive tournament of order 3. A $TT_3$-packing of a digraph $D$ is a set of arc-disjoint copies of $TT_3$ subgraphs of $D$.\n\nLet $f(n)$ be the minimum size of a $TT_3$-packing over all tournaments of order $n$. The conjecture and its tightness say $f(n)= \\left \\lceil n(n-1)/6 - n/3\\right\\rceil$.\n\nThe best lower bound for $f(n)$ so far is due to Kabiya and Yuster [KY] proved that $f(n) > \\frac{41}{300} n^2(1+o(1))$.\n\nBibliography:\n[KY] M. Kabiya and R. Yuster. Packing transitive triples in a tournament. Ann. Comb. 12 (2008), no. 3, 291–-306.\n\n*[Y] R. Yuster. The number of edge-disjoint transitive triples in a tournament. Discrete Math. 287 (2004). no. 1-3,187--191.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Minimum number of arc-disjoint transitive subtournaments of order 3 in a tournament\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3133, "problem_number": "OPG-57613", "title": "Imbalance conjecture", "statement": "Conjecture Suppose that for all edges $e\\in E(G)$ we have $imb(e)>0$. Then $M_{G}$ is graphic.", "background": "Source: Open Problem Garden. Original node ID: 57613. URL: http://www.openproblemgarden.org/op/imbalance_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/imbalance_conjecture\n- Author(s): Kozerenko, Sergiy\n- Subject(s): Graph Theory\n- Keywords: edge imbalance; graphic sequences\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 24th, 2013 by Sergiy Kozerenko\n\nProblem-page discussion:\nConsider simple undirected graph $G$ and let $e=uv\\in E(G)$.\n\nThe imbalance of the edge $e$ defined as $imb(e)=|deg(u)-deg(v)|$.\n\nThe multiset of all edge imbalances of $G$ is denoted by $M_{G}$.\n\nNote, that conjecture is verified for all such graphs with $\\leq 9$ vertices.\n\nBibliography:\n*Graphs with graphic imbalance sequences\n\nBibliography links:\n- Graphs with graphic imbalance sequences: http://mathoverflow.net/questions/140819/graphs-with-graphic-imbalance-sequences\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Imbalance conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3134, "problem_number": "OPG-59908", "title": "Fractional Hadwiger", "statement": "Conjecture For every graph $G$,\n(a) $\\chi_f(G)\\leq\\text{had}(G)$\n(b) $\\chi(G)\\leq\\text{had}_f(G)$\n(c) $\\chi_f(G)\\leq\\text{had}_f(G)$.", "background": "Source: Open Problem Garden. Original node ID: 59908. URL: http://www.openproblemgarden.org/op/fractional_hadwiger.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/fractional_hadwiger\n- Author(s): Harvey, Daniel J.; Reed, Bruce A.; Seymour, Paul D.; Wood, David R.\n- Subject(s): Graph Theory\n- Keywords: fractional coloring, minors\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: March 16th, 2014 by David Wood\n\nProblem-page discussion:\nHere $\\chi$ is the chromatic number, $\\chi_f$ is the fractional chromatic number, $\\text{had}$ is the Hadwiger number, and $\\text{had}_f$ is the fractional Hadwiger number (which was recently introduced independently by Fox [F] and Pedersen [P]).\n\nIt is well known and easily proved (see [HW]) that\n$\\chi_f(G)\\leq\\chi(G)\\text{ and }\\text{had}(G)\\leq\\text{had}_f(G)\\leq\\text{tw}(G)+1,$\nwhere $\\text{tw}(G)$ is the treewidth of $G$.\n\nHadwiger's famous conjecture, $\\chi(G)\\leq\\text{had}(G)$, bridges the gap in the above inequalities. The above conjectures therefore are weaker than Hadwiger's conjecture. Note that Conjecture (a) implies Conjecture (c), and Conjecture (b) implies Conjecture (c).\n\nNote that Reed and Seymour [RS] proved that $\\chi_f(G)\\leq2\\,\\text{had}(G)$.\n\nConjecture (a) is due to Reed and Seymour [RS]. Conjecture (b) is due to Harvey and Wood [HW]. Conjecture (c) is independently due to Harvey and Wood [HW] and Pedersen [P].\n\nPedersen [P] presents a natural equivalent formulation of Conjecture (c).\n\nBibliography:\n*[HW] Daniel J. Harvey, David R. Wood, Parameters tied to treewidth. arXiv:1312.3401, 2013.\n\n[F] Jacob Fox. Constructing dense graphs with sublinear Hadwiger number. J. Combin. Theory Ser. B (to appear).\n\n*[P] Anders Sune Pedersen. Contributions to the Theory of Colourings, Graph Minors, and Independent Sets, PhD thesis, Department of Mathematics and Computer Science University of Southern Denmark, 2011.\n\n*[RS] Bruce A. Reed, Paul D. Seymour, Fractional colouring and Hadwiger's conjecture. J. Combin. Theory Ser. B, 74(2), 147-152.\n\nBibliography links:\n- arXiv:1312.3401: http://www.arxiv.org/abs/1312.3401\n- Constructing dense graphs with sublinear Hadwiger number: http://www.arxiv.org/abs/1108.4953\n- Contributions to the Theory of Colourings, Graph Minors, and Independent Sets: http://www.imada.sdu.dk/%7Easp/Thesis_2ed.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"Fractional Hadwiger\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3135, "problem_number": "OPG-59952", "title": "Chromatic Number of Common Graphs", "statement": "Question Do common graphs have bounded chromatic number?", "background": "Source: Open Problem Garden. Original node ID: 59952. URL: http://www.openproblemgarden.org/op/chromatic_number_of_common_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/chromatic_number_of_common_graphs\n- Author(s): Hatami, H; Hladký, J.; Kráľ, D.; Norine, S.; Razborov, A.\n- Subject(s): Graph Theory\n- Keywords: common graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 15th, 2014 by David Wood\n\nProblem-page discussion:\nA graph $H$ is common if the sum of the number of copies of $H$ in a graph $G$ and the number in the complement of $G$ is asymptotically minimised by taking $G$ to be a random graph (see [HHKNR] for a formal definition).\n\nGoodman proved that $K_3$ is common [G]. Erdös [E] conjectured that every complete graph is common. Later, this conjecture was extended to all graphs by Burr and Rosta [BR]. Sidorenko [S89] disproved Burr and Rosta’s conjecture by showing that a triangle with a pendant edge is not common. Conjectures of Erdös and Simonovits [ES] and Sidorenko [S91,S93] imply that every bipartite graph is common. Disproving the first conjecture of Erdös, Thomason proved that $K_4$ is not common [T]. More generally, Jagger, Šťovíček and Thomason proved that no common graph contains $K_4$ as a subgraph [JST]. The 5-wheel is an example of a 4-chromatic common graph [HHKNR].\n\nBibliography:\n[BR] Burr, S. A. and Rosta, V. On the Ramsey multiplicities of graphs: Problems and recent results. J. Graph Theory 4 (1980) 347–361.\n\n[E] Paul Erdös, On the number of complete subgraphs contained in certain graphs. Magyar Tud. Akad. Mat. Kutato ́ Int. Ko ̈zl. 7 (1962) 459–464.\n\n[ES] Paul Erdös and Miklós Simonovits (1984) Cube-supersaturated graphs and related problems. In Progress in Graph Theory: Waterloo, Ont., 1982, Academic Press, pp. 203–218.\n\n*[HHKNR] H. Hatami, J. Hladký, D. Kráľ, S. Norine, A. Razborov: Non-three-colorable common graphs exist, Combinatorics, Probability and Computing 21 (2012), 734–742.\n\n[JST] Jagger, Chris; Šťovíček, Pavel; Thomason, Andrew. Multiplicities of subgraphs. Combinatorica 16 (1996) 123–141.\n\n[S89] Sidorenko, A. Cycles in graphs and functional inequalities. Mat. Zametki 46 (1989) 72–79, 104.\n\n[S91] Sidorenko, A. Inequalities for functionals generated by bipartite graphs. Diskret. Mat. 3 (1991) 50–65.\n\n[S93] Sidorenko, A. A correlation inequality for bipartite graphs. Graphs Combin. 9 (1993) 201–204.\n\n[T] Thomason, Andrew. A disproof of a conjecture of Erdo ̋s in Ramsey theory, J. London Math. Soc. (2) 39 (1989) 246–255.\n\nRelated:\nRelated problems\nSidorenko's Conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Chromatic Number of Common Graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3136, "problem_number": "OPG-59997", "title": "Circular flow numbers of $r$-graphs", "statement": "A nowhere-zero $r$-flow $(D(G),\\phi)$ on $G$ is an orientation $D$ of $G$ together with a function $\\phi$ from the edge set of $G$ into the real numbers such that $1 \\leq |\\phi(e)| \\leq r-1$, for all $e \\in E(G)$, and $\\sum_{e \\in E^+(v)}\\phi(e) = \\sum_{e \\in E^-(v)}\\phi(e), \\textrm{ for all } v \\in V(G)$.\n\nA $(2t+1)$-regular graph $G$ is a $(2t+1)$-graph if $|\\partial_G(X)| \\geq 2t+1$ for every $X \\subseteq V(G)$ with $|X|$ odd.\n\nConjecture Let $t > 1$ be an integer. If $G$ is a $(2t+1)$-graph, then $F_c(G) \\leq 2 + \\frac{2}{t}$.", "background": "Source: Open Problem Garden. Original node ID: 59997. URL: http://www.openproblemgarden.org/op/circular_flow_numbers_of_r_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/circular_flow_numbers_of_r_graphs\n- Author(s): Steffen, Eckhard\n- Subject(s): Graph Theory\n- Keywords: flow conjectures; nowhere-zero flows\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 6th, 2015 by Eckhard Steffen\n\nProblem-page discussion:\nSince every $(2t+1)$-regular class 1 graph is a $(2t+1)$-graph, the truth of this conjecture would imply the truth of the conjecture on the circular flow number of regular class 1 graphs. If it is true for even $t$, say $t=2t'$, then Jaeger's modular orientation conjecture is true for $(4t'+1)$-regular graphs and hence, by a result of Jaeger, it would imply the truth of Tutte's 5-flow conjecture. For $t=2$ it is Tutte's 3-flow conjecture.\n\nBibliography:\n*[ES_2015]E. Steffen, Edge-colorings and circular flow numbers on regular graphs, J. Graph Theory 79, 1–7, 2015\n\nRelated:\nRelated problems\n3-flow conjecture\nJaeger's modular orientation conjecture\nCircular flow number of regular class 1 graphs\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Circular flow numbers of $r$-graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3137, "problem_number": "OPG-60027", "title": "3-Decomposition Conjecture", "statement": "Conjecture (3-Decomposition Conjecture) Every connected cubic graph $G$ has a decomposition into a spanning tree, a family of cycles and a matching.", "background": "Source: Open Problem Garden. Original node ID: 60027. URL: http://www.openproblemgarden.org/op/3_decomposition_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/3_decomposition_conjecture\n- Author(s): Arthur; Hoffmann-Ostenhof\n- Subject(s): Graph Theory\n- Keywords: cubic graph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: January 24th, 2017 by arthur\n\nProblem-page discussion:\nWe state the conjecture in a more precise manner:\n\nLet $G$ be a connected cubic graph. Then $G$ contains a spanning tree $H_1$, a $2$-regular subgraph $H_2$ and a matching $H_3$ (where only $H_3$ and not $H_1$ or $H_2$ may be empty) such that $E(H_1) \\cup E(H_2) \\cup E(H_3) = E(G)$ and $E(H_i) \\cap E(H_j) =\\emptyset$ for every $\\{i,j\\} \\subseteq \\{1,2,3\\}$ with $i\\not=j$.\n\nThe conjecture holds for all hamiltionian cubic graphs and for all connected planar cubic graphs, see [1] and see also [7].\n\nEvery cubic graph G which has a spanning tree T such that every vertex of T has degree three or one (such spanning tree T is called a HIST) obviously satisfies this conjecture. But not every connected cubic graph has a HIST, see [2].\n\nThe 3-Decomposition Conjecture has been shown to be equivalent to the following conjecture:\n\nConjecture (2-Decomposition Conjecture) Let $G$ be connected graph where every vertex has degree two or three. Suppose that for every cycle $C$ of $G$, $G-E(C)$ is disconnected, then $G$ has a decomposition into a spanning tree $T$ and a matching $M$, i.e $G-M=T$.\n\nNote that every cycle $C$ which passes through a vertex of degree two satisfies the condition that G-E(C) is disconnected.\n\nRemark: The 3-Decomposition Conjecture has also been shown to hold for other classes of cubic graphs, see for instance [3,4]. A survey on the 3-Decompostion conjecture has been given by the author 2015 in Pilsen (at that time the planar case was still open) see iti.zcu.cz/plzen15/talks/1-2a-Arthur-Survey_decomposition.ppt (and press play if you find the play button). Note that there are several papers on the problem whether a planar graph $G$ has a matching $M$ such that $G-M$ is acyclic, see for instance [6].\n\nBibliography:\n[1] Arthur Hoffmann-Ostenhof, Tomáš Kaiser, Kenta Ozeki, \\arXiv[Decomposing planar cubic graphs] 1609.05059 [math.CO]\n[2] Arthur Hoffmann-Ostenhof, Kenta Ozeki, \\arXiv[On HISTs in Cubic Graphs] 1507.07689 [math.CO]\n[3] F. Abdolhosseini, S. Akbari, H. Hashemi, M.S. Moradian, \\arXiv[Hoffmann-Ostenhof's conjecture for traceable cubic graphs] 1607.04768[math.CO]\n[4] Anna Bachstein, Dong Ye (talk): www.rwoodroofe.math.msstate.edu/workshop2014/bachstein_slides.pdf\n[5] Arthur Hoffmann-Ostenhof (talk): www.iti.zcu.cz/plzen15/talks/1-2a-Arthur-Survey_decomposition.ppt\n[6] Yingqian Wang, Qijun Zhang, Discrete Mathematics 311 (2011) 844–849, Decomposing a planar graph with girth at least 8 into a forest and a matching\n[7] Kenta Ozeki, Dong Ye, Decomposing plane cubic graphs, European Journal of Combinatorics 52 (2016) 40-46.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"3-Decomposition Conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3138, "problem_number": "OPG-60029", "title": "Cycle Double Covers Containing Predefined 2-Regular Subgraphs", "statement": "Conjecture Let $G$ be a $2$-connected cubic graph and let $S$ be a $2$-regular subgraph such that $G-E(S)$ is connected. Then $G$ has a cycle double cover which contains $S$ (i.e all cycles of $S$ ).", "background": "Source: Open Problem Garden. Original node ID: 60029. URL: http://www.openproblemgarden.org/op/cycle_double_covers_containing_predefined_2_regular_subgraphs.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cycle_double_covers_containing_predefined_2_regular_subgraphs\n- Author(s): Arthur; Hoffmann-Ostenhof\n- Subject(s): Graph Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 21st, 2017 by arthur\n\nProblem-page discussion:\nUsed definitions in the above conjecture: a \"cycle\" is a connected 2-regular subgraph, a \"cycle double cover\" of a graph $G$ is a set of cycles of $G$ such that every edge of $G$ is contained in precisely two cycles of the set. This conjecture has been motivated by Theorem 3, respectively, Theorem 4 in www.arxiv.org/abs/1711.10614. A weaker conjecture (Conjecture 14) has been stated in \"Snarks with special spanning trees\" (see www.arxiv.org/abs/1706.05595).\n\nRelated:\nRelated problems\nCycle double cover conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Cycle Double Covers Containing Predefined 2-Regular Subgraphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3139, "problem_number": "OPG-60030", "title": "Monochromatic vertex colorings inherited from Perfect Matchings", "statement": "Conjecture For which values of $n$ and $d$ are there bi-colored graphs on $n$ vertices and $d$ different colors with the property that all the $d$ monochromatic colorings have unit weight, and every other coloring cancels out?", "background": "Source: Open Problem Garden. Original node ID: 60030. URL: http://www.openproblemgarden.org/op/monochromatic_vertex_colorings_inherited_from_perfect_matchings.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/monochromatic_vertex_colorings_inherited_from_perfect_matchings\n- Subject(s): Graph Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: March 4th, 2019 by Mario Krenn\n\nProblem-page discussion:\nBackground: This and many related questions are directly inspired from quantum physics, and their solutions would directly contribute to new understanding in quantum physics.\n\nBi-Colored Graph: A bi-colored weighted graph $G=(V(G),E(G))$, on $n$ vertices with $d$ colors is an undirected, not necessarily simple graph where there is a fixed ordering of the vertices $V(G)=v_1, \\ldots, v_n$ and to each edge $e \\in E(G)$ there is a complex weight $w_e$ and an ordered pair of (not necessarily different) colors $(c_1(e),c_2(e))$ associated with it from the $d$ possible colors. We say that an edge is monochromatic if the associated pair of colors are not different, otherwise the edge is bi-chromatic. Moreover, if $e$ is an edge incident to the vertices $v_i,v_j \\in V(G)$ with $i2$ is a connected cubic graph admitting a $3$-edge coloring. Then there is an edge $e \\in E(G)$ such that the cubic graph homeomorphic to $G-e$ has a $3$-edge coloring.", "background": "Source: Open Problem Garden. Original node ID: 60046. URL: http://www.openproblemgarden.org/op/3_edge_coloring_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/3_edge_coloring_conjecture\n- Author(s): Arthur; Hoffmann-Ostenhof\n- Subject(s): Graph Theory\n- Keywords: 3-edge coloring; 4-flow; removable edge\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Prize listed: no\n- Posted: April 28th, 2020 by arthur\n\nProblem-page discussion:\nReformulation via 4-flows:\n\nConjecture Suppose $G$ is a cubic graph with a nowhere-zero $4$-flow, then there is an edge $e \\in E(G)$ such that $G-e$ has a nowhere-zero $4$-flow.\n\nComments:\n- June 23rd, 2022 | Anonymous | Context: Is this conjecture missing some greater context? It seems obviously false on its own\n- June 22nd, 2022 | Anonymous | question: wouldn't removing any edge from a cubic graph make the graph not cubic?\n- August 3rd, 2021 | Anonymous | A counterexample?: What would be the cubic graph homeomorphic to K4-e? I think I can show there does not exist a cubic graph homeomorphic to K4-e. If so, this would seem to contradict the conjecture's claim.\n- November 13th, 2020 | Anonymous | Is there yet any progress on this problem?: Hello, I would like to know whether anybody made any progress on this. I tried to google and found nothing. Also, why is there nothing in Bibliography of this problem? Is there any paper involving or proposing it? Thanks in advance\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"3-Edge-Coloring Conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3142, "problem_number": "OPG-60055", "title": "Chromatic number of $\\frac{3}{3}$-power of graph", "statement": "Let $G$ be a graph and $m,n\\in \\mathbb{N}$. The graph $G^{\\frac{m}{n}}$ is defined to be the $m$-power of the $n$-subdivision of $G$. In other words, $G^{\\frac{m}{n}}=(G^{\\frac{1}{n}})^m$.\n\nConjecture Let $G$ be a graph with $\\Delta(G)\\geq 2$. Then $\\chi(G^{\\frac{3}{3}})\\leq 2\\Delta(G)+1$.", "background": "Source: Open Problem Garden. Original node ID: 60055. URL: http://www.openproblemgarden.org/op/chromatic_number_of_frac_3_3_power_of_graph.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/chromatic_number_of_frac_3_3_power_of_graph\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 20th, 2023 by Iradmusa\n\nBibliography:\n[1] Mahsa Mozafari-Nia and M. N. Iradmusa, Simultaneous coloring of vertices and incidences of graphs, Australasian Journal of Combinatorics, Vol. 85, Mo. 3, pp. 287-307, 2023.\n\n[2] Mahsa Mozafari-Nia and M. N. Iradmusa, Simultaneous coloring of vertices and incidences of outerplanar graphs, Electronic Journal of Graph Theory and Applications, Vol.11, No.1, pp.245-262, 2023.\n\n[3] Mahsa Mozafari-Nia and M. N. Iradmusa, A note on coloring of 3/3-power of subquartic graphs, Australasian Journal of Combinatorics, Vol. 79, No. 3, pp. 454-460, 2021.\n\n[4] M. N. Iradmusa, A short proof of 7-colorability of 3/3-power of subcubic graphs, Iranian Journal of Science and Technology, Transactions A: Science, Vol. 44, No. 1, pp. 225-226, 2020.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Chromatic number of $\\frac{3}{3}$-power of graph\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3143, "problem_number": "OPG-160", "title": "57-regular Moore graph?", "statement": "Question Does there exist a 57-regular graph with diameter 2 and girth 5?", "background": "Source: Open Problem Garden. Original node ID: 160. URL: http://www.openproblemgarden.org/op/57_regular_moore_graph.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/57_regular_moore_graph\n- Author(s): Hoffman, Alan J.; Singleton, Robert R.\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: cage; Moore graph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 18th, 2007 by mdevos\n\nProblem-page discussion:\nA Moore graph is a graph with diameter $d$ and girth $2d+1$. It is known that every Moore graph is regular [S] (even distance-regular), and two (rather trivial) families of such graphs are provided by complete graphs and odd cycles. In one of the founding papers in the subject of algebraic graph theory, Hoffman and Singleton [HS] proved that every $r$-regular Moore graph with diameter 2 must have $r \\in \\{2,3,7,57\\}$. For $r=2$ and $r=3$ such graphs exist, are unique, and are familiar: the pentagon, and the Petersen graph. For $r=7$ Hoffman and Singleton constructed such a graph - now known as the Hoffman-Singleton graph, but for $r=57$ we are still uncertain whether such a graph exists.\n\nThe pentagon, the Petersen graph, and the Hoffman-Singleton graph are all very highly symmetric graphs, and much of the interest in these objects is related to exceptional phenomena in small finite groups. In contrast to this, Higman proved that a 57-regular Moore graph cannot be vertex transitive (see [C]). In some sense, this is indication that even if a 57-regular Moore graph exists, it will be of less interest than its younger siblings. Nevertheless, as a lingering problem left by one of the first papers in algebraic graph theory, this is viewed as an important question.\n\nThere are some easily established properties of a 57-regular Moore graph. For instance it must have 3250 vertices and independence number at most 400. However it seems not nearly enough is known to narrow the search sufficiently.\n\nBibliography:\n[C] P. J. Cameron, Automorphisms of graphs in: Selected topics in graph theory, Volume 2, eds. L. W. Beineke and R. J. Wilson (Academic Press, London) 1983, pp. 89-127. MathSciNet\n\n* [HS] A. J. Hoffman and R. R. Singleton, On Moore graphs with diameters 2 and 3. IBM J. Res. Develop. 4 (1960) 497--504. MathSciNet\n\n[S] R. R. Singleton, There is no irregular Moore graph. American Mathematical Monthly 75, vol 1 (1968) 42–43. MathSciNet\n\nSource links:\n- regular: http://en.wikipedia.org/wiki/regular graph\n- diameter: http://en.wikipedia.org/wiki/diameter (graph theory)\n- girth: http://en.wikipedia.org/wiki/girth\n\nDiscussion links:\n- Moore graph: http://en.wikipedia.org/wiki/Moore graph\n- Petersen graph: http://en.wikipedia.org/wiki/Petersen graph\n- Hoffman-Singleton graph: http://en.wikipedia.org/wiki/Hoffman-Singleton graph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0797250\n- On Moore graphs with diameters 2 and 3: http://www.research.ibm.com/journal/rd/045/ibmrd0405H.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0140437\n- There is no irregular Moore graph: http://www.jstor.org/view/00029890/di991528/99p1853x/0\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0225679\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"57-regular Moore graph?\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3144, "problem_number": "OPG-161", "title": "Hamiltonian paths and cycles in vertex transitive graphs", "statement": "Problem Does every connected vertex-transitive graph have a Hamiltonian path?", "background": "Source: Open Problem Garden. Original node ID: 161. URL: http://www.openproblemgarden.org/op/hamiltonian_paths_and_cycles_in_vertex_transitive_graphs.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hamiltonian_paths_and_cycles_in_vertex_transitive_graphs\n- Author(s): Lovasz, Laszlo\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: cycle; hamiltonian; path; vertex-transitive\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 18th, 2007 by mdevos\n\nProblem-page discussion:\nThe question posed here is due to Lovasz [L], but the general problem of finding Hamiltonian paths and cycles in highly symmetric graphs is much older. Knuth has traced it back to bell ringing, and it appears again in gray codes and in the knight's tour of a chessboard.\n\nVertex-transitive graphs are, of course, very special, very well-behaved graphs, and it seems unsurprising that many of them have Hamiltonian cycles. What is surprising is that there are only five connected ones known which do not have Hamiltonian cycles. This list consists of the complete graph on 2 vertices, the Petersen graph, Coxeter's graph, and the graphs obtained from Petersen and Coxeter by truncating every vertex (inflate each vertex to a triangle). In particular, we do not know of a vertex transitive graph without a Hamiltonian path.\n\nInterestingly, there seems to be considerable disagreement among experts as to what the answer will be. On one hand, there does not appear to be any particular reason why vertex-transitive graphs should almost always have Hamiltonian cycles. On the other hand, such graphs have been studied and searched for at great length, and so far every one investigated with the exception of the five listed above has proved to have a Hamiltonian cycle. Babai formulated the following conjecture which is in quite sharp contrast to the problem above.\n\nConjecture (Babai [B96]) There exists $\\epsilon > 0$ so that there are infinitely many connected vertex-transitive graphs $G$ with longest cycle of length $<(1-\\epsilon)|V(G)|$.\n\nFor general vertex-transitive graphs, very little is known. Babai [B79] has shown that a vertex-transitive graph on $n$ vertices has a cycle of length $\\ge \\sqrt{3n}$, but (though a very clever arguement) this is obviously quite far from the conjecture. Considerable attention has been given to the special case of Cayley graphs. Here we have the following conjecture.\n\nConjecture Every connected Cayley graph (apart from $K_2$ ) has a Hamiltonian cycle.\n\nThe above conjecture is not difficult to prove for abelian groups. Witte [W] proved it for $p$-groups, and it has also been established for certain special types of generating sets. Two other results of note are a theorem of Pak-Radocic [PR] showing that every group $G$ has a generating set of size $\\le \\log_2(|G|)$ for which the corresponding Cayley graph is Hamiltonian, and a theorem of Krivelevich-Sudakov [KS] showing that almost surely taking a random set of $\\log^5(|G|)$ elements of $G$ as generators yields a Hamiltonian graph.\n\nBibliography:\n[B79] L. Babai, Long cycles in vertex-transitive graphs. J. Graph Theory 3 (1979), no. 3, 301--304. MathSciNet\n\n[B96] L. Babai, Automorphism groups, isomorphism, reconstruction, in Handbook of Combinatorics, Vol. 2, Elsevier, 1996, 1447-1540. MathSciNet\n\n[KS] M. Krivelevich and B. Sudakov, Sparse pseudo-random graphs are Hamiltonian. J. Graph Theory 42 (2003), no. 1, 17--33. MathSciNet\n\n[L] L. Lov\\'{a}sz, \"Combinatorial structures and their applications\", (Proc. Calgary Internat. Conf., Calgary, Alberta, 1969), pp. 243-246, Problem 11, Gordon and Breach, New York, 1970.\n\n[PR] I. Pak and R. Radocic, Hamiltonian paths in Cayley graphs, preprint\n\n[W] D. Witte, Cayley digraphs of prime-power order are Hamiltonian. J. Combin. Theory Ser. B 40 (1986), no. 1, 107--112. MathSciNet\n\n[WG] D. Witte and J.A. Gallian, A survey: Hamiltonian cycles in Cayley graphs. Discrete Math. 51 (1984), no. 3, 293--304. MathSciNet\n\nSource links:\n- vertex-transitive graph: http://en.wikipedia.org/wiki/vertex-transitive graph\n- Hamiltonian path: http://en.wikipedia.org/wiki/Hamiltonian path\n\nDiscussion links:\n- bell ringing: http://en.wikipedia.org/wiki/bell ringing\n- gray codes: http://en.wikipedia.org/wiki/gray codes\n- knight's tour: http://en.wikipedia.org/wiki/knight's tour\n- Petersen graph: http://en.wikipedia.org/wiki/Petersen graph\n- Coxeter's graph: http://mathworld.wolfram.com/CoxeterGraph.html\n- Cayley graphs: http://en.wikipedia.org/wiki/cayley graph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0542553\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1373683\n- Sparse pseudo-random graphs are Hamiltonian: http://www.math.princeton.edu/%7Ebsudakov/pseudo-hamiltonian.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1943104\n- Hamiltonian paths in Cayley graphs: http://www-math.mit.edu/%7Epak/hamcayley8.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0830597\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0762322\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamiltonian paths and cycles in vertex transitive graphs\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3145, "problem_number": "OPG-345", "title": "Triangle free strongly regular graphs", "statement": "Problem Is there an eighth triangle free strongly regular graph?", "background": "Source: Open Problem Garden. Original node ID: 345. URL: http://www.openproblemgarden.org/op/triangle_free_strongly_regular_graphs.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/triangle_free_strongly_regular_graphs\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: strongly regular; triangle free\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 28th, 2007 by mdevos\n\nProblem-page discussion:\nA regular graph $G$ is strongly regular if there exist integers $\\lambda, \\mu$ so that every pair of adjacent vertices have exactly $\\lambda$ common neighbors, and every pair of nonadjacent vertices have exactly $\\mu$ common neighbors. To eliminate degeneracies, we shall further assume that $\\mu \\ge 1$. If $G$ is $k$-regular and $|V(G)| = n$, then we say that $G$ is a $(n,k,\\lambda,\\mu)$ strongly regular graph.\n\nThere are exactly seven triangle-free strongly regular graphs known: The five cycle, the Petersen Graph, The Clebsch Graph, the Hoffman-Singleton Graph, The Gewirtz Graph, the Higman-Sims Graph, and a $(77,16,0,4)$ strongly regular subgraph of the Higman-Sims graph. Every Moore Graph of diameter 2 is a triangle-free strongly regular graph, so if there is a 57-regular Moore Graph of diameter 2, this would add another to the list.\n\nSee Andries Brouwer's graph descriptions for more on these graphs.\n\nBibliography:\n[G] C. D. Godsil, Problems in Algebraic Combinatorics, Electronic Journal of Combinatorics, Volume 2, F1\n\nDiscussion links:\n- strongly regular: http://en.wikipedia.org/wiki/strongly regular graph\n- Petersen Graph: http://en.wikipedia.org/wiki/Petersen Graph\n- Hoffman-Singleton Graph: http://en.wikipedia.org/wiki/Hoffman-Singleton Graph\n- Moore Graph: http://en.wikipedia.org/wiki/Moore graph\n- 57-regular Moore Graph: http://www.openproblemgarden.org/?q=op/57_regular_moore_graph\n- Andries Brouwer's graph descriptions: http://www.win.tue.nl/%7Eaeb/drg/graphs/index.html\n\nBibliography links:\n- Problems in Algebraic Combinatorics: http://www.combinatorics.org/Volume_2/PDFFiles/v2i1f1.pdf\n\nComments:\n- May 30th, 2020 | Anonymous | Higman-Sims Graph: In fact, all (seven) known primitive triangle-free strongly regular graphs are actual *subgraphs* of the Higman-Sims graph (which btw was first constructed by Dale Mesner). A Moore graph of degree 57 would of course break this mold.\n- February 23rd, 2012 | Anonymous | Reply..: I hate math..Lol _________________________________________________________________________________________ toll free number\n- February 8th, 2012 | Anonymous | The complement of $2 K_n$: The complement of $2 K_n$ is triangle free srg with parameters $(2n,n,0,n)$. Probably this infinite case should be excluded from the conjecture.\n- April 21st, 2011 | Anonymous | If the number of the: If the number of the vertices are even, we can determine regular triangle free graphs up to half the number of vertices of any degree. However, this does not hold true if the vertices numbers are odd. Some of the examples of even vertices graphs are Petersen graph (10 vertices), Heawood graph (14 vertices), Clebsch graph (16 vertices), Pappus graph (18 vertices) and odd vertices graph includes Schläfli graph (27 vertices), Perkel graph (57 vertices). 800 numbers\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Triangle free strongly regular graphs\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3146, "problem_number": "OPG-348", "title": "Half-integral flow polynomial values", "statement": "Let $\\Phi(G,x)$ be the flow polynomial of a graph $G$. So for every positive integer $k$, the value $\\Phi(G,k)$ equals the number of nowhere-zero $k$-flows in $G$.\n\nConjecture $\\Phi(G,5.5) > 0$ for every 2-edge-connected graph $G$.", "background": "Source: Open Problem Garden. Original node ID: 348. URL: http://www.openproblemgarden.org/op/half_integral_flow_polynomial_values.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/half_integral_flow_polynomial_values\n- Author(s): Mohar, Bojan\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: nowhere-zero flow\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 31st, 2007 by mohar\n\nProblem-page discussion:\nBy Seymour's 6-flow theorem, $\\Phi(G,k) > 0$ for every 2-edge-connected graph $G$ and every integer $k\\ge6$.\n\nIt would be interesting to find any non-integer rational number $x>5$ so that $\\Phi(G,x) > 0$ for every 2-edge-connected graph $G$. It is known that zeros of flow polynomials are dense in the complex plane.\n\nSource links:\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Half-integral flow polynomial values\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3147, "problem_number": "OPG-372", "title": "Ramsey properties of Cayley graphs", "statement": "Conjecture There exists a fixed constant $c$ so that every abelian group $G$ has a subset $S \\subseteq G$ with $-S = S$ so that the Cayley graph ${\\mathit Cayley}(G,S)$ has no clique or independent set of size $> c \\log |G|$.", "background": "Source: Open Problem Garden. Original node ID: 372. URL: http://www.openproblemgarden.org/op/ramsey_properties_of_cayley_graphs.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/ramsey_properties_of_cayley_graphs\n- Author(s): Alon, Noga\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: Cayley graph; Ramsey number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 10th, 2007 by mdevos\n\nProblem-page discussion:\nThe classic bounds from Ramsey theory show that every $n$ vertex graph must have either a clique or an independent set of size $c \\log n$ and further random graphs almost surely have this property (using different values of $c$ ). The above conjecture asserts that every group has a Cayley graph with similar behavior.\n\nImproving upon some earlier results of Agarwal et. al. [AAAS], Green [G] proved that there exists a constant $c$ so that whenever a set $S \\subseteq {\\mathbb Z}_n$ is chosen at random, and we form the graph with vertex set ${\\mathbb Z}_n$ and two vertices $i$, $j$ joined if $i+j \\in S$, then this graph almost surely has both maximum clique size and maximum independent size $O(\\log n)$. The reader should note that such graphs are not generally Cayley graphs - although the definition is similar.\n\nAs a word of caution, Green [G] also shows that a randomly chosen subset of the group ${\\mathbb Z}_2^n$ almost surely has both max. clique and max. independent set of size $\\Theta( \\log N \\log \\log N )$ where $N = 2^n$.\n\nBibliography:\n[AAAS] P. K. Agarwal, N. Alon, B. Aronov, S. Suri, Can visibility graphs be represented compactly? Discrete Comput. Geom. 12 (1994), no. 3, 347--365. MathSciNet\n\n*[C] Problem BCC14.6 from the BCC Problem List (edited by Peter Cameron)\n\n[G] B. Green, Counting sets with small sumset, and the clique number of random Cayley graphs, Combinatorica 25 (2005), no. 3, 307--326. MathSciNet\n\nSource links:\n- Cayley graph: http://en.wikipedia.org/wiki/cayley graph\n\nBibliography links:\n- Can visibility graphs be represented compactly?: http://www.math.tau.ac.il/%7Enogaa/PDFS/main.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1298916\n- BCC Problem List: http://www.maths.qmul.ac.uk/%7Epjc/bcc/allprobs.pdf\n- Counting sets with small sumset, and the clique number of random Cayley graphs: http://arxiv.org/PS_cache/math/pdf/0304/0304183v2.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2141661\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 17.\n\nAttempt notes:\nTarget:\nMake progress on \"Ramsey properties of Cayley graphs\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3148, "problem_number": "OPG-407", "title": "Laplacian Degrees of a Graph", "statement": "Conjecture If $G$ is a connected graph on $n$ vertices, then $c_k(G) \\ge d_k(G)$ for $k = 1, 2, \\dots, n-1$.", "background": "Source: Open Problem Garden. Original node ID: 407. URL: http://www.openproblemgarden.org/op/laplacian_degrees_of_a_graph.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/laplacian_degrees_of_a_graph\n- Author(s): Guo, Ji-Ming\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: degree sequence; Laplacian matrix\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 22nd, 2007 by Robert Samal\n\nProblem-page discussion:\n(Reproduced from [M].)\n\nLet $L = D - A$ be the Laplacian matrix of a graph $G$ of order $n$. Let $t_k$ be the $k$-th largest eigenvalue of $L$ ( $k = 1,\\dots,n$ ). For the purpose of this problem, we call the number $$c_k = c_k(G) = t_k + k - 2$$the$k$-th Laplacian degree of$G$. In addition to that, let$d_k(G)$be the$k$-th largest (usual) degree in$G$. It is known that every connected graph satisfies$c_k(G) \\ge d_k(G)$for$k = 1$[GM],$k = 2$[LP] and for$k = 3$ [G].\n\nBibliography:\n[GM] R. Grone, R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math.7 (1994) 221-229. MathSciNet\n\n[LP] J.S. Li, Y.L. Pan, A note on the second largest eigenvalue of the Laplacian matrix of a graph, Linear Multilin. Algebra 48 (2000) 117-121. MathSciNet\n\n*[G] J.-M. Guo, On the third largest Laplacian eigenvalue of a graph, Linear Multilin. Algebra 55 (2007) 93-102. MathSciNet\n\n[M] B. Mohar, Problem of the Month\n\nDiscussion links:\n- Laplacian matrix: http://en.wikipedia.org/wiki/Laplacian matrix\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1271994\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1813439\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2281876\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P0612_LaplacianDegrees.html\n\nComments:\n- February 29th, 2008 | Anonymous | Proved: Proved by Willem Haemers and Andries Brouwer, see guo.pdf.\n- September 13th, 2007 | Gordon Royle | Equality?: Any conjectures about the structure of the graphs for which equality holds (for any particular value of k)?\n\nGordon Royle\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 20.\n\nAttempt notes:\nTarget:\nMake progress on \"Laplacian Degrees of a Graph\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3149, "problem_number": "OPG-824", "title": "Cores of strongly regular graphs", "statement": "Question Does every strongly regular graph have either itself or a complete graph as a core?", "background": "Source: Open Problem Garden. Original node ID: 824. URL: http://www.openproblemgarden.org/op/cores_of_strongly_regular_graphs.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cores_of_strongly_regular_graphs\n- Author(s): Cameron, Peter J.; Kazanidis, Priscila A.\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: core; strongly regular\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 16th, 2008 by mdevos\n\nProblem-page discussion:\nIf true, this curious question indicates a very interesting property of strongly regular graphs. While on the surface, there would appear to be no particular reason for it to hold, it has already been verified for a number of interesting classes of graphs. Cameron and Kazanidis [CK] showed that it holds for rank-3 graphs, while Godsil and Royle [GR] have showed that it holds for point graphs of generalized quadrangles, block graphs of Steiner systems and orthogonal arrays with sufficiently many points, and for all strongly regular graphs on at most 36 vertices.\n\nBibliography:\n*[CK] P. J. Cameron and P. A. Kazanidis, Cores of symmetric graphs, J. Australian Math. Soc., to appear.\n\n[GR] C. Godsil and G.F. Royle, Cores of Geometric Graphs\n\nSource links:\n- strongly regular graph: http://en.wikipedia.org/wiki/strongly regular graph\n- core: http://en.wikipedia.org/wiki/core (graph theory)\n\nBibliography links:\n- Cores of Geometric Graphs: http://xxx.tau.ac.il/pdf/0806.1300v1\n\nComments:\n- October 18th, 2011 | Anonymous | Correction: I believe you mean \"Godsil and Royle\", not \"Gordon and Royle\".\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Cores of strongly regular graphs\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3150, "problem_number": "OPG-36880", "title": "Does the chromatic symmetric function distinguish between trees?", "statement": "Problem Do there exist non-isomorphic trees which have the same chromatic symmetric function?", "background": "Source: Open Problem Garden. Original node ID: 36880. URL: http://www.openproblemgarden.org/op/does_the_symmetric_chromatic_function_distinguish_trees.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/does_the_symmetric_chromatic_function_distinguish_trees\n- Author(s): Stanley, Richard P.\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: chromatic polynomial; symmetric function; tree\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 25th, 2009 by mdevos\n\nProblem-page discussion:\nStanley [S] introduced the following symmetric function associated with a graph. Let $x_1,x_2,\\ldots$ be commuting indeterminates, and for every graph $G=(V,E)$ let ${\\mathcal C}_G$ be the set of all proper colorings $f: V \\rightarrow {\\mathbb N}$. Then the chromatic symmetric function is defined to be\n$$\nX_G = \\sum_{f \\in {\\mathcal C}_G} \\prod_{v \\in V} x_{f(v)}.\n$$\n So, the coefficient of a term $x_1^{d_1} x_2^{d_2} \\ldots$ in $X_G$ is precisely the number of proper colorings of $G$ where color $i$ appears exactly $d_i$ times. It is immediate that $X_G$ is homogeneous of degree $|V|$ and is symmetric.\n\nIf we set $x_1,x_2,\\ldots,x_k = 1$ and $x_{k+1}, x_{k+2} \\ldots = 0$ and evaluate, we get the number of proper colorings of $G$ using the colors $1,2,\\ldots,k$. Therefore, the chromatic symmetric function contains all of the information of the chromatic polynomial. In fact, the chromatic symmetric function contains strictly more information about the graph, since there exist examples of graphs which have distinct chromatic symmetric functions but have the same chromatic polynomial.\n\nThis natural problem of Stanley remains wide open. It has recently been established for some special classes of trees, namely caterpillars and spiders [MMW].\n\nBibliography:\n[MMW] J. Martin, M. Morin, and J. D. Wagner, On distinguishing trees by their chromatic symmetric functions. J. Combin. Theory Ser. A 115 (2008), no. 2, 237–253. MathSciNet\n\n*[S] R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166–194.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2382514\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Does the chromatic symmetric function distinguish between trees?\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3151, "problem_number": "OPG-164", "title": "Graham's conjecture on tree reconstruction", "statement": "Problem for every graph $G$, we let $L(G)$ denote the line graph of $G$. Given that $G$ is a tree, can we determine it from the integer sequence $|V(G)|, |V(L(G))|, |V(L(L(G)))|, \\ldots$?", "background": "Source: Open Problem Garden. Original node ID: 164. URL: http://www.openproblemgarden.org/op/grahams_conjecture_on_tree_reconstruction.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/grahams_conjecture_on_tree_reconstruction\n- Author(s): Graham, Ronald L.\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: reconstruction; tree\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 18th, 2007 by mdevos\n\nProblem-page discussion:\nGraph reconstruction is a notoriously difficult subject. This conjecture is an unusual type of reconstruction problem where our class of graphs is very limited - just trees, but we are also given relatively little information - just a sequence of integers.\n\nBibliography:\n[GR] C. Godsil and G. Royle, Algebraic graph theory. Graduate Texts in Mathematics, 207. Springer-Verlag, New York, 2001 (page 18).\n\nSource links:\n- line graph: http://en.wikipedia.org/wiki/line graph\n\nComments:\n- April 14th, 2009 | Anonymous | Reference: Could someone pleas give a proper reference? If I'm not mistaken, the problem is just *mentioned* in G&R, without references (anyway, I didn't find any).\n- December 22nd, 2008 | Anonymous | Graph theory: Does for any tree T there exist n that L^n(T) is a regular graph? Or perhaps for all graph?\n- January 16th, 2013 | leshabirukov | No: Consider L^i(T) is a graph with \"even triangle\"(triangle with even degrees of vertices) subgraph. Edges of even triangle produce new even triangle in L^(i+1)(T). And if there is an odd degree vertex adjacent to parent triangle, there would be another one adjacent to child. So, irregular subgraph remains.\n- June 10th, 2010 | Anonymous | No: Let G be a star graph of order 5. Then L(G) = C_4. Note that L(C_4) = C_4.\n- January 8th, 2013 | Anonymous | If G is a star graph of: If G is a star graph of order 5, then L(G) = K_5.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Graham's conjecture on tree reconstruction\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3152, "problem_number": "OPG-801", "title": "Nearly spanning regular subgraphs", "statement": "Conjecture For every $\\epsilon > 0$ and every positive integer $k$, there exists $r_0 = r_0(\\epsilon,k)$ so that every simple $r$-regular graph $G$ with $r \\ge r_0$ has a $k$-regular subgraph $H$ with $|V(H)| \\ge (1- \\epsilon) |V(G)|$.", "background": "Source: Open Problem Garden. Original node ID: 801. URL: http://www.openproblemgarden.org/op/nearly_spanning_regular_subgraphs.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/nearly_spanning_regular_subgraphs\n- Author(s): Alon, Noga; Mubayi, Dhruv\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: regular; subgraph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 22nd, 2008 by mdevos\n\nProblem-page discussion:\nPetersen's theorem asserts that every regular graph of even degree contains a 2-factor (i.e. a spanning 2-regular subgraph). Iterating this easy result we find that for any pair of positive even integers $k,r$, every $r$-regular graph has a spanning $k$-regular subgraph. The cases when either $k$ or $r$ is odd are considerably more complicated. There are some nice general results (see [AFK]) which show that every regular graph of sufficiently high degree contains a $k$-regular subgraph. However these theorems give no bound on the size of this subgraph.\n\nFor $k=1$ this conjecture is an easy consequence of Vizing's Theorem. Indeed, this theorem implies that every $d$-regular graph $G$ has a 1-regular subgraph $H$ with $|V(H)| \\ge (1 - \\frac{1}{d+1}) |V(G)|$ (just choose a largest color class from a $(d+1)$-edge coloring). Alon [A] proved the conjecture for $k=2$ with the help of two famous results on permanents: the Minc Conjecture (proved by Bregman), and the van der Waerden conjecture (proved by Falikman and Egorichev). It is open for all $k \\ge 3$.\n\nBibliography:\n*[A] N. Alon, Problems and results in extremal combinatorics, J, Discrete Math. 273 (2003), 31-53.\n\n[AFK] N. Alon, S. Friedland and G. Kalai, Regular subgraphs of almost regular graphs, J. Combinatorial Theory, Ser. B 37(1984), 79-91.\n\nBibliography links:\n- Problems and results in extremal combinatorics: http://www.math.tau.ac.il/%7Enogaa/PDFS/extremal1.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Nearly spanning regular subgraphs\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3153, "problem_number": "OPG-826", "title": "Complete bipartite subgraphs of perfect graphs", "statement": "Problem Let $G$ be a perfect graph on $n$ vertices. Is it true that either $G$ or $\\bar{G}$ contains a complete bipartite subgraph with bipartition $(A,B)$ so that $|A|, |B| \\ge n^{1 - o(1)}$?", "background": "Source: Open Problem Garden. Original node ID: 826. URL: http://www.openproblemgarden.org/op/complete_bipartite_subgraphs_of_perfect_graphs.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/complete_bipartite_subgraphs_of_perfect_graphs\n- Author(s): Fox, Jacob\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: perfect graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 17th, 2008 by mdevos\n\nProblem-page discussion:\nEvery perfect graph on $n$ vertices either has a clique or an independent set of size $\\ge n^{1/2}$, so weakening the bound on $|A|$, $|B|$ to $\\lfloor \\frac{1}{2} n^{1/2} \\rfloor$ gives a true statement. Jacob Fox [F] has proved that every comparability graph $G$ on $n$ vertices has a complete bipartite subgraph of size $\\ge c \\frac{n}{\\log n}$, and (up to the constant) this is best possible.\n\nBibliography:\n[F] J. Fox, A Bipartite Analogue of Dilworth’s Theorem, Order 23 (2006), 197-209.\n\nBibliography links:\n- A Bipartite Analogue of Dilworth’s Theorem: http://www.princeton.edu/%7Ejacobfox/papers/bipdil.pdf\n\nComments:\n- January 19th, 2021 | Anonymous | Claimed to be solved: In this recent preprint on Paul Seymour's webpage: http://web.math.princeton.edu/~pds/papers/pure5/paper.pdf (not on ArXiv nor published yet)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Complete bipartite subgraphs of perfect graphs\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3154, "problem_number": "OPG-36933", "title": "Asymptotic Distribution of Form of Polyhedra", "statement": "Problem Consider the set of all topologically inequivalent polyhedra with $k$ edges. Define a form parameter for a polyhedron as $\\beta:= v/(k+2)$ where $v$ is the number of vertices. What is the distribution of $\\beta$ for $k \\to \\infty$?", "background": "Source: Open Problem Garden. Original node ID: 36933. URL: http://www.openproblemgarden.org/op/asymptotic_distribution_of_form_of_polyhedra.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/asymptotic_distribution_of_form_of_polyhedra\n- Author(s): Rüdinger, Andreas\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: polyhedral graphs, distribution\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 9th, 2009 by andreasruedinger\n\nProblem-page discussion:\nConsider the set of all topologically inequivalent polyhedra on a sphere with k edges (i.e. polyhedral graphs, Sloan Sequence A002840 ). Due to duality the distribution of the form parameter $\\beta:= v/(k+2)$ is symmetric about $\\beta=1/2$. Now a natural question is whether the distribution of beta tends to a limiting distribution when the number of edges tends to infinity. Is there any nontrivial limit theorem by means of rescaling? Some numerical values can be found on Counting Polyhedra suggesting that the distribution concentrates around $\\beta=1/2$.\n\nDiscussion links:\n- Sloan Sequence A002840: http://www.research.att.com/%7Enjas/sequences/A002840\n- Counting Polyhedra: http://home.att.net/%7Enumericana/data/polycount.htm\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Asymptotic Distribution of Form of Polyhedra\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3155, "problem_number": "OPG-37038", "title": "Domination in cubic graphs", "statement": "Problem Does every 3-connected cubic graph $G$ satisfy $\\gamma(G) \\le \\lceil |G|/3 \\rceil$?", "background": "Source: Open Problem Garden. Original node ID: 37038. URL: http://www.openproblemgarden.org/op/domination_in_cubic_graphs.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/domination_in_cubic_graphs\n- Author(s): Reed, Bruce A.\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: cubic graph; domination\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 19th, 2009 by mdevos\n\nProblem-page discussion:\nIf the girth of $G$ is sufficiently large, then $\\gamma(G)\\leq 0.2999|G|$ [KSV].\n\nBibliography:\n[KSV] Daniel Kral, Petr Skoda, Jan Volec: Domination number of cubic graphs with large girth\n\nBibliography links:\n- Domination number of cubic graphs with large girth: http://arxiv.org/abs/0907.1166\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Domination in cubic graphs\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3156, "problem_number": "OPG-37163", "title": "Friendly partitions", "statement": "A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.\n\nProblem Is it true that for every $r$, all but finitely many $r$-regular graphs have friendly partitions?", "background": "Source: Open Problem Garden. Original node ID: 37163. URL: http://www.openproblemgarden.org/op/friendly_partitions.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/friendly_partitions\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: edge-cut; partition; regular\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 8th, 2009 by mdevos\n\nProblem-page discussion:\nLet me say at the start, that I (M. DeVos) suspect this problem has been considered previously, so I await a more correct attribution.\n\nAn unfriendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in the opposite class as its own. It is an easy fact that every (finite) graph has an unfriendly partition; for instance, any maximum size edge-cut gives a partition with this property.\n\nFinding friendly partitions appears to be considerably more difficult. Perhaps one reason why is that there exist graphs without unfriendly partitions. For instance, $K_{2n}$ and $K_{2n+1,2n+1}$ have no unfriendly partitions. However, it appears possible that the only graphs which fail to have friendly partitions are fairly dense.\n\nWhen $r=3$, the above problem is fairly easy to solve, as it reduces to the problem of finding two vertex disjoint cycles. Every cubic graph other than $K_4$ or $K_{3,3}$ has two disjoint cycles, and thus has a friendly partition. The case when $r=4$ is also not terribly complicated. However, the next step up, $r=5$ looks like a tricky problem which requires something new.\n\nRelated:\nRelated problems\nUnfriendly partitions\n\nComments:\n- August 5th, 2020 | Anonymous | Name of the problem: In the bibliography, this is known as Satisfactory graph partition or Internal graph partition.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Friendly partitions\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3157, "problem_number": "OPG-46708", "title": "Subgraph of large average degree and large girth.", "statement": "Conjecture For all positive integers $g$ and $k$, there exists an integer $d$ such that every graph of average degree at least $d$ contains a subgraph of average degree at least $k$ and girth greater than $g$.", "background": "Source: Open Problem Garden. Original node ID: 46708. URL: http://www.openproblemgarden.org/op/subgraph_of_large_average_degree_and_large_average_degree.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/subgraph_of_large_average_degree_and_large_average_degree\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Basic Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 5th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture is true for regular graphs as observed by Alon (see [KO]). The case $g\\leq 4$ was proved in [KO].\n\nBibliography:\n[KO] D. Kühn and D. Osthus, Every graph of sufficiently large average degree contains a C4-free subgraph of large average degree, Combinatorica, 24 (2004), 155-162.\n\n*[T] C. Thomassen, Girth in graphs, J. Combin. Theory B 35 (1983), 129–141.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Subgraph of large average degree and large girth.\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3158, "problem_number": "OPG-56028", "title": "Almost all non-Hamiltonian 3-regular graphs are 1-connected", "statement": "Conjecture Denote by $NH(n)$ the number of non-Hamiltonian 3-regular graphs of size $2n$, and similarly denote by $NHB(n)$ the number of non-Hamiltonian 3-regular 1-connected graphs of size $2n$.\n\nIs it true that $\\lim\\limits_{n \\rightarrow \\infty} \\displaystyle\\frac{NHB(n)}{NH(n)} = 1$?", "background": "Source: Open Problem Garden. Original node ID: 56028. URL: http://www.openproblemgarden.org/op/almost_all_non_hamiltonian_3_regular_graphs_are_1_connected.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/almost_all_non_hamiltonian_3_regular_graphs_are_1_connected\n- Author(s): Haythorpe, Michael\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: Hamiltonian, Bridge, 3-regular, 1-connected\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: August 23rd, 2013 by mhaythorpe\n\nProblem-page discussion:\nA stronger version of this conjecture asks whether it is also the case that $\\displaystyle\\frac{NHB(n)}{NH(n)} > \\displaystyle\\frac{NHB(k)}{NH(k)}$ for all $n > k$.\n\nExperimental data was given by Filar et al [FHN] demonstrating that the strong conjecture is satisfied for all $n \\leq 12$, and with sampled data provided for $n = 20$ and $n = 25$. No further results have been forthcoming.\n\nThe experimental data can be viewed at http://dx.doi.org/10.7151/dmgt.1485\n\nPackers And Movers Chandigarh\nPackers And Movers Hyderabad\nPackers And Movers Bangalore\n\nBibliography:\n[FHN] Jerzy A Filar, Giang T Nguyen, Michael Haythorpe, \"A conjecture on the prevalence of cubic bridge graphs\", Discussiones Mathematicae Graph Theory 30(1):175--179 (2010).\n\nDiscussion links:\n- Packers And Movers Chandigarh: http://www.packersandmoverschandigarh.co.in\n- Packers And Movers Hyderabad: http://www.packersandmoversinhyderabad.co.in\n- Packers And Movers Bangalore: http://www.packersandmoversinbangalore.co.in\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Almost all non-Hamiltonian 3-regular graphs are 1-connected\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3159, "problem_number": "OPG-146", "title": "Partitioning edge-connectivity", "statement": "Question Let $G$ be an $(a+b+2)$-edge-connected graph. Does there exist a partition $\\{A,B\\}$ of $E(G)$ so that $(V,A)$ is $a$-edge-connected and $(V,B)$ is $b$-edge-connected?", "background": "Source: Open Problem Garden. Original node ID: 146. URL: http://www.openproblemgarden.org/op/partitioning_edge_connectivity.\n\nSource subject path: Graph Theory > Basic Graph Theory > Connectivity.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partitioning_edge_connectivity\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Basic Graph Theory; Connectivity\n- Keywords: edge-coloring; edge-connectivity\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nBy the Nash-Williams/Tutte theorem ([NW] or [T]) on disjoint spanning trees, the above conjecture is true if $G$ is $2(a+b)$-edge-connected. This is the only partial result I know of. Here is a related conjecture.\n\nConjecture There exists a fixed integer $k$ so that every $k$-edge-connected graph $G=(V,E)$ has a subset of edges $S$ with the property that every edge-cut of $G$ has between $\\frac{1}{3}$ and $\\frac{2}{3}$ of its edges in $S$.\n\nThe values $\\frac{1}{3}$ and $\\frac{2}{3}$ are of no special importance in the above conjecture. Indeed, an affirmative answer to the above problem with $\\frac{1}{3}$ and $\\frac{2}{3}$ replaced by $\\frac{1}{t}$ and $1 - \\frac{1}{t}$ for any $t > 0$ would still be valuable - and in particular, would imply the 2+epsilon flow conjecture.\n\nDefinition: Let $G=(V,E)$ be a graph and let $P=\\{E_1,E_2,...,E_t\\}$ be a partition of $E$. We say that $P$ is $k$-courteous if $G \\setminus E_i$ is $k$-edge-connected for every $1 \\le i \\le t$.\n\nProblem What is the smallest integer $t$ so that every 3-edge-connected graph has a 2-courteous coloring of size $t$?\n\nIt is known (see [DJS]) that $4 \\le t \\le 10$. It would be quite interesting if the truth were in fact $t=4$. An improvement on the current upper bound would have some consequences for certain flow problems and cycle-cover problems. In general, one may define a function $H: {\\mathbb Z}^2 \\rightarrow {\\mathbb Z} \\cup \\{\\infty\\}$ so that $H(a,b)$ is the smallest integer $t$ (or $\\infty$ if none exists) so that every $a$-edge-connected graph has a $b$-courteous coloring of size $t$. It is known (see [DJS]) that $H(2k+2,2k+1) = \\infty$, and that $2k+1 < H(2k+1,2k) < C 100^k$. Two special cases when better values are known are $2 < H(4,2) < 5$ and $5 < H(5,4) < 31$.\n\nBibliography:\n[DJS] M. DeVos, T. Johnson, P.D. Seymour, Cut-coloring and circuit covering\n\n[Ed] J. Edmonds, Minimum Partition of a Matriod into Independent Subsets, J. Res. Nat. Bur. Standards 69B (1965) 67-72. MathSciNet\n\n[NW] C.S.J.A. Nash-Williams, Edge Disjoint Spanning Trees of Finite Graphs, J. London Math. Soc. 36 (1961) 445-450. MathSciNet\n\n[T] W.T. Tutte, On the problem of decomposing a graph into n connected factors, J. London Math. Soc. 36 (1961), 221-230. MathSciNet\n\nSource links:\n- edge-connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n\nDiscussion links:\n- edge-cut: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- 2+epsilon flow conjecture: http://www.openproblemgarden.org/?q=op/2_epsilon_flow_conjecture\n\nBibliography links:\n- Cut-coloring and circuit covering: http://www.math.princeton.edu/%7Epds/papers/cutcolouring/paper.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0190025\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0133253\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0140438\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 32.\n\nAttempt notes:\nTarget:\nMake progress on \"Partitioning edge-connectivity\" in Graph Theory; Basic Graph Theory; Connectivity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3160, "problem_number": "OPG-56233", "title": "Kriesell's Conjecture", "statement": "Conjecture Let $G$ be a graph and let $T\\subseteq V(G)$ such that for any pair $u,v\\in T$ there are $2k$ edge-disjoint paths from $u$ to $v$ in $G$. Then $G$ contains $k$ edge-disjoint trees, each of which contains $T$.", "background": "Source: Open Problem Garden. Original node ID: 56233. URL: http://www.openproblemgarden.org/op/kriesells_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Connectivity.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/kriesells_conjecture\n- Author(s): Kriesell, Matthias\n- Subject(s): Graph Theory; Basic Graph Theory; Connectivity\n- Keywords: Disjoint paths; edge-connectivity; spanning trees\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 25th, 2013 by Jon Noel\n\nProblem-page discussion:\nThis problem was featured as unsolved problem #22 in Bondy and Murty's book \"Graph Theory\" [BM].\n\nSee also a posting on the open problem forum of the Egerváry Research Group on Combinatorial Optimization.\n\nBibliography:\n[BM] J. A. Bondy and U. S. R. Murty. Graph theory, volume 244 of Graduate Texts in Mathematics. Springer, New York, 2008.\n\nDiscussion links:\n- posting: http://lemon.cs.elte.hu/egres/open/Kriesell's_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Kriesell's Conjecture\" in Graph Theory; Basic Graph Theory; Connectivity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3161, "problem_number": "OPG-137", "title": "Cycle double cover conjecture", "statement": "Conjecture For every graph with no bridge, there is a list of cycles so that every edge is contained in exactly two.", "background": "Source: Open Problem Garden. Original node ID: 137. URL: http://www.openproblemgarden.org/op/cycle_double_cover_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cycle_double_cover_conjecture\n- Author(s): Seymour, Paul D.; Szekeres, George\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cover; cycle\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nThis beautiful conjecture was made independently by Szekeres and Seymour in the 70's and is now widely considered to be among the most important open problems in graph theory. Note the similarity between this conjecture and the Berge-Fulkerson conjecture on perfect matchings. Attempts to prove this conjecture have lead to a variety of conjectured strengthenings, which appear on other pages. See: The circular embedding conjecture, The five cycle double cover conjecture, The faithful cover conjecture, and Decomposing Eulerian graphs.\n\nIf a graph $G$ has a nowhere-zero 4-flow then it follows from a result of Tutte that $G$ satisfies the above conjecture. Thus, by Jaeger's 4-flow theorem [J], the above conjecture is true for every 4-edge-connected graph. A cubic graph has a nowhere-zero 4-flow if and only if it is 3-edge-colorable, so the above conjecture is also true for 3-edge-colorable cubic graphs. In general, it follows from vertex splitting arguments that problem may be reduced to cubic graphs which are not 3-edge-colorable.\n\nFor a general graph $G$ with no cut-edge, Bermond, Jackson and Jaeger [BJJ] used Jaeger's 8-flow theorem [J] to prove that $G$ has a list of circuits so that every edge is contained in exactly four. Fan [F] used Seymour's 6-flow theorem [S81] to prove that G has a list of circuits so that every edge is contained in exactly six.\n\nLet $G$ be a directed graph and let $C$ be a circuit (not necessarily a directed circuit) of $G$. If we choose a direction to travel around $C$, then every edge of $C$ is either traversed forward or backward. The following strengthening of the cycle double cover conjecture takes directions into account.\n\nConjecture (The oriented cycle double cover conjecture) If $G$ is an orientation of a bridgeless graph, then there is a list $L$ of circuits of $G$ with directions so that every edge of $G$ is traversed forward by exactly one circuit in $L$ and backward by exactly one circuit in $L$.\n\nTutte also showed that every graph with a nowhere-zero 4-flow satisfies this conjecture. Thus, as above this conjecture is true for 4-edge-connected graphs and for 3-edge-colorable cubic graphs.\n\nIt was mentioned above that for a general graph $G$ with no bridge, there is a list of circuits containing every edge exactly four times. By taking two copies of each circuit in this list and giving them opposite directions, we have a list of circuits so that every edge is traversed forward and backward exactly four times. Luis Goddyn and I (M. DeVos) have observed that the same ideas used in Fan's article [Fa] can be used to construct a list of circuits with directions so that every edge is traversed forward and backward exactly three times. The following natural question seems to be open.\n\nConjecture (The oriented cycle four cover conjecture) If $G$ is an orientation of a bridgeless graph, then there is a list $L$ of circuits of $G$ with directions so that every edge of $G$ is traversed forward by exactly two circuits in $L$ and backward by exactly two circuits in $L$.\n\nSince every graph with a nowhere-zero 4-flow has a list of circuits with directions traversing every edge forward and backward exactly once, the above conjecture would follow from The three 4-flows conjecture.\n\nBibliography:\n[AGZ] B. Alspach, L. Goddyn, and C-Q Zhang, Graphs with the circuit cover property, Trans. Amer. Math. Soc., 344 (1994), 131-154. MathSciNet\n\n[BJJ] J.C. Bermond, B. Jackson, and F. Jaeger, Shortest covering of graphs with cycles, J. Combinatorial Theory Ser. B 35 (1983), 297-308. MRhref{0735197}\n\n[DJS] M. DeVos, T. Johnson, P.D. Seymour, Cut-coloring and circuit covering\n\n[F] G. Fan, Integer flows and cycle covers, J. Combinatorial Theory Ser. B 54 (1992), 113-122. MathSciNet\n\n[FZ] G. Fan and C.Q. Zhang, Circuit decompositions of Eulerian graphs, J. Combinatorial Theory Ser. B 78 (2000), 1-23. MathSciNet\n\n[FG] X. Fu and L. Goddyn, Matroids with the circuit cover property, Europ. J. Combinatorics 20 (1999), 61-73. MathSciNet\n\n[J] F. Jaeger, Flows and Generalized Coloring Theorems in Graphs, J. Combinatorial Theory Ser. B 26 (1979) 205-216. MathSciNet\n\n[Ki] P.A. Kilpatrick, Tutte's First Colour-Cycle Conjecture, Thesis, Cape Town (1975).\n\n[Sz] G. Szekeres, Polyhedral decompositions of cubic graphs. Bull. Austral. Math. Soc. 8, 367-387. MathSciNet\n\n[S91] P.D. Seymour, Nowhere-Zero 6-Flows, J. Combinatorial Theory Ser. B 30 (1981) 130-135. MathSciNet\n\n[S79] P.D. Seymour, Sums of circuits in Graph Theory and Related Topics edited by J.A. Bondy and U.S.R. Murty, Academic Press, New York/Berlin (1979), 341-355. MathSciNet\n\n[S95] P.D. Seymour, Nowhere-Zero Flows, in Handbook of Combinatoircs, edited by R. Graham, M. Grotschel and L. Lovasz, (1995) 289-299. MathSciNet\n\n[T54] W.T. Tutte, A Contribution on the Theory of Chromatic Polynomials, Canad. J. Math. 6 (1954) 80-91. MathSciNet\n\n[T66] W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combinatorial Theory 1 (1966) 15-50. MathSciNet\n\nSource links:\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n\nDiscussion links:\n- Berge-Fulkerson conjecture: http://www.openproblemgarden.org/?q=op/the_berge_fulkerson_conjecture\n- The circular embedding conjecture: http://www.openproblemgarden.org/?q=op/the_circular_embedding_conjecture\n- The five cycle double cover conjecture: http://www.openproblemgarden.org/?q=op/m_n_cycle_covers\n- The faithful cover conjecture: http://www.openproblemgarden.org/?q=op/faithful_cycle_covers\n- Decomposing Eulerian graphs: http://www.openproblemgarden.org/?q=op/decomposing_eulerian_graphs\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n- cubic graph: http://en.wikipedia.org/wiki/cubic graph\n- edge-colorable: http://en.wikipedia.org/wiki/edge coloring\n\nBibliography links:\n- Graphs with the circuit cover property: http://www.jstor.org/view/00029947/di981444/98p0199p/0\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1181180\n- Cut-coloring and circuit covering: http://www.math.princeton.edu/%7Epds/papers/cutcolouring/paper.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1142267\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1737620\n- Matroids with the circuit cover property: http://www.math.sfu.ca/%7Egoddyn/Papers/9531-matroids-circuit-cover.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1669600\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0532588\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0325438\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0615308\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0538060\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1373660\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0061366\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0194363\n\nComments:\n- February 7th, 2012 | Andrew King | Claimed solution in the affirmative: Alexander Souza has offered for review on arXiv a constructive solution in the affirmative.\n\nhttp://arxiv.org/abs/1202.0569\n\nEdit: Unless we are mistaken, the Petersen graph is a counterexample to Theorem 2. Furthermore there is an unresolved problem with Lemma 9.\n- October 6th, 2011 | khaniki | question: should the cycles be distinct??\n- February 6th, 2012 | Andrew King | No; consider the example of: No; consider the example of a graph which is itself a cycle. This has a unique cycle double cover, and the cycles are not distinct.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Cycle double cover conjecture\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3162, "problem_number": "OPG-138", "title": "The circular embedding conjecture", "statement": "Conjecture Every 2-connected graph may be embedded in a surface so that the boundary of each face is a cycle.", "background": "Source: Open Problem Garden. Original node ID: 138. URL: http://www.openproblemgarden.org/op/the_circular_embedding_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_circular_embedding_conjecture\n- Author(s): Haggard, Gary\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cover; cycle\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nThis conjecture implies the cycle double cover conjecture, since the list of cycles which bound faces covers each edge exactly twice. Let $G$ be a cubic graph, let $L$ be a list of cycles covering every edge of $G$ exactly two times, and form a topological space by gluing a disc to each circuit in $L$. This space is a surface, and every face is bounded by a cycle. Thus, the circular embedding conjecture and the cycle double cover conjecture are equivalent for cubic graphs. For general graphs, this construction may fail since the neighborhood of a vertex may not be a disc (it could be a pinchpoint).\n\nA stronger variant of this conjecture asserts that it is possible to find an embedding as above with the added condition that the dual graph is 5-colorable. This variant implies The five cycle double cover conjecture since the circuits bounding faces of a given color class may be grouped into a cycle. Next we state a different strengthening which asserts that we may find an embedding as above into an orientable surface.\n\nConjecture (The oriented circular embedding conjecture) Every 2-connected graph may be embedded in an orientable surface so that the boundary of each face is a circuit.\n\nIf this conjecture is true, then the oriented cycle double cover conjecture (see cycle double cover) is also true, since the list of circuits bounding faces all traversed in the clockwise direction cover each edge exactly once in each direction (since the surface is orientable, we may specify a global clockwise orientation). As was the case above, the oriented circular embedding conjecture is equivalent to the oriented cycle double cover conjecture for cubic graphs. Also as above, there is a strengthening of this conjecture which asserts that the graph may be embedded so that the dual graph is 5-colorable. If true, this would imply The orientable five cycle double cover conjecture.\n\nBibliography:\n[H] G. Haggard, Edmonds characterization of disc embeddings. Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977), pp. 291--302. MathSciNet\n\nSource links:\n- connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- embedded: http://en.wikipedia.org/wiki/graph embedding\n\nDiscussion links:\n- cycle double cover conjecture: http://www.openproblemgarden.org/?q=op/cycle_double_cover_conjecture\n- five cycle double cover conjecture: http://www.openproblemgarden.org/?q=op/m_n_cycle_covers\n- cycle double cover: http://www.openproblemgarden.org/?q=op/cycle_double_cover_conjecture\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0491201\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"The circular embedding conjecture\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3163, "problem_number": "OPG-139", "title": "(m,n)-cycle covers", "statement": "Conjecture Every bridgeless graph has a (5,2)-cycle-cover.", "background": "Source: Open Problem Garden. Original node ID: 139. URL: http://www.openproblemgarden.org/op/m_n_cycle_covers.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/m_n_cycle_covers\n- Author(s): Celmins, Uldis A.; Preissmann, Myriam\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cover; cycle\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: If $G=(V,E)$ is a graph, a binary cycle of $G$ is a set $C \\subseteq E$ such that every vertex of the graph $(V,C)$ has even degree. An $(m,n)$-cycle-cover of $G$ is a list $L$ consisting of $m$ cycles so that every edge of $G$ is contained in exactly $n$ of these cycles.\n\nSince every binary cycle can be written as a disjoint union of edge sets of ordinary cycles, the above conjecture is a strengthening of the cycle double cover conjecture. For positive integers $m,n$ it is natural to ask what family of graphs have $(m,n)$-cycle-covers. The following chart gives some information about this question for small values of $m$ and $n$. A \"yes\" in the $(m,n)$ box indicates that every graph with no cut-edge has an $(m,n)$-cycle-cover. A \"no\" indicates that no graph has an $(m,n)$-cycle-cover. A more detailed explanation of the entries in this chart appears below it.\n\nm\n\nn\n\n2 3 4\n\n5 6 7 8 9 10\n\n11\n2 Eulerian NZ 4-flow NZ 4-flow\n\n5CDC conj open\n4 no Eulerian\n\n5 post. sets B-F conj yes [BJJ] yes\n6\n\nno Eulerian 7 post. sets?? yes [F]\n\nyes\n\nWe did not include odd values of n, since any graph with an $(m,n)$-cycle-cover for an odd integer $n$ must be Eulerian. The entry \"NZ 4-flow\" is short for nowhere-zero 4-flow. Thus, our chart indicates that ( $G$ has a nowhere-zero 4-flow) if and only if ( $G$ has a $(3,2)$-cycle-cover) if and only if ( $G$ has a $(4,2)$-cycle-cover). These equivalences were discovered by Tutte [Tu].\n\nTwo of the $(m,n)$ boxes are conjectures. The 5CDC conj is the 5 cycle double cover conjecture and the B-F conjecture is the Berge-Fulkerson conjecture. In both of these cases, the conjecture is equivalent to the assertion that every graph with no cut-edge has an $(m,n)$-cycle-cover (i.e. it would be accurate to put a \"yes\" in the corresponding. box). For emphasis, we state the Berge-Fulkerson conjecture again below in this new form.\n\nConjecture (The Berge-Fulkerson conjecture) Every graph with no cut-edge has a (6,4)-cycle-cover.\n\nThe fact that the above conjecture is equivalent to the usual statement of the Berge-Fulkerson conjecture was discovered by Jaeger [J]. For cubic graphs this equivalence is easy to see, since $M_1,\\ldots,M_6$ satisfy the Berge-Fulkerson conjecture if and only if $E\\M_1,\\ldots,E\\M_6$ is a $(6,4)$-cycle-cover. By Jaeger's argument, the weak Berge-Fulkerson conjecture is equivalent to the statement that there exists a fixed integer $k$ so that every bridgeless graph has a $(3k,2k)$-cycle-cover.\n\nA postman set is a set of edges $J$ such that $E(G)\\J$ is a cycle. The entry \"k post. sets\" in the $(k,k-1)$ box of the above chart indicates that a graph G has a $(k,k-1)$-cycle-cover if and only if it is possible to partition the edges of $G$ into $k$ postman sets. This equivalence follows immediately from the definition. Rizzi's Packing postman sets conjecture is thus equivalent to the following conjecture on cycle-covers.\n\nConjecture (the packing postman sets conjecture) If every odd edge-cut of $G$ has size $\\ge 2k+1$, then $G$ has a $(2k+1,2k)$-cycle-cover.\n\nNext we turn our attention to orientable cycle covers. If $H$ is a directed graph a map $\\phi:E(H) \\rightarrow \\{-1,0,1\\}$ is a 2-flow or an oriented cycle if at every vertex of $H$, the sum of $\\phi$ on the incoming edges is equal to the sum of $\\phi$ on the outgoing edges. It is easy to see that the support of a 2-flow is always a cycle. Furthermore, for any oriented cycle, there is a list $L$ of edge-disjoint circuits with directions so that an edge $e$ is forward (backward) in a circuit of $L$ if and only if $\\phi(e)=1$ ( $\\phi(e)=-1$ ). So as in the unoriented case, an oriented cycle may be viewed as the edge-disjoint union of oriented circuits. For an even integer $n$, a $(m,n)$-oriented-cycle-cover of a graph $G$ is a list of $m$ oriented cycles so that every edge of $G$ appears as a forward edge $n/2$ times and a backward edge $n/2$ times. The following conjecture is the common generalization of the orientable cycle double cover conjecture and the five cycle double cover conjecture. It is due to Archdeacon and Jaeger.\n\nConjecture (The orientable five cycle double cover conjecture) Every graph without a cut-edge has a (5,2)-oriented-cycle-cover.\n\nConsiderably less is known about $(m,n)$-oriented-cycle-covers. We sumarize some of what is known for small values of $m$ and $n$ below.\n\nm\n\nn\n\n2\n\n3 4 5 6 7 8\n\n9 10 11\n2 Eulerian\n\nNZ 3-flow NZ 4-flow O5CDC conj open\n4\n\nno Eulerian??? conj.\n\nopen\n6 no Eulerian???? yes [DG]\n\nEvery graph with an $(m,n)$-cycle-cover also has a $(2m,2n)$-oriented-cycle-cover obtained by taking two copies of each cycle with opposite orientations. Thus, by Bermond, Jackson, and Jaeger's $(7,4)$-cycle-cover theorem, every bridgeless graph with no has a $(14,8)$-oriented-cycle-cover. DeVos and Goddyn have observed that Seymour's 6-flow theorem can be used to construct an $(11,6)$-oriented-cycle-cover for every bridgeless graph. By combining these, we find that for every even integer $n \\ge 10$ there exists an $m$ so that every bridgeless graph has an $(m,n)$-oriented-cycle-cover. This question is still open for $n=2,4,10$.\n\nThe following conjecture appears in the above chart.\n\nConjecture (The orientable eight cycle four cover conjecture) Every graph with no cut-edge has a (8,4)-oriented-cycle-cover.\n\nThis conjecture may be viewed as a sort of oriented version of the Berge-Fulkerson conjecture. To see this analogy, note that ( $G$ has a nowhere-zero 4-flow) if and only if ( $G$ has a $(3,2)$-cycle-cover) if and only if ( $G$ has a $(4,2)$-oriented-cycle-cover). The Berge-Fulkerson conjecture and the above conjecture assert respectively that every bridgeless graph has a $(6,4)$-cycle-cover and a $(8,4)$-oriented-cycle-cover (i.e. a cover with double the parameters which are equivalent to a nowhere-zero 4-flow). As with most of the conjectures in this area, the above conjecture is trivially true for graphs with nowhere-zero 4-flows and it holds for the Petersen graph.\n\nBibliography:\n[A] D. Archdeacon, Face coloring of embedded graphs. J. Graph Theory, 8(1984), 387-398.\n\n[BJJ] J.C. Bermond, B. Jackson, and F. Jaeger, Shortest covering of graphs with cycles, J. Combinatorial Theory Ser. B 35 (1983), 297-308. MathSciNet\n\n*[C] A. U. Celmins, On cubic graphs that do not have an edge-3-colouring, Ph.D. Thesis, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada, 1984.\n\n[F] G. Fan, Integer flows and cycle covers, J. Combinatorial Theory Ser. B 54 (1992), 113-122. MathSciNet\n\n[J] F. Jaeger, Flows and Generalized Coloring Theorems in Graphs, J. Combinatorial Theory Ser. B 26 (1979) 205-216. MathSciNet\n\n[J88] F. Jaeger, Nowhere zero flow problems. Selected Topics in Graph Theory 3 (L.W.Beineke and R.J.Wilson eds.), Academic Press, London (1988), 71-95.\n\n*[P] M. Preissmann, Sur les colorations des arêtes des graphes cubiques, Thèse de 3ème cycle, Grenoble (1981).\n\n[T54] W.T. Tutte, A Contribution on the Theory of Chromatic Polynomials, Canad. J. Math. 6 (1954) 80-91. MathSciNet\n\n[T66] W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combinatorial Theory 1 (1966) 15-50. MathSciNet\n\nSource links:\n- bridgeless: http://en.wikipedia.org/wiki/bridge (graph theory)\n\nDiscussion links:\n- Eulerian: http://en.wikipedia.org/wiki/eulerian graph\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n- Berge-Fulkerson conjecture: http://www.openproblemgarden.org/?q=op/the_berge_fulkerson_conjecture\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0735197\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1142267\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0532588\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0061366\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0194363\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 35.\n\nAttempt notes:\nTarget:\nMake progress on \"(m,n)-cycle covers\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3164, "problem_number": "OPG-140", "title": "Faithful cycle covers", "statement": "Conjecture If $G = (V,E)$ is a graph, $p: E \\rightarrow {\\mathbb Z}$ is admissable, and $p(e)$ is even for every $e \\in E(G)$, then $(G,p)$ has a faithful cover.", "background": "Source: Open Problem Garden. Original node ID: 140. URL: http://www.openproblemgarden.org/op/faithful_cycle_covers.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/faithful_cycle_covers\n- Author(s): Seymour, Paul D.\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cover; cycle\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: Let $G=(V,E)$ be a graph and let $p:E \\rightarrow {\\mathbb Z}$. A list $L$ of cycles of $G$ is a faithful (cycle) cover of $(G,p)$ if every edge $e$ of $G$ occurs in exactly $p(e)$ cycles of $L$. Thus, the cycle double cover conjecture is equivalent to the statement that $(G,2)$ has a faithful cover for every graph $G$ with no bridge. We define the map $p$ to be admissable if $p(e)$ satisfies the following properties:\n\ni $p$ is nonnegative.\n\nii $p(S)$ is even for every edge-cut $S$.\n\niii $p(e) \\le p(S)/2$ for every edge-cut $S$ and edge $e \\in S$.\n\nIt is easy to see that $G$ has a faithful cover only if $p$ is admissable. However, the converse is false. A counterexample is obtained by taking the Petersen graph, putting weight $2$ on the edges of a perfect matching, and $1$ elsewhere.\n\nMore generally, for a graph G=(V,E), one may consider the vector space of real numbers indexed by E. We associate every circuit C with its incidence vector. Most of the basic questions about this space are solved. Seymour [S] has shown that a vector p can be written as a nonnegative rational combination of cycles if and only if it satisfies conditions (i) and (iii) in the definition of admissable. It is an easy exercise to show that for a 3-edge-connected graph G, a vector p can be written as an integer combination of cycles if and only if p satisfies (ii) in the definition of admissable. Seymour's conjecture is equivalent to the statement that every admissable map may be realized as a half-integer combination of circuits. Note the similarity of this to The Berge-Fulkerson conjecture.\n\nThe most interesting result about faithful covers is a theorem of Alspach, Goddyn, and Zhang which resolved a conjecture of Seymour. They prove that whenever $G$ has no minor isomorphic to Petersen, every admissable map has a corresponding faithful cover. For a general graph $G$ with no bridge, Bermond, Jackson, and Jaeger [BJJ] proved that $(G,4)$ has a faithful cover and Fan [F] proveed that $(G,6)$ has a faithful cover. DeVos, Johnson, and Seymour [DJS] proved that $(G,p)$ has a faithful cover whenever $p$ is admissable and there is a nonnegative integer $k$ such that $32k+83 < p(e) < 36k+88$ holds for every edge $e$. However, little else seems to be known. In particular, it does not appear to be known if there exist integers $a,b$ with $a-b$ arbitrarily large so that $(G,p)$ has a faithful cover whenever $p$ is an admissable function taking on only the values $a,b$. Such a result would appear to require an idea not contained in any of the aforementioned papers.\n\nThe analogous problem for oriented circuit covers does not appear to be very promising. It is easy to see that for an orientation of a series parallel graph G and a map $p:E(G) \\rightarrow G$ which satisfies the obvious conditions, that $(G,p)$ will have a circuit cover using every edge in its given direction. However, even with a $K_4$ minor, there is a great deal of forcing, and nothing much looks like it would be true.\n\nBibliography:\n\\[AGZ] B. Alspach, L. Goddyn, and C-Q Zhang, Graphs with the circuit cover property, Trans. Amer. Math. Soc., 344 (1994), 131-154. MathSciNet\n\n[BJJ] J.C. Bermond, B. Jackson, and F. Jaeger, Shortest covering of graphs with cycles, J. Combinatorial Theory Ser. B 35 (1983), 297-308. MathSciNet\n\n[DJS] M. DeVos, T. Johnson, P.D. Seymour, Cut-coloring and circuit covering\n\n[F] G. Fan, Integer flows and cycle covers, J. Combinatorial Theory Ser. B 54 (1992), 113-122. MathSciNet\n\n[S] P.D. Seymour, Sums of circuits in Graph Theory and Related Topics edited by J.A. Bondy and U.S.R. Murty, Academic Press, New York/Berlin (1979), 341-355. MathSciNet\n\nDiscussion links:\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n- edge-cut: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- The Berge-Fulkerson conjecture: http://www.openproblemgarden.org/?q=op/the_berge_fulkerson_conjecture\n- minor: http://en.wikipedia.org/wiki/minor (graph theory)\n- Petersen: http://en.wikipedia.org/wiki/petersen graph\n\nBibliography links:\n- Graphs with the circuit cover property: http://www.jstor.org/view/00029947/di981444/98p0199p/0\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1181180\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0735197\n- Cut-coloring and circuit covering: http://www.math.princeton.edu/%7Epds/papers/cutcolouring/paper.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1142267\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0538060\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 26.\n\nAttempt notes:\nTarget:\nMake progress on \"Faithful cycle covers\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3165, "problem_number": "OPG-141", "title": "Decomposing eulerian graphs", "statement": "Conjecture If $G$ is a 6-edge-connected Eulerian graph and $P$ is a 2-transition system for $G$, then $(G,P)$ has a compaible decomposition.", "background": "Source: Open Problem Garden. Original node ID: 141. URL: http://www.openproblemgarden.org/op/decomposing_eulerian_graphs.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_eulerian_graphs\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cover; cycle; Eulerian\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: Let $G$ be an Eulerian graph and for every vertex $v$, let $P(v)$ be a partition of the edges incident with $v$. We call $P$ a transition system. If every member of $P(v)$ has size at most $k$ (for every $v$ ), then we call $P$ a $k$-transition sytem. A compatible decomposition of $(G,P)$ is a list of edge-disjoint cycles $C_1,\\ldots,C_n$ with union $G$ so that every $C_i$ contains at most one edge from every member of $P(v)$.\n\nLet $G$ be a graph and let $G'$ be the graph obtained from $G$ by replacing each edge $e$ of G by two edges $e',e\"$ in parallel. Let $P$ be the 2-transition system of $G$ with ${e',e\"} \\in P(v)$ whenever $e'$ and $e\"$ are incident with $v$. Now, $G'$ is an Eulerian graph and every compatible decomposition of $(G',P)$ gives a cycle double cover of $G$. Since the cycle double cover conjecture can be reduced to graphs which are 3-edge-connected, the above conjecture would imply the cycle double cover conjecture.\n\nWe define a transition system $P$ to be admissable if every member of $P(v)$ contains no more than half of the edges in any edge-cut. It is easy to see that if there is a compatible decomposition of $(G,P)$, then $P$ must be admissable. The converse of this is not true; There is an admissable 2-transition system of the graph $K_5$ which does not admit a compatible decomposition. Recently, G. Fan and C.Q. Zhang [FZ] have proved that $(G,P)$ does have a compatible decomposition whenever $P$ is admissable and $G$ has no $K_5$ minor. This result imporoved upon an earlier theorem of Fleischner and Frank [FF]. Very recently, I have proved a weak version of the above conjecture, by showing that $(G,P)$ also has a compatible decomposition when P is a 2-transition system and G is 80-edge-connected. I'd quite like to see an improvement on this bound. Here is a related conjecture.\n\nConjecture (Sabidussi) Let $W$ be an Euler tour of the graph $G$. If $G$ has no vertex of degree two, then there is a cycle decomposition of $G$, say $F$, so that no two consecutive edges of $W$ are in a common circuit of $F$.\n\nIf $W$ is given by $v_1,e_1,v_2,e_2,...,e_{m-1},v_m$ then we may form a 2-transition system $P$ by putting $\\{e_{i-1},e_i\\}$ in $P(v_i)$ for every $i$ (working modulo $m$ ). Now a compatible decomposition of $(G,P)$ is precisely a cycle decomposition of $G$ satisfying the above conjecture. Thus, Sabidussi's conjecture is equivalent to the assertion that $(G,P)$ has a compatible decomposition whenever $G$ has no vertex of degree two and $P$ is a 2-transition system which comes from an Euler tour.\n\nLet $G$ be a directed Eulerian graph and for every vertex $v$, let $P(v)$ be a partition of the edges incident with $v$ into pairs so that every in-edge is paired with an out-edge. We define a compatible decomposition to be a decomposition of $G$ into directed circuits so that every directed circuit contains at most one edge from every member of $P(v)$. Our current techniques don't seem to shed any light on the problem of finding compatible decompositions for Eulerian digraphs. Next I pose a very basic question which is still open.\n\nProblem (DeVos) Does there exist a fixed integer $k$ such that $(G,P)$ has a compatible decomposition whenever $G$ is a $k$-edge-connected directed Eulerian graph and $P$ is a 2-transition system?\n\nSource links:\n- edge-connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- Eulerian graph: http://en.wikipedia.org/wiki/eulerian graph\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 23.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposing eulerian graphs\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3166, "problem_number": "OPG-385", "title": "Barnette's Conjecture", "statement": "Conjecture Every 3-connected cubic planar bipartite graph is Hamiltonian.", "background": "Source: Open Problem Garden. Original node ID: 385. URL: http://www.openproblemgarden.org/op/barnettes_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/barnettes_conjecture\n- Author(s): Barnette, David W.\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: bipartite; cubic; hamiltonian\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 12th, 2007 by Robert Samal\n\nProblem-page discussion:\n(Originally appeared in [B], this discussion appears as [M].)\n\n- It is known that this is not true if you remove the \"bipartite\" condition, but the smallest 3-connected cubic planar graph which is not Hamiltonian has 38 vertices.\n\n- Holton, Manvel, and McKay [HMM] proved (using computers) that all graphs having fewer than 66 vertices satisfy the conjecture.\n\n- [A communication by Robert Aldred, Gunnar Brinkmann, and Brendan McKay (December 2002):]\n\nA paper of Holton, Manvel and McKay [HMM] proved Barnette's conjecture for up to 64 vertices, inclusive. This is to announce that the conjecture remains true up to 84 vertices, inclusive. The method used was the same as in the 1985 paper, but took advantage of two developments. One was the new program plantri (Brinkmann and McKay, to be published) which can generate the required graphs without isomorphs at more than $100\\,000$ per second. The other was the advance in computers. Total cpu time was about 3 years, almost all of it taken in finding hamiltonian cycles. Specifically, for all 3-connected cubic planar bipartite graphs up to 60 vertices, and those up to 64 vertices not having a 4-face adjacent to two others, we found a hamiltonian cycle using $x$ and avoiding $y$ for each pair of edges $x$ and $y$. There are over $10^{10}$ such graphs. By a theorem of Kelman's, one can build a counterexample to Barnette's conjecture when one has a 3-connected cubic planar bipartite graph with this property: for some two edges $x$ and $y$ on the same face, there is no hamiltonian cycle that uses $x$ and avoids $y$. We did not find any such graph even where $x$ and $y$ are not required to be on the same face. Perhaps the path to finding a counterexample is to strengthen Kelman's method to some more complicated condition involving 3 or more edges, as then it is more likely to fail on a smaller size.\n\nThere is another conjecture of Barnette (checked by Brendan McKay and Gunnar Brinkmann up to 250 vertices).\n\nConjecture Every planar cubic 3-connected graph with faces only of sizes 3, 4, 5, and 6 is Hamiltonian.\n\nBibliography:\n*[B] David W. Barnette, Conjecture 5, Recent progress in combinatorics (ed. W. T. Tutte), Academic Press, New York (1969) 343, MathSciNet\n\n[HMM] Derek A.Holton, Bennet Manvel, Brendan D. McKay, Hamiltonian cycles in cubic 3-connected bipartite planar graphs, J. Combin. Theory Ser. B 38 (1985) 279-297. MathSciNet\n\n[M] B. Mohar, Problem of the Month\n\nDiscussion links:\n- plantri: http://cs.anu.edu.au/%7Ebdm/plantri/\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0250896\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0796604\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P4BarnetteConjecture.html\n\nComments:\n- August 9th, 2011 | Anonymous | Copper Basin Construction: This is nice idea. This is to announce that the conjecture remains true up to 84 vertices, inclusive. The method used was the same as in the 1985 paper, but took advantage of two developments. Thank you.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Barnette's Conjecture\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3167, "problem_number": "OPG-480", "title": "r-regular graphs are not uniquely hamiltonian.", "statement": "Conjecture If $G$ is a finite $r$-regular graph, where $r > 2$, then $G$ is not uniquely hamiltonian.", "background": "Source: Open Problem Garden. Original node ID: 480. URL: http://www.openproblemgarden.org/op/uniquely_hamiltonian_graphs.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/uniquely_hamiltonian_graphs\n- Author(s): Sheehan, John\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: hamiltonian; regular; uniquely hamiltonian\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 24th, 2007 by Robert Samal\n\nProblem-page discussion:\n(Reproduced from [M].)\n\nA graph $G$ is said to be uniquely hamiltonian if it contains precisely one Hamiltonian cycle.\n\nThis conjecture has been proved for all odd values of $r$ [T] and for all even values of $r > 23$ [H]. By Petersen's theorem, it would suffice to prove it for $r = 4$.\n\nBibliography:\n[H] P. Haxell, Oberwolfach reports, 2006.\n\n[M] Bojan Mohar, Problem of the Month\n\n*[S] John Sheehan: The multiplicity of Hamiltonian circuits in a graph. Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), pp. 477-480. Academia, Prague, 1975, MathSciNet\n\n[T] A.G. Thomason, Hamiltonian cycles and uniquely edge colourable graphs. Advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977). Ann. Discrete Math. 3 (1978), Exp. No. 13, 3 pp.\n\nDiscussion links:\n- Hamiltonian cycle: http://en.wikipedia.org/wiki/hamiltonian cycle\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P0703_HamiltonicityInfinite.html\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0398896\n\nComments:\n- February 3rd, 2022 | Anonymous | Is this question still open?: I think, the autor answers the question in this article: Uniqueness of maximal dominating cycles in 3-regular graphs and of hamiltonian cycles in 4-regular graphs (https://doi.org/10.1002/jgt.3190180503). (And the conjecture is fals for all even values.) Am I wrong?\n- June 20th, 2022 | Anonymous | The construction in that: The construction in that paper has parallel edges, so it is not a counter example. As far as I am aware, Sheehan's conjecture is still open.\n- May 3rd, 2022 | Anonymous | The conjecture is only for: The conjecture is only for simple graphs. The paper you mention gives counterexamples that have multiple edges.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"r-regular graphs are not uniquely hamiltonian.\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3168, "problem_number": "OPG-485", "title": "Hamiltonian cycles in line graphs", "statement": "Conjecture Every 4-connected line graph is hamiltonian.", "background": "Source: Open Problem Garden. Original node ID: 485. URL: http://www.openproblemgarden.org/op/hamiltonian_cycles_in_line_graphs.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hamiltonian_cycles_in_line_graphs\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: hamiltonian; line graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 24th, 2007 by Robert Samal\n\nProblem-page discussion:\n\n- It is known that if $G$ is 4-edge-connected, then its line graph $L(G)$ is hamiltonian.\n- Thomassen's is a special case of a conjecture due to Matthews and Sumner: every 4-connected claw-free graph is hamiltonian.\n- However, by a result of Ryjacek [R] conjectures of Thomassen and of Matthews and Sumner are equivalent.\n- Moreover [R], one may restrict to 4-connected line graphs of triangle-free graphs.\n\nBibliography:\n[R] Zdenek Ryjacek: On a closure concept in claw-free graphs. J. Combin. Theory Ser. B 70 (1997), no. 2, 217--224, MathSciNet\n\n*[T] Carsten Thomassen, Reflections on graph theory, J. Graph Theory 10 (1986) 309-324, MathSciNet\n\nSource links:\n- line graph: http://en.wikipedia.org/wiki/line graph\n- hamiltonian: http://en.wikipedia.org/wiki/Hamilton cycle\n\nDiscussion links:\n- claw-free: http://en.wikipedia.org/wiki/Claw_(graph_theory)\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1459867\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0856118\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamiltonian cycles in line graphs\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3169, "problem_number": "OPG-500", "title": "Geodesic cycles and Tutte's Theorem", "statement": "Problem If $G$ is a $3$-connected finite graph, is there an assignment of lengths $\\ell: E(G) \\to \\mathb R^+$ to the edges of $G$, such that every $\\ell$-geodesic cycle is peripheral?", "background": "Source: Open Problem Garden. Original node ID: 500. URL: http://www.openproblemgarden.org/op/geodesic_cycles_and_tuttes_theorem.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/geodesic_cycles_and_tuttes_theorem\n- Author(s): Georgakopoulos, Agelos; Sprüssel, Philipp\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cycle space; geodesic cycles; peripheral cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: August 4th, 2007 by Agelos\n\nProblem-page discussion:\nA cycle $C$ is $\\ell$-geodesic if for every two vertices $x,y$ on $C$ there is no $x$- $y$ ~path in $G$ shorter, with respect to $\\ell$, than both $x$- $y$ ~arcs on $C$.\n\nIt is not hard to prove [GS] that for every finite graph $G$ and every assignment of edge lengths $\\ell: E(G) \\to \\mathb R^+$ the $\\ell$-geodesic cycles of $G$ generate its cycle space. Thus, a positive answer to the problem would imply a new proof of Tutte's classical theorem [T] that the peripheral cycles of a $3$-connected finite graph generate its cycle space.\n\nBibliography:\n*[GS] Angelos Georgakopoulos, Philipp Sprüssel: Geodesic topological cycles in locally finite graphs. Preprint 2007.\n\n[T] W.T. Tutte, How to draw a graph. Proc. London Math. Soc. 13 (1963), 743–768.\n\nSource links:\n- peripheral: http://en.wikipedia.org/wiki/peripheral cycle\n\nBibliography links:\n- Geodesic topological cycles in locally finite graphs: http://www.math.uni-hamburg.de/home/georgakopoulos/geo.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Geodesic cycles and Tutte's Theorem\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3170, "problem_number": "OPG-638", "title": "Jones' conjecture", "statement": "For a graph $G$, let $cp(G)$ denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let $cc(G)$ denote the cardinality of a minimum feedback vertex set (set of vertices $X$ so that $G-X$ is acyclic).\n\nConjecture For every planar graph $G$, $cc(G)\\leq 2cp(G)$.", "background": "Source: Open Problem Garden. Original node ID: 638. URL: http://www.openproblemgarden.org/op/jones_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/jones_conjecture\n- Author(s): Kloks, Ton; Lee, Chuan-Min; Liu, Jiping\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cycle packing; feedback vertex set; planar graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 9th, 2007 by cmlee\n\nProblem-page discussion:\nIn [KLL], the authors mention that there exists a family of nonplanar graphs for which $cc(G) = \\Theta( cp(G) \\log cp(G) )$, so no such result could hold for general graphs. They also point out that the conjecture is tight for wheels, and they prove it for the special case of outerplanar graphs.\n\nBibliography:\n*[KLL] Ton Kloks, Chuan-Min Lee, and Jiping Liu, New Algorithms for $k$-Face Cover, $k$-Feedback Vertex Set, and $k$-Disjoint Cycles on Plane and Planar Graphs, in Proceedings of the 28th International Workshop on Graph-Theoretic Concepts in Computer Science (WG2002), LNCS 2573, pp. 282--295, 2002.\n\nComments:\n- December 5th, 2019 | David Wood | Proved for subcubic planar: Proved for subcubic planar graphs by Marthe Bonamy, François Dross, Tomáš Masařík, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk [https://arxiv.org/abs/1912.01570].\n- October 29th, 2007 | Anonymous | Why Jones'?: Does anyone know why this is called Jones' Conjecture?\n- November 16th, 2007 | Anonymous | Reply: Why Jones'?: I am Jones. My Taiwanese name is Chuan-Min Lee. This conjecture came up when I was working on it with Ton Kloks and Jiping Liu. I used the name \"Jones\" instead of my Taiwanese name for ease of communication.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Jones' conjecture\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3171, "problem_number": "OPG-700", "title": "Chords of longest cycles", "statement": "Conjecture If $G$ is a 3-connected graph, every longest cycle in $G$ has a chord.", "background": "Source: Open Problem Garden. Original node ID: 700. URL: http://www.openproblemgarden.org/op/chords_of_longest_cycles.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/chords_of_longest_cycles\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: chord; connectivity; cycle\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 12th, 2007 by mdevos\n\nProblem-page discussion:\nA chord of a cycle $C$ is an edge $e$ so that $e \\not\\in E(C)$, but both ends of $e$ are in $V(C)$. Longest cycles are of great interest in basic graph theory, and this appealing conjecture suggests a very simple property they should share - at least in 3-connected graphs.\n\nIn dense graphs, this conjecture is easy to verify - for instance, if $G$ is Hamiltonian, it is trivially true. More interestingly, Thomassen [T] proved that his conjecture is true for cubic graphs using a clever sufficient condition for Hamiltonicity (based on Thomason's lollipop method) combined with a pretty theorem of Fleischner and Steibitz (cycle plus triangles graphs are 3-colorable).\n\nOther work on this conjecture has focused on graphs embedded in surfaces. Zhang [Z] has proved the conjecture for planar graphs of minimum degree four, Li and Zhang have proved the conjecture for graphs in the projective plane of minimum degree four [LZ1] and for 4-connected graphs in the klein bottle or torus [LZ2]. Finally, Kawarabayashi, Niu, and Zhang [KNZ] have shown the conjecture for 4-connected graphs on a fixed surface with sufficiently high face-width.\n\nBibliography:\n[KNZ] K. Kawarabayashi, J. Niu, C. Q. Zhang, Chords of longest circuits in locally planar graphs. European J. Combin. 28 (2007), no. 1, 315--321. MathSciNet\n\n[LZ1] X. Li, C. Q. Zhang, Chords of longest circuits in 3-connected graphs. Discrete Math. 268 (2003), no. 1-3, 199--206. MathSciNet\n\n[LZ2] X. Li, C. Q. Zhang, Chords of longest circuits of graphs embedded in torus and Klein bottle. J. Graph Theory 43 (2003), no. 1, 1--23. MathSciNet.\n\n[T2] C. Thomassen, Chords of longest cycles in cubic graphs. J. Combin. Theory Ser. B 71 (1997), no. 2, 211--214. MathSciNet.\n\n[Z] C. Q. Zhang, Longest cycles and their chords. J. Graph Theory 11 (1987), no. 4, 521--529. MathSciNet.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2261821\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1983278\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1974479\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1483476\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0917199\n\nComments:\n- October 14th, 2023 | Robert Samal | New partial results: New partial results for this appear in\n\nCarsten Thomassen: Chords in longest cycles, JCTB 129 (2018) 148-157\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Chords of longest cycles\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3172, "problem_number": "OPG-2095", "title": "Hamiltonicity of Cayley graphs", "statement": "Question Is every Cayley graph Hamiltonian?", "background": "Source: Open Problem Garden. Original node ID: 2095. URL: http://www.openproblemgarden.org/op/hamiltonicity_of_cayley_graphs.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hamiltonicity_of_cayley_graphs\n- Author(s): Rapaport-Strasser, E.\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: September 25th, 2008 by tchow\n\nProblem-page discussion:\nThis problem seems to have been first considered by Rapaport-Strasser [R]. Lovasz [L] conjectured more generally that every vertex-transitive graph is Hamiltonian. For a survey of results up to 1996, see [CG]. Although many specific Cayley graphs have been shown to be Hamiltonian, there are few general results. One exception is the theorem by Pak and Radoičić [PR] that every finite group $G$ with at least three elements has a generating set $S$ of size $|S| \\le \\log_2|G|$, such that the corresponding Cayley graph is Hamiltonian.\n\nBibliography:\n[CG] S. J. Curran, J. A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs—a survey, Discrete Math. 156 (1996), 1–18.\n\n[L] L. Lovasz, Problem 11, in “Combinatorial structures and their applications,” University of Calgary, Calgary, Alberta, Canada (1970), Gordon and Breach, New York.\n\n[PR] I. Pak and R. Radoičić, Hamiltonian paths in Cayley graphs, preprint.\n\n*[R] E. Rapaport-Strasser, Cayley color groups and Hamilton lines, Scripta Math. 24 (1959), 51–58.\n\nSource links:\n- Cayley graph: http://en.wikipedia.org/wiki/Cayley graph\n\nBibliography links:\n- Hamiltonian paths in Cayley graphs: http://www.math.umn.edu/%7Epak/hamcayley8.pdf\n\nComments:\n- December 15th, 2010 | Anonymous | Hamiltonicity of dence Cayley graphs: Christofides, Hladky, and Mathe (http://arxiv.org/abs/1008.2193) proved using the Regularity Method the case when the vertex-transitive graph is sufficiently dense.\n- September 21st, 2009 | Anonymous | [PR] paper: First, link to PR paper has moved with Pak homepage. Second, it is now published in Discrete Math.\n- January 8th, 2009 | Anonymous | Is this at all relevant?: Not sure but wondering if this link is at all relevant to the problem:\n\nhttp://books.google.com/books?id=aSyXqtfOuU4C&pg=PA49&dq=Cayley+graph&num=100&client=firefox-a#PPA49,M1\n\nThanks,\n\n- Farley\n- January 8th, 2009 | md | graphs vs. digraphs: The book you link to has examples of directed Cayley graphs with no (directed) Hamiltonian cycle. The existence of such graphs is interesting and relevant, but does not give a counterexample to the problem stated here.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamiltonicity of Cayley graphs\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3173, "problem_number": "OPG-37241", "title": "Strong 5-cycle double cover conjecture", "statement": "Conjecture Let $C$ be a circuit in a bridgeless cubic graph $G$. Then there is a five cycle double cover of $G$ such that $C$ is a subgraph of one of these five cycles.", "background": "Source: Open Problem Garden. Original node ID: 37241. URL: http://www.openproblemgarden.org/op/strong_5_cycle_double_cover_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/strong_5_cycle_double_cover_conjecture\n- Author(s): Arthur; Hoffmann-Ostenhof\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cycle cover\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: August 3rd, 2010 by arthur\n\nProblem-page discussion:\nA cycle in $G$ is meant to be a $2$-regular subgraph of $G$. A five cycle double cover of $G$ is a set of five cycles of $G$ such that every edge of $G$ is contained in exactly two of these cycles.\n\nThis conjecture is a combination and thus strengthening of the $5$-cycle double cover conjecture and the strong cycle double cover conjecture.\n\nRelated:\nRelated problems\nCycle double cover conjecture\n\nComments:\n- November 23rd, 2011 | niekeaerts | ?: 3-connected Counterexample: $|VG|=8$\n$|EG|=12$\n$VG=\\{a,b,c,d,e,f,g,h\\}$\nAdjacency matrix (in alphabetical order):\n$$\n\\left( \\begin{array}{llllllll } 0&1&1&0&0&0&0&1 & 1&0&1&1&0&0&0&0 & 1&1&0&0&1&0&0&0 & 0&1&0&0&1&1&0&0 & 0&0&1&1&0&0&1&0 & 0&0&0&1&0&0&1&1 & 0&0&0&0&1&1&0&1 & 1&0&0&0&0&1&1&0 \\end{array}\\right)\n$$\n\nFor the explanation, see the following comment. -----------\n\nIt could be so that I am making a mistake, if so, please explain my mistake to me.\nI came to this point by simple trial and error.\nI would like to upload a simple picture, but I seem to be a little lost on how to do this.\n\nNieke Aerts\n- November 24th, 2011 | Robert Samal | Not a counterexample: Dear Nieke,\n\nunfortunately, your graph is definitely not a counterexample. I could not follow your explanations, let me instead show, why your graph with the indicated cycle has the desired 5-CDC.\n\nYour graph is planar, 2-edge-connected and the circuit C is a boundary of one of the faces. It turns out that for all such instances the conjecture is true: consider all face boundaries -- a collection of circuits. Now a proper 4-coloring of the dual graph splits the circuits into four cycles, the given circuit is contained in one of them. (The fifth cycle can be empty in this case.)\n\nThink about it, and if you still believe you have a counterexample, post again. For now, I am not reading the other comments, as they prove something that turns out to be false:-).\n\nBest wishes, Robert\n- November 24th, 2011 | niekeaerts | Indeed, no counterexamples: Dear Robert,\n\nThanks for your reply.\n\nI was thinking that the cycles have to be connected, which is obviously not true. So therefore my thought-to-be-counterexamples weren't correct. Thanks for the help! I wouldn't mind deleting all those comments, but I am not sure how to:)\n\nBest, Nieke\n- November 23rd, 2011 | niekeaerts | Explanation of the 3-connected counterexample: Consider the circuit $a,b,d,f,h,a$ to be color 1. And assume there is a 5-cycle cover containing this circuit as one of the cycles. We distinguish the cycles by color.\nThen $(a,b)$ and $(a,c)$ are colored with the same color (color 2) in the second cycle covering them, and similarly $(a,b)$ and $(a,h)$ have the same color (color 3) in the second cycle covering them. (As otherwise $(a,c)$ is colored twice by the cycle $(a,c,a)$ which, if allowed, quickly shows necessity of 6 colors)\n- November 23rd, 2011 | niekeaerts | Explanation of the 3-connected counterexample (Part II): Now $(b,c)$ still needs to be covered twice, which cannot be done by 1 color as then again one needs 6 colors. One of the colors has to be color 2, otherwise this will leave $b$ towards $a$ and $d$ and therefore there will be no escape possibility from $b$ for the two new colors. So the triangle $a,b,c$ is colored with color 2. By symmetry the triangle $f,g,h$ is also one color cycle (color 4). Now $(c,e)$, $(d,e)$ and $(e,g)$ are not colored and therefore need to be in the cycle of color 3 and the cycle of color 5. But then $e$ has degree 3 in both cycles which is a contradiction.\nSo a 5-cycle double cover containing this cycle does not exist.\n- November 23rd, 2011 | niekeaerts | ?: Counterexample for a graph with a 2-cut: $|VG|=8$\n$|EG|=12$\n$VG=\\{a,b,c,d,e,f,g,h\\}$\nAdjacency matrix (in alphabetical order):\n$$\n\\left( \\begin{array}{llllllll } 0&1&1&1&0&0&0&0 & 1&0&1&1&0&0&0&0 & 1&1&0&0&1&0&0&0 & 1&1&0&0&0&1&0&0 & 0&0&1&0&0&0&1&1 & 0&0&0&1&0&0&1&1 & 0&0&0&0&1&1&0&1 & 0&0&0&0&1&1&1&0 \\end{array}\\right)\n$$\n\nConsider the circuit $a,c,e,g,f,d,a$ to be color 1, now on either side of the 2-cut, we need three more colors, of which only one color can serve both (due to the 2-cut), so we need at least 6 colors (6 cycles).\n\n-----------\nIt could be so that I am making a mistake, if so, please explain my mistake to me.\nI came to this point by simple trial and error.\nI would like to upload a simple picture, but I seem to be a little lost on how to do this.\n\nNieke Aerts\n- November 23rd, 2011 | niekeaerts | Consider the circuit to be: Consider the circuit $a,b,d,f,h,a$ to be color 1. And assume there is a 5-cycle cover containing this circuit as one of the cycles. We distinguish the cycles by color.\nThen $(a,b)$ and $(a,c)$ are colored with the same color (color 2) in the second cycle covering them, and similarly $(a,c),(a,h)$ have the same color (color 3) in the second cycle covering them. (As otherwise $(a,c)$ is colored twice by the cycle $(a,c,a)$ which, if allowed, quickly shows necessity of 6 colors)\n- November 23rd, 2011 | niekeaerts | !!Ignore previous comment!!: Sorry, I pressed reply at the wrong statement.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 26.\n\nAttempt notes:\nTarget:\nMake progress on \"Strong 5-cycle double cover conjecture\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3174, "problem_number": "OPG-46584", "title": "Decomposing an eulerian graph into cycles.", "statement": "Conjecture Every simple eulerian graph on $n$ vertices can be decomposed into at most $\\frac{1}{2}(n-1)$ cycles.", "background": "Source: Open Problem Garden. Original node ID: 46584. URL: http://www.openproblemgarden.org/op/decomposing_an_eulerian_graph_into_cycles.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_an_eulerian_graph_into_cycles\n- Author(s): Hajós, G.\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture is tight because a complete graph on $2k+1$ vertices cannot be covered by less than $k$ cycles.\n\nThere is a similar conjecture about decomposition of a connected graph into paths.\n\nBibliography:\n* [L] L. Lovász, On covering of graphs. In Theory of Graphs (Proc. Colloq., Tihany, 1966), 231--236. Academic Press, New York, 1968.\n\nRelated:\nRelated problems\nDecomposing a connected graph into paths.\n\nDiscussion links:\n- decomposition of a connected graph into paths: http://www.openproblemgarden.org/?q=node/46583\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposing an eulerian graph into cycles.\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3175, "problem_number": "OPG-46606", "title": "Decomposing an eulerian graph into cycles with no two consecutives edges on a prescribed eulerian tour.", "statement": "Conjecture Let $G$ be an eulerian graph of minimum degree $4$, and let $W$ be an eulerian tour of $G$. Then $G$ admits a decomposition into cycles none of which contains two consecutive edges of $W$.", "background": "Source: Open Problem Garden. Original node ID: 46606. URL: http://www.openproblemgarden.org/op/decomposing_an_eulerian_graph_into_cycles_with_no_two_consecutives_edges_on_a_prescirbed_eulerian_tour.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_an_eulerian_graph_into_cycles_with_no_two_consecutives_edges_on_a_prescirbed_eulerian_tour\n- Author(s): Sabidussi, Gert\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nKotzig proved the converse statement:\n\nTheorem Let $G$ be an eulerian graph of minimum degree $4$ and let $(C_1, \\dots, C_p)$ be a decomposition into cycles of $G$. Then $G$ admits an eulerian tour such that none of the $C_i$, $1\\leq i\\leq p$, contains two consecutive edges of $W$.\n\nFor more details on eulerian graphs, see [F].\n\nBibliography:\n*[F] H. Fleischner. Eulerian Graphs and Related Topics. Part 1. Vol. 1. Annals of Discrete Mathematics, Vol. 45, (1990) North-Holland, Amsterdam.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposing an eulerian graph into cycles with no two consecutives edges on a prescribed eulerian tour.\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3176, "problem_number": "OPG-47285", "title": "Every prism over a 3-connected planar graph is hamiltonian.", "statement": "Conjecture If $G$ is a $3$-connected planar graph, then $G\\square K_2$ has a Hamilton cycle.", "background": "Source: Open Problem Garden. Original node ID: 47285. URL: http://www.openproblemgarden.org/op/every_prism_over_a_3_connected_planar_graph_is_hamiltonian.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/every_prism_over_a_3_connected_planar_graph_is_hamiltonian\n- Author(s): Kaiser, Tomás; Král, Daniel; Rosenfeld, Moshe; Ryjácek, Zdenek; Voss, Heinz-Jürgen\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 11th, 2013 by fhavet\n\nProblem-page discussion:\nThe Cartesian product $G\\square K_2$ is called the prism over $G$.\n\nRosenfeld and Barnette [RB] showed that the Four-Colour Theorem implies that cubic planar 3-connected graphs have hamiltonian prisms. Fleischner [F] found a proof avoiding the use of the Four Colour Theorem. Eventually, Paulraja [P] showed that planarity is inessential here: The prism over any 3-connected cubic graph has a Hamilton cycle.\n\nClearly, if $G$ is hamiltonian, then $G\\square K_2$ is also hamiltonian. A classical theorem of Tutte [T] states that all 4-connected planar graphs are hamiltonian. There are well-known examples of non-hamiltonian 3-connected planar graphs.\n\nBibliography:\n[F] H. Fleischner, The prism of a 2-connected, planar, cubic graph is hamiltonian (a proof independent of the four colour theorem), in Graph theory in memory of G. A. Dirac, Volume 41 of Ann. Discrete Math., 1989), 141–170.\n\n*[KKRRV] T. Kaiser, D. Kráľ, M. Rosenfeld, Z. Ryjáček, H.-J. Voss, Hamilton cycles in prisms, Journal of graph theory 56 (2007), 249-269.\n\n[P] P. Paulraja, “A characterization of hamiltonian prisms”, J. Graph Theory 17 (1993) 161–171.\n\n[RB] M. Rosenfeld and D. Barnette, Hamiltonian circuits in certain prisms, Discrete Math. 5 (1973) 389–394.\n\n[T] W. T. Tutte, A theorem on planar graphs, Trans. Amer. Math. Soc. 82 (1956) 99–116.\n\nRelated:\nRelated problems\nBarnette's Conjecture\n\nDiscussion links:\n- Cartesian product: http://en.wikipedia.org/wiki/Cartesian product of graphs\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Every prism over a 3-connected planar graph is hamiltonian.\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3177, "problem_number": "OPG-47294", "title": "4-connected graphs are not uniquely hamiltonian", "statement": "Conjecture Every $4$-connected graph with a Hamilton cycle has a second Hamilton cycle.", "background": "Source: Open Problem Garden. Original node ID: 47294. URL: http://www.openproblemgarden.org/op/4_connected_graphs_are_not_uniquely_hamiltonian.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/4_connected_graphs_are_not_uniquely_hamiltonian\n- Author(s): Fleischner, Herbert\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 11th, 2013 by fhavet\n\nBibliography:\n*[F] H. Fleischner, Uniquely Hamiltonian graphs of minimum degree four,, J. Graph Theory, to appear.\n\nRelated:\nRelated problems\nr-regular graphs are not uniquely hamiltonian.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"4-connected graphs are not uniquely hamiltonian\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3178, "problem_number": "OPG-47356", "title": "Hamilton decomposition of prisms over 3-connected cubic planar graphs", "statement": "Conjecture Every prism over a $3$-connected cubic planar graph can be decomposed into two Hamilton cycles.", "background": "Source: Open Problem Garden. Original node ID: 47356. URL: http://www.openproblemgarden.org/op/decomposing_the_prism_of_a_3_connected_cubic_planar_graphs_in_hamilton_cycles.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_the_prism_of_a_3_connected_cubic_planar_graphs_in_hamilton_cycles\n- Author(s): Alspach, Brian; Rosenfeld, Moshe\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 12th, 2013 by fhavet\n\nProblem-page discussion:\nThe prism over a graph $G$ is the Cartesian product $G\\square K_2$.\n\nRosenfeld and Barnette [RB] deduced from the Four-Colour Theorem that the prism over cubic planar 3-connected $G$ has a Hamilton cycle $C$. The graph $G\\setminus E(C)$ is cycle factor (spanning union of cycles). The conjecture says that one can choses $C$ so that the cycle factor $G\\setminus E(C)$ has a unique cycle, that is a Hamilton cycle.\n\nBibliography:\n*[AR] B. Alspach and M. Rosenfeld, On Hamilton decompositions of prisms over simple $3$-polytopes. Graphs Combin. 2 (1986), 1--8.\n\n[RB] M. Rosenfeld and D. Barnette, Hamiltonian circuits in certain prisms, Discrete Math. 5 (1973) 389–394.\n\nRelated:\nRelated problems\nEvery prism over a 3-connected planar graph is hamiltonian.\n\nDiscussion links:\n- Cartesian product: http://en.wikipedia.org/wiki/Cartesian product of graphs\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamilton decomposition of prisms over 3-connected cubic planar graphs\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3179, "problem_number": "OPG-142", "title": "The Berge-Fulkerson conjecture", "statement": "Conjecture If $G$ is a bridgeless cubic graph, then there exist 6 perfect matchings $M_1,\\ldots,M_6$ of $G$ with the property that every edge of $G$ is contained in exactly two of $M_1,\\ldots,M_6$.", "background": "Source: Open Problem Garden. Original node ID: 142. URL: http://www.openproblemgarden.org/op/the_berge_fulkerson_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Matchings.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_berge_fulkerson_conjecture\n- Author(s): Berge, Claude; Fulkerson, Delbert R.\n- Subject(s): Graph Theory; Basic Graph Theory; Matchings\n- Keywords: cubic; perfect matching\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nThis conjecture is due to Berge and Fulkerson, and appears first in [F] (see [S79b]).\n\nIf $G$ is 3-edge-colorable, then we may choose three perfect matchings $M_1,M_2,M_3$ so that every edge is in exactly one. Taking each of these twice gives us 6 perfect matchings with the properties described above. Thus, the above conjecture holds trivially for 3-edge-colorable graphs. There do exist bridgeless cubic graphs which are not 3-edge-colorable (for instance the Petersen graph), but the above conjecture asserts that every such graph is close to being 3-edge-colorable.\n\nDefinition: An $r$-graph is an $r$-regular graph $G$ on an even number of vertices with the property that every edge-cut which separates $V(G)$ into two sets of odd cardinality has size at least $r$.\n\nObserve that a cubic graph is a 3-graph if and only if it has no bridge. If G is an $r$-regular graph which has an $r$-edge-coloring, then every color class is a perfect matching, so $|V(G)|$ must be even, and every color must appear in every edge-cut which separates $V(G)$ into two sets of odd size, so every edge-cut of this form must have size at least $r$. Thus, every $r$-edge-colorable $r$-regular graph is an $r$-graph. In a sense, we may view the $r$-graphs as the $r$-regular graphs which have the obvious necessary conditions to be $r$-edge-colorable. Seymour [S79b] defined $r$-graphs and offered the following generalization of the Berge-Fulkerson conjecture.\n\nConjecture (The generalized Berge-Fulkerson conjecture (Seymour)) Let $G$ be an $r$-graph. Then there exist $2r$ perfect matchings $M_1,\\ldots,M_{2r}$ of $G$ with the property that every edge of $G$ is contained in exactly two of $M_1,\\ldots,M_{2r}$.\n\nMore generally, for a graph $G=(V,E)$, one may consider the vector space of real numbers indexed by $E$. We associate every perfect matching $M$ with its characteristic vector. In this context, the Berge-Fulkerson conjecture asserts that for every 3-graph, the vector which is identically 1 may be written as a half-integer combination of perfect matchings. Edmonds matching polytope theorem [E] gives a complete characterization of what vectors in ${\\mathbb R}^E$ which can be written as a nonnegative real combination of perfect matchings. A particular consequence of this theorem is that the vector which is identically 1 can be written as a nonnegative rational combination of perfect matchings if G is an $r$-graph. It follows from this that for every $r$-graph $G$, there is a list of perfect matchings $M_1,\\ldots,M_{kr}$ so that every edge is contained in exactly $k$ of them. Unfortunately, the particular $k$ depends on the graph. The following weak version of the Berge-Fulkerson conjecture asserts that this dependence is inessential.\n\nConjecture (the weak Berge-Fulkerson conjecture) There exists a positive integer $k$ with the following property. Every 3-graph $G$ has a list of $3k$ perfect matchings such that every edge of $G$ is contained in exactly $k$ of them.\n\nThere is a fascinating theorem of Lovasz [L] that characterizes which vectors in ${\\mathbb Z}^E$ can be written as an integer combination of perfect matchings. However, very little is known about nonnegative integer combinations of perfect matchings. In particular, if the Berge-Fulkerson conjecture is true, then for every 3-graph $G=(V,E)$, there is a list of 5 perfect matchings with union $E$ (take any 5 of the 6 perfect matchings given by the conjecture). The following weakening of this (suggested by Berge) is still open.\n\nConjecture There exists a fixed integer $k$ such that the edge set of every 3-graph can be written as a union of $k$ perfect matchings.\n\nAnother consequence of the Berge-Fulkerson conjecture would be that every 3-graph has 3 perfect matchings with empty intersection (take any 3 of the 6 perfect matchings given by the conjecture). The following weakening of this (also suggested by Berge) is still open.\n\nConjecture There exists a fixed integer $k$ such that every 3-graph has a list of $k$ perfect matchings with empty intersection.\n\nBibliography:\n[E] J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices, J. Res. Nat. Bur Stand B, Math & Math Phys. 69B (1965), 125-130.\n\n[F] D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Math. Programming 1 (1971) 168-194. MathSciNet\n\n[KKN] T. Kaiser, D. Kral, and S. Norine, Unions of perfect matchings in cubic graphs\n\n[L] L. Lovasz, Matching structure and the matching lattice, J. Combin. Theory Ser. B 43 (1987), 187-222. MathSciNet\n\n[R] R. Rizzi, Indecomposable r-graphs and some other counterexamples, J. Graph Theory 32 (1999), 1-15. MathSciNet\n\n[S79a] P.D. Seymour, \"Some unsolved problems on one-factorizations of graphs\", Graph theory and Related Topics, J.A. Bondy and U.S.R. Murty (Editors), Academic, New York (1979), 367-368.\n\n[S79b] P.D. Seymour, On multi-colourings of cubic graphs, and conjectures of Fulkerson and Tutte, Proc. London Math Soc. 38 (1979), 423-460. MathSciNet\n\nSource links:\n- bridgeless: http://en.wikipedia.org/wiki/bridge (graph theory)\n- cubic: http://en.wikipedia.org/wiki/cubic graph\n- perfect matchings: http://en.wikipedia.org/wiki/matching\n\nDiscussion links:\n- edge-colorable: http://en.wikipedia.org/wiki/edge coloring\n- Petersen graph: http://en.wikipedia.org/wiki/Petersen graph\n- regular: http://en.wikipedia.org/wiki/regular graph\n- edge-cut: http://en.wikipedia.org/wiki/connectivity (graph theory)\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0294149\n- Unions of perfect matchings in cubic graphs: http://www.math.princeton.edu/%7Esnorin/papers/union-en.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0904405\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1704172\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0532981\n\nComments:\n- April 4th, 2011 | Louis Esperet | equivalent conjecture: It was recently proved by G Mazzuocolo (The Equivalence of Two Conjectures of Berge and Fulkerson, J. Graph Theory (2010) doi:10.1002/jgt.20545), that Berge-Fulkerson is indeed equivalent to the statement that for any 3-graph G, the edge-set of G can be covered by 5 perfect matchings. This might be worth mentioning it.\n- February 25th, 2009 | Andrew King | Covering with perfect matchings: It is not hard to show that you can cover the edges of a bridgeless cubic graph with $\\log(n)$ perfect matchings. Is there some smaller-order function that suffices?\n- February 25th, 2009 | mdevos | I believe this is best known: The $log(n)$ bound you mention (a consequence of Edmond's perfect matching polytope theorem) is, to my knowledge, the best known lower bound.\n- October 16th, 2011 | Emilio Brazil | Reference for log(n) bound: I'm working with bridgeless cubic graphs and this bound is a good theoretical bound for my purposes. I'd like to know references for this $\\log(n)$ bound.\n- November 23rd, 2011 | Andrew King | log(n) bound: The proof is basically the same as Lovász' proof that $\\chi(G) \\leq \\log(n)\\cdot (\\chi_f(G)+1)$: It is well-known that the fractional chromatic number of a bridgeless cubic graph is 3. Therefore there is a probability distribution on the perfect matchings of $G$ such that given a random matching from this distribution, an edge $e$ is hit with probability 1/3.\n\nNow let $k$ be $\\log_{3/2}(n)+1$ and take $k$ perfect matchings from this distribution, and consider the probability that an edge $e$ is in none of them. This is $(2/3)^k < 1/n$, so by the union bound, the probability that every edge is in one of the matchings is greater than $1- n(1/n) = 0$. Therefore there is some choice of $k$ perfect matchings that cover the graph.\n- November 22nd, 2011 | Anonymous | Best known upper bound: I believe that the best known upper bound is given here http://arxiv.org/abs/1111.1871\n- March 2nd, 2009 | Andrew King | Matchings and odd cuts: I like thinking about this problem in terms of fractional colourings. A roughly equivalent proof (to the one you mentioned) is as follows. In a fractional 3-edge-colouring, every matching must be a perfect matching. The weights on the matchings, when divided by 3, give you a probability distribution. If you pick $3 \\log(n)$ matchings from this distribution, the union bound tells you that with positive probability you hit every edge. This method is powerful enough to prove that the fractional and integer chromatic numbers are always within a factor of $\\log(n)$ of one another.\n\nHere are two even weaker questions:\n\nIs there some $k$ such that any bridgeless cubic graph contains $k$ perfect matchings whose intersection does not contain an odd cut?\n\nDoes every bridgeless cubic graph contain two perfect matchings that together hit no 5-cut 8 or 10 times?\n\nI suspect both of these are open as well.\n- March 4th, 2009 | mdevos | Sure: Okay, sure, but the proof that there exists a fractional 3-edge-coloring is either a corollary of Edmond's Theorem or something essentially equivalent to it (such as the approach Seymour uses in his paper on r-graphs). In short, I agree that we are talking about essentially the same argument.\n\nI believe that your first question is open. It is a well known conjecture (elsewhere on this site) that there should be 2 perfect matchings whose intersection does not contain an odd cut.\n\nThe second question has been resolved (using Edmond's theorem) by Kaiser, Kral, and Norine and I will add a link to the paper in the reference section.\n- October 14th, 2011 | Anonymous | proof of log(n): Do you know a reference for a proof this log(n) boundary? I looked for it in text book but I don't have any clue.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 24.\n\nAttempt notes:\nTarget:\nMake progress on \"The Berge-Fulkerson conjecture\" in Graph Theory; Basic Graph Theory; Matchings, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3180, "problem_number": "OPG-543", "title": "The intersection of two perfect matchings", "statement": "Conjecture Every bridgeless cubic graph has two perfect matchings $M_1$, $M_2$ so that $M_1 \\cap M_2$ does not contain an odd edge-cut.", "background": "Source: Open Problem Garden. Original node ID: 543. URL: http://www.openproblemgarden.org/op/intersecting_two_perfect_matchings.\n\nSource subject path: Graph Theory > Basic Graph Theory > Matchings.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/intersecting_two_perfect_matchings\n- Author(s): Macajova, Edita; Skoviera, Martin\n- Subject(s): Graph Theory; Basic Graph Theory; Matchings\n- Keywords: cubic; nowhere-zero flow; perfect matching\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 30th, 2007 by mdevos\n\nProblem-page discussion:\nLet $G = (V,E)$ be a bridgeless cubic graph. A binary cycle (henceforth called cycle) is a set $C \\subseteq E$ so that every vertex of $(V,C)$ has even degree (equivalently, a cycle is any member of the binary cycle space). A postman join is a set $J \\subseteq E$ so that $E \\setminus J$ is a cycle. Note that since $G$ is cubic, every perfect matching is a postman join. Next we state a well-known theorem of Jaeger in three equivalent forms.\n\nTheorem (Jaeger's 8-flow theorem)\n\n- $G$ has a nowhere-zero flow in the group ${\\mathbb Z}_2^3$.\n- $G$ has three cycles $C_1,C_2,C_3$ so that $C_1 \\cup C_2 \\cup C_3 = E$.\n- $G$ has three postman joins $J_1,J_2,J_3$ so that $J_1 \\cap J_2 \\cap J_3 = \\emptyset$.\n\nThe last of these statements is interesting, since The Berge Fulkerson Conjecture (if true) implies the following:\n\nConjecture $G$ has three perfect matchings $M_1,M_2,M_3$ so that $M_1 \\cap M_2 \\cap M_3= \\emptyset$.\n\nSo, we know that $G$ has three postman joins $J_1,J_2,J_3$ with empty intersection, and it is conjectured that $J_1,J_2,J_3$ may be chosen so that each is a perfect matching, but now we see two statements in between the theorem and the conjecture. Namely, is it true that $J_1,J_2,J_3$ may be chosen so that one is a perfect matching? or two? The first of these was solved recently.\n\nTheorem (Macajova, Skoviera) $G$ has two postman sets $J_1,J_2$ and one perfect matching $M$ so that $M \\cap J_1 \\cap J_2 = \\emptyset$\n\nThe second of these asks for two perfect matchings $M_1,M_2$ and one postman join $J$ so that $M_1 \\cap M_2 \\cap J = \\emptyset$. It is an easy exercise to show that a set $S \\subseteq E$ contains a postman join if an only if $S$ has nonempty intersection with every odd edge-cut. Therefore, finding two perfect matchings and one postman join with empty common intersection is precisely equivalent to the conjecture at the start of this page - find two perfect matchings whose intersection contains no odd edge-cut.\n\nBibliography:\n* Edita Macajova, Martin Skoviera, Fano colourings of cubic graphs and the Fulkerson conjecture. Theoret. Comput. Sci. 349 (2005), no. 1, 112--120. MathSciNet\n\nRelated:\nRelated problems\nThe Berge-Fulkerson conjecture\n\nDiscussion links:\n- The Berge Fulkerson Conjecture: http://www.openproblemgarden.org/?q=node/142\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2183473\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 24.\n\nAttempt notes:\nTarget:\nMake progress on \"The intersection of two perfect matchings\" in Graph Theory; Basic Graph Theory; Matchings, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3181, "problem_number": "OPG-600", "title": "Matchings extend to Hamiltonian cycles in hypercubes", "statement": "Question Does every matching of hypercube extend to a Hamiltonian cycle?", "background": "Source: Open Problem Garden. Original node ID: 600. URL: http://www.openproblemgarden.org/op/matchings_extends_to_hamilton_cycles_in_hypercubes.\n\nSource subject path: Graph Theory > Basic Graph Theory > Matchings.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/matchings_extends_to_hamilton_cycles_in_hypercubes\n- Author(s): Ruskey, Frank; Savage, Carla\n- Subject(s): Graph Theory; Basic Graph Theory; Matchings\n- Keywords: Hamiltonian cycle; hypercube; matching\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: September 28th, 2007 by Jirka\n\nProblem-page discussion:\nThis question is due to Ruskey and Savage and appears in [RS] (page 19, question 3). The answer is positive for $d$-cube if $d \\le 4$.\n\nFink [F] proved Kreweras' conjecture [K] which asserts that every perfect matching of hypercube extends to a Hamiltonian cycle.\n\nBibliography:\n[RS] F. Ruskey and C. D. Savage, SIAM Journal on Discrete Mathematics 6, No.1 (1993) 152-166. download\n\n[F] J. Fink. Perfect matchings extend to Hamilton cycles in hypercubes. J. Comb. Theory, Ser. B, 97(6):1074-1076, 2007. download\n\n[K] G. Kreweras, Matchings and Hamiltonian cycles on hypercubes, Bull. Inst. Combin. Appl. 16 (1996), 87--91.\n\nSource links:\n- matching: http://en.wikipedia.org/wiki/matching\n- hypercube: http://en.wikipedia.org/wiki/hypercube\n- Hamiltonian cycle: http://en.wikipedia.org/wiki/Hamiltonian path\n\nDiscussion links:\n- perfect matching: http://en.wikipedia.org/wiki/matching\n\nBibliography links:\n- download: http://www4.ncsu.edu/%7Esavage/AVAILABLE_FOR_MAILING/transposition_matching.ps\n- download: http://kam.mff.cuni.cz/%7Efink/publications/kreweras1.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Matchings extend to Hamiltonian cycles in hypercubes\" in Graph Theory; Basic Graph Theory; Matchings, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3182, "problem_number": "OPG-743", "title": "Random stable roommates", "statement": "Conjecture The probability that a random instance of the stable roommates problem on $n \\in 2{\\mathbb N}$ people admits a solution is $\\Theta( n ^{-1/4} )$.", "background": "Source: Open Problem Garden. Original node ID: 743. URL: http://www.openproblemgarden.org/op/random_stable_roommates.\n\nSource subject path: Graph Theory > Basic Graph Theory > Matchings.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/random_stable_roommates\n- Author(s): Mertens, Stephan\n- Subject(s): Graph Theory; Basic Graph Theory; Matchings\n- Keywords: stable marriage; stable roommates\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 26th, 2008 by mdevos\n\nProblem-page discussion:\nA system of preferences for a graph $G$ is a family $\\{ >_v \\}_{v \\in V(G)}$ so that every $>_v$ is a linear ordering of the neighbors of the vertex $v$. We say that $v$ prefers $u$ to $u'$ if $u >_v u'$. A perfect matching $M$ in $G$ is stable if there do not exist $uv,u'v' \\in M$ so that $u$ prefers $v'$ to $v$ and $v'$ prefers $u$ to $u'$.\n\nA famous theorem of Gale-Shapley [GS] proves that every system of preferences on a complete bipartite graph $K_{n,n}$ admits a stable perfect matching. Indeed, they provide an amusing algorithm to construct one. On complete graphs, this problem is known as either the homosexual stable marriage problem, or more commonly, the stable roommate problem. Here there does not always exist a solution (that is, a stable perfect matching), but Irving [I] constructed an algorithm which runs in polynomial time, and outputs a solution if one exists.\n\nLet $P_n$ denote the probability that a random instance of the stable roommates problem has a solution (so the above conjecture asserts that $P_n = \\Theta( n^{-1/4}$ ). The following are the best known asymptotic bounds for $P_n$ (with $n$ even) and hold for $n$ sufficiently large. The lower bound is due to Pittel [P] and the upper bound to Pittel and Irving [IP]\n\n$$\n\\frac{2 e ^{3/2} }{ \\sqrt{\\pi n}} \\le P_n \\le \\frac{\\sqrt{e}}{2}\n$$\n\nMertens [M] did an extensive Monte-Carlo simulation to obtain the above conjecture. Indeed, by guessing at the constant he even offers the stronger conjecture $P_n \\simeq e \\sqrt{ \\frac{2}{\\pi} } n ^{-1/4}$.\n\nBibliography:\n[GS] D. Gale D and L. S. Shapley, College admissions and the stability of marriage, Am. Math. Mon. 69 9-15.\n\n[I] R. W. Irving, An efficient algorithm for the stable roommates problem, J. Algorithms 6 577-95.\n\n[IP] B. Pittel and R. W. Irving, An upper bound for the solvability of a random stable roommates instance, Random Struct. Algorithms 5 465-87.\n\n*[M] S. Mertens, Random stable matchings, J. Stat. Mech. Theory Exp. 2005, no. 10 MathSciNet\n\n[P] B. Pittel, The 'stable roommates' problem with random preferences, Ann. Probab. 21 1441-77\n\nBibliography links:\n- Random stable matchings: http://www.iop.org/EJ/article/1742-5468/2005/10/P10008/jstat5_10_p10008.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2185394\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Random stable roommates\" in Graph Theory; Basic Graph Theory; Matchings, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3183, "problem_number": "OPG-153", "title": "Highly connected graphs with no K_n minor", "statement": "Problem Is it true for all $n \\ge 0$, that every sufficiently large $n$-connected graph without a $K_n$ minor has a set of $n-5$ vertices whose deletion results in a planar graph?", "background": "Source: Open Problem Garden. Original node ID: 153. URL: http://www.openproblemgarden.org/op/high_connectivity_no_k_n.\n\nSource subject path: Graph Theory > Basic Graph Theory > Minors.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/high_connectivity_no_k_n\n- Author(s): Thomas, Robin\n- Subject(s): Graph Theory; Basic Graph Theory; Minors\n- Keywords: connectivity; minor\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 10th, 2007 by mdevos\n\nProblem-page discussion:\nA famous conjecture of Jorgensen asserts that every 6-connected graph without a $K_6$-minor is apex (planar plus one vertex). If true, Jorgensen's conjecture does not generalize (naively) to higher connectivities, since for sufficiently large $n$, there do exist $n$-connected graphs which are not close to planar in the sense we are considering (many more than $n-5$ vertices must be deleted to leave a planar graph). This conjecture of Thomas asserts that all such graphs are small in size.\n\nFor $n \\le 6$ this conjecture is true. For $n \\le 4$ this conjecture is trivial, since any graph without a $K_4$-minor is planar. The $n=5$ case follows from a theorem of Wagner which gives a construction for all graphs without $K_5$-minors (and from which it follows that every 4-connected graph with no $K_5$ minor is planar). The $n=6$ case was recently resolved by DeVos, Hegde, Kawarabayashi, Norine, Thomas, and Wollan. The difficulties associated with finding $K_n$ minors in graphs make this conjecture appear daunting, but if true, it would yield powerful insight into the structure of graphs.\n\nSource links:\n- connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- $K_n$: http://en.wikipedia.org/wiki/complete graph\n- minor: http://en.wikipedia.org/wiki/minor (graph theory)\n- planar graph: http://en.wikipedia.org/wiki/planar graph\n\nDiscussion links:\n- conjecture of Jorgensen: http://www.openproblemgarden.org/?q=op/jorgensens_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Highly connected graphs with no K_n minor\" in Graph Theory; Basic Graph Theory; Minors, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3184, "problem_number": "OPG-154", "title": "Jorgensen's Conjecture", "statement": "Conjecture Every 6-connected graph without a $K_6$ minor is apex (planar plus one vertex).", "background": "Source: Open Problem Garden. Original node ID: 154. URL: http://www.openproblemgarden.org/op/jorgensens_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Minors.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/jorgensens_conjecture\n- Author(s): Jorgensen, Leif K.\n- Subject(s): Graph Theory; Basic Graph Theory; Minors\n- Keywords: connectivity; minor\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 10th, 2007 by mdevos\n\nProblem-page discussion:\nFor $n \\le 5$, the class of graphs with no $K_n$ minor is very well understood. Simple graphs without $K_3$ minors are forests. Graphs without $K_4$ minors are called series-parallel graphs, and have a simple construction. Finally, Wagner [W] obtained a construction for all graphs without $K_5$ minors. For $n \\ge 6$, an explicit characterization of those graphs without $K_n$ minors appears hopeless. The graph minors project of Robertson and Seymour give a rough structure theorem for such classes, but much remains unknown. In particular, this conjecture and Thomas' conjecture highly connected graphs with no $K_n$-minor suggest interesting properties of highly connected graphs without $K_n$ minors which appear quite difficult to resolve.\n\nPart of the interest in graphs without $K_n$ minors stems from Hadwiger's conjecture (every loopless graph without a $K_{n+1}$ minor is $n$-colorable). Indeed, Wagner's work on graphs with no $K_5$ minor was done while studying the $n=4$ case of Hadwiger. More recently, Robertson, Seymour, and Thomas [RST] proved Hadwiger's conjecture for $n=5$, and in doing so came somewhat close to proving Jorgensoen's conjecture. The thrust of their argument is to prove that any minimal counterexample to Hadwiger for $n=5$ is apex. However, in doing so, they exploit both connectivity and coloring properties of a minimal counterexample. It would appear to be difficult to modify their argument to prove Jorgensen's conjecture.\n\nDeVos, Hegde, Kawarabayashi, Norine, Thomas, and Wollan proved this conjecture true for all sufficiently large graphs [KNTWa,KNTWb].\n\nBibliography:\n[RST] N. Robertson, P. D. Seymour, R. Thomas, Hadwiger's conjecture for $K\\sb 6$-free graphs. Combinatorica 13 (1993), no. 3, 279-361. MathSciNet\n\n[W] K. Wagner Uber eine Eigenschaft der ebenen Komplexe, Math. Ann 114 (1937) 570-590. MathSciNet\n\n[KNTWa] Ken-ichi Kawarabayashi, Serguei Norine, Robin Thomas, Paul Wollan. $K_6$ minors in 6-connected graphs of bounded tree-width. J. Combinatorial Theory, Series B, 136:1--32, 2019\n\n[KNTWb] Ken-ichi Kawarabayashi, Serguei Norine, Robin Thomas, Paul Wollan. $K_6$ minors in large 6-connected graphs. J. Combinatorial Theory, Series B, 129:158-203, 2019.\n\nSource links:\n- connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- $K_6$: http://en.wikipedia.org/wiki/complete graph\n- minor: http://en.wikipedia.org/wiki/minor (graph theory)\n\nDiscussion links:\n- series-parallel graphs: http://en.wikipedia.org/wiki/series-parallel graph\n- highly connected graphs with no $K_n$-minor: http://www.openproblemgarden.org/?q=op/high_connectivity_no_k_n\n\nBibliography links:\n- Hadwiger's conjecture for $K\\sb 6$-free graphs: http://www.math.gatech.edu/%7Ethomas/PAP/hadwiger.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1238823\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1513158\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"Jorgensen's Conjecture\" in Graph Theory; Basic Graph Theory; Minors, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3185, "problem_number": "OPG-729", "title": "Seagull problem", "statement": "Conjecture Every $n$ vertex graph with no independent set of size $3$ has a complete graph on $\\ge \\frac{n}{2}$ vertices as a minor.", "background": "Source: Open Problem Garden. Original node ID: 729. URL: http://www.openproblemgarden.org/op/seagull_problem.\n\nSource subject path: Graph Theory > Basic Graph Theory > Minors.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/seagull_problem\n- Author(s): Seymour, Paul D.\n- Subject(s): Graph Theory; Basic Graph Theory; Minors\n- Keywords: coloring; complete graph; minor\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: January 6th, 2008 by mdevos\n\nProblem-page discussion:\nThis conjecture is significant because it is an interesting unproved consequence of Hadwiger's conjecture (this implication is proved next). In fact, some experts have suggested that this problem might hold the key to finding a counterexample to Hadwiger. I (M. DeVos) have attributed this conjecture to Seymour, but I believe that it may have been independently suggested by Mader and by others. Its curious title will be explained later in this discussion.\n\nHadwiger's conjecture (every loopless graph with chromatic number $\\ge n$ has $K_n$ as a minor) is one of the outstanding problems in graph theory. This conjecture has been resolved for small values of $n$; when $n \\le 4$ it is relatively easy, for $n=5,6$ it has been proven to be equivalent to the Four color theorem. The Seagull problem concerns the other extreme - when the size of the chromatic number is close to the order of the graph. If $G$ is an $n$ vertex graph with no independent set of size 3, then $\\chi(G) \\ge \\frac{n}{2}$ since each color class has size $\\le 2$. If Hadwiger's conjecture holds for $G$, it must then have a minor which is a complete graph on $\\ge \\frac{n}{2}$ vertices. This is precisely the statement of the Seagull problem.\n\nThe (essentially) best known bound for the conjecture is that every $n$ vertex graph $G$ with no independent set of size 3 has a complete graph on $\\ge \\frac{n}{3}$ vertices as a minor. This argument is where the name of this conjecture arises. Let us call a seagull of $G$ an induced subgraph which is a 2-edge path (such a subgraph may be drawn suggestively as a seagull). Then, for every seagull $S$ and every vertex $v$ not in $S$, there must be an edge between $v$ and one of the two endpoints of $S$ (this follows from the assumption that $G$ has no independent set of size 3). This feature makes seagulls especially useful for constructing complete graphs as minors - as we now demonstrate. Choose a maximal collection ${\\mathcal S}$ of pairwise disjoint seagulls of $G$. The graph $G'$ obtained from $G$ by deleting every vertex which appears in a seagull in ${\\mathcal S}$ cannot have any two vertices at distance 2 from one another (since this would yield a seagull), so $G'$ must be a disjoint union of complete graphs. Since $G'$ has no independent set of size 3, it is in fact a disjoint union of at most two complete graphs. By deleting every vertex in the smaller complete subgraph of $G'$ from $G$ and then contracting both edges in every seagull in ${\\mathcal S}$, we obtain a complete graph minor of $G$ with size $\\ge \\frac{n}{3}$.\n\nKawarabayashi, Plummer, and Toft have improved this bound slightly by showing that $G$ must have a complete graph minor of size $\\ge \\frac{n + \\omega(G)}{3}$, but it looks very difficult to get much more out of this type of argument.\n\nBibliography:\n[KPT] K. Kawarabayashi, M. Plummer, and B. Toft, Improvements of the theorem of Duchet and Meynial's theorem on Hadwiger's Conjecture, J. Combin. Theory Ser. B. 95 (2005) 152-167.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Seagull problem\" in Graph Theory; Basic Graph Theory; Minors, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3186, "problem_number": "OPG-37379", "title": "Forcing a $K_6$-minor", "statement": "Conjecture Every graph with minimum degree at least 7 contains a $K_6$-minor.\n\nConjecture Every 7-connected graph contains a $K_6$-minor.", "background": "Source: Open Problem Garden. Original node ID: 37379. URL: http://www.openproblemgarden.org/op/forcing_a_k_6_minor.\n\nSource subject path: Graph Theory > Basic Graph Theory > Minors.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/forcing_a_k_6_minor\n- Author(s): Barát,János; Joret, Gwenaël; Wood, David R.\n- Subject(s): Graph Theory; Basic Graph Theory; Minors\n- Keywords: connectivity; graph minors\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: January 16th, 2012 by David Wood\n\nProblem-page discussion:\nThe first conjecture implies the second.\n\nWhether the second conjecture is true was first asked in [KT]. Both conjectures were stated in [BJW].\n\nThe second conjecture is implied by Jørgensen’s conjecture, which asserts that every $6$-connected $K_6$-minor-free graph is apex (which have minimum degree at most $6$ and are thus not $7$-connected). Since Jørgensen’s conjecture is true for sufficiently large graphs [KNTWa,KNTWb], the second conjecture is true for sufficiently large graphs.\n\nBibliography:\n*[BJW] János Barát, Gwenaël Joret, David R. Wood. Disproof of the List Hadwiger Conjecture, Electronic J. Combinatorics 18:P232, 2011.\n\n*[KT] Ken-ichi Kawarabayashi and Bjarne Toft. Any 7-chromatic graph has $K_7$ or $K_{4,4}$ as a minor. Combinatorica 25 (3), 327–353, 2005.\n\n[KNTWa] Ken-ichi Kawarabayashi, Serguei Norine, Robin Thomas, Paul Wollan. $K_6$ minors in $6$-connected graphs of bounded tree-width. http://arxiv.org/abs/1203.2171\n\n[KNTWb] Ken-ichi Kawarabayashi, Serguei Norine, Robin Thomas, Paul Wollan. $K_6$ minors in large $6$-connected graphs. http://arxiv.org/abs/1203.2192\n\nRelated:\nRelated problems\nJorgensen's Conjecture\n\nDiscussion links:\n- Jørgensen’s conjecture: http://www.openproblemgarden.org/?q=node/154\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Forcing a $K_6$-minor\" in Graph Theory; Basic Graph Theory; Minors, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3187, "problem_number": "OPG-59911", "title": "Forcing a 2-regular minor", "statement": "Conjecture Every graph with average degree at least $\\frac{4}{3}t-2$ contains every 2-regular graph on $t$ vertices as a minor.", "background": "Source: Open Problem Garden. Original node ID: 59911. URL: http://www.openproblemgarden.org/op/forcing_a_2_regular_minor.\n\nSource subject path: Graph Theory > Basic Graph Theory > Minors.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/forcing_a_2_regular_minor\n- Author(s): Reed, Bruce A.; Wood, David R.\n- Subject(s): Graph Theory; Basic Graph Theory; Minors\n- Keywords: minors\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: March 16th, 2014 by David Wood\n\nProblem-page discussion:\nReed and Wood [RW] explained that a result of Corradi and Hajnal [CH] implies that if $H$ is the graph consisting of $k$ disjoint triangles, then every graph with average degree at least $4k-2$ contains $H$ as a minor. Moreover, the bound of $4k-2$ is best possible since the complete bipartite graph $K_{2k-1,n}$ contains no $H$-minor, but has average degree tending to $4k-2$ (as $n\\rightarrow\\infty$ ). Thus the conjecture would generalise this result.\n\nUpdate: There has been a lot of recent progress on this conjecture [HW,CNLWY].\n\nBibliography:\n[CH] Keresztely Corradi and Andras Hajnal. On the maximal number of independent circuits of a graph. Acta Math. Acad. Sci. Hungar., 14:423–443, 1963.\n\n*[RW] Bruce Reed and David R. Wood. Forcing a sparse minor, arXiv:1402.0272, 2013.\n\n[HW] Daniel J. Harvey and David R. Wood. Cycles of given size in a dense graph. SIAM J. Discrete Math. 29.4:2336–2349, 2015.\n\n[CNLWY] E. Csóka, S. Norin, I. Lo, H. Wu and L. Yepremyan. The extremal function for disconnected minors. J. Comb. Theory B 126 (2017), 162-174.\n\nBibliography links:\n- 1402.0272: http://www.arxiv.org/abs/1402.0272\n- Cycles of given size in a dense graph: http://dx.doi.org/10.1137/15M100852X\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Forcing a 2-regular minor\" in Graph Theory; Basic Graph Theory; Minors, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3188, "problem_number": "OPG-46583", "title": "Decomposing a connected graph into paths.", "statement": "Conjecture Every simple connected graph on $n$ vertices can be decomposed into at most $\\frac{1}{2}(n+1)$ paths.", "background": "Source: Open Problem Garden. Original node ID: 46583. URL: http://www.openproblemgarden.org/op/decomposing_a_connected_graph_into_paths.\n\nSource subject path: Graph Theory > Basic Graph Theory > Paths.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_a_connected_graph_into_paths\n- Author(s): Gallai, Tibor\n- Subject(s): Graph Theory; Basic Graph Theory; Paths\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture is tight because a complete graph on $n$ vertices cannot be covered by less than $(n+1)/2$ cycles.\n\nThere is a similar conjecture about decomposition of an eulerian graph into cycles.\n\nBibliography:\n* [L] L. Lovász, On covering of graphs. In Theory of Graphs (Proc. Colloq., Tihany, 1966), 231--236. Academic Press, New York, 1968.\n\nRelated:\nRelated problems\nDecomposing an eulerian graph into cycles.\n\nDiscussion links:\n- decomposition of an eulerian graph into cycles: http://www.openproblemgarden.org/?q=node/46584\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposing a connected graph into paths.\" in Graph Theory; Basic Graph Theory; Paths, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3189, "problem_number": "OPG-46613", "title": "Partition of a cubic 3-connected graphs into paths of length 2.", "statement": "Problem Does every $3$-connected cubic graph on $3k$ vertices admit a partition into $k$ paths of length $2$?", "background": "Source: Open Problem Garden. Original node ID: 46613. URL: http://www.openproblemgarden.org/op/partition_of_a_cubic_3_connected_graphs_into_paths_of_length_2.\n\nSource subject path: Graph Theory > Basic Graph Theory > Paths.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partition_of_a_cubic_3_connected_graphs_into_paths_of_length_2\n- Author(s): Kelmans, Alexander K.\n- Subject(s): Graph Theory; Basic Graph Theory; Paths\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nMore generally, the following question is posed.\n\nProblem Does every $3$-connected cubic graph on at least $3k$ vertices contain $k$ pairwise vertex-disjoint paths of length $2$?\n\nIn [K1], Kelmans gave a construction that provided infi nitely many 2-connected graphs for which the above statement is false.\n\nBibliography:\n[K1] Alexander K. Kelmans, Packing 3-vertex paths in 2-connected graphs\n\n*[K2] Alexander K. Kelmans, On $\\Lambda$--Packing in 3--connected Graphs, RUTCOR Research Report 23--2005, Rutgers University. See also Packing 3-vertex Paths In Cubic 3-connected Graphs\n\nBibliography links:\n- Packing 3-vertex paths in 2-connected graphs: http://www.arxiv.org/abs/0712.4151\n- Packing 3-vertex Paths In Cubic 3-connected Graphs: http://www.arxiv.org/abs/0910.2766v2\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Partition of a cubic 3-connected graphs into paths of length 2.\" in Graph Theory; Basic Graph Theory; Paths, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3190, "problem_number": "OPG-170", "title": "Linial-Berge path partition duality", "statement": "Conjecture The minimum $k$-norm of a path partition on a directed graph $D$ is no more than the maximal size of an induced $k$-colorable subgraph.", "background": "Source: Open Problem Garden. Original node ID: 170. URL: http://www.openproblemgarden.org/op/linial_berge_path_partition_duality.\n\nSource subject path: Graph Theory > Coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/linial_berge_path_partition_duality\n- Author(s): Berge, Claude; Linial, Nathan\n- Subject(s): Graph Theory; Coloring\n- Keywords: coloring; directed path; partition\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 27th, 2007 by berger\n\nProblem-page discussion:\nDefinitions: Let $D$ be a directed graph. A path partition of $D$ is a set of vertex disjoint paths in it (some might be singletons), covering all vertices. Let $k$ be a positive integer. The $k$ norm of a path partition is the sum of $\\min\\{|V(P)|,k\\}$ for all paths $P$ in it.\n\nThis conjecture is known for acyclic graphs and for $k =1,2$.\n\nComments:\n- September 19th, 2007 | Anonymous | Berge-Linial conjecture: There is a typo in the formulation:\n\nReplace \"maximum k-norm\" by \"minimum k-norm\".\n\nThanks! Best, Andras Sebo\n- September 19th, 2007 | mdevos | Thanks Andras!: Thanks much for the correction, I've updated the problem.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Linial-Berge path partition duality\" in Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3191, "problem_number": "OPG-562", "title": "Three-chromatic (0,2)-graphs", "statement": "Question Are there any (0,2)-graphs with chromatic number exactly three?", "background": "Source: Open Problem Garden. Original node ID: 562. URL: http://www.openproblemgarden.org/op/three_chromatic_0_2_graphs.\n\nSource subject path: Graph Theory > Coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/three_chromatic_0_2_graphs\n- Author(s): Payan, Charles\n- Subject(s): Graph Theory; Coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 7th, 2007 by Gordon Royle\n\nProblem-page discussion:\nA (0,2)-graph is a graph such that every pair of distinct vertices has either 0 or 2 common neighbours. It is fairly easy to see that a (0,2)-graph is necessarily regular and a variety of other properties can be shown to hold. Although (0,2)-graphs with chromatic number 2, 4 and 5 are known it is open as to whether there can be any (0,2)-graphs with chromatic number exactly three.\n\nBibliography:\n*[P] Payan, Charles: On the chromatic number of cube-like graphs, Discrete Math. 103 (1992), no. 3, 271--277.\n\nComments:\n- February 6th, 2013 | Anonymous | Finite Three-Chromatic (0,2)-graphs: An infinite three-chromatic (0, 2)-graph is easy to construct. See Payan.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Three-chromatic (0,2)-graphs\" in Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3192, "problem_number": "OPG-815", "title": "Total Colouring Conjecture", "statement": "Conjecture A total coloring of a graph $G = (V,E)$ is an assignment of colors to the vertices and the edges of $G$ such that every pair of adjacent vertices, every pair of adjacent edges and every vertex and incident edge pair, receive different colors. The total chromatic number of a graph $G$, $\\chi\"(G)$, equals the minimum number of colors needed in a total coloring of $G$. It is an old conjecture of Behzad that for every graph $G$, the total chromatic number equals the maximum degree of a vertex in $G$, $\\Delta(G)$ plus one or two. In other words,\n$$\n\\chi\"(G)=\\Delta(G)+1\\ \\ or \\ \\ \\Delta(G)+2.\n$$", "background": "Source: Open Problem Garden. Original node ID: 815. URL: http://www.openproblemgarden.org/op/behzads_conjecture.\n\nSource subject path: Graph Theory > Coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/behzads_conjecture\n- Author(s): Behzad, M.\n- Subject(s): Graph Theory; Coloring\n- Keywords: Total coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 4th, 2008 by Iradmusa\n\nProblem-page discussion:\nThe lower bound $\\Delta(G)+1$ is trivial by looking at the number of colours required on a vertex of maximum degree and its incident edges. It is easy to prove $\\chi\"(G)\\leq 2\\Delta(G)+2$. Molloy and Reed [MR] showed that there exists a constant $C$ such that $\\chi\"(G)\\leq Delta(G)+C$ for every graph $G$.\n\nThe Edge list coloring conjecture would imply that $\\chi\"(G)\\leq \\Delta(G)+3$.\n\nThe Total Colouring Conjecture was proved for $\\Delta(G)=3$ by Rosenfeld [R] and also by Vijayaditya [V], and for $\\Delta(G)\\in\\{4,5\\}$ by Kostochka [K1,K2,K3]; in fact the proof for $\\Delta(G)=5$ holds for multigraphs.\n\nThe Conjecture has also been established for many graph classes. For every planar graph G with $\\Delta(G) \\geq 7$, the following clever argument proves it. By the 4 Color Theorem, we can color the vertices with the colors 1, 2, 3, 4. By a result of Sanders and Zhao [SZ], we can color the edges of the graph with the colors $3, 4, \\ldots, \\Delta(G) + 1, \\Delta(G) + 2$. Uncolor each edge that was colored 3 or 4. Note that each uncolored edge has exactly two colors from $\\{1,2,3,4\\}$ forbidden. Hence, each uncolored edge has at least two colors available. Note that the uncolored edges induce a disjoint union of paths and even cycles. Thus, by a special case of a theorem of Erdos, Rubin, and Taylor [ERT], we can color the edges from their lists of two available colors each.\n\nBibliography:\n*[B] M. Behzad, Graphs and their chromatic numbers, Ph.D. Thesis, Michigan State University, 1965.\n\n[ERT] P. Erdos, A.L. Rubin, and H. Taylor, Choosability in graph, Cong. Numer. 26, 125-157, 1979.\n\n[K1] A.V. Kostochka, The total coloring of a multigraph with maximal degree 4. Discrete Math. 17, 161-163, 1977.\n\n[K2] A.V. Kostochka, Upper bounds of chromatic functions of graph (in Russian). Ph.D. Thesis, Novosibirsk, 1978.\n\n[K3] A.V. Kostochka, Exact upper bound for the total chromatic number of a graph (in Russian). In: Proc. 24th Int. Wiss. Koll., Tech Hochsch. Ilmenau, 1979, pages 33-36, 1979.\n\n[MR] M. Molloy and B.Reed. A bound on the total chromatic number. Combinatorica, 18(2), 241-280, 1998.\n\n[R] M. Rosenfeld, On the total coloring of certain graphs. Israel J. Math. 9, 396-402, 1971.\n\n[SZ] D.P. Sanders and Y. Zhao, Planar Graphs of Maximum Degree Seven are Class 1. J. Comb. Theory B. 83, 201-212, 2001.\n\n[V] N. Vijayaditya, On total chormatic number of a graph. J. London Math. Soc. (2) 3, 405-408, 1971.\n\nRelated:\nRelated problems\nEdge list coloring conjecture\nList Total Colouring Conjecture\n\nDiscussion links:\n- Edge list coloring conjecture: http://www.openproblemgarden.org/?q=node/2110\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Total Colouring Conjecture\" in Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3193, "problem_number": "OPG-830", "title": "4-regular 4-chromatic graphs of high girth", "statement": "Problem Do there exist 4-regular 4-chromatic graphs of arbitrarily high girth?", "background": "Source: Open Problem Garden. Original node ID: 830. URL: http://www.openproblemgarden.org/op/high_girth_low_degree_4_chromatic_graphs.\n\nSource subject path: Graph Theory > Coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/high_girth_low_degree_4_chromatic_graphs\n- Author(s): Grunbaum, Branko\n- Subject(s): Graph Theory; Coloring\n- Keywords: coloring; girth\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 18th, 2008 by mdevos\n\nProblem-page discussion:\nGrunbaum conjectured that for every $m$, there exist $m$-regular $m$-chromatic graphs of arbitrarily high girth. However, this was shown dramatically false by Johansson, who proved that every triangle free graph $G$ with maximum degree $\\Delta$ satisfies $\\chi(G) \\le C \\frac{\\Delta}{\\log \\Delta}$ for some fixed constant $C$. Neverless, some interesting smaller cases of Grunbaum's conjecture, such as the one highlighted above, might still be true.\n\nThere are only a few 4-regular 4-chromatic graphs of girth $\\ge 4$ which are known. These include the Chvatal graph, Brinkmann graph (discovered independently by Kostochka), and Grunbaum graph. To the best of my (M. DeVos') knowledge, this might be the full list of such graphs.\n\nThere do exist 4-chromatic graphs of minimum degree $\\le 6$ and arbitrarily high girth, but it is open wether there exist 4-chromatic graphs of minimum degree 5 and arbitrary girth.\n\nDiscussion links:\n- Chvatal graph: http://mathworld.wolfram.com/ChvatalGraph.html\n- Brinkmann graph: http://mathworld.wolfram.com/BrinkmannGraph.html\n- Grunbaum graph: http://mathworld.wolfram.com/GruenbaumGraphs.html\n\nComments:\n- January 29th, 2012 | Anonymous | The list of known graphs is not full: Your list of 4 regular 4 chromatic graphs with girth >= 4 is not complete. Check \"A note on 4-regular 4 chromatic graphs with girth 4\" http://math.nju.edu.cn/~zkmfl/kmzhang/41-60/Zhang44.pdf. Girth 5 is known too.\n- November 13th, 2008 | Anonymous | What about higher degree?: What maximum degree is needed in a triangle-free graph (or higher girth) with chromatic number 5? What about chromatic number 6 and 7? Are good bounds known on these maximum degrees?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"4-regular 4-chromatic graphs of high girth\" in Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3194, "problem_number": "OPG-46533", "title": "Coloring the union of degenerate graphs", "statement": "Conjecture The union of a $1$-degenerate graph (a forest) and a $2$-degenerate graph is $5$-colourable.", "background": "Source: Open Problem Garden. Original node ID: 46533. URL: http://www.openproblemgarden.org/op/coloring_the_union_of_degenerate_graphs.\n\nSource subject path: Graph Theory > Coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/coloring_the_union_of_degenerate_graphs\n- Author(s): Tarsi, Michael\n- Subject(s): Graph Theory; Coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 3rd, 2013 by fhavet\n\nProblem-page discussion:\nA graph is $k$-degenerate if it can be reduced to $K_1$ (the graph with a unique vertex) by repeatedly deleting vertices of degree at most $k$. A $1$-degenerate graph $G_1$ admits a proper $2$-colouring $c_1$, and a $2$-degenerate graph $G_2$ admits a proper $3$-colouring $c_2$. Thus, $(c_1,c_2)$ is a proper $6$-colouring of $G_1$ and $G_2$.\n\nThe conjecture is tigth because $K_5$ is the union of a $1$-degenerate graph and a $2$-degenerate graph.\n\nBased on a decompostion of the complete graph, Klein and Schönheim [KlSc93] generalised this conjecture to $(m_1, \\dots, m_s)$-composed graphs, which are unions of $s$ graphs $G_1, \\dots, G_s$ such that $G_i$ is $m_i$-degenerate, $1\\leq i\\leq s$.\n\nConjecture Every $(m_1, \\dots, m_s)$-composed graph is $\\left(\\sum_{i=1}^s m_i+\\bigg\\lfloor\\frac{1}{2}\\bigg(1+\\sqrt{1+8\\sum_{1\\leq i Coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/list_total_colouring_conjecture\n- Author(s): Borodin, Oleg V.; Kostochka, Alexandr V.; Woodall, Douglas R.\n- Subject(s): Graph Theory; Coloring\n- Keywords: list coloring; Total coloring; total graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 29th, 2013 by Jon Noel\n\nProblem-page discussion:\nThe list chromatic number of a graph $G$, denoted $\\chi_\\ell(G)$, is defined here. Given a multigraph $H$, the total graph $T(H)$ of $H$ is a graph on vertex set $V(T(H)):=V(H)\\cup E(H)$ where\n\n- two elements of $V(H)$ are adjacent in $T(H)$ if and only if they are adjacent in $H$;\n- two elements of $E(H)$ are adjacent in $T(H)$ if and only if they share an endpoint;\n- an element of $V(H)$ is adjacent to an element of $E(H)$ in $T(H)$ if it is incident with it.\n\nThis problem is related to the List (Edge) Colouring Conjecture as well as the Total Colouring Conjecture.\n\nKostochka and Woodall [KW] conjectured that $\\chi_\\ell(G^2)=\\chi(G^2)$ for every graph $G$; this was known as the List Square Colouring Conjecture. It is stronger than the List Total Colouring Conjecture since, given a multigraph $H$, the total graph of $H$ can be obtained by subdividing each edge of $H$ and taking the square. Moreover, the graph obtained from $H$ by subdividing each edge is bipartite and one part of the bipartition consists of vertices of degree $2$. Thus, the List Total Colouring Conjecture corresponds to this (very) special case of the List Square Colouring Conjecture.\n\nHowever, the List Square Colouring Conjecture is not true in general. For a family of counterexamples, see the paper of Kim and Park [KP].\n\nBibliography:\n*[BKW] O. V. Borodin, A. V. Kostochka, and D. R. Woodall. List edge and list totalcolourings of multigraphs. J. Combin. Theory Ser. B, 71(2):184–204, 1997.\n\n[KW] A. V. Kostochka and D. R. Woodall. Choosability conjectures and multicircuits. Discrete Math., 240(1-3):123–143, 2001.\n\n[KP] Seog-Jin Kim and Boram Park: Counterexamples to the List Square Coloring Conjecture, submitted.\n\nRelated:\nRelated problems\nEdge list coloring conjecture\nTotal Colouring Conjecture\nChoosability of Graph Powers\n\nDiscussion links:\n- here: http://en.wikipedia.org/wiki/List coloring\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"List Total Colouring Conjecture\" in Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3196, "problem_number": "OPG-143", "title": "Petersen coloring conjecture", "statement": "Conjecture Let $G$ be a cubic graph with no bridge. Then there is a coloring of the edges of $G$ using the edges of the Petersen graph so that any three mutually adjacent edges of $G$ map to three mutually adjancent edges in the Petersen graph.", "background": "Source: Open Problem Garden. Original node ID: 143. URL: http://www.openproblemgarden.org/op/petersen_coloring_conjecture.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/petersen_coloring_conjecture\n- Author(s): Jaeger, Francois\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: cubic; edge-coloring; Petersen graph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nThis extrordainary conjecture asserts that in a very strong sense, every bridgeless cubic graph has all of the cycle-space properties posessed by the Petersen graph. If true, this conjecture would imply both The Berge-Fulkerson conjecture and The five cycle double cover conjecture.\n\nIf $G$ is a graph and $C \\subseteq E(G)$ we say that $C$ is a binary cycle if every vertex in the graph $(V(G),C)$ has even degree. If $H$ is a graph and $f: E(G) \\rightarrow E(H)$ is a map, we say that $f$ is cycle-continuous if the pre-image of every binary cycle is a binary cycle. The following conjecture is an equivalent reformulation of the Petersen coloring conjecture.\n\nConjecture (Petersen coloring conjecture (2)) Every bridgeless graph has a cycle-continuous mapping to the Petersen graph.\n\nSource links:\n- cubic: http://en.wikipedia.org/wiki/cubic graph\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n- Petersen: http://en.wikipedia.org/wiki/petersen graph\n\nDiscussion links:\n- The Berge-Fulkerson conjecture: http://www.openproblemgarden.org/?q=op/the_berge_fulkerson_conjecture\n- The five cycle double cover conjecture: http://www.openproblemgarden.org/?q=op/m_n_cycle_covers\n\nComments:\n- November 8th, 2011 | Anonymous | Question: For which bridgeless cubic graphs has this been checked for?\n- November 24th, 2011 | Robert Samal | Re:: It is trivially true for all that are 3-edge-colorable -- which is the vast majority. Among the rest, I checked it using computer and lists of snarks for all graphs upto 34 vertices. (And some more -- e.g. all flower-snarks.)\n\nBest wishes, Robert\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Petersen coloring conjecture\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3197, "problem_number": "OPG-144", "title": "Packing T-joins", "statement": "Conjecture There exists a fixed constant $c$ (probably $c=1$ suffices) so that every graft with minimum $T$-cut size at least $k$ contains a $T$-join packing of size at least $(2/3)k-c$.", "background": "Source: Open Problem Garden. Original node ID: 144. URL: http://www.openproblemgarden.org/op/packing_t_joins.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/packing_t_joins\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: packing; T-join\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nDefinitions: A graft consists of a graph $G=(V,E)$ together with a distinguished set $T \\subseteq V$ of even cardinality. A $T$-cut is an edge-cut $\\delta(X)$ of $G$ with the property that $|X \\cap T|$ is odd. A $T$-join is a set $S \\subseteq E$ with the property that a vertex of $(V,S)$ has odd degree if and only if it is in $T$. A $T$-join packing is a set of pairwise disjoint T-joins.\n\nIt is an easy fact that every $T$-join and every $T$-cut intersect in an odd number of elements. It follows easily from this that the maximum size of a $T$-join packing is always less than or equal to the minimum size of a $T$-cut. There is a simple example of a graft with $|T|=4$ with minimum $T$-cut size $k$ which contains only $(2/3)k$ disjoint T-joins. The above conjecture asserts that this is essentially the worst case. DeVos and Seymour [DS] have obtained a partial result toward the above conjecture, proving that every graft with minimum $T$-cut size $k$ contains a $T$-join packing of size at least the floor of $(1/3)k$.\n\nDefinition: We say that a graft $G$ is an $r$-graph if $G$ is $r$-regular, $T=V$, and every $T$-cut of G has size at least $r$.\n\nConjecture (Rizzi) If $G$ is an $r$-graph, then $G$ contains a $T$-join packing of size at least $r-2$.\n\nIn an $r$-graph, every perfect matching is a $T$-join, so the above conjecture is true with room to spare for $r$-graphs which are $r$-edge-colorable. Indeed, Seymour had earlier conjectured that every $r$-graph contains $r-2$ disjoint perfect matchings. This however was disproved by Rizzi [R] who constructed for every $r>2$ an $r$-graph in which every two perfect matchings intersect. Rizzi suggested the above problem as a possible fix for Seymour's conjecture. DeVos and Seymour have proved that every $r$-graph has a $T$-join packing of size at least the floor of $r/2$.\n\nDefinition: Let $G$ be a graph and let $T$ be the set of vertices of $G$ of odd degree. A $T$-join of $(G,T)$ is defined to be a postman set.\n\nNote that when $T$ is the set of vertices of odd degree, a cocycle of $G$ is a $T$-cut if and only if it has odd size. Rizzi has shown that the following conjecture is equivalent to the above conjecture in the special case when $r$ is odd.\n\nConjecture (The packing postman sets conjecture (Rizzi)) If every odd edge-cut of $G$ has size $>2k+1$ then the edges of $G$ may be partitioned into $2k+1$ postman sets.\n\nThe Petersen graph (or more generally any non $(2k+1)$-edge-colorable $(2k+1)$-graph) shows that the above conjecture would be false with the weaker assumption that every odd edge-cut has size $>2k$. The following conjecture asserts that odd edge-cut size $>2k$ is enough (for the same conclusion) if we assume in addition that G has no Petersen minor.\n\nConjecture (Conforti, Johnson) If $G$ has no Petersen minor and every odd edge-cut of $G$ has size $>2k$ then the edges of $G$ may be partitioned into $2k+1$ postman sets.\n\nGerard Cornuejols [C] has kindly offered $5000 for a solution to this conjecture. However, it will be tough to find a quick proof since this conjecture does imply the 4-color theorem. Robertson, Seymour, Sanders, and Thomas [RSST] have proved the above conjecture for cubic graphs. Conforti and Johnson [CJ] proved it under the added hypothesis that G has no 4-wheel minor.\n\nBibliography:\n[CJ] M. Conforti and E.L. Johnson, Two min-max theorems for graphs noncontractible to a four wheel, preprint.\n\n[C] G. Cornuejols, Combinatorial Optimization, packing and covering, SIAM, Philadelphia (2001).\n\n[R] R. Rizzi, Indecomposable r-Graphs and Some Other Counterexamples, J. Graph Theory 32 (1999) 1-15. MathSciNet\n\n[RSST] N. Robertson, D.P. Sanders, P.D. Seymour, and R. Thomas, A New Proof of the Four-Color Theorem, Electron. Res. Announc., Am. Math. Soc. 02, no 1 (1996) 17-25.\n\n[S] P.D. Seymour, Some Unsolved Problems on One-Factorizations of Graphs, in Graph Theory and Related Topics, edited by J.A. Bondy and U.S.R. Murty, Academic Press, New York 1979) 367-368.\n\nDiscussion links:\n- edge-cut: http://en.wikipedia.org/wiki/connectivity (graph theory)\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1704172\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 25.\n\nAttempt notes:\nTarget:\nMake progress on \"Packing T-joins\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3198, "problem_number": "OPG-145", "title": "Acyclic edge-colouring", "statement": "Conjecture Every simple graph with maximum degree $\\Delta$ has a proper $(\\Delta+2)$-edge-colouring so that every cycle contains edges of at least three distinct colours.", "background": "Source: Open Problem Garden. Original node ID: 145. URL: http://www.openproblemgarden.org/op/acyclic_edge_coloring.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/acyclic_edge_coloring\n- Author(s): Fiamcik, Jozef\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: edge-coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nAn edge-colouring with the property that every cycle contains edges of at least three distinct colours is called an acyclic edge-colouring. It is known (see [AMR]) that every graph of maximum degree $\\Delta$ has an acyclic edge-colouring of size $O(\\Delta )$. The best upper bound so far is $4\\Delta -4$ and is due to Esperet and Parreau [EP]. It is also known (see [ASZ]) that this conjecture is true for graphs with girth at least $C \\Delta \\log(\\Delta )$ (for some fixed constant $C$ ).\n\nBibliography:\n[AMR] N. Alon, C. McDiarmid and B. Reed, Acyclic colouring of graphs, Random Structures and Algorithms 2 (1991), 277-288. MathSciNet\n\n[ASZ] N. Alon, B. Sudakov and A. Zaks, Acyclic edge-colorings of graphs, J. Graph Theory 37 (2001), 157-167. MathSciNet\n\n[EP] L. Esperet and A. Parreau, Acyclic edge-coloring using entropy compression, arXiv:1206.1535 [math.CO].\n\nSource links:\n- edge-colouring: http://en.wikipedia.org/wiki/edge-coloring\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1109695\n- Acyclic edge-colorings of graphs: http://www.math.tau.ac.il/%7Enogaa/PDFS/asz2.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1837021\n- Acyclic edge-coloring using entropy compression: http://www.arxiv.org/abs/1206.1535v3\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Acyclic edge-colouring\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3199, "problem_number": "OPG-182", "title": "A generalization of Vizing's Theorem?", "statement": "Conjecture Let $H$ be a simple $d$-uniform hypergraph, and assume that every set of $d-1$ points is contained in at most $r$ edges. Then there exists an $r+d-1$-edge-coloring so that any two edges which share $d-1$ vertices have distinct colors.", "background": "Source: Open Problem Garden. Original node ID: 182. URL: http://www.openproblemgarden.org/op/a_generalization_of_vizings_theorem.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_generalization_of_vizings_theorem\n- Author(s): Rosenfeld, Moshe\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: edge-coloring; hypergraph; Vizing\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 11th, 2007 by mdevos\n\nProblem-page discussion:\nVizing's Theorem is equivalent to the above statement for $d=2$. For higher dimensions, this problem looks difficult since the main tool used in the proof of Vizing's theorem (Kempe chains) do not appear to work.\n\nSource links:\n- uniform: http://en.wikipedia.org/wiki/hypergraph\n\nComments:\n- June 23rd, 2009 | Anonymous | Reference: Could someone please add a reference? There should be some paper (or a conference talk?) where Rosenfeld proposed the conjecture.\n\n-DOT\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"A generalization of Vizing's Theorem?\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3200, "problem_number": "OPG-388", "title": "List colorings of edge-critical graphs", "statement": "Conjecture Suppose that $G$ is a $\\Delta$-edge-critical graph. Suppose that for each edge $e$ of $G$, there is a list $L(e)$ of $\\Delta$ colors. Then $G$ is $L$-edge-colorable unless all lists are equal to each other.", "background": "Source: Open Problem Garden. Original node ID: 388. URL: http://www.openproblemgarden.org/op/list_colorings_of_edge_critical_graphs.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/list_colorings_of_edge_critical_graphs\n- Author(s): Mohar, Bojan\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: edge-coloring; list coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 12th, 2007 by Robert Samal\n\nProblem-page discussion:\n(Reproduced from [M].)\n\nA graph $G$ is said to be $\\Delta$-edge-critical if it is not $\\Delta$-edge-colorable but every edge-deleted subgraph is $\\Delta$-edge-colorable. (Here $\\Delta$ is the maximum degree of $G$.)\n\nBibliography:\n*[M] B. Mohar, Problem of the Month\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P7ListEdgeCriticalGraphs.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"List colorings of edge-critical graphs\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3201, "problem_number": "OPG-624", "title": "Universal Steiner triple systems", "statement": "Problem Which Steiner triple systems are universal?", "background": "Source: Open Problem Garden. Original node ID: 624. URL: http://www.openproblemgarden.org/op/universal_steiner_triple_systems.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/universal_steiner_triple_systems\n- Author(s): Grannell, Mike; Griggs, Terry; Knor, Martin; Skoviera, Martin\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: cubic graph; Steiner triple system\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 5th, 2007 by macajova\n\nProblem-page discussion:\nA cubic graph $G$ is $S$-edge-colorable for a Steiner triple system $S$ if its edges can be colored with the points of $S$ in such a way that the points assigned to three edges sharing a vertex form a triple in $S$.\n\nA Steiner triple system $S$ is called universal if any (simple) cubic graph is $S$-colorable.\n\nIt is easy to see that if $S_3$ denotes the trivial Steiner triple system with three points and one triple, then $S_3$-colorable graphs are precisely (cubic) edge-3-colorable graphs. For the same reason, any cubic edge-3-colorable graph is $S$-colorable for any Steiner triple system (with at least one edge). Thus, the study of $S$-colorings may be viewed as an attempt to understand snarks.\n\nIt is not hard to see, that a graph is Fano-colorable iff it has a nowhere-zero 8-flow. Thus (by Jaeger's result) Fano plane is \"almost universal\": it is possible to use it to color any bridgeless cubic graph (but it doesn't work for any graph with a bridge).\n\nGrannell et al. [GGKS] constructed a universal Steiner triple system of order 381. Holroyd, Skoviera [HS] proved that neither projective nor affine Steiner triple systems are universal. Kral et al. [KMPS] proved that any non-affine non-projective non-trivial point-transitive Steiner triple system is universal.\n\nBibliography:\n*[GGKS] M.J. Grannell, T.S. Griggs, M. Knor, M. Skoviera, A Steiner triple system which colours all cubic graphs, J. Graph Theory 46 (2004), 15--24. MathSciNet\n\n[HS] F. Holroyd and M. Skoviera, Colouring of cubic graphs by Steiner triple systems, J.~Combin. Theory Ser. B 91 (2004), 57--66.\n\n[KMPS] D. Kral, E. Macajova, A. Por, J.-S. Sereni, Characterization results for Steiner triple systems and their application to edge-colorings of cubic graphs, preprint.\n\nSource links:\n- Steiner triple systems: http://mathworld.wolfram.com/SteinerTripleSystem.html\n\nDiscussion links:\n- snarks: http://en.wikipedia.org/wiki/snark (graph theory)\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2051465\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Universal Steiner triple systems\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3202, "problem_number": "OPG-2110", "title": "Edge list coloring conjecture", "statement": "Conjecture Let $G$ be a loopless multigraph. Then the edge chromatic number of $G$ equals the list edge chromatic number of $G$.", "background": "Source: Open Problem Garden. Original node ID: 2110. URL: http://www.openproblemgarden.org/op/edge_list_coloring_conjecture.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/edge_list_coloring_conjecture\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 25th, 2008 by tchow\n\nProblem-page discussion:\nThe list edge chromatic number of $G$ is also known as the list chromatic index, the edge choosability, or the edge choice number of $G$. It is the list chromatic number of the line graph of $G$. Similarly, the edge chromatic number of $G$ is also known as the edge chromatic index, and it is the chromatic number of the line graph of $G$. The chromatic number of a graph is always less than or equal to the list chromatic number; the two quantities differ in general, but the conjecture says that they coincide for line graphs. Sometime the conjecture is simply referred to as the list coloring conjecture, although this is perhaps poor terminology since there exist other conjectures about list coloring.\n\nPerhaps the most famous partial result is Galvin's theorem [G] that the conjecture holds for bipartite multigraphs. Galvin's result settled the well-known Dinitz conjecture in the affirmative.\n\nThe conjecture has been attributed to many different people. See [JT, Problem 12.20] for a history of the problem up to 1995.\n\nBibliography:\n[G] Fred Galvin, The list chromatic index of a bipartite multigraph. J. Combin. Theory Ser. B 63 (1995), 153–158.\n\n[JT] Tommy R. Jensen and Bjarne Toft, Graph Coloring Problems. New York: Wiley-Interscience, 1995.\n\nRelated:\nRelated problems\nRota's basis conjecture\nList Total Colouring Conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Edge list coloring conjecture\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3203, "problem_number": "OPG-2226", "title": "Seymour's r-graph conjecture", "statement": "An $r$-graph is an $r$-regular graph $G$ with the property that $|\\delta(X)| \\ge r$ for every $X \\subseteq V(G)$ with odd size.\n\nConjecture $\\chi'(G) \\le r+1$ for every $r$-graph $G$.", "background": "Source: Open Problem Garden. Original node ID: 2226. URL: http://www.openproblemgarden.org/op/seymours_r_graph_conjecture.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/seymours_r_graph_conjecture\n- Author(s): Seymour, Paul D.\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: edge-coloring; r-graph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 3rd, 2008 by mdevos\n\nProblem-page discussion:\nThis conjecture is among the most important unsolved problems in edge coloring. It is very close in nature to Goldberg's Conjecture, and is also closely related to Rizzi's packing postman sets conjecture (see packing T-joins).\n\nIf $G$ is an $r$-regular graph and there exists $X \\subseteq V(G)$ with $|X|$ odd and $|\\delta(X)| < r$, then it is immediate that $G$ is not $r$-edge-colourable, since every perfect matching must use at least one edge from $\\delta(X)$. This is in some sense the only obvious obstruction to $r$-edge-colorability that we know of. So, $r$-graphs are the $r$-regular graphs which do not fail to be $r$-edge-colorable for this obvious reason. Not every $r$-graph is $r$-edge-colorable, for instance Petersen's graph is a 3-graph which is not 3-edge-colorable. However, this conjecture asserts that all such graphs are still $(r+1)$-edge-colorable.\n\nThis conjecture has been proved for $r \\le 11$ by Nishizeki and Kashiwagi [NK].\n\nBibliography:\n[NK] T. Nishizeki and K. Kashiwagi, An upper bound on the chromatic index of multigraphs. Graph theory with applications to algorithms and computer science (Kalamazoo, Mich., 1984), 595--604, Wiley-Intersci. Publ., Wiley, New York, 1985. MathSciNet.\n\n*[S] P.D. Seymour, On multicolourings of cubic graphs, and conjectures of Fulkerson and Tutte. Proc. London Math. Soc. (3) 38 (1979), no. 3, 423--460. MathSciNet.\n\nRelated:\nRelated problems\nPacking T-joins\nGoldberg's conjecture\n\nDiscussion links:\n- Goldberg's Conjecture: http://www.openproblemgarden.org/?q=node/2242\n- packing T-joins: http://www.openproblemgarden.org/?q=node/144\n- Petersen's graph: http://en.wikipedia.org/wiki/Petersen's graph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0812694\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0532981\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Seymour's r-graph conjecture\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3204, "problem_number": "OPG-2242", "title": "Goldberg's conjecture", "statement": "The overfull parameter is defined as follows:\n$$\nw(G) = \\max_{H \\subseteq G} \\left\\lceil \\frac{ |E(H)| }{ \\lfloor \\tfrac{1}{2} |V(H)| \\rfloor} \\right\\rceil.\n$$\n\nConjecture Every graph $G$ satisfies $\\chi'(G) \\le \\max\\{ \\Delta(G) + 1, w(G) \\}$.", "background": "Source: Open Problem Garden. Original node ID: 2242. URL: http://www.openproblemgarden.org/op/goldbergs_conjecture.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/goldbergs_conjecture\n- Author(s): Goldberg, Mark K.\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: edge-coloring; multigraph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 4th, 2008 by mdevos\n\nProblem-page discussion:\nThis important problem remains open despite considerable attention. The same conjecture was independently discovered by Andersen and Seymour.\n\nVizing's Theorem, one of the cornerstones of graph colouring, shows that $\\chi'(G) \\le \\Delta(G) + 1$ for every simple graph $G$. So, in particular, every simple graph satisfies Goldberg's conjecture. Graphs with parallel edges need not satisfy Vizing's bound. For instance, if $G$ is the graph obtained from a triangle by adding an extra $k-1$ edges in parallel with each existing one, then $\\Delta(G) = 2k$ but $\\chi'(G) = 3k$. More generally, if $H$ is a subgraph of $G$, then every colour can appear on at most $\\lfloor \\frac{1}{2}|V(H)| \\rfloor$ edges of $H$, so $\\chi'(G) \\ge |E(H)| / \\lfloor \\tfrac{1}{2} |V(H)| \\rfloor$. Thus, $w(G)$, our overfull parameter, is a natural lower bound on $\\chi'(G)$, and Goldberg's conjecture asserts that whenever $\\chi'(G)$ exceeds $\\Delta(G)+1$, then it is equal to this lower bound.\n\nAlthough the statement of the conjecture may appear to be the most natural formulation, there are a couple of related conjectures with similar lower bounds. For instance, Seymour's r-graph conjecture is equivalent to the statement that $\\chi'(G) \\le \\max \\{\\Delta(G), w(G) \\} + 1$. Goldberg also conjectured that $\\chi'(G) \\le \\max\\{ \\Delta(G), w(G) + 1\\}$.\n\nIn addition to simple graphs, Goldberg's Conjecture is known to hold for any graph $G$ which satisfies one of the following\n\n- $\\Delta(G) \\le 11$\n- $G$ has no minor isomorphic to $K_5$ minus an edge.\n- $\\Delta(G)$ is sufficiently large in comparison with $|V(G)|$.\n\n$\\quad$\n\nPackers And Movers Chandigarh\nPackers And Movers Hyderabad\nPackers And Movers Bangalore\n\nBibliography:\n*[G] M. K. Goldberg, Multigraphs with a chromatic index that is nearly maximal. (Russian) A collection of articles dedicated to the memory of Vitaliĭ Konstantinovič Korobkov. Diskret. Analiz No. 23 (1973), 3--7, 72. MathSciNet\n\nRelated:\nRelated problems\nSeymour's r-graph conjecture\n\nDiscussion links:\n- Seymour's r-graph conjecture: http://www.openproblemgarden.org/?q=node/2226\n- Packers And Movers Chandigarh: http://www.packersandmoverschandigarh.co.in\n- Packers And Movers Hyderabad: http://www.packersandmoversinhyderabad.co.in\n- Packers And Movers Bangalore: http://www.packersandmoversinbangalore.co.in\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0354429\n\nComments:\n- February 6th, 2009 | Anonymous | Latest Developments: Stiebitz et al(2006), Yu(2008) and Kurt(2009) has separately shown $\\chi'(G) \\geq \\Delta+\\sqrt{\\Delta/2}$ implies Goldberg Conjecture. While Yu's method gives a methodological approach to the general problem, Kurt provides a very short and elementary proof.\n\nScheide (2008) has shown $\\chi'(G)>\\frac{15}{14}\\Delta+\\frac{12}{14}$ implies the Goldberg Conjecture.\n\nKurt(2009) has shown $\\chi'(G)>\\frac{17}{16}\\Delta+\\frac{14}{16}$ implies the Goldberg Conjecture.\n- June 5th, 2013 | SAC | Different Goldberg Conjecture?: http://plms.oxfordjournals.org/content/106/4/703\n\n--Stephen\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 23.\n\nAttempt notes:\nTarget:\nMake progress on \"Goldberg's conjecture\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3205, "problem_number": "OPG-46475", "title": "Strong edge colouring conjecture", "statement": "A strong edge-colouring of a graph $G$ is a edge-colouring in which every colour class is an induced matching; that is, any two vertices belonging to distinct edges with the same colour are not adjacent. The strong chromatic index $s\\chi'(G)$ is the minimum number of colours in a strong edge-colouring of $G$.\n\nConjecture $$s\\chi'(G) \\leq \\frac{5\\Delta^2}{4}, \\text{if$\\Delta$is even,}$$$$s\\chi'(G) \\leq \\frac{5\\Delta^2-2\\Delta +1}{4},&\\text{if$\\Delta$is odd.}$$", "background": "Source: Open Problem Garden. Original node ID: 46475. URL: http://www.openproblemgarden.org/op/strong_edge_colouring_conjecture.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/strong_edge_colouring_conjecture\n- Author(s): Erdos, Paul; Nesetril, Jaroslav\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 1st, 2013 by fhavet\n\nProblem-page discussion:\nThe conjectured bounds would be sharp. When $D$ is even, expanding each vertex of a $5$-cycle into a stable set of size $\\Delta/2$ yields such a graph with $5\\Delta^2/4$ edges in which the largest induced matching has size $1$. A similar construction achieves the bound when $\\Delta$ is odd.\n\nGreedy colouring the edges yields $s\\chi'(G) \\leq 2\\Delta(\\Delta-1)+1$. Using probabilistic methods, Molloy and Reed~[MoRe97] proved that there is a positive constant $\\epsilon$ such that, for sufficiently large $\\Delta$, every graph with maximum degree $\\Delta$ has strong chromatic index at most $(2-\\epsilon)\\Delta^2$.\n\nThe greedy bound proves the conjecture for $\\Delta \\leq 2$. For $\\Delta =3$, the conjectured bound of 10 was proved independently by Hor\\'ak, He, and Trotter[HHT] and by Andersen [A]. For $\\Delta=4$, the conjectured bound is 20, and Cranston [C] proved that 22 colours suffice.\n\nFor a bipartite graph $G$, Faudree et al. [FGST] conjectured that $s\\chi'(G)\\leq \\Delta^2(G)$. This is implied by the stronger conjecture due to Kaiser.\n\nConjecture Let $G=((A_1,A_2),E)$ be a bipartite graph such that every vertex in $A_1$ has degree at most $\\Delta_1$ and every vertex in $A_2$ has degree at most $\\Delta_2$. Then $s\\chi'(G)\\leq \\Delta_1\\Delta_2$.\n\nBibliography:\n[A] L. D. Andersen. The strong chromatic index of a cubic graph is at most 10. Discrete Math., 108(1-3):231--252, 1992.\n\n[C] D. W. Cranston. Strong edge-coloring of graphs with maximum degree 4 using 22 colors. Discrete Math., 306(21):2772--2778, 2006.\n\n[FGST] R. J. Faudree, A. Gyárfás, R. H. Schelp, and Zs. Tuza. Induced matchings in bipartite graphs. Discrete Math., 78(1-2):83--87, 1989.\n\n[HHT] P. Horák, Q. He, and W. T. Trotter. Induced matchings in cubic graphs. J. Graph Theory, 17(2):151--160, 1993.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 27.\n\nAttempt notes:\nTarget:\nMake progress on \"Strong edge colouring conjecture\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3206, "problem_number": "OPG-125", "title": "Cores of Cayley graphs", "statement": "Conjecture Let $M$ be an abelian group. Is the core of a Cayley graph (on some power of $M$ ) a Cayley graph (on some power of $M$ )?", "background": "Source: Open Problem Garden. Original node ID: 125. URL: http://www.openproblemgarden.org/op/cores_of_cayley_graphs.\n\nSource subject path: Graph Theory > Coloring > Homomorphisms.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cores_of_cayley_graphs\n- Author(s): Samal, Robert\n- Subject(s): Graph Theory; Coloring; Homomorphisms\n- Keywords: Cayley graph; core\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 6th, 2007 by Robert Samal\n\nProblem-page discussion:\nEven the case $M=\\mathbb{Z}_2$ is open. In this case, Cayley graphs on some power of $\\mathbb{Z}_2$ are called cube-like graphs, they have been introduced by Lov\\'asz as an example of graphs, for which every eigenvalue is an integer.\n\nSo, in this case we ask, whether a core of each cube-like graph is a cube-like graph.\n\nSource links:\n- core: http://en.wikipedia.org/wiki/core (graph theory)\n- Cayley graph: http://en.wikipedia.org/wiki/Cayley graph\n\nComments:\n- February 29th, 2012 | Anonymous | Who first conjectured this?: Who first conjectured that the core of a cubelike graph is cubelike?\n- February 22nd, 2008 | Gordon Royle | Question needs refining...: As stated, the conjecture is false in an uninteresting way... it is possible for a Cayley graph of Z_15 have a 5-cycle as a core.... So if we take M = Z_15 then the result is false.\n\nPerhaps the question should either (a) be restricted to elementary abelian groups or (b) have the conclusion being that the core of a Cayley graph on M must be a Cayley graph on N where N is a (group) homomorphic image of M.\n\nGordon Royle http://people.csse.uwa.edu.au/gordon\n- March 1st, 2008 | Robert Samal | Re: Question needs refining...: That is true. I was mainly thinking about $M={\\mathbb Z}_p$ (for a prime $p$ ). However, your suggestion (b) looks sensible as well.\n\nThanks for your comment!\n\nRobert Samal\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Cores of Cayley graphs\" in Graph Theory; Coloring; Homomorphisms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3207, "problem_number": "OPG-167", "title": "Pentagon problem", "statement": "Question Let $G$ be a 3-regular graph that contains no cycle of length shorter than $g$. Is it true that for large enough~ $g$ there is a homomorphism $G \\to C_5$?", "background": "Source: Open Problem Garden. Original node ID: 167. URL: http://www.openproblemgarden.org/op/pentagon_problem.\n\nSource subject path: Graph Theory > Coloring > Homomorphisms.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/pentagon_problem\n- Author(s): Nesetril, Jaroslav\n- Subject(s): Graph Theory; Coloring; Homomorphisms\n- Keywords: cubic; homomorphism\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 24th, 2007 by Robert Samal\n\nProblem-page discussion:\nThis question was asked by Nesetril at numerous problem sessions (and also appears as [N]). By Brook's theorem any triangle-free cubic graph is 3-colorable. Does a stronger assumption on girth of the graph imply stronger coloring properties?\n\nThis problem is motivated by complexity considerations [GHN] and also by exploration of density of the homomorphism order: We write $G \\prec H$ if there is a homomorphism $G \\to H$ but there is no homomorphism $H \\to G$. It is known that whenever $G \\prec H$ holds and $H$ ~is not bipartite, there is a graph~ $K$ satisfying $G \\prec K \\prec H$. A negative solution to the Pentagon problem would have the following density consequence: for each cubic graph~ $H$ for which~ $C_5 \\prec H$ holds, there exists a cubic graph~ $K$ satisfying $C_5 \\prec K \\prec H$ (see [N]).\n\nIf we replaced $C_5$ in the statement of the problem by a longer odd cycle, we would get a stronger statement. It is known that no such strenghthening is true. This was proved by Kostochka, Nesetril, and Smolikova [KNS] for $C_{11}$ (hence for all $C_l$ with $l \\ge 11$ ), by Wanless and Wormald [WW] for $C_9$, and recently by Hatami [H] for $C_7$. Each of these results uses probabilistic arguments (random regular graphs), no constructive proof is known.\n\nHaggkvist and Hell [HH] proved that for every integer~ $g$ there is a graph~ $U_g$ with odd girth at least~ $g$ (that is, $U_g$ does not contain odd cycle of length less than~ $g$ ) such that every cubic graph of odd girth at least~ $g$ maps homomorphically to~ $U_g$. Here, the graph~ $U_g$ may have large degrees. This leads to a weaker version of the Pentagon problem:\n\nQuestion Is it true that for every $k$ there exists a cubic graph $H_k$ of girth~ $k$ and an integer~ $g$ such that every cubic graph of girth at least~ $g$ maps homomorphically to~ $H_k$?\n\nA particular question in this direction: does a high-girth cubic graph map to the Petersen graph?\n\nAs an approach to this, we mention a result of DeVos and Samal [DS]: a cubic graph of girth at least~ $17$ admits a homomorphism to the Clebsch graph. In context of the Pentagon problem, the following reformulation is particularly appealing: If $G$ ~is a cubic graph of girth at least~ $17$, then there is a cut-continuous mapping from~ $G$ to~ $C_5$; that is, there is a mapping $f: E(G) \\to E(C_5)$ such that for any cut $X \\subseteq E(C_5)$ the preimage $f^{-1}(X)$ is a cut. (Here by cut we mean the edge-set of a spanning bipartite subgraph. A more thorough exposition of cut-continuous mappings can be found in~[DNR].)\n\nBibliography:\n[DNR] Matt DeVos, Jaroslav Nesetril, and Andre Raspaud, On edge-maps whose inverse preverses flows and tensions, Graph Theory in Paris: Proceedings of a Conference in Memory of Claude Berge, Birkhäuser 2006.\n\n[DS] Matt DeVos and Robert Samal, High girth cubic graphs map to the Clebsch graph\n\n[GHN] Anna Galluccio, Pavol Hell, and Jaroslav Nesetril, The complexity of $H$-colouring of bounded degree graphs, Discrete Math. 222 (2000), no.~1-3, 101--109, MathSciNet\n\n[HH] Roland Haggkvist and Pavol Hell, Universality of $A$-mote graphs, European J. Combin. 14 (1993), no.~1, 23--27.\n\n[H] Hamed Hatami, Random cubic graphs are not homomorphic to the cycle of size~7, J. Combin. Theory Ser. B 93 (2005), no.~2, 319--325, MathSciNet\n\n[KNS] Alexandr~V. Kostochka, Jaroslav Nesetril, and Petra Smolikova, Colorings and homomorphisms of degenerate and bounded degree graphs, Discrete Math. 233 (2001), no.~1-3, 257--276, Fifth Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications, (Prague, 1998), MathSciNet\n\n*[N] Jaroslav Nesetril, Aspects of structural combinatorics (graph homomorphisms and their use), Taiwanese J. Math. 3 (1999), no.~4, 381--423, MathSciNet\n\n[WW] I.M. Wanless and N.C. Wormald, Regular graphs with no homomorphisms onto cycles, J. Combin. Theory Ser. B 82 (2001), no.~1, 155--160, MathSciNet\n\nSource links:\n- homomorphism: http://en.wikipedia.org/wiki/graph_homomorphism\n\nDiscussion links:\n- cubic graph: http://en.wikipedia.org/wiki/cubic graph\n- girth: http://en.wikipedia.org/wiki/girth\n\nBibliography links:\n- High girth cubic graphs map to the Clebsch graph: http://www.arxiv.org/abs/math.CO/0602580\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1771392\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2117942\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2002c:05077\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1730980\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2002a:05221\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 25.\n\nAttempt notes:\nTarget:\nMake progress on \"Pentagon problem\" in Graph Theory; Coloring; Homomorphisms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3208, "problem_number": "OPG-412", "title": "Mapping planar graphs to odd cycles", "statement": "Conjecture Every planar graph of girth $\\ge 4k$ has a homomorphism to $C_{2k+1}$.", "background": "Source: Open Problem Garden. Original node ID: 412. URL: http://www.openproblemgarden.org/op/mapping_planar_graphs_to_odd_cycles.\n\nSource subject path: Graph Theory > Coloring > Homomorphisms.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/mapping_planar_graphs_to_odd_cycles\n- Author(s): Jaeger, Francois\n- Subject(s): Graph Theory; Coloring; Homomorphisms\n- Keywords: girth; homomorphism; planar graph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 24th, 2007 by mdevos\n\nProblem-page discussion:\nThis conjecture is Jaeger's modular orientation conjecture restricted to planar graphs and then dualized. To see this duality, first note that circular coloring and circular flows are dual for planar graphs, and then observe that $(2k+1)$-orientations are equivalent to $2 + \\frac{1}{k}$-flows and $2 + \\frac{1}{k}$-colorings are equivalent to homomorphisms to $C_{2k+1}$. So if $G$ and $G^*$ are dual planar graphs, then we have the following equivalences.\n\n- $G$ has a $(2k+1)$-orientation.\n- $G$ has a $2 + \\frac{1}{k}$-flow.\n- $G^*$ has a $2 + \\frac{1}{k}$-coloring.\n- $G^*$ has a homomorphism to $C_{2k+1}$.\n\nThere is an easy family of graphs which show that the above conjecture (if true) is best possible. Let $H_k$ be the graph obtained from an odd circuit of length $4k-1$ by adding a new vertex $u$ joined to every existing vertex by a path of length $2k-1$. Now, $H_k$ is a planar graph of girth $4k-1$, but there is no homomorphism from $H_k$ to $C_{2k+1}$. To see the latter claim, suppose (for a contradiction) that such a homomorphsim $f$ exists, let $C$ be the unique circuit of $H_k \\setminus u$ and let $a=f(u)$. Now, no vertex in $C$ can map to $a$ since every such vertex is distance $2k-1$ from $u$. However we must then have a homomorphism from $C$ to $C_{2k+1} \\setminus a$, which is impossible since $C$ is an odd circuit and $C_{2k+1} \\setminus u$ is bipartite.\n\nThe k=1 case of the above conjecture asserts that every (loopless) triangle free planar graph has a homomorphism to the triangle. In other words, every (loopless) triangle free planar graph is 3-colorable. This is a well known theorem of Grotszch. For every k>1, the above conjecture is still open. Actually, I think this conjecture is already quite interesting for k=2. One reason is that this case of the conjecture implies the 5-color theorem for planar graphs. To see this implication, suppose that the above conjecture is true for k=2, let G be a simple loopless planar graph, and let G' be the graph obtained from G by subdividing each edge two times. Now, G' has girth at least 9, so by our assumption there is a homomorphism from G' to C_5. It is easy to see that adjacent vertices of G must map to different vertices of C_5 under this homomorphism. Thus, we have a proper 5-coloring of G as desired.\n\nLet us call a homomorphism to $C_{2k+1}$ a $C_{2k+1}$-coloring. It is quite easy to show that every planar graph of girth > 10k has a $C_{2k+1}$-coloring. This follows from a simple degeneracy argument: Every such (nonempty) graph must have a either a vertex of degree $\\le 1$, or a path $P$ of length $2k-1$ all of whose internal vertices have degree two. Both of these configurations are reducible, in the sense that we may delete either a vertex of degree $\\le 1$ or the interior vertices of $P$ and then extend any $C_{2k+1}$-coloring of the resulting graph to a $C_{2k+1}$-coloring of the original. By more complicated, but similar degeneracy arguments, we can approach this conjecture. To my knowledge, the best result to date is as follows.\n\nTheorem (Borodin, Kim, Kostochka, West) Every planar graph of girth $\\ge \\frac{20k-2}{3}$ has a homomorphism to $C_{2k+1}$.\n\nFor the special case of the conjecture when $k=2$, Matt DeVos and Adam Deckelbaum have an unpublished improvement showing that every planar graph with odd girth $\\ge 11$ has a homomorphism to $C_5$.\n\nBibliography:\n[BKKW] O. V. Borodin, S. J. Kim, A. V. Kostochka, D. B. West, Homomorphisms from sparse graphs with large girth. Dedicated to Adrian Bondy and U. S. R. Murty. J. Combin. Theory Ser. B 90 (2004), no. 1, 147--159. MathSciNet\n\n[Ja] F. Jaeger, On circular flows in graphs in Finite and Infinite Sets, volume 37 of Colloquia Mathematica Societatis Janos Bolyai, edited by A. Hajnal, L. Lovasz, and V.T. Sos. North-Holland (1981) 391-402.\n\n[Zh] X. Zhu, Circular chromatic number of planar graphs of large odd girth, Electronic Journal of Combinatorics Vol. 8 no. 1 (2001).\n\nRelated:\nRelated problems\nJaeger's modular orientation conjecture\n\nDiscussion links:\n- Jaeger's modular orientation conjecture: http://www.openproblemgarden.org/?q=node/130\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2041323\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 23.\n\nAttempt notes:\nTarget:\nMake progress on \"Mapping planar graphs to odd cycles\" in Graph Theory; Coloring; Homomorphisms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3209, "problem_number": "OPG-434", "title": "Weak pentagon problem", "statement": "Conjecture If $G$ is a cubic graph not containing a triangle, then it is possible to color the edges of $G$ by five colors, so that the complement of every color class is a bipartite graph.", "background": "Source: Open Problem Garden. Original node ID: 434. URL: http://www.openproblemgarden.org/op/weak_pentagon_problem.\n\nSource subject path: Graph Theory > Coloring > Homomorphisms.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/weak_pentagon_problem\n- Author(s): Samal, Robert\n- Subject(s): Graph Theory; Coloring; Homomorphisms\n- Keywords: Clebsch graph; cut-continuous mapping; edge-coloring; homomorphism; pentagon\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 13th, 2007 by Robert Samal\n\nProblem-page discussion:\nThis conjecture has several reformulations: the conclusion of the conjecture can be replaced by either of the following:\n\n- $G$ has a homomorphism to the Clebsch graph.\n- there is a cut-continuous mapping from $G$ to $C_5$.\n\nFor the latter variant, few definitions are in place. A cut-continuous mapping from a graph~ $G$ to a graph~ $H$ is a mapping $f: E(G) \\to E(H)$ such that the preimage of every cut in~ $H$ is a cut in~ $G$. Here, by a cut in~ $H$ we mean the edge-set of a spanning bipartite subgraph of~ $H$---less succinctly, it is the set of all edges leaving some subset of vertices of~ $H$.\n\nCut-continuous mappings are closely related with graph homomorphisms (see [DNR], [S]). In particular, every homomorphism from~ $G$ to~ $H$ naturally induces a cut-continuous mapping from~ $G$ to~ $H$; thus, the presented conjecture can be thought of as a weaker version of Nesetril's Pentagon problem.\n\nWe mention a generalization of the conjecture, that deals with longer cycles/larger number of colors. The $n$-dimensional projective cube, denoted $PQ_n$, is the simple graph obtained from the $(n+1)$-dimensional cube~ $Q_{n+1}$ by identifying pairs of antipodal vertices (vertices that differ in all coordinates). Note that $PQ_4$ is the Clebsch graph.\n\nQuestion What is the largest integer $k$ with the property that all cubic graphs of sufficiently high girth have a homomorphism to $PQ_{2k}$?\n\nAgain, the question has several reformulations due to the following simple proposition.\n\nProposition For every graph $G$ and nonnegative integer $k$, the following properties are equivalent.\n\n- There exists a coloring of~ $E(G)$ by $2k+1$ colors so that the complement of every color class is a bipartite graph.\n- $G$ has a homomorphism to $PQ_{2k}$\n- $G$ has a cut-continuous mapping to~ $C_{2k+1}$\n\nThere are high-girth cubic graphs with the largest cut of size less then $0.94\\cdot |E|$. Such graphs do not admit a homomorphism to $PQ_{2k}$ for any $k \\ge 8$, so there is indeed some largest integer~ $k$ in the above question. To bound this largest~ $k$ from below, recall that every cubic graph maps homomorphically to $K_4 = PQ_2$. Moreover, it is known [DS] that cubic graphs of girth at least 17 admit a homomorphism to $PQ_4$ (the Clebsch graph). This shows $k\\ge 2$ (and also provides a support for the main conjecture).\n\nBibliography:\n[DNR] Matt DeVos, Jaroslav Nesetril and Andre Raspaud: On edge-maps whose inverse preserves flows and tensions, \\MRref{MR2279171}\n\n*[DS] Matt Devos, Robert Samal: \\arXiv[High Girth Cubic Graphs Map to the Clebsch Graph}{math.CO/0602580}\n\n[S] Robert Samal, On XY mappings, PhD thesis, Charles University 2006, tech. report\n\nRelated:\nRelated problems\nPentagon problem\n\nDiscussion links:\n- Clebsch graph: http://en.wikipedia.org/wiki/Gallery_of_named_graphs\n- graph homomorphisms: http://en.wikipedia.org/wiki/graph homomorphism\n- Pentagon problem: http://www.openproblemgarden.org/?q=node/167\n- girth: http://en.wikipedia.org/wiki/girth\n\nBibliography links:\n- tech. report: http://kam.mff.cuni.cz/%7Ekamserie/serie/clanky/2006/s772.ps\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Weak pentagon problem\" in Graph Theory; Coloring; Homomorphisms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3210, "problem_number": "OPG-37232", "title": "Algorithm for graph homomorphisms", "statement": "Question\n\nIs there an algorithm that decides, for input graphs $G$ and $H$, whether there exists a homomorphism from $G$ to $H$ in time $O(c^{|V(G)|+|V(H)|})$ for some constant $c$?", "background": "Source: Open Problem Garden. Original node ID: 37232. URL: http://www.openproblemgarden.org/op/algorithm_for_graph_homomorphisms.\n\nSource subject path: Graph Theory > Coloring > Homomorphisms.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/algorithm_for_graph_homomorphisms\n- Author(s): Fomin, Fedor V.; Heggernes, Pinar; Kratsch, Dieter\n- Subject(s): Graph Theory; Coloring; Homomorphisms\n- Keywords: algorithm; Exponential-time algorithm; homomorphism\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 8th, 2010 by jfoniok\n\nProblem-page discussion:\nAn affirmative answer is known in several cases: if $H=K_k$ (graph coloring) [L], [BH], [K]; if $H$ has bounded treewidth [FHK]; if $H$ has bounded cliquewidth [W].\n\nBibliography:\n[BH] Andreas Björklund, Thore Husfeldt: Inclusion--Exclusion Algorithms for Counting Set Partitions, Proc. FOCS'06 (2006).\n\n*[FHK] Fedor V. Fomin, Pinar Heggernes, Dieter Kratsch: Exact Algorithms for Graph Homomorphisms, Theory Comput. Syst. 41 (2007), no. 2, 381--393. MathSciNet\n\n[K] Mikko Koivisto: An $O^\\ast(2^n)$ Algorithm for Graph Coloring and Other Partitioning Problems via Inclusion--Exclusion, Proc. FOCS'06 (2006).\n\n[L] Eugene L. Lawler: A note on the complexity of the chromatic number problem, Information Processing Lett. 5 (1976), no. 3, 66--67. MathSciNet\n\n[W] Magnus Wahlström: New Plain-Exponential Time Classs for Graph Homomorphism, CSR2009, LNCS5675 (2009), 346--355.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2329330\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0464675\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Algorithm for graph homomorphisms\" in Graph Theory; Coloring; Homomorphisms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3211, "problem_number": "OPG-37619", "title": "Circular choosability of planar graphs", "statement": "Let $G = (V, E)$ be a graph. If $p$ and $q$ are two integers, a $(p,q)$-colouring of $G$ is a function $c$ from $V$ to $\\{0,\\dots,p-1\\}$ such that $q \\le |c(u)-c(v)| \\le p-q$ for each edge $uv\\in E$. Given a list assignment $L$ of $G$, i.e.~a mapping that assigns to every vertex $v$ a set of non-negative integers, an $L$-colouring of $G$ is a mapping $c: V \\to N$ such that $c(v)\\in L(v)$ for every $v\\in V$. A list assignment $L$ is a $t$- $(p,q)$-list-assignment if $L(v) \\subseteq \\{0,\\dots,p-1\\}$ and $|L(v)| \\ge tq$ for each vertex $v \\in V$. Given such a list assignment $L$, the graph G is $(p,q)$- $L$-colourable if there exists a $(p,q)$- $L$-colouring $c$, i.e. $c$ is both a $(p,q)$-colouring and an $L$-colouring. For any real number $t \\ge 1$, the graph $G$ is $t$- $(p,q)$-choosable if it is $(p,q)$- $L$-colourable for every $t$- $(p,q)$-list-assignment $L$. Last, $G$ is circularly $t$-choosable if it is $t$- $(p,q)$-choosable for any $p$, $q$. The circular choosability (or circular list chromatic number or circular choice number) of G is $$cch(G):= \\inf\\{t \\ge 1: G \\text{ is circularly$t$-choosable}\\}.$$\n\nProblem What is the best upper bound on circular choosability for planar graphs?", "background": "Source: Open Problem Garden. Original node ID: 37619. URL: http://www.openproblemgarden.org/op/circular_choosability_of_planar_graphs.\n\nSource subject path: Graph Theory > Coloring > Homomorphisms.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/circular_choosability_of_planar_graphs\n- Author(s): Mohar, Bojan\n- Subject(s): Graph Theory; Coloring; Homomorphisms\n- Keywords: choosability; circular colouring; planar graphs\n- Importance: Low ✭\n- Recommended for undergraduates: no\n- Posted: August 23rd, 2012 by rosskang\n\nProblem-page discussion:\nThe problem was first posed in 2003 by Mohar (Problem 4 of link*) who suggested the answer should be between 4 and 5.\n\nSome time later, Havet, Kang, Müller, and Sereni [HKMS] showed that in fact the answer is somewhere between 6 and 8. The upper bound extends a celebrated planar choosability proof due to Thomassen [T]. The lower bound is by way of an elementary, though rather large, construction.\n\nBibliography:\n[HKMS] F. Havet, R. J. Kang, T. Müller, and J.-S. Sereni. Circular choosability. J. Graph Theory 61 (2009), no. 4, 241--270.\n\n[T] C. Thomassen. Every planar graph is 5-choosable. J. Combinatorial Theory B 62 (1994) 180--181\n\nDiscussion links:\n- link: http://www.fmf.uni-lj.si/%7Emohar/Problems/P0201ChoosabilityCircular.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 23.\n\nAttempt notes:\nTarget:\nMake progress on \"Circular choosability of planar graphs\" in Graph Theory; Coloring; Homomorphisms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3212, "problem_number": "OPG-439", "title": "Graceful Tree Conjecture", "statement": "Conjecture All trees are graceful", "background": "Source: Open Problem Garden. Original node ID: 439. URL: http://www.openproblemgarden.org/op/graceful_tree_conjecture.\n\nSource subject path: Graph Theory > Coloring > Labeling.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/graceful_tree_conjecture\n- Subject(s): Graph Theory; Coloring; Labeling\n- Keywords: combinatorics; graceful labeling\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 13th, 2007 by kintali\n\nProblem-page discussion:\nLabel the vertices of a simple undirected graph $G(V,E)$ (where $|V| = n$ and $|E| = m$ ) with integers from $0$ to $m$. Now label each edge with absolute difference of the labels of its incident vertices. The labeling is said to be graceful if the edges are labelled $1$ through $m$ inclusive (with no number repeated).\n\nA graph is called graceful if it has at least one such labeling. This labeling was originally introduced in 1967 by Rosa. The name graceful labeling was coined later by Golomb.\n\nGracefully labeled graphs serve as models in a wide range of applications including coding theory and communication network addressing.\n\nThe graceful labeling problem is to determine which graphs are graceful. It is conjectured (by Kotzig, Ringel and Rosa) that all trees are graceful.\n\nDespite numerous (more than 200) publications on graceful labeling for over three decades, only a very restricted classes of trees (and also of some other graphs) have been shown to be graceful. These restricted classes include paths, stars, complete bipartite graphs, prism graphs, wheel graphs, caterpillar graphs, olive trees, and symmetrical trees.\n\nComments:\n- November 29th, 2020 | Anonymous | Bibliography: Many, many references on the topic appear in Joseph Gallian's \"A Dynamic Survey of Graph Labeling\", section 2 ( https://www.combinatorics.org/ds6 ).\n- September 29th, 2014 | Anonymous | Graceful Tree examples: If you want to develop intuition for a proof, feel free to use this program I wrote (http://bl.ocks.org/NPashaP/7683252). Good luck.\n- November 30th, 2009 | Anonymous | applications of graceful graphs: can you explain some applications of graceful graphs\n- June 12th, 2010 | Anonymous | Apllication of graceful labeling: I want the application of graceful label\n- November 30th, 2009 | Anonymous | graph theory: Can you pls send some applications of gracefulgraphs\n- May 5th, 2009 | Anonymous | Tentative proof: I just saw this paper on arxiv, entitled \"A complete proof of The Graceful Tree Conjecture using the concept of Edge Degree\".\n\nI'm surprised to see such a short proof for such a long-standing open problem, but surely people who are a lot more into the subject than I will be able to provide more constructive comments on the paper.\n- May 6th, 2009 | Robert Samal | Re: Tentative proof: It's a bit worrisome that this is already the eight version on the arXiv... this probably means that the previous seven versions had some sort of flaw in it... I haven't read the paper though...\n- April 14th, 2009 | Anonymous | injective labeling: Rosa (1967) required the labeling to be injective.\n- August 1st, 2007 | Anonymous | Err..: The complete bipartite graph isn't a tree.\n- August 2nd, 2007 | Anonymous | not a problem: graceful labellings are defined for arbitrary graphs, not just trees.\n- August 2nd, 2007 | Anonymous | no duh..: I was just pointing out the statement\n\n\"...only a very restricted classes of *trees* have been shown to be graceful. *These* restricted classes include...the complete bipartite graph\".\n\ndoesn't make sense.\n- August 4th, 2007 | rs | Re: wording: Truly, this was unfortunate choice of wording; I corrected it. (Btw to provide a graceful labeling for complete bipartite graphs is quite an easy exercise.)\n- April 12th, 2008 | Anonymous | graceful labeling of complete bipartite graphs: how to give a graceful labeling to complete bipartite graph. Can u suggest some efficient algorithm or scheme for that Plz email it to rocky_blt@yahoo.co.in if u have any...\n\nthanx\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Graceful Tree Conjecture\" in Graph Theory; Coloring; Labeling, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3213, "problem_number": "OPG-37323", "title": "Good Edge Labelings", "statement": "Question What is the maximum edge density of a graph which has a good edge labeling?\n\nWe say that a graph is good-edge-labeling critical, if it has no good edge labeling, but every proper subgraph has a good edge labeling.\n\nConjecture For every $c<4$, there is only a finite number of good-edge-labeling critical graphs with average degree less than $c$.", "background": "Source: Open Problem Garden. Original node ID: 37323. URL: http://www.openproblemgarden.org/op/good_edge_labelings.\n\nSource subject path: Graph Theory > Coloring > Labeling.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/good_edge_labelings\n- Author(s): Araújo, Julio; Cohen, Nathann; Giroire, Frédéric; Havet, Frédéric\n- Subject(s): Graph Theory; Coloring; Labeling\n- Keywords: good edge labeling, edge labeling\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 30th, 2011 by DOT\n\nProblem-page discussion:\nLet $G$ be a finite undirected simple graph. A good edge labeling of $G$ is an assignment of distinct numbers to the edges such that every cycle has at least two local maxima. (The distinctness of the labels is required only to make the term `local maximum' unambiguous.)\n\nEquivalently, a labeling of the edges is good, if for every pair of distinct vertices $u,v$, there is at most one increasing path from $u$ to $v$.\n\nHaving a good edge labeling is inherited by subgraphs.\n\nIt is easy to verify that the graphs $K_3$ and $K_{2,3}$ have no good edge labeling. In [ACGH2] an infinite class of graphs without good edge labelings is given, none of whom is a subgraph of the other. In [BFT] contains an example of a minimal graph without good edge labeling which as average degree < 3 (thus refuting an earlier conjecture saying that a good-edge-labeling critical graph with average degree less than three is either $K_3$ or $K_{2,3}$ ). In that same paper it is shown that every such graph must have girth at most 4.\n\nGood edge labeling of graphs was introduced in [BCP] in the context of the so-called Routing and Wavelength Assignment (RWA) problem. The problems above are proposed in [ACGH1] and [ACGH2]. There the algorithmic problem of determining whether a graph has a good edge labeling is shown to be NP-hard. Moreover, the authors also prove that every planar graph with girth at least six has a good edge labeling.\n\nBibliography:\n[BCP] J-C. Bermond, M. Cosnard, and S. Pérennes. Directed acyclic graphs with unique path property. Technical report 6932, INRIA, May 2009\n\n[ACGH1] J. Araújo, N. Cohen, F. Giroire, F. Havet. Good edge-labelling of graphs. (English summary) LAGOS'09—V Latin-American Algorithms, Graphs and Optimization Symposium, 275–280, Electron. Notes Discrete Math., 35, Elsevier Sci. B. V., Amsterdam, 2009. MathSciNet\n\n[ACGH2*] J. Araujo, N. Cohen, F. Giroire, and F. Havet. Good edge-labelling of graphs. Research Report 6934, INRIA, 2009.\n\n[BFT] M. Bode, B. Farzad, D.O. Theis. Good edge-labelings and graphs of girth at least 5. (arXiv:1109.1125)\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2579442\n\nComments:\n- February 16th, 2013 | Anonymous | First question (almost) solved: Mehrabian, Mitsche, and Pralat showed that any $n$-vertex graph with a good edge-labelling has at most $n \\log_2 n$ edges, and that for each $n$ there is an $n$-vertex graphs with a good edge-labelling having $n \\log_2 n - O(n)$ edges.\n\nhttp://arxiv.org/abs/1211.2641\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Good Edge Labelings\" in Graph Theory; Coloring; Labeling, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3214, "problem_number": "OPG-126", "title": "5-flow conjecture", "statement": "Conjecture Every bridgeless graph has a nowhere-zero 5-flow.", "background": "Source: Open Problem Garden. Original node ID: 126. URL: http://www.openproblemgarden.org/op/5_flow_conjecture.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/5_flow_conjecture\n- Author(s): Tutte, William T.\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: cubic; nowhere-zero flow\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nFor planar graphs, this theorem follows from flow/coloring duality, and the Five color theorem (every loopless planar graph is 5-colorable). In light of this, we may view this conjecture as a widesweeping generalization of the 5-color-theorem. The Petersen graph does not have a nowhere-zero 4-flow, which shows that this conjecture (if true) is best possible.\n\nIt is far from obvious that there should exist a fixed number $k$ so that every bridgeless graph has a nowhere-zero $k$-flow. Indeed, this weaker conjecture was also made by Tutte, but was resolved by Kilpatrick [K] and independently Jaeger [J], who both proved that bridgeless graphs have nowhere-zero 8-flows. Seymour [S] improved upon this result by showing that bridgeless graphs have nowhere-zero 6-flows.\n\nBibliography:\n[J] F. Jaeger, Flows and Generalized Coloring Theorems in Graphs, J. Combinatorial Theory Ser. B 26 (1979) 205-216. MathSciNet\n\n[K] P.A. Kilpatrick, Tutte's First Colour-Cycle Conjecture, Thesis, Cape Town (1975).\n\n[S] P.D. Seymour, Nowhere-Zero 6-Flows, J. Combinatorial Theory Ser. B 30 (1981) 130-135. MathSciNet\n\n[T54] W.T. Tutte, A Contribution on the Theory of Chromatic Polynomials, Canad. J. Math. 6 (1954) 80-91. MathSciNet\n\n[Tt66] W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combinatorial Theory 1 (1966) 15-50. MathSciNet\n\nSource links:\n- bridgeless: http://en.wikipedia.org/wiki/bridge (graph theory)\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nDiscussion links:\n- planar graphs: http://en.wikipedia.org/wiki/planar graphs\n- flow/coloring duality: http://en.wikipedia.org/wiki/nowhere-zero flows\n- Five color theorem: http://en.wikipedia.org/wiki/Five color theorem\n- Petersen graph: http://en.wikipedia.org/wiki/Petersen graph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0532588\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0615308\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0061366\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0194363\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"5-flow conjecture\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3215, "problem_number": "OPG-127", "title": "4-flow conjecture", "statement": "Conjecture Every bridgeless graph with no Petersen minor has a nowhere-zero 4-flow.", "background": "Source: Open Problem Garden. Original node ID: 127. URL: http://www.openproblemgarden.org/op/4_flow_conjecture.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/4_flow_conjecture\n- Author(s): Tutte, William T.\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: minor; nowhere-zero flow; Petersen graph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nIt is a consequence of a theorem of Tutte that a cubic graph has a nowhere-zero 4-flow if and only if it is 3-edge-colorable. Thus, the 4-flow conjecture implies that every bridgeless cubic graph with no Petersen minor is 3-edge-colorable (another conjecture of Tutte). Note that the Four Color Theorem is equivalent to the assertion that planar cubic graphs without bridges are 3-edge-colorable, so even this weaker conjecture is a strengthening of the Four Color Theorem. This weaker conjecture was recently proved by Robertson, Seymour, and Thomas [RST]. Their proof involves a reduction to the case of nearly planar graphs, and then an application of 4-color-theorem type techniques (computer assisted) to color these graphs.\n\nMost conjectures about flows can be easily reduced to the case of cubic graphs by splitting arguments. The idea is to take a vertex $v$ incident with edges $e_1,\\ldots,e_k$ and \"split\" $v$, that is, replace $v$ by two new vertices $v_1$ and $v_2$, and for every edge $e_i$ join it to either $v_1$ or $v_2$ (sometimes the edge $v_1 v_2$ is also added). For instance, this technique can be used to reduce the general 5-flow conjecture down to the special case of cubic graphs. Unfortunately, that technique does not apply here, since splitting a vertex may introduce a Petersen minor.\n\nPetersen's graph is not an apex graph (deleting any vertex still leaves a nonplanar graph). It follows that no apex graph can have a Petersen minor, so the above conjecture implies that every bridgeless apex graph has a nowhere-zero 4-flow. By splitting the vertices which lie in the plane this can be reduced to the special case where all vertices which lie in the plane have degree 3. This is then equivalent to the following old conjecture of Gr\\\"{o}tzsch.\n\nConjecture (Gr\\\"{o}tzsch) If $G$ is a 2-connected connected planar graph of maximum degree 3, then $G$ is 3-edge-colorable unless it has exactly one vertex of degree 2.\n\nBibliography:\n[AH] K. Appel, W. Haken, Every Planar Map is Four Colorable, Bull. Amer. Math. Soc. 82 (1976) 711-712. MathSciNet\n\n[RSST] N. Robertson, D.P. Sanders, P.D. Seymour, and R. Thomas, A New Proof of the Four-Color Theorem, Electron. Res. Announc., Am. Math. Soc. 02, no 1 (1996) 17-25. MathSciNet\n\n[RST] N. Robertson, P.D. Seymour, and R. Thomas, Tutte's edge-colouring conjecture. J. Combin. Theory Ser. B 70 (1997), no. 1, 166--183. MathSciNet\n\n[Tut54] W.T. Tutte, A Contribution on the Theory of Chromatic Polynomials, Canad. J. Math. 6 (1954) 80-91. MathSciNet\n\n[Tut66] W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combinatorial Theory 1 (1966) 15-50. MathSciNet\n\nSource links:\n- bridgeless: http://en.wikipedia.org/wiki/bridge (graph theory)\n- Petersen: http://en.wikipedia.org/wiki/petersen graph\n- minor: http://en.wikipedia.org/wiki/minor (graph theory)\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nDiscussion links:\n- cubic graph: http://en.wikipedia.org/wiki/cubic graph\n- edge-colorable: http://en.wikipedia.org/wiki/edge coloring\n- Four Color Theorem: http://en.wikipedia.org/wiki/Four Color Theorem\n- planar: http://en.wikipedia.org/wiki/planar graph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0424602\n- A New Proof of the Four-Color Theorem: http://www.ams.org/era/1996-02-01/S1079-6762-96-00003-0/S1079-6762-96-00003-0.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1405965\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1441265\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0061366\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0194363\n\nComments:\n- April 11th, 2009 | Anonymous | fix the problm: fix the problm\n- April 12th, 2009 | Anonymous | What is wrong?: Perhaps you could elaborate.. what needs to be fixed?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"4-flow conjecture\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3216, "problem_number": "OPG-128", "title": "3-flow conjecture", "statement": "Conjecture Every 4-edge-connected graph has a nowhere-zero 3-flow.", "background": "Source: Open Problem Garden. Original node ID: 128. URL: http://www.openproblemgarden.org/op/3_flow_conjecture.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/3_flow_conjecture\n- Author(s): Tutte, William T.\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: nowhere-zero flow\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nGrotzsch proved that every triangle free (and loopless) planar graph is 3-colorable. By flow/coloring duality, this is equivalent to the statement that every 4-edge-connected planar graph has a nowhere-zero 3-flow. The 3-flow conjecture asserts that this is still true without the assumption of planarity.\n\nJaeger proved that 4-edge-connected graphs have nowhere-zero 4-flows, but very little is known about nowhere-zero 3-flows. In particular, the following weak version of the 3-flow conjecture is still wide open.\n\nConjecture (The weak 3-flow conjecture (Jaeger)) There exists a fixed integer $k$ so that every $k$-edge-connected graph has a nowhere-zero 3-flow.\n\nLai and Zhang [LZ] have proved that if $G$ has $n$ vertices and edge-connectivity at least $4 \\log_2(n)$ then $G$ has a nowhere-zero 3-flow. A similar result (edge connectivity at least $4 \\log(n) + 2$ ) also follows from a theorem of Alon, Linial, and Meshulam [ALM] on additive bases of vector spaces.\n\nBibliography:\n[ALM] N. Alon, N. Linial, and R. Meshulam Additive Bases of Vector Spaces over Prime Fields J. Combinatorial Theory Ser. A 57 (1991), 203-210.\n\n[J] F. Jaeger, Flows and Generalized Coloring Theorems in Graphs, J. Combinatorial Theory Ser. B 26 (1979) 205-216.\n\n[LZ] H.J. Lai and C.Q. Zhang, Nowhere-Zero 3-Flows of Highly Connected Graphs, Discrete Math 110 (1992) 179-183.\n\n[T54] W.T. Tutte, A Contribution on the Theory of Chromatic Polynomial, Canad. J. Math. 6 (1954) 80-91.\n\n[T66] W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combinatorial Theory 1 (1966) 15-50.\n\nSource links:\n- edge-connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nDiscussion links:\n- planar graph: http://en.wikipedia.org/wiki/planar graph\n- colorable: http://en.wikipedia.org/wiki/graph coloring\n- flow/coloring duality: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nComments:\n- April 12th, 2011 | Flo Pfender | weak 3-flow conjecture proved: Carsten Thomassen recently (Christmas 2010) proved the weak 3-flow conjecture for k=8.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"3-flow conjecture\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3217, "problem_number": "OPG-130", "title": "Jaeger's modular orientation conjecture", "statement": "Conjecture Every $4k$-edge-connected graph can be oriented so that ${\\mathit indegree}(v) - {\\mathit outdegree}(v) \\cong 0$ (mod $2k+1$ ) for every vertex $v$.", "background": "Source: Open Problem Garden. Original node ID: 130. URL: http://www.openproblemgarden.org/op/jaegers_modular_orientation_conjecture.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/jaegers_modular_orientation_conjecture\n- Author(s): Jaeger, Francois\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: nowhere-zero flow; orientation\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nJaeger called an orientation with the above property a modular $(2k+1)$-orientation, and observed that a graph has a modular $(2k+1)$-orientation if and only if it has a $(2+\\frac{1}{k})$-flow. Thus, this conjecture may be seen as a sharp form of the 2+epsilon flow conjecture. For k=1, this problem is precisely the 3-flow conjecture, and for k=2, Jaeger showed that this conjecture (if true) would imply the 5-flow conjecture. If true, this conjecture would be best possible for every value of k.\n\nThe restriction of this conjecture to planar graphs is open, and has a dual formulation. See Mapping planar graphs to odd cycles.\n\nBibliography:\n[J] F. Jaeger, On circular flows in graphs. Finite and infinite sets, Vol. I, II (Eger, 1981), 391--402, Colloq. Math. Soc. János Bolyai, 37, North-Holland, Amsterdam, 1984.. MathSciNet\n\nRelated:\nRelated problems\nMapping planar graphs to odd cycles\n\nSource links:\n- edge-connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n\nDiscussion links:\n- 2+epsilon flow conjecture: http://www.openproblemgarden.org/?q=op/2_epsilon_flow_conjecture\n- 3-flow conjecture: http://www.openproblemgarden.org/?q=op/3_flow_conjecture\n- 5-flow conjecture: http://www.openproblemgarden.org/?q=op/5_flow_conjecture\n- Mapping planar graphs to odd cycles: http://www.openproblemgarden.org/?q=node/412\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0818250\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Jaeger's modular orientation conjecture\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3218, "problem_number": "OPG-131", "title": "Bouchet's 6-flow conjecture", "statement": "Conjecture Every bidirected graph with a nowhere-zero $k$-flow for some $k$, has a nowhere-zero $6$-flow.", "background": "Source: Open Problem Garden. Original node ID: 131. URL: http://www.openproblemgarden.org/op/bouchets_6_flow_conjecture.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/bouchets_6_flow_conjecture\n- Author(s): Bouchet, Andre\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: bidirected graph; nowhere-zero flow\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: A bidirected graph is a graph in which every edge has two arrowheads, one next to each endpoint. If the edge $e$ has ends $u$ and $v$, then the arrowheads nearest $u$ and $v$ may point either toward $u$ or toward $v$ (giving four possibilities in all). If $G$ is a bidirected graph, a $k$-flow of G is a map $\\phi:E(G)\\to \\{-(k-1),...,-1,0,1,...,k-1\\}$ with the property that at every vertex, the sum of $\\phi$ on the edges whose ends at $v$ are directed into $v$ is equal to the sum of $\\phi$ on the edges whose ends at $v$ are directed out of $v$. We say that $\\phi$ is nowhere-zero if $\\phi(e) \\neq 0$ for every $e \\in E(G)$ (see nowhere-zero flows).\n\nA bidirected Orientation of the Petersen graph\n\nFlows on bidirected graphs arise naturally as duals of local-tensions on a non-orientable surface. For more on this relationship, see [B]. Bouchet proved that the above conjecture is true with 6 replaced by 216, and exhibited a bidirected Petersen graph as above which shows that 6 is the best value possible. Zyka [Z] and independently Fouquet improved upon this result proving that the above conjecture is true with 6 replaced by 30. Khelladi [K] proved that for 4-connected graphs, the above conjecture is true with 6 replaced by 18. DeVos [D] proved that the above conjecture holds with 6 replaced by 12, and showed that every 4-edge-connected bidirected graph with a nowhere-zero integer flow also has a nowhere-zero 4-flow.\n\nBibliography:\n[B] A. Bouchet, Nowhere-Zero Integral Flows on a Bidirected Graph, J. Combinatorial Theory Ser. B 34 (1983) 279-292. MathSciNet\n\n[D] M. DeVos, Flows on Bidirected Graphs, preprint.\n\n[K] A. Khelladi, Nowhere-Zero Integral Chains and Flows in Bidirected Graphs, J. Combinatorial Theory Ser. B 43 (1987) 95-115. MathSciNet\n\n[Z] O. Zyka, Bidirected Nowhere-Zero Flows, Thesis, Charles University, Praha (1988).\n\nDiscussion links:\n- nowhere-zero flows: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0714451\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0897242\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Bouchet's 6-flow conjecture\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3219, "problem_number": "OPG-132", "title": "The three 4-flows conjecture", "statement": "Conjecture For every graph $G$ with no bridge, there exist three disjoint sets $A_1,A_2,A_3 \\subseteq E(G)$ with $A_1 \\cup A_2 \\cup A_3 = E(G)$ so that $G \\setminus A_i$ has a nowhere-zero 4-flow for $1 \\le i \\le 3$.", "background": "Source: Open Problem Garden. Original node ID: 132. URL: http://www.openproblemgarden.org/op/three_4_flows_conjecture.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/three_4_flows_conjecture\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: nowhere-zero flow\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nA graph $G$ has a nowhere-zero 4-flow if and only if there exist disjoint sets $A_1,A_2,A_3 \\subseteq E(G)$ with $A_1 \\cup A_2 \\cup A_3 = E(G)$ so that $G\\A_i$ has a nowhere-zero 2-flow for $1 \\le i \\le 3$. Thus, the above conjecture is true with room to spare for such graphs. Since every 4-edge-connected graph and every 3-edge-colorable cubic graph has a nowhere-zero 4-flow, this conjecture is automatically true for these families. As with the 5-flow conjecture or the cycle double cover conjecture, establishing this conjecture comes down to proving it for cubic graphs which are not 3-edge-colorable.\n\nThis conjecture is a consequence of the Petersen coloring conjecture, and it implies the Orientable cycle four cover conjecture. The latter implication follows immediately from the fact that every graph with a nowhere-zero 4-flow has an orientable cycle double cover. Actually, it is possible that for every graph $G$ with no cut-edge, there exist disjoint sets $A_B_1,B_2 \\subseteq E(G)$ with $A \\cup B_1 \\cup B_2 = E(G)$ and so that $G\\B_1$ and $G\\B_2$ have nowhere-zero 3-flows and $G\\A$ has a nowhere-zero 2-flow. The Petersen graph has such a decomposition ( $B_1$ and $B_2$ should be alternate edges of some 8-circuit) and so does every graph with a nowhere-zero 4-flow. If this stronger statement is true, then it would imply the oriented eight cycle four cover conjecture.\n\nBibliography:\n[J] F. Jaeger, On circular flows in graphs. Finite and infinite sets, Vol. I, II (Eger, 1981), 391--402, Colloq. Math. Soc. János Bolyai, 37, North-Holland, Amsterdam, 1984.. MathSciNet\n\nSource links:\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nDiscussion links:\n- edge-colorable: http://en.wikipedia.org/wiki/graph coloring\n- cubic: http://en.wikipedia.org/wiki/cubic graph\n- 5-flow conjecture: http://www.openproblemgarden.org/?q=op/5_flow_conjecture\n- cycle double cover conjecture: http://www.openproblemgarden.org/?q=op/cycle_double_cover_conjecture\n- Petersen coloring conjecture: http://www.openproblemgarden.org/?q=op/petersen_coloring_conjecture\n- Orientable cycle four cover conjecture: http://www.openproblemgarden.org/?q=op/cycle_double_cover_conjecture\n- Petersen graph: http://en.wikipedia.org/wiki/petersen graph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0818250\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"The three 4-flows conjecture\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3220, "problem_number": "OPG-134", "title": "A homomorphism problem for flows", "statement": "Conjecture Let $M,M'$ be abelian groups and let $B \\subseteq M$ and $B' \\subseteq M'$ satisfy $B=-B$ and $B' = -B'$. If there is a homomorphism from $Cayley(M,B)$ to $Cayley(M',B')$, then every graph with a B-flow has a B'-flow.", "background": "Source: Open Problem Garden. Original node ID: 134. URL: http://www.openproblemgarden.org/op/a_homomorphism_problem_for_flows.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_homomorphism_problem_for_flows\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: homomorphism; nowhere-zero flow; tension\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition:Let $G$ be a directed graph, Let $M$ be an abelian group, and let $B$ be a subset of $M$ such that $B=-B$. We say that a flow or a tension $\\phi:E(G) \\rightarrow M$ is a $B$-flow or a $B$-tension if the range is a subset of $B$. If $\\phi$ is a $B$-flow ( $B$-tension) of $G$ and we reverse the direction of the edge $e$, then we may obtain a new $B$-flow ( $B$-tension) by changing $\\phi(e)$ to $-\\phi(e)$. Thus, the existence of a $B$-flow or $B$-tension does not depend on the orientation, and we say that an undirected graph has a $B$-flow or a $B$-tension if some (and thus every) orientation of it admits such a map. We define the Cayley graph $Cayley(M,B)$ to be the simple graph with vertex set $M$ in which two vertices $u,v$ are joined by an edge if and only if $u-v \\in B$.\n\nIt is well known that a graph has a $B$-tension if and only if it has a homomorphism to $Cayley(M,B)$. So, if $M,M',B,B'$ are as in the conjecture and there is a homomorphism from $Cayley(M,B)$ to $Cayley(M',B')$, then every graph G with a $B$-tension has a $B'$-tension. This follows from the previous sentence and the fact that the composition of two homomorphisms is another homomorphism. In essence, the above conjecture states that the same equivalence should hold true for flows.\n\nIf $H$ and $H^*$ are directed planar dual graphs (each edge of $H^*$ crosses left to right over the corresponding edge of $H$ ), then a map $\\phi:E(H) \\to M$ is a tension if and only if the dual map $\\phi^*:E(H^*) \\to M$ ( $\\phi^*$ is given by the rule $\\phi^*(e^*)=\\phi(e)$ ) is a flow of $H^*$. Thus, planar duality exchanges flows and tensions. For two undirected planar dual graphs, $G$ and $G^*$ we have that G has a $B$-flow if and only if $G^*$ has a $B$-tension. It follows from this duality and the observation from the previous paragraph, that the above conjecture is true for planar graphs.\n\nThis conjecture is also known in the special case when $B=M\\setminus \\{0\\}$ and $B'=M'\\setminus \\{0\\}$. In this case, $Cayley(M,B)$ and $Cayley(M',B')$ are the complete graphs on $|M|$ and $|M'|$ vertices respectively, so there is a homomorphism from $Cayley(M,B)$ to $Cayley(M',B')$ if and only if $|M'|$ is greater than or equal to $|M|$. Thus, in this case the conjecture is equivalent to the assertion that every graph with a nowhere-zero $M$-flow also has a nowhere-zero $M'$-flow if $|M'|$ is at least $|M|$. This statement is true by a result of Tutte.\n\nSource links:\n- homomorphism: http://en.wikipedia.org/wiki/graph homomorphism\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 31.\n\nAttempt notes:\nTarget:\nMake progress on \"A homomorphism problem for flows\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3221, "problem_number": "OPG-135", "title": "Real roots of the flow polynomial", "statement": "Conjecture All real roots of nonzero flow polynomials are at most 4.", "background": "Source: Open Problem Garden. Original node ID: 135. URL: http://www.openproblemgarden.org/op/real_roots_of_the_flow_polynomial.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/real_roots_of_the_flow_polynomial\n- Author(s): Welsh, Dominic J. A.\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: flow polynomial; nowhere-zero flow\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nFor every graph $G$, let $P_G$ be the chromatic polynomial of $G$ and let $Q_G$ be the flow polynomial of $G$. If $G$ is loopless, then $P_G(k)>0$ for all sufficiently large integers $k$ (as $P_G(k)$ = # of k-colorings of $G$ ). It follows from Seymour's 6-flow theorem that if $G$ has no bridge, then $Q_G(k)>0$ for all integers $k>5$ (as $Q_G(k)$ = # of nowhere-zero flows in the group of integers modulo $k$ ). It is natural to ask if all real roots of these polynomials are small. For the chromatic polynomial, $P_G$, this is not the case. There exist graphs with chromatic number 3 for which $P_G$ has arbitrarily large real roots. The above conjecture asserts that the flow polynomial exhibits the opposite behavior. One word of caution, it is known that the set of roots of flow polynomials is dense in the complex plane.\n\nBibliography:\n[S] P.D. Seymour, Nowhere-Zero 6-Flows, J. Combinatorial Theory Ser. B 30 (1981) 130-135. MathSciNet\n\nDiscussion links:\n- chromatic polynomial: http://en.wikipedia.org/wiki/graph coloring\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n- nowhere-zero flows: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0615308\n\nComments:\n- December 17th, 2010 | Robert Samal | Not bounded by 5, either: A preprint by Jesper L. Jacobsen and Jesus Salas claims that there are graphs with roots of their flow polynomial being above 5. The generalized Petersen graphs G(7n,7) are claimed to have roots of flow polynomial that accumulate at approximately $5.23$.\n\nI suppose this makes the original conjecture truly false. An interesting variant, though, is to find out, if all roots of flow polynomials are $\\le 6$. (Thanks to Bojan Mohar for pointing out the paper to me.)\n- August 21st, 2009 | Gordon Royle | Welsh's conjecture is false: Welsh's conjecture on flow roots is false. In fact, many cubic graphs with reasonably large girth and enough vertices have flow roots between 4 and 5, and it is almost certain that we can find graphs with flow roots arbitrarily close to 5.\n\nHowever I strongly believe that \"All real roots of nonzero flow polynomials are at most FIVE\".\n\nSee my recent survey article \"Recent results on chromatic and flow roots of graphs and matroids, Surveys in Combinatorics 2009\" for more detail.\n\nGordon Royle\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Real roots of the flow polynomial\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3222, "problem_number": "OPG-136", "title": "Unit vector flows", "statement": "Conjecture For every graph $G$ without a bridge, there is a flow $\\phi: E(G) \\rightarrow S^2 = \\{ x \\in {\\mathbb R}^3: |x| = 1 \\}$.\n\nConjecture There exists a map $q:S^2 \\rightarrow \\{-4,-3,-2,-1,1,2,3,4\\}$ so that antipodal points of $S^2$ receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero.", "background": "Source: Open Problem Garden. Original node ID: 136. URL: http://www.openproblemgarden.org/op/unit_vector_flows.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/unit_vector_flows\n- Author(s): Jain, Kamal\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: nowhere-zero flow\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nThe main interest in these two conjectures is that together they imply Tutte's 5-flow conjecture. This follows easily from the fact that the 5-flow conjecture can be reduced to cubic graphs without bridges, and for such a graph $G$, the composition of the maps $\\phi$ and $q$ (given by the above conjectures) is a nowhere-zero 5-flow.\n\nThere are a couple of easy partial results toward the first conjecture which follow from well-known flow/cycle-cover results. First, Tutte showed that every graph with a nowhere-zero 4-flow has a list of three 2-flows $f_1,f_2,f_3: E(G) \\to \\{-1,0,1\\}$ so that every edge is in the support of exactly two of these flows. Combining these flows and normalizing appropriately gives an $S^2$-flow. Bermond, Jackson, and Jaeger [BJJ] showed that every graph with no bridge has a list of seven 2-flows so that every edge is in the support of exactly four of these flows. Combining these and normalizing appropriately gives an $S^6$-flow.\n\nIt seems likely that a graph has an $S^1$-flow if and only if it has a nowhere-zero 3-flow. The \"if\" direction of this implication isn't hard to show and the \"only if\" direction looks quite possible.\n\nA dual concept to that of a flow is that of a tension. Observe that a graph $G$ has a $S^n$ tension if and only if can be embedded in ${\\mathbb R}^{n+1}$ so that all edges are unit length line segments. Such embeddings have received some attention over the years. In particular, there is considerable interest in finding the best possible upper bound on the chromatic number of graphs which embed in ${\\mathbb R}^2$ in this manner. This is Hadwinger-Nelson problem on coloring the plane.\n\nBibliography:\n[BJJ] J.C. Bermond, B. Jackson, and F. Jaeger, Shortest covering of graphs with cycles, J. Combinatorial Theory Ser. B 35 (1983), 297-308. MRhref{0735197}\n\n[T54] W.T. Tutte, A Contribution on the Theory of Chromatic Polynomials, Canad. J. Math. 6 (1954) 80-91. MathSciNet\n\n[T66] W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combinatorial Theory 1 (1966) 15-50. MathSciNet\n\nSource links:\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n\nDiscussion links:\n- Tutte's 5-flow conjecture: http://www.openproblemgarden.org/?q=op/5_flow_conjecture\n- cubic: http://en.wikipedia.org/wiki/cubic graph\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0061366\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0194363\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"Unit vector flows\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3223, "problem_number": "OPG-323", "title": "Antichains in the cycle continuous order", "statement": "If $G$, $H$ are graphs, a function $f: E(G) \\rightarrow E(H)$ is called cycle-continuous if the pre-image of every element of the (binary) cycle space of $H$ is a member of the cycle space of $G$.\n\nProblem Does there exist an infinite set of graphs $\\{G_1,G_2,\\ldots \\}$ so that there is no cycle continuous mapping between $G_i$ and $G_j$ whenever $i \\neq j$?", "background": "Source: Open Problem Garden. Original node ID: 323. URL: http://www.openproblemgarden.org/op/antichains_in_the_cycle_continuous_order.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/antichains_in_the_cycle_continuous_order\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: antichain; cycle; poset\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 12th, 2007 by mdevos\n\nProblem-page discussion:\nThe definition of a cycle-continuous mapping is based on some work of Jaeger, and the most interesting question on this subject is undoubtedly Jaeger's Petersen coloring conjecture.\n\nLet us define a relation on the set of all finite graphs with at least one edge by the rule $G>H$ if there is a cycle-continuous mapping from $G$ to $H$. It is not difficult to verify that $>$ is a quasi order (reflexive and transitive). In this order, every Eulerian graph dominates every other graph, and every graph with a cut edge is dominated by every other graph.\n\nLet $A_i$ be the graph on two vertices with $i$ parallel edges. Then $A_3 < A_5 < A_7 <...$ with all the inequalities strict, so this sequence is an infinite chain. Very little else seems to be known about this order. In particular, the problem highlighted above - does there exist an infinite antichain? remains open.\n\nDiscussion links:\n- Petersen coloring conjecture: http://www.openproblemgarden.org/?q=op/petersen_coloring_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"Antichains in the cycle continuous order\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3224, "problem_number": "OPG-59994", "title": "Circular flow number of regular class 1 graphs", "statement": "A nowhere-zero $r$-flow $(D(G),\\phi)$ on $G$ is an orientation $D$ of $G$ together with a function $\\phi$ from the edge set of $G$ into the real numbers such that $1 \\leq |\\phi(e)| \\leq r-1$, for all $e \\in E(G)$, and $\\sum_{e \\in E^+(v)}\\phi(e) = \\sum_{e \\in E^-(v)}\\phi(e), \\textrm{ for all } v \\in V(G)$. The circular flow number of $G$ is inf $\\{ r | G$ has a nowhere-zero $r$-flow $\\}$, and it is denoted by $F_c(G)$.\n\nA graph with maximum vertex degree $k$ is a class 1 graph if its edge chromatic number is $k$.\n\nConjecture Let $t \\geq 1$ be an integer and $G$ a $(2t+1)$-regular graph. If $G$ is a class 1 graph, then $F_c(G) \\leq 2 + \\frac{2}{t}$.", "background": "Source: Open Problem Garden. Original node ID: 59994. URL: http://www.openproblemgarden.org/op/circular_flow_number_of_regular_class_1_graphs.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/circular_flow_number_of_regular_class_1_graphs\n- Author(s): Steffen, Eckhard\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: nowhere-zero flow, edge-colorings, regular graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 5th, 2015 by Eckhard Steffen\n\nProblem-page discussion:\nThe conjecture is true for $t=1$, i.e. for cubic graphs. It says, that the circular flow number of $(2t+1)$-regular class 1 graphs is bounded by the circular flow number of the complete graph on $2t+2$ vertices.\n\nBibliography:\n[ES_2001] E. Steffen, Circular flow numbers of regular multigraphs, J. Graph Theory 36, 24 – 34 (2001)\n\n*[ES_2015] E. Steffen, Edge-colorings and circular flow numbers on regular graphs, J. Graph Theory 79, 1–7, 2015\n\nRelated:\nRelated problems\n(2 + epsilon)-flow conjecture\nJaeger's modular orientation conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 17.\n\nAttempt notes:\nTarget:\nMake progress on \"Circular flow number of regular class 1 graphs\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3225, "problem_number": "OPG-171", "title": "Strong colorability", "statement": "Let $r$ be a positive integer. We say that a graph $G$ is strongly $r$-colorable if for every partition of the vertices to sets of size at most $r$ there is a proper $r$-coloring of $G$ in which the vertices in each set of the partition have distinct colors.\n\nConjecture If $\\Delta$ is the maximal degree of a graph $G$, then $G$ is strongly $2 \\Delta$-colorable.", "background": "Source: Open Problem Garden. Original node ID: 171. URL: http://www.openproblemgarden.org/op/strong_colorability.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/strong_colorability\n- Author(s): Aharoni, Ron; Alon, Noga; Haxell, Penny E.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: strong coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 27th, 2007 by berger\n\nProblem-page discussion:\nHaxell proved that if $\\Delta$ is the maximal degree of a graph $G$, then $G$ is strongly $3 \\Delta - 1$-colorable. She later proved that the strong chromatic number $\\chi_S$ is at most $(2.75+\\epsilon)\\Delta$ for sufficiently large $\\Delta$ depending on $\\epsilon$. Aharoni, Berger, and Ziv proved the fractional relaxation.\n\nComments:\n- November 19th, 2009 | Andrew King | Recent progress: Haxell proved that $(2.75 + \\epsilon)\\Delta$ is sufficient for sufficiently large $\\Delta$ depending on $\\epsilon$. (JGT, 2008)\n\nAharoni, Berger, and Ziv proved the fractional relaxation, i.e. that with partition cliques of size $2\\Delta$ we have a fractional $2\\Delta$ colouring. (Combinatorica, 2007)\n- July 24th, 2007 | Anonymous | Problem Solved!: See \"On the Strong Chromatic Number of Graphs\" by Maria Axenovich and Ryan Martin (2006)\n- July 25th, 2007 | Anonymous | only partly solved: The paper cited (available at http://orion.math.iastate.edu/axenovic/Papers/Martin_Strong.pdf) only resolves the above conjecture for graphs G which have maximum degree at least |V(G)|/6.\n- September 3rd, 2007 | Anonymous | That theorem has been proved: That theorem has been proved in an even older paper by another set of authors.\n\nAfter a quick search I found that paper at http://abel.math.umu.se/~klasm/Uppsatser/factor.pdf\n- September 3rd, 2007 | mdevos | still only a partial solution: Once again, the paper cited offers only a partial solution. Quoting from the paper, \"From Theorem 1.1, we conclude that the strong chromatic number can be bounded by $2\\Delta(G)$ if $|V(G)| \\le 6\\Delta(G)$. This result should not be compared with the more complete and difficult result of Alon.\" Here, the difficult result of Alon is the theorem that there exists a fixed constant $K$ so that every graph of maximum degree $\\Delta$ is strongly $K \\Delta$-colorable.. precisely the result which is sharpened by this conjecture.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Strong colorability\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3226, "problem_number": "OPG-335", "title": "Reed's omega, delta, and chi conjecture", "statement": "For a graph $G$, we define $\\Delta(G)$ to be the maximum degree, $\\omega(G)$ to be the size of the largest clique subgraph, and $\\chi(G)$ to be the chromatic number of $G$.\n\nConjecture $\\chi(G) \\le \\ceil{\\frac{1}{2}(\\Delta(G)+1) + \\frac{1}{2}\\omega(G)}$ for every graph $G$.", "background": "Source: Open Problem Garden. Original node ID: 335. URL: http://www.openproblemgarden.org/op/reeds_omega_delta_and_chi_conjecture.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/reeds_omega_delta_and_chi_conjecture\n- Author(s): Reed, Bruce A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 22nd, 2007 by mdevos\n\nProblem-page discussion:\nPerhaps the two most trivial bounds on $\\chi(G)$ are $\\chi(G) \\ge \\omega(G)$ and $\\chi(G) \\le \\Delta(G) + 1$. The above conjecture roughly asserts that the (rounded-up) average of $\\Delta(G)+1$ and $\\omega(G)$ should again be an upper bound on $\\chi(G)$.\n\nThe conjecture is easy to verify when $\\omega(G)$ is very large. It is trivial when $\\omega(G) \\ge \\Delta(G)$, and it follows from Brook's theorem if $\\omega(G) = \\Delta(G)-1$. On the other hand, if $\\omega(G) = 2$, so $G$ is triangle free, then the conjecture is also true for $\\Delta$ sufficiently large. Indeed, Johannsen proved the much stronger fact that there exists a fixed constant $c$ so that $\\chi(G) \\le \\frac{c \\Delta(G)}{\\log \\Delta(G)}$ for every triangle free graph $G$.\n\nReed showed that the conjecture holds when $\\Delta(G) = |V(G)| - 1$ by way of matching theory. More interestingly, he proved (using probabilistc methods) that the conjecture is true provided that $\\Delta$ is sufficiently large, and $\\omega$ is sufficiently close to $\\Delta$. More precisely, he proves the following:\n\nTheorem There exists a fixed constant $\\Delta_0$ such that for every $\\Delta \\ge \\Delta_0$, if $G$ is a graph of maximum degree $\\Delta$ with no clique of size $>k$ for some $k \\ge (1 - \\frac{1}{70000000}) \\Delta$ then $\\chi(G) \\le \\frac{\\Delta + 1 + k}{2}$.\n\nIt is known that the conjecture is true fractionally (that is with $\\chi(G)$ replaced by $\\chi_f(G)$, the fractional chromatic number of~ $G$ ).\n\nBibliography:\n*[R] B. Reed, $\\omega, \\Delta$, and $\\chi$, J. Graph Theory 27 (1998) 177-212.\n\nSource links:\n- clique: http://en.wikipedia.org/wiki/clique (graph theory)\n\nDiscussion links:\n- fractional chromatic number: http://en.wikipedia.org/wiki/fractional chromatic number\n\nComments:\n- September 18th, 2007 | Anonymous | The statement of the: The statement of the conjecture is slightly incorrect. Instead of the +1 at the end, there should simply be a round-up. The conjecture is true for line graphs and quasi-line graphs, graphs with independence number 2, and any graph on I believe 12 vertices.\n\nAn outright proof of the result for triangle-free graphs would be very nice. Lovasz' result on splitting graphs is not quite enough in this case.\n- September 24th, 2007 | Robert Samal | changed: Thanks for the comment. I changed the statement to the conjectured (slightly stronger) version with round-up.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 24.\n\nAttempt notes:\nTarget:\nMake progress on \"Reed's omega, delta, and chi conjecture\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3227, "problem_number": "OPG-401", "title": "Circular coloring triangle-free subcubic planar graphs", "statement": "Problem Does every triangle-free planar graph of maximum degree three have circular chromatic number at most $\\frac{20}{7}$?", "background": "Source: Open Problem Garden. Original node ID: 401. URL: http://www.openproblemgarden.org/op/circular_chromatic_number_of_triangle_free_planar_graphs.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/circular_chromatic_number_of_triangle_free_planar_graphs\n- Author(s): Ghebleh, Mohammad; Zhu, Xuding\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: circular coloring; planar graph; triangle free\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 20th, 2007 by mdevos\n\nProblem-page discussion:\nThroughout, we let $\\chi_c(G)$ denote the circular chromatic number of the graph $G$.\n\nA well-known Question of Nesetril asks if $\\chi_c(G) \\le \\frac{5}{2}$ for all cubic graphs $G$ of sufficiently high girth. A conjecture of Jaeger asserts that $\\chi_c(G) \\le 2 + \\frac{1}{k}$ for every planar graph $G$ of girth $4k+1$. There are numerous partial results on these problems, and there are many interesting questions concerning the circular chromatic numbers of restricted families of graphs. Here we are restricted to planar graphs of girth $\\ge 4$ with maximum degree $\\le 3$. The dodecahedron lives in this class and has $\\chi_c = \\frac{20}{7}$. It remains unclear if anyone else in this class might have $\\chi_c$ larger.\n\nA related conjecture of X. Zhu asserts that for every triangle-free planar graph $G$ with $\\Delta(G)\\le 4$ and $|V(G)|<3k$ one has $\\chi_c(G)\\le 3-1/k$.\n\nRelated:\nRelated problems\nPentagon problem\nJaeger's modular orientation conjecture\n\nDiscussion links:\n- circular chromatic number: http://en.wikipedia.org/wiki/circular coloring\n- Question of Nesetril: http://www.openproblemgarden.org/?q=node/167\n- conjecture of Jaeger: http://www.openproblemgarden.org/?q=node/130\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Circular coloring triangle-free subcubic planar graphs\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3228, "problem_number": "OPG-494", "title": "Oriented chromatic number of planar graphs", "statement": "An oriented colouring of an oriented graph is assignment $c$ of colours to the vertices such that no two arcs receive ordered pairs of colours $(c_1,c_2)$ and $(c_2,c_1)$. It is equivalent to a homomorphism of the digraph onto some tournament of order $k$.\n\nProblem What is the maximal possible oriented chromatic number of an oriented planar graph?", "background": "Source: Open Problem Garden. Original node ID: 494. URL: http://www.openproblemgarden.org/op/oriented_chromatic_number_of_planar_graphs.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/oriented_chromatic_number_of_planar_graphs\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: oriented coloring; oriented graph; planar graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 4th, 2007 by Robert Samal\n\nProblem-page discussion:\nRaspaud and Sopena [RS] showed using Borodin's result about acyclic chromatic number of planar graphs, that every planar oriented graph has oriented chromatic number at most 80. (Their motivation came from a work of Courcelle [C] concerning the monadic second-order logic of graphs. That, however, deals with a stronger variant of coloring.)\n\nOn the other hand, Marshall [M] showed that there is an oriented planar graph with oriented chromatic number at least~17.\n\nBibliography:\n[C] B. Courcelle, The monadic second order logic of graphs VI: On several representations of graphs by relational structures, Discrete Appl. Math. 54 ( 1994),\n\n[M] T. H. Marshall. On $\\cal P$-universal graphs. Research Report 2001-510, KAM-DIMATIA Series, 2001.\n\n*[RS] A. Raspaud and E. Sopena. Good and semi-strong colorings of oriented planar graphs. Inform. Process. Lett., 51(4):171–174, 1994. MathSciNet\n\nSource links:\n- oriented chromatic number: http://en.wikipedia.org/wiki/Oriented_coloring\n\nDiscussion links:\n- acyclic chromatic number: http://en.wikipedia.org/wiki/acyclic coloring\n\nBibliography links:\n- On $\\cal P$-universal graphs: http://kam.mff.cuni.cz/%7Ekamserie/serie/clanky/2001/s510.ps\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1294309\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Oriented chromatic number of planar graphs\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3229, "problem_number": "OPG-550", "title": "Coloring and immersion", "statement": "Conjecture For every positive integer $t$, every (loopless) graph $G$ with $\\chi(G) \\ge t$ immerses $K_t$.", "background": "Source: Open Problem Garden. Original node ID: 550. URL: http://www.openproblemgarden.org/op/coloring_and_immersion.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/coloring_and_immersion\n- Author(s): Abu-Khzam, Faisal N.; Langston, Michael A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: coloring; complete graph; immersion\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: September 4th, 2007 by mdevos\n\nProblem-page discussion:\nLet $G$ be a graph and let $uv, vw \\in E(G)$. The operation of deleting the edges $uv$ and $vw$ and then adding a new edge between $v$ and $w$ is called a split. We say that a graph $G$ immerses a graph $H$ if a graph isomorphic to $H$ may be obtained from $G$ by repeatedly making splits and deleting vertices and edges.\n\nThe graph containment relations of minor and topological minor have been thoroughly studied with respect to graph coloring. In particular, there are two famous conjectures: Hajos conjectured that every graph with chromatic number $\\ge t$ contains a subdivision of the complete graph $K_t$ as a subgraph. Hadwiger conjectured that every graph with chromatic number $\\ge t$ contains $K_t$ as a minor. While Hajos' Conjecture is false for $t \\ge 8$ (indeed it is actually false on average), Hadwiger's Conjecture remains open, and is one of the outstanding problems in Graph Theory.\n\nOn the other hand, the relationship between graph coloring and immersions seems to have been largely ignored until Abu-Khzam and Langston made the above conjecture. In addition to formulating this conjecture, they proved it for $t \\le 4$ and showed that a minimal counterexample to it must be 4-connected and $t$-edge-connected. Recently, DeVos, Kawarabayashi, Mohar, and Okamura have resolved the conjecture for $t \\le 7$.\n\nBibliography:\n* Faisal N. Abu-Khzam and Michael A. Langston, Graph Coloring and the Immersion Order\n\nBibliography links:\n- Graph Coloring and the Immersion Order: http://www.cs.utk.edu/%7Elangston/projects/papers/conjecture.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Coloring and immersion\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3230, "problem_number": "OPG-616", "title": "Coloring the Odd Distance Graph", "statement": "The Odd Distance Graph, denoted ${\\mathcal O}$, is the graph with vertex set ${\\mathbb R}^2$ and two points adjacent if the distance between them is an odd integer.\n\nQuestion Is $\\chi({\\mathcal O}) = \\infty$?", "background": "Source: Open Problem Garden. Original node ID: 616. URL: http://www.openproblemgarden.org/op/coloring_the_odd_distance_graph.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/coloring_the_odd_distance_graph\n- Author(s): Rosenfeld, Moshe\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: coloring; geometric graph; odd distance\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 3rd, 2007 by mdevos\n\nProblem-page discussion:\nThis question is a relative of the famous problem about coloring the Unit Distance Graph (the graph on ${\\mathbb R}^2$ where two points are adjacent if the distance between them is 1). See Moshe's online lecture Famous and lesser known problems in “elementary” combinatorial geometry and number theory at time 15:20 for a nice introduction.\n\nPerhaps the first property of ${\\mathcal O}$ to determine is the size of the largest complete subgraph (were ${\\mathcal O}$ to contain arbitrarily large complete subgraphs, its chromatic number would be $\\infty$ ). It is obvious that ${\\mathcal O}$ contains triangles, but perhaps surprisingly, it does not contain a complete subgraph on four vertices. In other words, there do not exist four points in ${\\mathbb R}^2$ so that all pairwise distances are odd. This was a problem on the Putnam Exam in 1993, and is proved by Rosenfeld in [R1] and [R2].\n\nA natural strengthening of the above question is to ask if there exists a proper $n$-coloring $f: V({\\mathcal O}) \\rightarrow \\{1,2,\\ldots,n\\}$ so that $f^{-1}(\\{i\\})$ is a measurable set for every $i$. Such colorings are called measurable colorings, and interestingly, the Odd Distance Graph has no finite measurable coloring. This follows from immediately from a theorem of Furstenberg, Katznelson and Weiss [FKW] which asserts that for every measurable subset $A \\subseteq {\\mathbb R}^2$ with positive upper density, there exists a number $r$ so that $A$ contains a pair of points at distance $r'$ for every $r' > r$. This theorem has a number of independent proofs, see also Falconer and Marstrand [FM], Bourgain [Bo], and Bukh [Bu].\n\nAll that seems to be known about the (usual) chromatic number of ${\\mathcal O}$ is that $\\chi({\\mathcal O}) \\ge 5$.\n\nBibliography:\n[Bo] J. Bourgain, A Szemerédi type theorem for sets of positive density in $R\\sp k$. Israel J. Math. 54 (1986), no. 3, 307--316. MathSciNet\n\n[Bu] B. Bukh, Measurable sets with excluded distances.\n\n[FM] K. J. Falconer and J. M. Marstrand, Plane sets with positive density at infinity contain all large distances. Bull. London Math. Soc. 18 (1986), no. 5, 471--474. MathSciNet\n\n[FKM] H. Furstenberg, Y. Katznelson, and B. Weiss, Ergodic theory and configurations in sets of positive density. Mathematics of Ramsey theory, 184--198, Algorithms Combin., 5, Springer, Berlin, 1990. MathSciNet\n\n[R1] M. Rosenfeld, Odd integral distances among points in the plane. Geombinatorics 5 (1996), no. 4, 156--159. MathSciNet\n\n[R2] M. Rosenfeld Famous and lesser known problems in “elementary” combinatorial geometry and number theory (video lecture - time 15:20)\n\nDiscussion links:\n- Famous and lesser known problems in “elementary” combinatorial geometry and number theory: http://videolectures.net/sicgt07_rosenfeld_falkp\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0853455\n- Measurable sets with excluded distances: http://arxiv.org/pdf/math/0703856\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0847986\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1083601\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1380145\n- Famous and lesser known problems in “elementary” combinatorial geometry and number theory: http://videolectures.net/sicgt07_rosenfeld_falkp\n\nComments:\n- March 19th, 2012 | Anonymous | The Odd-Distance Graph: In the book Research Problems in Discrete Geometry, on page 252, it is stated that every K_4 free graph is a subgraph of the odd distance graph. We just proved that W_5, the five wheel is not a subgraph of the odd distance graph. I further believe that there are triangle free graphs that are not subgraphs of the odd distance graph, and even graphs with large girth.\n- May 8th, 2008 | JSteinhardt | Flaw: Actually, it appears there is a flaw with the below proof. The spectral bound on the chromatic number assumes a measurable coloring, as we want to say the following:\n\n$$2(\\chi-1)\\lambda_{\\min}||f||^2 & = & \\sum_{i,j=1}^{\\chi} \\lambda_{\\min}||f_i-f_j||^2$$$$\\leq \\sum_{i,j=1}^{\\chi} \\langle f_i-f_j, B(f_i-f_j) \\rangle$$$$= \\sum_{i,j=1}^{\\chi} \\langle f_i,Bf_i \\rangle + \\langle f_j,Bf_j \\rangle - 2\\langle f_i,Bf_j \\rangle$$$$= -2\\sum_{i,j=1}^{\\chi} \\langle f_i,Bf_j \\rangle$$$$= -2\\langle\\sum_{i=1}^{\\chi} f_i,B(\\sum_{i=1}^{\\chi} f_i\\rangle$$$$= -2\\langle f,Bf \\rangle$$$$= -2\\lambda ||f||^2$$\n\nbut this assumes that each $f_i$ is Lesbegue integrable, which in turn requires measurable coloring classes.\n- May 4th, 2008 | JSteinhardt | Solution: I believe I have a solution. I will sketch it here. (Sorry, it's broken up into three posts because I cannot figure out how to post something more than 1000 characters...but I have seen longer solutions posted elsewhere so I assume it's okay; if not, I apologize.)\n\nConsider the operator $B_{a}: L^2(R^2) \\to L^2(R^2)$ defined by\n\n$$(B_{a}f)(x,y) = \\int_{-\\pi}^{\\pi} \\sum_{k=0}^{\\infty} a^{-k} f(x+(2k+1)\\cos(t),y+(2k+1)\\sin(t)) dt$$\n\nThis is in some sense a weighting of the adjacency operator. We can then prove the result (well-known for finite graphs) that $\\chi(O) \\geq 1-\\frac{\\lambda_{\\max}}{\\lambda_{\\min}}$, where $\\lambda_{\\max},\\lambda{\\min}$ are the sup and inf of the spectrum of $B_{a}$.\n\nWe note that the eigenfunctions of $B_{a}$ are simply the exponential maps $f_{(r,s)}(x,y) = e^{i(rx+sy)}$.\n- May 4th, 2008 | JSteinhardt | Solution (continued): We see that the eigenvalue of the eigenfunction $f_{(r,s)}$ is given by\n\n$$\\lambda_{(r,s)} = \\int_{-\\pi}^{\\pi} \\sum_{k=0}^{\\infty} a^{-k} e^{i(2k+1)(r\\cos(t)+s\\sin(t))} dt = \\int_{-\\pi}^{\\pi} \\sum_{k=0}^{\\infty} a^{-k} e^{i(2k+1)\\sqrt{r^2+s^2}\\cos(t+\\phi)} dt$$\n\nfor an appropriately chosen $\\phi$. Thus we need only actually consider $\\lambda_{(r,0)}$, which we from now on denote $\\lambda(r)$. Then some calculation gives us that the integral is\n\n$$\\int_{-\\pi}^{\\pi} \\frac{a(a-1)\\cos(r\\cos(t))}{(a-1)^2+4a\\sin^2(r\\cos(t))} dt$$\n\nWe can show that (and this will suffice) that when $a$ is near $1$,\n\n$$\\int_{0}^{\\frac{\\pi}{2}} \\frac{(a-1)\\cos(r\\cos(t))}{(a-1)^2+4a\\sin^2(r\\cos(t))} dt \\geq -4(a-1)^{-\\frac{3}{4}}-\\frac{\\pi}{2}$$\n- May 4th, 2008 | JSteinhardt | Solution (final part): Let $h$ be the function we are integrating. Let $R_k$ denote the region for which $|h(t)| \\geq 1$ and that contains the value of $t$ where $\\cos(t) = \\frac{k\\pi}{r}$. Then we note that $|\\int_{R_k} h(x) dx| > |\\int_{R_{k-1}} h(x) dx|$ and the sines of the integral alternate, so we can just calculate the first one and everything else will be bounded (in particular by $\\frac{\\pi}{2}$ ). With a bit of Taylor approximation, we can bound the size of each $R_k$ by $\\frac{4\\sqrt[4]{a-1}}{\\sqrt{r}}$, and noting that $h$ is always positive for $r \\leq \\frac{\\pi}{2}$, we can replace the $\\sqrt{r}$ with $1$ and then bound $h$ by $\\frac{1}{a-1}$. This gives us the bound we claimed above and we are done.\n\nJacob Steinhardt\n- October 31st, 2007 | Anonymous | Circular chromatic number of the odd distance graph: The proof for $\\chi(G)\\geq 5$ has been recently extended to $\\chi_c(G)\\geq 5$, which implies the previous result, where $\\chi_c$ is the circular chromatic number.\n\nNicolas Roussel.\n- November 7th, 2007 | Anonymous | Correction of previous comment: The proof is actually for $\\chi_c(G)\\geq 4.5$.\n\nNR.\n- November 24th, 2007 | Anonymous | Subgraph construction: For any rational $r\\in[4,4.5)$, there is a subgraph $H_r$ of the odd-distance graph with $\\chi_c(H_r)=r$\n\n[1] Pan Zhi-Shi, Roussel Nicolas, Subgraphs of the odd-distance graph with given circular chromatic number, manuscript\n\nNR.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 67.\n\nAttempt notes:\nTarget:\nMake progress on \"Coloring the Odd Distance Graph\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3231, "problem_number": "OPG-771", "title": "Partial List Coloring", "statement": "Conjecture Let $G$ be a simple graph with $n$ vertices and list chromatic number $\\chi_\\ell(G)$. Suppose that $0\\leq t\\leq \\chi_\\ell$ and each vertex of $G$ is assigned a list of $t$ colors. Then at least $\\frac{tn}{\\chi_\\ell(G)}$ vertices of $G$ can be colored from these lists.", "background": "Source: Open Problem Garden. Original node ID: 771. URL: http://www.openproblemgarden.org/op/partial_list_coloring.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partial_list_coloring\n- Author(s): Albertson, Michael O.; Grossman, Sara; Haas, Ruth\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: list assignment; list coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 5th, 2008 by Iradmusa\n\nProblem-page discussion:\nAlbertson, Grossman, and Haas introduce this interesting question in [AGH], and prove some partial results. For instance, they show that under the above assumptions, at least $(1 - (\\frac{ \\chi(G) - 1}{\\chi(G)} )^t) \\cdot n$ vertices of $G$ can be colored from the lists.\n\nBibliography:\n*[AGH] M. Albertson, S. Grossman and R. Haas, Partial list colouring, Discrete Math., 214(2000), pp. 235-240.\n\nBibliography links:\n- Partial list colouring: http://maven.smith.edu/%7Ealbertson/ruth.ps\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Partial List Coloring\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3232, "problem_number": "OPG-788", "title": "Partial List Coloring", "statement": "Let $G$ be a simple graph, and for every list assignment $\\mathcal{L}$ let $\\lambda_{\\mathcal{L}}$ be the maximum number of vertices of $G$ which are colorable with respect to $\\mathcal{L}$. Define $\\lambda_t = \\min{ \\lambda_{\\mathcal{L}} }$, where the minimum is taken over all list assignments $\\mathcal{L}$ with $|\\mathcal{L}| = t$ for all $v \\in V(G)$.\n\nConjecture [2] Let $G$ be a graph with list chromatic number $\\chi_\\ell$ and $1\\leq r\\leq s\\leq \\chi_\\ell$. Then\n$$\n\\frac{\\lambda_r}{r}\\geq\\frac{\\lambda_s}{s}.\n$$", "background": "Source: Open Problem Garden. Original node ID: 788. URL: http://www.openproblemgarden.org/op/partial_list_coloring_0.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partial_list_coloring_0\n- Author(s): Iradmusa, Moharram\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: list assignment; list coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 12th, 2008 by Iradmusa\n\nProblem-page discussion:\nAs you see this conjecture in the special case $s=\\chi_\\ell$, is the conjecture of Albertson, Grossman and Haas [1]: $\\lambda_t\\geq\\frac{tn}{\\chi_\\ell}$ for any $0\\leq t\\leq \\chi_\\ell$.\n\nBibliography:\n[1] M. Albertson, S. Grossman and R. Haas, Partial list colouring, Discrete Math., 214(2000), pp. 235-240.\n\n[2] Moharram N. Iradmusa, A Note on Partial List Colorings, Australasian Journal of Combinatorics, Vol.46, 2010, $19-24$.\n\nRelated:\nRelated problems\nPartial List Coloring\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Partial List Coloring\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3233, "problem_number": "OPG-806", "title": "Hedetniemi's Conjecture", "statement": "Conjecture If $G,H$ are simple finite graphs, then $\\chi(G \\times H) = \\min \\{ \\chi(G), \\chi(H) \\}$.\n\nHere $G \\times H$ is the tensor product (also called the direct or categorical product) of $G$ and $H$.", "background": "Source: Open Problem Garden. Original node ID: 806. URL: http://www.openproblemgarden.org/op/hedetniemis_conjecture.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hedetniemis_conjecture\n- Author(s): Hedetniemi, Stephen T.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: categorical product; coloring; homomorphism; tensor product\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 25th, 2008 by mdevos\n\nProblem-page discussion:\nThis beautiful and seemingly innocent conjecture asserts a deep and important property of graph coloring. It is undoubtedly one of the most significant unsolved problems in graph coloring and graph homomorphisms.\n\nWe write $G \\rightarrow H$ if there is a homomorphism from $G$ to $H$. The graph $G \\times H$ has a two natural projection maps (projecting onto either the first or second coordinate), and these maps are homomorphisms to $G$ and to $H$. So, in short, $G \\times H \\rightarrow G$ and $G \\times H \\rightarrow H$. A graph is $n$-colorable if and only if it has a homomorphism to $K_n$. Combining this with the transitivity of $\\rightarrow$ we find that $\\chi(G \\times H) \\le \\min\\{ \\chi(G), \\chi(H) \\}$ (indeed, if $\\chi(G) = n$, then $G \\times H \\rightarrow G$ and $G \\rightarrow K_n$, so $G \\times H \\rightarrow K_n$- equivalently, $G \\times H$ is $n$-colorable). So, the hard direction of Hedetniemi's Conjecture is to prove that $\\chi(G \\times H) \\ge \\min \\{ \\chi(G), \\chi(H) \\}$.\n\nLet's define $P(n)$ to be the proposition that $\\chi(G \\times H) \\ge n$ whenever $\\chi(G) \\ge n$ and $\\chi(H) \\ge n$. Then the above conjecture is equivalent to the statement that $P(n)$ holds for every positive integer $n$. Now $P(1)$ holds trivially and $P(2)$ follows from the observation that the product of two graphs each of which contains an edge is a graph which contains an edge. The next case is quite easy too, if $\\chi(G) \\ge 3$ and $\\chi(H) \\ge 3$, then both $G$ and $H$ contain an odd cycle. Since the product of two odd cycles contains an odd cycle, this shows $\\chi(G \\times H) \\ge 3$. The next case up, $P(4)$ was proved by El-Zahar and Sauer by way of a beautiful argument. It is open for all higher values.\n\nA key tool in the proof of El-Zahar and Sauer is the use of exponential graphs. For any pair of graphs $G, H$ the exponential graph $G^H$ is a graph whose vertex set consists of all mappings $f: V(H) \\rightarrow V(G)$. Two vertices $f,g$ are adjacent if $f(x)g(y)$ is an edge of $G$ whenever $xy$ is an edge of $H$. It is easy to see the relevance of $K_n^G$ to this problem. If we have an $n$-coloring $f$ of $G \\times H$, then for every vertex $x \\in V(H)$, there is a mapping $f_x: V(G) \\rightarrow V(K_n)$ given by $f_x(v) = f(v,x)$. This associates each $x \\in V(H)$ with a vertex in $K_n^G$. Now it is easy to verify that whenever $x,y$ are adjacent vertices in $H$, the maps $f_x$ and $f_y$ are adjacent in $K_n^G$. Rather more surprisingly, Hedetniemi's conjecture may be reformulated as follows:\n\nConjecture (version 2 of Hedetniemi) If $\\chi(G) > n$, then $K_n^G$ is $n$-colorable.\n\nThe following conjecture asserts that Hedetniemi's conjecture still holds with circular chromatic number instead of the usual chromatic number. Here $\\chi_c(G)$ is the circular chromatic number of $G$. Since $\\chi(G) = \\lceil \\chi_c(G) \\rceil$ this is a generalization of the original conjecture.\n\nConjecture (Zhu) If $G$ and $H$ are finite simple graphs then $\\chi_c(G \\times H) = \\min\\{ \\chi_c(G), \\chi_c(H) \\}$.\n\nA graph $G$ has circular chromatic number $\\frac{n}{k}$ for positive integers $n,k$ if and only if $G$ has a homomorphism to the graph $K_{n/k}$. This is a graph whose vertex set consists of $n$ vertices cyclically ordered, with two vertices adjacent if they are distance $\\ge k$ apart in the cyclic ordering. So again, we may state this conjecture in terms of homomorphisms to graphs of the form $K_{n/k}$. More generally, let us call a graph $K$ multiplicative if $G \\times H \\rightarrow K$ implies either $G \\rightarrow K$ or $H \\rightarrow K$. Now Hedetniemi's conjecture asserts that every $K_n$ is multiplicative and Zhu's conjecture asserts that every $K_{n/k}$ is multiplicative. With this terminology, El-Zahar and Sauer proved that $K_3$ is multiplicative. A clever generalization of their argument due to Haggkvist, Hell, Miller and Neumann Lara showed that every odd cycle is multiplicative. Recently, Tardif bootstrapped this theorem with the help of a couple of interesting operators on the category of graphs to prove the $K_{n/k}$ is multiplicative whenever $n/k < 4$. Ignoring trivial cases and equivalences, these are essentially the only graphs known to be multiplicative.\n\nIt might be tempting to hope that all graphs are multiplicative, but this is false. To construct a non-multiplicative graph, take two graphs $G,H$ with the property that $G \\not\\rightarrow H$ and $H \\not\\rightarrow G$ (for instance $K_3$ and the Grotzsch Graph). Now $G \\times H$ is not multiplicative since $G \\not\\rightarrow G \\times H$ and $H \\not\\rightarrow G \\times H$, but $G \\times H \\rightarrow G \\times H$. It seems that there is no general conjecture as to what graphs are multiplicative. Some other Cayley graphs look like reasonable candidates to me (M. DeVos), but I haven't any evidence one way or the other.\n\nPoljak and Rodl defined the function $f(n) = \\min \\{ \\chi(G \\times H): \\chi(G) = n = \\chi(H) \\}$. So, Hedetniemi's conjecture is equivalent to $f(n) = n$. Using an interesing inequality relating the chromatic number of a digraph $D$ to the chromatic number of a type of line graph of $D$, they were able to prove the following quite surprising result: Either $f$ is bounded by $9$ or $\\lim_{n \\rightarrow \\infty} f(n) = \\infty$.\n\nThere are a number of interesting partial results not mentioned here, and the reader is encouraged to see the survey article by Zhu.\n\nSource links:\n- tensor product: http://en.wikipedia.org/wiki/tensor product of graphs\n\nDiscussion links:\n- homomorphism: http://en.wikipedia.org/wiki/graph homomorphism\n- circular chromatic number: http://en.wikipedia.org/wiki/circular coloring\n\nComments:\n- March 1st, 2020 | Anonymous | Yaroslav Shitov made some: Yaroslav Shitov made some significant breakthroughs in this area - https://arxiv.org/abs/1905.02167.\n- February 27th, 2013 | Anonymous | Lovasz Theta: I believe that Robert Samal has conjectured a version of this for the Lovasz $\\vartheta$ function, i.e. that\n$$\n\\bar{\\vartheta}(G \\times H) = \\min\\{\\bar{\\vartheta}(G), \\bar{\\vartheta}(H)\\}\n$$\n where $\\bar{\\vartheta}(G):= \\vartheta(\\overline{G})$. I can't find it in the Garden, but it is in this presentation by Samal: http://iuuk.mff.cuni.cz/research/cmi/cmi-I-Samal.pdf\n- July 23rd, 2013 | Robert Samal | Re: Lovasz Theta: In fact, for the Lovasz $\\vartheta$ function (of the complement of the graph) it is a theorem, see arXiv:1305.5545.\n- October 18th, 2011 | Anonymous | Fractional version is true: It is probably worth mentioning that Zhu recently proved the fractional version of this conjecture: that $\\chi_f(G \\times H) = \\min\\{\\chi_f(G),\\chi_f(H)\\}$.\n\nXuding Zhu, The fractional version of Hedetniemi’s conjecture is true, European Journal of Combinatorics, Volume 32, Issue 7, October 2011, Pages 1168-1175, ISSN 0195-6698, 10.1016/j.ejc.2011.03.004. (http://www.sciencedirect.com/science/article/pii/S0195669811000552)\n- November 8th, 2010 | Jon Noel | Very interesting Conjecture.: Very interesting Conjecture.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 68.\n\nAttempt notes:\nTarget:\nMake progress on \"Hedetniemi's Conjecture\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3234, "problem_number": "OPG-1764", "title": "Counting 3-colorings of the hex lattice", "statement": "Problem Find $\\lim_{n \\rightarrow \\infty} (\\chi( H_n, 3)) ^{ 1 / |V(H_n)| }$.", "background": "Source: Open Problem Garden. Original node ID: 1764. URL: http://www.openproblemgarden.org/op/counting_3_colorings_of_the_hex_lattice.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/counting_3_colorings_of_the_hex_lattice\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: coloring; Lieb's Ice Constant; tiling; torus\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 5th, 2008 by mdevos\n\nProblem-page discussion:\nWe'll begin by putting in place the necessary notation. Let ${\\mathcal T}$ be the regular triangular tiling of the plane. For every $n \\ge 1$ there is a regular map which triangulates the torus, denoted $T_n$, which may be obtained from a regular hexagonal piece of ${\\mathcal T}$ of side-length $n$ by identifying points on opposite edges of this hexagon. Let $H_n$ be the dual of $T_n$ (on the torus). Then $H_n$ is a regular map on the torus - a hexagonal tiling. One last definition: for any graph $G$ and any positive integer $k$ we let $\\chi(G,k)$ denote the number of proper $k$-coloring of $G$.\n\nA famous theorem of Lieb [L] shows that $\\lim_{n \\rightarrow \\infty} (\\chi(Q_n,3))^{1 / |V(Q_n)|} = (\\frac{4}{3})^{3/2}$ where $Q_n$ denotes the $n \\times n$ quadrangulation of the torus. This theorem is usually stated in terms of Eulerian orientations, and is of interest to physicists as the constant $(\\frac{4}{3})^{3/2}$ (called Lieb's Ice Constant) determines the \"residual entropy for square ice\".\n\nThomassen proved that every planar graph $G$ with girth $\\ge 5$ has exponentially many proper 3-colorings. More precisely, he showed that $(\\chi(G,3))^{ 1 / |V(G)| } \\ge 2 ^{1 / 10000}$. This gives a lower bound on the limit in the above problem (assuming it exists).\n\nBibliography:\n[L] E. H. Lieb, Exact Solution of the Problem of the Entropy of Two-Dimensional Ice. Phys. Rev. Lett. 18, 692-694, 1967.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Counting 3-colorings of the hex lattice\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3235, "problem_number": "OPG-2040", "title": "Circular colouring the orthogonality graph", "statement": "Let ${\\mathcal O}$ denote the graph with vertex set consisting of all lines through the origin in ${\\mathbb R}^3$ and two vertices adjacent in ${\\mathcal O}$ if they are perpendicular.\n\nProblem Is $\\chi_c({\\mathcal O}) = 4$?", "background": "Source: Open Problem Garden. Original node ID: 2040. URL: http://www.openproblemgarden.org/op/circular_colouring_the_orthogonality_graph.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/circular_colouring_the_orthogonality_graph\n- Author(s): DeVos, Matt; Ghebleh, Mohammad; Goddyn, Luis A.; Mohar, Bojan; Naserasr, Reza\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: circular coloring; geometric graph; orthogonality\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 23rd, 2008 by mdevos\n\nProblem-page discussion:\nIn the problem statement, $\\chi_c$ denotes the circular chromatic number.\n\nColoring properties of ${\\mathcal O}$ are, rather surprisingly, of interest in quantum mechanics. If the spins of certain particles are measured in three orthogonal directions, then these measurements always return one $0$ and two values which are $\\pm 1$. If such a particle has \"decided\" in advance how it will respond to any possible measurement, then the set of directions in which it will respond $0$ must be an independent set in the orthogonality graph ${\\mathcal O}$ which meets every triangle. Kochen and Specker have shown that ${\\mathcal O}$ (even certain finite subgraphs of it) does not have any independent set meeting every triangle, thus exhibiting a rather mysterious property of these particles. In some sense, if the person doing the measurement has the free will to decide in which directions to measure, then the particle must have some free will to decide how it will respond.\n\nThe property that ${\\mathcal O}$ has no independent set which meets every triangle shows that $\\chi({\\mathcal O}) \\ge 4$. On the other hand, if we center a regular octahedron at the origin, and assign a color to each line $L$ depending on which pair of opposite faces it passes through (if $L$ meets more than one pair of opposite faces, just choose one) we get a proper 4-coloring of ${\\mathcal O}$. Therefore, $\\chi({\\mathcal O}) = 4$.\n\nThese bounds prove that $3 \\le \\chi_c({\\mathcal O}) \\le 4$. By investigating certain finite subgraphs of ${\\mathcal O}$, DeVos, Ghebleh, Goddyn, Mohar, and Naserasr have shown that $\\chi_c({\\mathcal O}) \\ge 3.5$.\n\nRelated:\nRelated problems\nPartitioning the Projective Plane\nThe Double Cap Conjecture\n\nDiscussion links:\n- circular chromatic number: http://en.wikipedia.org/wiki/circular coloring\n\nComments:\n- September 28th, 2009 | Anonymous | Still open?: Does anyone know if this problem is still open?\n- September 28th, 2009 | md | pretty sure: I am pretty confident this is still open. Apart from this one many-author paper, I don't think it has received any significant attention.\n- November 4th, 2008 | Anonymous | This is problem 10769 of the: This is problem 10769 of the American Mathematical Monthly. The solution appeared in AMM 108 (2001), 774-775. See also http://tph.tuwien.ac.at/~svozil/publ/blatter.htm\n\nChristian Blatter\n- November 4th, 2008 | mdevos | not exactly!: The problem you mention, 10769 of the AMM, concerns the chromatic number of the orthogonality graph. The problem here concerns the circular chromatic number of the orthogonality graph, and I believe it is still open.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Circular colouring the orthogonality graph\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3236, "problem_number": "OPG-34839", "title": "Double-critical graph conjecture", "statement": "A connected simple graph $G$ is called double-critical, if removing any pair of adjacent vertexes lowers the chromatic number by two.\n\nConjecture $K_n$ is the only $n$-chromatic double-critical graph", "background": "Source: Open Problem Garden. Original node ID: 34839. URL: http://www.openproblemgarden.org/op/double_critical_graph_conjecture.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/double_critical_graph_conjecture\n- Author(s): Erdos, Paul; Lovasz, Laszlo\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: coloring; complete graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: January 18th, 2009 by DFR\n\nProblem-page discussion:\nThis conjecture is a special case of a more general problem by Erdos and Lovasz proposed in 1966. It has been independently proven for the case where $\\chi(G) = 5$ by Mozhan [3] and Stiebitz [4].\n\nBibliography:\n*[1] P. Erdos, Problem 2, in: Theory of Graphs (Proc. Colloq., Tihany, 1966), Academic Press, New York, 1968, p. 361.\n\n[2] F. Chung, R. Graham, Erdos on graphs: His legacy of unsolved problems, A K Peters, Wellesley, Massachusetts, 1998.\n\n[3] N. N. Mozhan, On double critical graphs with the chromatic number five, Metody Diskretb. Anal. 46 (1987) 50-59.\n\n[4] M. Stiebitz, $K_5$ is the only double-critical $5$-chromatic graph, Discrete Math. 64 (1987) 91-93.\n\nSource links:\n- chromatic number: http://en.wikipedia.org/wiki/chromatic number\n\nComments:\n- November 30th, 2009 | Anonymous | Statement needs connected assumption: G needs to be assumed connected in the statement. Otherwise the disjoint union of K_k and a vertex (say) would be a counterexample.\n- May 29th, 2009 | Anonymous | missing connectivity condition: As it is now the conjecture is false -- take the disjoint union of a clique and a vertex. Just need to add \"connected\" to the definition of doubly critical.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Double-critical graph conjecture\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3237, "problem_number": "OPG-36907", "title": "Bounding the chromatic number of triangle-free graphs with fixed maximum degree", "statement": "Conjecture A triangle-free graph with maximum degree $\\Delta$ has chromatic number at most $\\ceil{\\frac{\\Delta}{2}}+2$.", "background": "Source: Open Problem Garden. Original node ID: 36907. URL: http://www.openproblemgarden.org/op/bounding_the_chromatic_number_of_triangle_free_graphs_with_fixed_maximum_degree.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/bounding_the_chromatic_number_of_triangle_free_graphs_with_fixed_maximum_degree\n- Author(s): Kostochka, Alexandr V.; Reed, Bruce A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: chromatic number; girth; maximum degree; triangle free\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 17th, 2009 by Andrew King\n\nProblem-page discussion:\nThis conjecture is a special case of Reed's $\\omega$, $\\Delta$, and $\\chi$ conjecture, which posits that for any graph, $\\chi \\leq \\lceil\\frac 12(\\Delta+1+\\omega)\\rceil$, where $\\omega$, $\\Delta$, and $\\chi$ are the clique number, maximum degree, and chromatic number of the graph respectively. Reed's conjecture is very easy to prove for complements of triangle-free graphs, but the triangle-free case seems challenging and interesting in its own right.\n\nThis conjecture is very much true for large values of $\\Delta$; Johansson proved that triangle-free graphs have chromatic number at most $\\frac{9\\Delta}{\\ln \\Delta}$. Surprisingly, the question appears to be open for every value of $\\Delta$ greater than four, up until Johansson's result implies the conjecture.\n\nKostochka previously proved that the chromatic number of a triangle-free graph is at most $\\frac{2\\Delta}{3}+2$, and he proved that for every $\\Delta \\geq 5$ there is a $g$ for which a graph of girth $g$ has chromatic number at most $\\frac{\\Delta}2+2$. Specifically, he showed that $g \\geq 4(\\Delta+2)\\ln \\Delta$ is sufficient. In [K] he posed the general problem: \"To find the best upper estimate for the chromatic number of the graph in terms of the maximal degree and density or girth.\"\n\nThe conjecture is implied by Brooks' Theorem for $\\Delta\\leq 5$. The three smallest open values of $\\Delta$ offer natural entry points to this problem. The easiest seems to be:\n\nProblem Does there exist a $6$-chromatic triangle-free graph of maximum degree 6?\n\nPerhaps looking at graphs of girth at least five would also be a good starting point.\n\nBibliography:\n[K] Kostochka, A. V., Degree, girth and chromatic number. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, pp. 679--696, Colloq. Math. Soc. János Bolyai, 18, North-Holland, Amsterdam-New York, 1978.\n\n*[R] Reed, B.A., $\\omega, \\Delta$, and $\\chi$, J. Graph Theory 27 (1998) 177-212.\n\nRelated:\nRelated problems\nReed's omega, delta, and chi conjecture\nGrunbaum's Conjecture\n\nComments:\n- May 16th, 2020 | Anonymous | Modifying the conjecture: From Reed's conjecture, it seems that the ceiling has to be replaced by flooring. Thanks.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Bounding the chromatic number of triangle-free graphs with fixed maximum degree\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3238, "problem_number": "OPG-36936", "title": "Graphs with a forbidden induced tree are chi-bounded", "statement": "Say that a family ${\\mathcal F}$ of graphs is $\\chi$-bounded if there exists a function $f: {\\mathbb N} \\rightarrow {\\mathbb N}$ so that every $G \\in {\\mathcal F}$ satisfies $\\chi(G) \\le f (\\omega(G))$.\n\nConjecture For every fixed tree $T$, the family of graphs with no induced subgraph isomorphic to $T$ is $\\chi$-bounded.", "background": "Source: Open Problem Garden. Original node ID: 36936. URL: http://www.openproblemgarden.org/op/graphs_with_a_forbidden_induced_tree_are_chi_bounded.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/graphs_with_a_forbidden_induced_tree_are_chi_bounded\n- Author(s): Gyarfas, Andras\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: chi-bounded; coloring; excluded subgraph; tree\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 16th, 2009 by mdevos\n\nProblem-page discussion:\nThis deep conjecture remains open despite considerable effort. Note that the conjecture would be false were the graph $T$ to be permitted to contain a cycle, since then the class would admit graphs of high girth (where $\\omega = 2$ ) and high chromatic number.\n\nIt is an easy exercise to prove this conjecture in the special case when $T$ is either a path or a star, but things get difficult from here. Kierstead and Penrice solved the special case when $T$ has radius 2, and Kierstead and Zhu solved the special case when $T$ has radius 3 and has the property that every vertex incident with the center vertex has degree 2.\n\nScott proved that the class of graphs which exclude all subdivisions of a fixed tree $T$ as induced subgraphs are $\\chi$-bounded. It follows from this that Gyarfas's conjecture also holds for subdivisions of stars.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Graphs with a forbidden induced tree are chi-bounded\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3239, "problem_number": "OPG-36939", "title": "Are vertex minor closed classes chi-bounded?", "statement": "Question Is every proper vertex-minor closed class of graphs chi-bounded?", "background": "Source: Open Problem Garden. Original node ID: 36939. URL: http://www.openproblemgarden.org/op/vertex_minor_closed_classes_are_chi_bounded.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/vertex_minor_closed_classes_are_chi_bounded\n- Author(s): Geelen, Jim\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: chi-bounded; circle graph; coloring; vertex minor\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 16th, 2009 by mdevos\n\nProblem-page discussion:\nWe say that a family of graphs ${\\mathcal F}$ is $\\chi$-bounded if there is a function $f: {\\mathbb N} \\rightarrow {\\mathbb N}$ so that $\\chi(G) \\le f( \\omega(G))$ for every $G \\in {\\mathcal F}$.\n\nIf $G$ is a simple graph, a vertex minor of $G$ is any graph which can be obtained by a sequence of the following operations:\n\n- delete a vertex\n- choose a vertex $v$ and complement the neighborhood of $v$ (i.e. whenever $u,w$ are neighbors of $v$, switch $u,w$ between adjacent/non-adjacent).\n\nDvorak and Kral [DK] showed that this conjecture is true for class of graphs of bounded rank-width, and the class of graphs having no vertex-minor isomorphic to the wheel $W_5$ on $6$ vertices.\n\nBibliography:\n[DK] Z. Dvorak and D. Král. Classes of graphs with small rank decompositions are χ-bounded. European J. Combin., 33(4):679–-683, 2012. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR3350076\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Are vertex minor closed classes chi-bounded?\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3240, "problem_number": "OPG-37907", "title": "Mixing Circular Colourings", "statement": "Question Is $\\mathfrak{M}_c(G)$ always rational?", "background": "Source: Open Problem Garden. Original node ID: 37907. URL: http://www.openproblemgarden.org/op/mixing_circular_colourings_0.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/mixing_circular_colourings_0\n- Author(s): Brewster, Richard C.; Noel, Jonathan A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: discrete homotopy; graph colourings; mixing\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: September 22nd, 2012 by Jon Noel\n\nProblem-page discussion:\nGiven a proper $k$-colouring $f$ of a graph $G$, consider the following 'mixing process:'\n\n- choose a vertex $v\\in V(G)$;\n- change the colour of $v$ (if possible) to yield a different $k$-colouring $f'$ of $G$.\n\nA natural problem arises: Can every $k$-colouring of $G$ be generated from $f$ by repeatedly applying this process? If so, we say that $G$ is $k$-mixing.\n\nThe problem of determining if a graph is $k$-mixing and several related problems have been studied in a series of recent papers [1,2,4-6]. The authors of [4] provide examples which show that a graph can be $k$-mixing but not $k'$-mixing for integers $k' > k$. For example, given $m\\geq3$ consider the bipartite graph $L_m$ which is obtained by deleting a perfect matching from $K_{m,m}$. It is an easy exercise to show that for $L_m$ is $k$-mixing if and only if $k\\geq3$ and $k\\neq m$. This example motivates the following definition.\n\nDefinition Define the mixing threshold of $G$ to be $$\\mathfrak{M}(G):= \\min\\{\\ell\\in\\mathbb{N}: G\\text{ is }k\\text{-mixing whenever } k\\geq\\ell\\}.$$\n\nAn analogous definition can be made for circular colouring. Recall, a $(k,q)$-colouring of a graph is a mapping $f:V(G)\\to \\{0,1,\\dots,k-1\\}$ such that if $uv\\in E(G)$, then $q\\leq |f(u)-f(v)|\\leq k-q$. As with ordinary colourings, we say that a graph $G$ is $(k,q)$-mixing if all $(k,q)$-colourings of $G$ can be generated from a single $(k,q)$-colouring $f$ by recolouring one vertex at a time.\n\nDefinition Define the circular mixing threshold of $G$ to be $$\\mathfrak{M}_c(G):= \\inf\\{\\ell\\in\\mathbb{Q}: G\\text{ is }(k,q)\\text{-mixing whenever } k/q \\geq\\ell\\}.$$\n\nSeveral bounds on the circular mixing threshold are obtained in [3], including the following which relates the circular mixing threshold to the mixing threshold.\n\nTheorem (Brewster and Noel $[3)$ ] For every graph $G$, $$\\mathfrak{M}_c(G)\\leq\\max\\left\\{\\frac{|V(G)|+1}{2}, \\mathfrak{M}(G)\\right\\}.$$\n\nAs a corollary, we have the following: if $|V(G)|\\leq 2\\mathfrak{M}(G)-1$, then $\\mathfrak{M}_c(G)\\leq \\mathfrak{M}(G)$. However, examples in [3] show that the ratio $\\mathfrak{M}_c/\\mathfrak{M}$ can be arbitrarily large in general.\n\nRegarding the problem of determining if $\\mathfrak{M}_c$ is rational, it is worth mentioning that there are no known examples of graphs $G$ for which $\\mathfrak{M}_c(G)$ is not an integer.\n\nOther problems are also given in [3]. One can check that $\\mathfrak{M}_c(K_2) = 2$ and $\\mathfrak{M}(K_2) = 3$. However, the only graphs which are known to satisfy $\\mathfrak{M}_c < \\mathfrak{M}$ are in some sense related to $K_2$, eg. trees and complete bipartite graphs.\n\nQuestion Is there a non-bipartite graph $G$ such that $\\mathfrak{M}_c(G) < \\mathfrak{M}(G)$?\n\nUsing an example from [4], it is shown in [3] that if $m\\geq2$ is an integer, then there is a graph $G$ such that $\\mathfrak{M}_c(G) = \\chi(G) = m$ if and only if $m\\neq 3$. A natural question to ask is whether a similar result holds for the circular chromatic number. Again, certain bipartite graphs are an exception.\n\nQuestion Is there a non-bipartite graph $G$ such that $\\mathfrak{M}_c(G) =\\chi_c(G)$?\n\nAlso, the example of $K_2$ shows that the circular mixing threshold is, in general, not attained. However, the following problem is open.\n\nQuestion Is the circular mixing threshold always attained for non-bipartite graphs?\n\nFor more precise versions of the last three questions, see [3].\n\nBibliography:\n[1] P. Bonsma and L. Cereceda. Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theoret. Comput. Sci. 410 (2009), (50): 5215--5226.\n\n[2] P. Bonsma, L. Cereceda, J. van den Heuvel, and M. Johnson. Finding paths between graph colourings: Computational complexity and possible distances. Electronic Notes in Discrete Mathematics 29 (2007): 463--469.\n\n*[3] R. C. Brewster and J. A. Noel. Mixing Homomorphisms and Extending Circular Colourings. Submitted. pdf.\n\n[4] L. Cereceda, J. van den Heuvel, and M. Johnson. Connectedness of the graph of vertex-colourings. Discrete Math. 308 (2008), (5-6): 913--919.\n\n[5] L. Cereceda, J. van den Heuvel, and M. Johnson. Mixing 3-colourings in bipartite graphs. European J. Combin. 30 (2009), (7): 1593--1606.\n\n[6] L. Cereceda, J. van den Heuvel, and M. Johnson. Finding paths between 3-colorings. Journal of Graph Theory, 67 (2011), (1): 69--82.\n\n[7] J. A. Noel. \"Jonathan Noel - Mixing Circular Colourings.\" Webpage.\n\nBibliography links:\n- pdf: http://www.math.mcgill.ca/jnoel/pdf/Mixing Paper.pdf\n- Connectedness of the graph of vertex-colourings: http://www.sciencedirect.com/science/article/pii/S0012365X07005456\n- Webpage: http://people.maths.ox.ac.uk/noel/MixCirc.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 38.\n\nAttempt notes:\nTarget:\nMake progress on \"Mixing Circular Colourings\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3241, "problem_number": "OPG-44879", "title": "Choice Number of k-Chromatic Graphs of Bounded Order", "statement": "Conjecture If $G$ is a $k$-chromatic graph on at most $mk$ vertices, then $\\text{ch}(G)\\leq \\text{ch}(K_{m*k})$.", "background": "Source: Open Problem Garden. Original node ID: 44879. URL: http://www.openproblemgarden.org/op/choice_number_of_k_chromatic_graphs_of_bounded_order.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/choice_number_of_k_chromatic_graphs_of_bounded_order\n- Author(s): Noel, Jonathan A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: choosability; complete multipartite graph; list coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: February 2nd, 2013 by Jon Noel\n\nProblem-page discussion:\nFor integers $m,k\\geq1$, let $K_{m*k}$ denote the complete $k$-partite graph in which every part has size $m$.\n\nIn one of the original papers on choosability, Erdos, Rubin and Taylor [ERT] proved that $\\text{ch}(K_{2*k})=k$. Later, Ohba [Ohba] conjectured the following generalization: if $|V(G)|\\leq 2\\chi(G)+1$, then TeX Embedding failed!.} This was proved by Noel, Reed and Wu [NRW12].\n\nTheorem (Noel, Reed and Wu 2012) If $|V(G)|\\leq 2\\chi(G)+1$, then $\\text{ch}(G)=\\chi(G)$.\n\nThe above theorem implies that the above conjecture holds for $m=2$. That is, if $G$ is a $k$-chromatic graph on at most $2k$ vertices (in fact, at most $2k+1$ vertices), then $\\text{ch}(G)=k=\\text{ch}(K_{2*k})$.\n\nKierstead [Kie00] proved that $\\text{ch}(K_{3*k})=\\left\\lceil\\frac{4k-1}{3}\\right\\rceil$. This was generalized by Noel, West, Wu and Zhu [NWWZ13] to the following:\n\nTheorem (Noel, West, Wu and Zhu 2013) For every graph $G$,\n$$\n\\text{ch}(G)\\leq\\max\\left\\{\\chi(G),\\left\\lceil\\frac{|V(G)|+\\chi(G)-1}{3}\\right\\rceil\\right\\}.\n$$\n\nTherefore, if $G$ is a $k$-chromatic graph on at most $3k$ vertices, then $\\text{ch}(G)\\leq \\left\\lceil\\frac{4k-1}{3}\\right\\rceil=\\text{ch}(K_{3*k})$. This shows that the conjecture is true for $m=3$.\n\nRecently, Kierstead, Salmon and Wang [KSW14] proved the following:\n\nTheorem (Kierstead, Salmon and Wang 2014) $\\text{ch}(K_{4*k})=\\left\\lceil\\frac{3k-1}{2}\\right\\rceil$.\n\nHowever, it is not known whether the upper bound of $\\left\\lceil\\frac{3k-1}{2}\\right\\rceil$ holds for all $k$-chromatic graphs on at most $4k$ vertices. If true, it would verify the conjecture for $m=4$.\n\nThe following is a refinement of the conjecture.\n\nConjecture (Noel 2013) For $n\\geq k\\geq 1$ there is a graph $G_{n,k}$ such that\n\n- $G_{n,k}$ is a complete $k$-partite graph on $n$ vertices,\n- the stability number of $G_{n,k}$ is $\\left\\lceil n/k\\right\\rceil$, and\n- every $k$-chromatic graph $G$ on at most $n$ vertices satisfies $\\text{ch}(G)\\leq \\text{ch}(G_{n,k})$.\n\nBibliography:\n[Alo92] N. Alon. Choice numbers of graphs: a probabilistic approach. Combin. Probab. Comput., 1(2):107–114, 1992.\n\n[ERT80] P. Erdos, A. L. Rubin, and H. Taylor. Choosability in graphs. Congress. Numer., XXVI, pages 125–157, 1980.\n\n[Kie00] H. A. Kierstead. On the choosability of complete multipartite graphs with part size three. Discrete Math., 211(1-3):255–259, 2000.\n\n[KSW14] H. A. Kierstead, A. Salmon and R. Wang. On the Choice Number of Complete Multipartite Graphs With Part Size Four.\n\n*[Noe13] J. A. Noel. Choosability of Graphs With Bounded Order: Ohba's Conjecture and Beyond. Master's thesis, McGill University, Montreal. pdf\n\n[NRW12] J. A. Noel, B. A. Reed, and H. Wu. A Proof of a Conjecture of Ohba. Preprint, arXiv:1211.1999v1, November 2012. Webpage\n\n[NWWZ13] J. A. Noel, D. B. West, H. Wu, and X. Zhu. Beyond Ohba's Conjecture: A bound on the choice number of $k$-chromatic graphs with $n$ vertices. Preprint, arXiv:1308.6739v1, August 2013. pdf\n\n[Ohb02] K. Ohba. On chromatic-choosable graphs. J. Graph Theory, 40(2):130–135, 2002.\n\n[Yan03] D. Yang. Extension of the game coloring number and some results on the choosability of complete multipartite graphs. PhD thesis, Arizona State University, Tempe, Arizona, 2003.\n\nRelated:\nRelated problems\nOhba's Conjecture\nChoice number of complete multipartite graphs with parts of size 4\n\nBibliography links:\n- On the Choice Number of Complete Multipartite Graphs With Part Size Four: http://www.arxiv.org/abs/1407.3817\n- pdf: http://people.maths.ox.ac.uk/noel/Masters.pdf\n- Webpage: http://people.maths.ox.ac.uk/noel/Ohba.html\n- pdf: http://people.maths.ox.ac.uk/noel/Choosability.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 28.\n\nAttempt notes:\nTarget:\nMake progress on \"Choice Number of k-Chromatic Graphs of Bounded Order\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3242, "problem_number": "OPG-46575", "title": "Melnikov's valency-variety problem", "statement": "Problem The valency-variety $w(G)$ of a graph $G$ is the number of different degrees in $G$. Is the chromatic number of any graph $G$ with at least two vertices greater than $$\\ceil{ \\frac{\\floor{w(G)/2}}{|V(G)| - w(G)} } ~?$$", "background": "Source: Open Problem Garden. Original node ID: 46575. URL: http://www.openproblemgarden.org/op/melnikovs_valency_variety_problem.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/melnikovs_valency_variety_problem\n- Author(s): Melnikov, L. S.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Importance: Low ✭\n- Recommended for undergraduates: no\n- Posted: March 3rd, 2013 by asp\n\nProblem-page discussion:\nAccording to Jensen and Toft [JT, p. 90], the problem is due to Melnikov and was mentioned by Vizing [V] and Zykov [Z]. According to Zykov [Z], Melnikov showed that the suggested lower bound would be best possible.\n\nA best possible upper bound on the chromatic number in terms of $|V(G)|$ and $w(G)$ is $|V(G)| - \\floor{ w(G) / 2 }$ as proved by Nettleton [N] and Dirac [D].\n\nBibliography:\n[D] G. A. Dirac. Valency-variey and chromatic number of abstract graphs. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 13, 59--64, 1964.\n\n[JT] Tommy R. Jensen, Bjarne Toft: Graph Coloring Problems, Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons Inc., New York, 1995.\n\n[N] R. E. Nettleton. Some generalized theorems on connectivity. Canad. J. Math. 12, 546--554, 1960.\n\n*[V] V. G. Vizing. Some unsolved problems in graph theory (in Russian). Uspekhi Mat. Nauk. 23, 117--134, 1968. English translation in Russian Math. Surveys 23, 125--141.\n\n*[Z] A. A. Zykov. Problem 11. In: H. Sachs, H.-J. Voss, and H. Walther, editors, Beiträge zur Graphentheorie vorgetragen auf dem Internationalen Kolloquium in Manebach DDR vom 9.-12. Mai 1967, page 228. B. G. Teubner, 1968.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Melnikov's valency-variety problem\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3243, "problem_number": "OPG-46900", "title": "Earth-Moon Problem", "statement": "Problem What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon?", "background": "Source: Open Problem Garden. Original node ID: 46900. URL: http://www.openproblemgarden.org/op/earth_moon_problem.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/earth_moon_problem\n- Author(s): Ringel, G.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: March 6th, 2013 by fhavet\n\nProblem-page discussion:\nIn term of graphs, it can be rephrased as follows. What is the maximum chromatic number of a graph $G$ which is the union of two planar graphs (on the same vertex set)?\n\nIf a graph $G$ on $n$ vertices is the union of two planar graphs, then it has at most $2(3n-6)$ edges, and so it is has a vertex of degree at most $11$. An easy induction shows that $G$ is 12-colourable, as observed by Heawood [He]. Gardner [G] reported an example requiring 9 colours (the join of $C_5$ and $K_6$ ). It is not known if configurations exist requiring 10, 11, or 12 colours.\n\nMore generally, one may ask for the maximum chromatic number of the union of $k$ planar graphs.\n\nProblem What is the maximum chromatic number of a graph $G$ which is the union of $k$ planar graphs?\n\nThe same reasoning as above shows that $6k$ colours are always sufficient. The minimum number of planar graphs to decompose a complete graph [BW] gives a lower bound of $6k-2$ for $k \\ne 2$.\n\nBibliography:\n[BW] L. W. Beineke and A. T. White, Topological Graph Theory, Selected Topics in Graph Theory, L. W. Beineke and R. J. Wilson, eds., Academic Press, 15-50, 1978.\n\n[G] M. Gardner, Mathematical Recreations: The Coloring of Unusual Maps Leads Into Uncharted Territory. Sci. Amer. 242, 14-22, 1980.\n\n[He] P.J. Heawood, Map Colour Theorems. Quart. J. Pure Appl. Math. 24, 332-338, 1890.\n\n*[R] G. Ringel, Färbungsprobleme auf Flachen und Graphen. Berlin: Deutsche Verlag der Wissenschaften, 1959.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Earth-Moon Problem\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3244, "problem_number": "OPG-46940", "title": "Acyclic list colouring of planar graphs.", "statement": "Conjecture Every planar graph is acyclically 5-choosable.", "background": "Source: Open Problem Garden. Original node ID: 46940. URL: http://www.openproblemgarden.org/op/acyclic_list_colouring_of_planar_graphs.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/acyclic_list_colouring_of_planar_graphs\n- Author(s): Borodin, Oleg V.; Fon-Der-Flasss, D. G.; Kostochka, Alexandr V.; Raspaud, André; Sopena, Eric\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture would imply to celebrated 5-colour theorems: one due toBorodin [B] stating that every planar graph is acyclically 5-colourable, and one due to Thomassen [T] stating that every plaanar graph is 5-choosable. This two theorems are best possible, because there are planar graphs which are not acyclically 4-colourable and others which are not 4-choosable.\n\nBorodin et al. [BFKRS] showed that every planar graph is acyclically 7-choosable.\n\nBibliography:\n[B] O.V. Borodin, On acyclic colorings of planar graphs, Discrete Math. 25 (1979) 211-236.\n\n*[BFKRS] O.V. Borodin, D.G. Fon-Der Flaass, A.V. Kostochka, A. Raspaud, and E. Sopena, Acyclic list 7-coloring of planar graphs, J. of Graph Theory 40-2 (2002) 83-90.\n\n[T] C. Thomassen. Every planar graph is 5-choosable. J. Combin. Theory Ser. B, 62(1994), no. 1, 180–181.\n\n[V] M. Voigt. List colourings of planar graphs. Discrete Math., 120(1993):no. 1-3, 215–219.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Acyclic list colouring of planar graphs.\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3245, "problem_number": "OPG-47358", "title": "List chromatic number and maximum degree of bipartite graphs", "statement": "Conjecture There is a constant $c$ such that the list chromatic number of any bipartite graph $G$ of maximum degree $\\Delta$ is at most $c \\log \\Delta$.", "background": "Source: Open Problem Garden. Original node ID: 47358. URL: http://www.openproblemgarden.org/op/list_chromatic_number_and_maximum_degree_of_bipartite_graphs.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/list_chromatic_number_and_maximum_degree_of_bipartite_graphs\n- Author(s): Alon, Noga\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 12th, 2013 by fhavet\n\nProblem-page discussion:\nFor definitions and an introduction to list colouring, see the related Wikipedia page.\n\nAlon [A] showed that the list chromatic number of a graph (not necessarily bipartite) of maximum degree $\\Delta$ is at least $\\frac{1}{2}(1-o(1))\\log_2\\Delta$. Random bipartite graphs show that this is tight up to a multiplicative factor $(2+o(1))$.\n\nIt is not diffcult to see that the list chromatic number of any bipartite graph $G$ of maximum degree $\\Delta$ is at most $O(\\Delta/\\log \\Delta)$. It also follows a more general result of Johansson [J] on triangle-free graphs.\n\nBibliography:\n*[A] N. Alon, Degrees and choice numbers, Random Structures Algorithms, 16 (2000), 364--368.\n\n[AK] N. Alon and M. Krivelevich, The choice number of random bipartite graphs, Annals of Combi- natorics 2 (1998), 291-297.\n\n[J] A. Johansson. Asymptotic choice number for triangle free graphs. Technical Report 91–95, DIMACS, 1996.\n\nDiscussion links:\n- Wikipedia page: http://en.wikipedia.org/wiki/list coloring\n\nBibliography links:\n- Degrees and choice numbers: http://www.cs.tau.ac.il/%7Enogaa/PDFS/degch1.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"List chromatic number and maximum degree of bipartite graphs\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3246, "problem_number": "OPG-47470", "title": "Colouring the square of a planar graph", "statement": "Conjecture Let $G$ be a planar graph of maximum degree $\\Delta$. The chromatic number of its square is\n\n- at most $7$ if $\\Delta =3$,\n- at most $\\Delta+5$ if $4\\leq\\Delta\\leq 7$,\n- at most $\\left\\lfloor\\frac32\\,\\Delta\\right\\rfloor+1$ if $\\Delta\\ge8$.", "background": "Source: Open Problem Garden. Original node ID: 47470. URL: http://www.openproblemgarden.org/op/colouring_the_square_of_a_planar_graph.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/colouring_the_square_of_a_planar_graph\n- Author(s): Wegner\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 13th, 2013 by fhavet\n\nProblem-page discussion:\nThe square of a graph $G$ is the graph $G^2$ on the same set of vertices, in which two vertices are adjacent when their distance in $G$ is at most 2.\n\nWegner [W] also gave examples showing that these bounds would be tight. For $\\Delta\\geq 8$, they are the following.\n\nTight examples for Wegner's conjecture\n\nFor $4\\leq \\Delta \\leq 9$, the examples are planar graphs on $\\Delta+5$ with maximum degree $\\Delta$ whose square is a complete graph.\n\nThis conjecture has also been generalized to the list chromatic number.\n\nConjecture Let $G$ be a planar graph of maximum degree $\\Delta$. The list chromatic number of its square is\n\n- at most $7$ if $\\Delta =3$,\n- at most $\\Delta+5$ if $4\\leq\\Delta\\leq 7$,\n- at most $\\left\\lfloor\\frac32\\,\\Delta\\right\\rfloor+1$ if $\\Delta\\ge8$.\n\nCranston and Kim [CK] showed that the square of every connected graph (non necessarily planar) which is subcubic (i.e., with $\\Delta\\le3$ ) is 8-choosable, except for the Petersen graph. However, the 7-choosability of the square of subcubic planar graphs is still open.\n\nHavet et al. [HHMR] proved the conjecture asymptotically:\n\nTheorem The square of every planar graph $G$ of maximum degree $\\Delta$ has list chromatic number at most $(1+o(1))\\,\\frac32\\,\\Delta$.\n\nIn fact, they proved this results for more general classes of graph. This led them to pose the following problem.\n\nProblem Is it true that for every minor-closed family ${\\cal F}$ of graphs (with ${\\cal F}$ not the set of all graphs), we have $\\chi(G^2)\\le \\bigl(\\frac32+o(1)\\bigr) \\Delta(G)$ for all $G\\in{\\cal F}$?\n\nBibliography:\n[HHMR] F. Havet, J. van den Heuvel, C. McDiarmid, and B. Reed. List Colouring Squares of Planar Graphs. Research Report RR-6586, INRIA, July 2008.\n\n[CK] D. W. Cranston and S.-J. Kim. List-coloring the square of a subcubic graph, J. Graph Theory, 57(1):65--87, 2008.\n\n*[W] G. Wegner. Graphs with given diameter and a coloring problem. Technical report, 1977.\n\nDiscussion links:\n- square: http://en.wikipedia.org/wiki/Graph power\n- list chromatic number: http://en.wikipedia.org/wiki/list coloring\n\nBibliography links:\n- List Colouring Squares of Planar Graphs: http://hal.inria.fr/inria-00303303/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Colouring the square of a planar graph\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3247, "problem_number": "OPG-47511", "title": "Weighted colouring of hexagonal graphs.", "statement": "Conjecture There is an absolute constant $c$ such that for every hexagonal graph $G$ and vertex weighting $p:V(G)\\rightarrow \\mathbb{N}$, $$\\chi(G,p) \\leq \\frac{9}{8}\\omega(G,p) + c$$", "background": "Source: Open Problem Garden. Original node ID: 47511. URL: http://www.openproblemgarden.org/op/weighted_colouring_of_hexagonal_graphs.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/weighted_colouring_of_hexagonal_graphs\n- Author(s): McDiarmid, Colin; Reed, Bruce A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 13th, 2013 by fhavet\n\nProblem-page discussion:\nA hexagonal graph is an induced subgraph of the triangular lattice. The triangular lattice $TL$ may be described as follows. The vertices are all integer linear combinations $a\\mathbf{e_1} + b\\mathbf{e_2}$ of the two vectors $\\mathbf{e_1}=(1,0)$ and $\\mathbf{e_2}=(\\frac{1}{2}, \\frac{\\sqrt{3}}{2})$. Two vertices are adjacent when the Euclidean distance between them is 1.\n\nLet $G$ be a graph and $p$ a vertex weighting $p:V(G)\\rightarrow \\mathbb{N}$. The weighted clique number of $(G,p)$, denoted by $\\omega(G,p)$, is the maximum weight of a clique, that is $\\max \\{p(C) \\tq C \\mbox{ clique of } G\\}$, where $p(C)=\\sum_{v\\in C} p(v)$. A $k$-colouring of a $(G,p)$ is a mapping $C:V(G)\\ra {\\cal P}(\\{1, \\dots, k\\})$ such that for every vertex $v\\in V(G)$, $|C(v)|=p(v)$ and for all edge $uv\\in E(G)$, $C(u)\\cap C(v)=\\emptyset$. The chromatic number of $(G,p)$, denoted by $\\chi(G,p)$, is the least integer $t$ such that $(G,p)$ admits a $t$-colouring.\n\nThe conjecture would be tight because of $C_9$ the cycle of length 9. The maximum size of stable set in $C_9$ is $4$. Thus $\\chi(C_9,\\mathbf{k})\\geq 9k/4$ and $\\omega(G,\\mathbf{k})=2k$, where $\\mathbf{k}$ is the all $k$ function.\n\nMcDiarmid and Reed [MR] proved that $\\chi(G,p)\\leq \\frac{4\\omega(G,p)+1}{3}$ for any hexagonal graph $G$ and vertex weighting $p$. Havet [H] proved that if a hexagonal graph $G$ is triangle-free, then $\\chi(G,p)\\leq\\frac{7}{6}\\omega(G,p) + 5$ (See also [SV]).\n\nThe conjecture would be implied by the following one, where $\\mathbf{4}$ is the all $4$ function.\n\nConjecture $\\chi(G,\\mathbf{4})\\leq 9$ for every hexagonal graph.\n\nSince $\\chi(G,\\mathbf{4}) \\geq 4|V(G)|/\\alpha(G)$, where $\\alpha(G)$ is the stability number (the maximum size of a stable set). A first step to this later conjecture would be to prove the following conjecture of McDiarmid.\n\nConjecture Let $G$ be a triangle-free hexagonal graph. $$\\alpha(G)\\geq \\frac{4}{9}|V(G)|$$\n\nBibliography:\n[H] F.Havet. Channel assignment and multicolouring of the induced subgraphs of the triangular lattice. Discrete Mathematics 233:219--231, 2001.\n\n*[MR] C. McDiarmid and B. Reed. Channel assignment and weighted coloring, Networks, 36:114--117, 2000.\n\n[SV] K. S. Sudeep and S. Vishwanathan. A technique for multicoloring triangle-free hexagonal graphs. Discrete Mathematics, 300(1-3), 256--259, 2005.\n\nDiscussion links:\n- stable set: http://en.wikipedia.org/wiki/Independent set (graph theory)\n\nBibliography links:\n- Channel assignment and multicolouring of the induced subgraphs of the triangular lattice: http://www.sciencedirect.com/science/article/pii/S0012365X00002417\n- A technique for multicoloring triangle-free hexagonal graphs: http://www.sciencedirect.com/science/article/pii/S0012365X05002694\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 35.\n\nAttempt notes:\nTarget:\nMake progress on \"Weighted colouring of hexagonal graphs.\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3248, "problem_number": "OPG-48583", "title": "Bounding the on-line choice number in terms of the choice number", "statement": "Question Are there graphs for which $\\text{ch}^{\\text{OL}}-\\text{ch}$ is arbitrarily large?", "background": "Source: Open Problem Garden. Original node ID: 48583. URL: http://www.openproblemgarden.org/op/bounding_the_on_line_choice_number_in_terms_of_the_choice_number.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/bounding_the_on_line_choice_number_in_terms_of_the_choice_number\n- Author(s): Zhu, Xuding\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: choosability; list coloring; on-line choosability\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: April 11th, 2013 by Jon Noel\n\nProblem-page discussion:\nWe let $\\text{ch}$ denote the (classical) choice number. For a definition of the on-line choice number of $G$ (denoted $\\text{ch}^{\\text{OL}}(G)$ ), see the following posting: On-Line Ohba's Conjecture.\n\nA result of Alon [Alo93] says that the choice number of a graph is bounded above and below by a function of the colouring number, defined as follows: $\\text{col}(G):=\\max\\{\\delta(H):H\\subseteq G\\}$.\n\nZhu [Zhu09] demonstrated that the on-line choice number is bounded above by the colouring number. By combining this with Alon's result, we have that there is a function $g$ such that $\\text{ch}^{\\text{OL}}(G)\\leq g(\\text{ch}(G))$ for every graph $G$. However, the function $g$ from Alon's result is exponential. In [Zhu09], Zhu asked if we can do better (polynomial? linear? etc).\n\nIt is known that there are graphs for which $\\text{ch}^{\\text{OL}}(G) = \\text{ch}(G) + 1$. Interestingly, as is mentioned in [CLM+13], it is not even known whether there is a graph $G$ such that $\\text{ch}^{\\text{OL}}(G)>\\text{ch}(G)+1$.\n\nThere are not many graphs for which the choice number (let alone the on-line choice number) is known exactly. For this reason, it seems that a natural starting point for this problem is to study the complete $k$-partite graph in which every part has size $3$, denoted $K_{3*k}$. Kierstead [Kie00] proved that $\\text{ch}(K_{3*k}) = \\left\\lceil\\frac{4k-1}{3}\\right\\rceil$. Kozik, Micek and Zhu proved that the $\\text{ch}^{\\text{OL}}(K_{3*k})\\leq\\frac{3k}{2}$.\n\nIt may be the case that $\\text{ch}^{\\text{OL}}(K_{3*k})>\\left\\lceil\\frac{4k-1}{3}\\right\\rceil$. Is it larger than $\\left\\lceil\\frac{4k-1}{3}\\right\\rceil+1$?\n\nBibliography:\n[Alo93] N. Alon. Restricted colorings of graphs. In Surveys in combinatorics, 1993 (Keele), volume 187 of London Math. Soc. Lecture Note Ser., pages 1–33. Cambridge Univ. Press, Cambridge, 1993.\n\n[CLM+13] J. Carraher, S. Loeb, T. Mahoney, G. Puleo, M.-T. Tsai, and D. West. Three Topics in Online List Coloring. Preprint, February 2013.\n\n[HWZ12] P. Huang, T. Wong, and X. Zhu. Application of polynomial method to on-line list colouring of graphs. European J. Combin., 33(5):872–883, 2012.\n\n[Kie00] H. A. Kierstead. On the choosability of complete multipartite graphs with part size three. Discrete Math., 211(1-3):255–259, 2000.\n\n[KKLZ12] S.-J. Kim, Y. S. Kwon, D. D.-F. Liu, and X. Zhu. On-line list colouring of complete multipartite graphs. Electron. J. Combin., 19(1):Paper 41, 13, 2012.\n\n[KMZ12] J. Kozik, P. Micek, and X. Zhu. Towards on-line Ohba’s conjecture. Preprint, arXiv:1111.5458v2, December 2012.\n\n[Sch09] U. Schauz. Mr. Paint and Mrs. Correct. Electron. J. Combin., 16(1):Research Paper 77, 18, 2009.\n\n[Sch10] U. Schauz. A paintability version of the combinatorial Nullstellensatz, and list colorings of k-partite k-uniform hypergraphs. Electron. J. Combin., 17(1):Research Paper 176, 13, 2010.\n\n*[Zhu09] X. Zhu. On-line list colouring of graphs. Electron. J. Combin., 16(1):Research Paper 127, 16, 2009.\n\nRelated:\nRelated problems\nOn-Line Ohba's Conjecture\nOhba's Conjecture\nChoice number of complete multipartite graphs with parts of size 4\n\nDiscussion links:\n- On-Line Ohba's Conjecture: http://www.openproblemgarden.org/op/on_line_ohbas_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Bounding the on-line choice number in terms of the choice number\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3249, "problem_number": "OPG-53019", "title": "Choosability of Graph Powers", "statement": "Question (Noel, 2013) Does there exist a function $f(k)=o(k^2)$ such that for every graph $G$,\n$$\n\\text{ch}\\left(G^2\\right)\\leq f\\left(\\chi\\left(G^2\\right)\\right)?\n$$", "background": "Source: Open Problem Garden. Original node ID: 53019. URL: http://www.openproblemgarden.org/op/choosability_of_graph_powers.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/choosability_of_graph_powers\n- Author(s): Noel, Jonathan A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: choosability; chromatic number; list coloring; square of a graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: July 13th, 2013 by Jon Noel\n\nProblem-page discussion:\nFor a survey of choosability, including relevant definitions, see [Noe] or click here.\n\nThe List Square Colouring Conjecture, due to Kostochka and Woodall [KW], states that $\\text{ch}\\left(G^2\\right) = \\chi\\left(G^2\\right)$ for every graph $G$. This was disproved by Kim and Park [KP], who proved that there is a sequence $\\{G_n\\}_n$ of graphs and a constant $c_1$ such that $\\chi\\left(G^2_n\\right)\\to\\infty$ and $\\text{ch}\\left(G_n^2\\right) \\geq c_1 \\chi\\left(G_n^2\\right)\\log\\left(\\chi\\left(G_n^2\\right)\\right)$ for all $n$. To obtain this lower bound from the construction of Kim and Park, one can apply the well-known result of Alon [Alo].\n\nIt may be the case that the correct upper bound for all graphs is of the same order of magnitude as the example in the result of Kim and Park.\n\nQuestion (Noel, 2013) Does there exist a positive constant $c_2$ such that every graph $G$ satisfies $\\text{ch}\\left(G^2\\right) \\leq c_2\\chi\\left(G^2\\right)\\log{\\chi\\left(G^2\\right)}$?\n\nBy calculating the clique number and maximum degree of $G^2$, one can easily show that $\\text{ch}\\left(G^2\\right)\\leq\\chi\\left(G^2\\right)^2$ (this observation is due to Young Soo Kwon), but it seems that no significantly better bound is known.\n\nProposition If $G$ contains an edge, then\n$$\n\\text{ch}\\left(G^2\\right)< \\chi\\left(G^2\\right)^2.\n$$\n\nProof We observe the following bounds:\n$$\n\\chi\\left(G^2\\right) \\geq \\omega\\left(G^2\\right) \\geq \\Delta(G)+1,\n$$\n\n$$\n\\text{ch}\\left(G^2\\right) \\leq \\Delta\\left(G^2\\right)+1 \\leq \\Delta(G)\\left(\\Delta(G)-1\\right) + \\Delta(G)+1 = \\Delta(G)^2+1.\n$$\n Therefore, since $\\Delta(G)>0$, we have\n$$\n\\text{ch}\\left(G^2\\right)\\leq \\Delta(G)^2+1 < \\left(\\Delta(G)+1\\right)^2 \\leq \\chi\\left(G^2\\right)^2.\n$$\n This completes the proof.\n\nThese questions are related to a problem of Zhu (see Doug West's webpage for more info) who asked whether there exists an integer $k$ such that for every graph $G$, we have that $G^k$ has choice number equal to chromatic number. This conjecture has been disproved independently by Kim, Kwon and Park [KKP] and Kosar, Petrickova, Reigniger and Yeager [KPRY]. The example of [KPRY] also yields, for every $k$, a sequence $\\{G_n\\}_n$ of graphs and a constant $c$ such that $\\chi\\left(G^k_n\\right)\\to\\infty$ and $\\text{ch}\\left(G_n^k\\right) \\geq c \\chi\\left(G_n^k\\right)\\log\\left(\\chi\\left(G_n^k\\right)\\right)$ for all $n$. They ask the following, more general, questions:\n\nQuestion (Kosar et al., 2013) Given $k\\geq2$, does there exist a function $f_k(x)=o(x^2)$ such that for every graph $G$,\n$$\n\\text{ch}\\left(G^k\\right)\\leq f_k\\left(\\chi\\left(G^k\\right)\\right)?\n$$\n\nTo our knowledge, it is not known whether there exists a function $f_k(x) = o(x^k)$ such that the same conclusion holds. (Intuitively, it seems that higher values of $k$ should yield a smaller separation between $\\text{ch}(G^k)$ and $\\chi(G^k)$; however, there seems to be no hard evidence to support this.)\n\nQuestion (Kosar et al., 2013) Given $k\\geq2$, does there exist a positive constant $c_k$ such that every graph $G$ satisfies $\\text{ch}\\left(G^k\\right) \\leq c_k\\chi\\left(G^k\\right)\\log{\\chi\\left(G^k\\right)}$? Moreover, can the constant $c_k$ be made independent of $k$?\n\nThese questions are also related to the so-called List Total Colouring Conjecture of Borodin, Kostochka and Woodall [BKW], which says that the total graph of a multigraph always satisfies $\\text{ch}=\\chi$. Given a multigraph $G$, the total graph of $G$ can be obtained by subdividing every edge of $G$ and then taking the square of the resulting graph.\n\nBibliography:\n[Alo] Noga Alon. Choice numbers of graphs: a probabilistic approach. Combin. Probab. Comput., 1(2):107–114, 1992.\n\n[BKW] Oleg V. Borodin, Alexandr V. Kostochka, and Douglas R. Woodall. List edge and list total colourings of multigraphs. J. Combin. Theory Ser. B, 71(2):184–204, 1997.\n\n[KP] Seog-Jin Kim and Boram Park: Counterexamples to the List Square Coloring Conjecture, submitted.\n\n[KKP] Seog-Jin Kim, Young Soo Kwon and Boram Park: Chromatic-choosability of the power of graphs.\n\n[KPRY] Nicholas Kosar, Sarka Petrickova, Benjamin Reiniger, Elyse Yeager: A note on list-coloring powers of graphs.\n\n[KW] Alexandr V. Kostochka and Douglas R. Woodall. Choosability conjectures and multicircuits, Discrete Math., 240 (2001), 123--143.\n\n[Noe] Jonathan A. Noel. Choosability of Graphs with Bounded Order: Ohba's Conjecture and Beyond, Master's thesis. McGill University (2013). pdf.\n\nRelated:\nRelated problems\nChoice number of complete multipartite graphs with parts of size 4\nChoice Number of k-Chromatic Graphs of Bounded Order\nList Total Colouring Conjecture\n\nDiscussion links:\n- click here: http://www.openproblemgarden.org/ http:/www.math.mcgill.ca/jnoel/pdf/Masters.pdf\n- Doug West's webpage: http://www.openproblemgarden.org/ http:/www.math.uiuc.edu/%7Ewest/regs/listkpow.html\n\nBibliography links:\n- Counterexamples to the List Square Coloring Conjecture: http://www.arxiv.org/abs/1305.2566\n- Chromatic-choosability of the power of graphs: http://www.arxiv.org/abs/1309.0888\n- A note on list-coloring powers of graphs: http://www.arxiv.org/abs/1309.7705\n- pdf: http://www.openproblemgarden.org/ http:/www.math.mcgill.ca/jnoel/pdf/Masters.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 32.\n\nAttempt notes:\nTarget:\nMake progress on \"Choosability of Graph Powers\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3250, "problem_number": "OPG-55984", "title": "Erdős–Faber–Lovász conjecture", "statement": "Conjecture If $G$ is a simple graph which is the union of $k$ pairwise edge-disjoint complete graphs, each of which has $k$ vertices, then the chromatic number of $G$ is $k$.", "background": "Source: Open Problem Garden. Original node ID: 55984. URL: http://www.openproblemgarden.org/op/erdos_faber_lovasz_conjecture.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/erdos_faber_lovasz_conjecture\n- Author(s): Erdos, Paul; Faber, Vance; Lovasz, Laszlo\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: chromatic number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: August 22nd, 2013 by Jon Noel\n\nProblem-page discussion:\nFrom [Erd81]: \"Faber, Lovász and I made this harmless looking conjecture at a party in Boulder Colorado in September 1972. Its diff\u001eculty was realised only slowly. I now off\u001ber 500 dollars for a proof or disproof. (Not long ago I only offered 50; the increase is not due to inflation but to the fact that I now think the problem is very diff\u001ecult. Perhaps I am wrong.)\"\n\nThe conjecture can be equivalently formulated in terms of seating assignments or hypergraph colouring; see Wikipedia or Doug West's Webpage.\n\nBibliography:\n[Erd81] P. Erdős. On the combinatorial problems which I would most like to see solved. Combinatorica, 1(1):25–42, 1981.\n\nRelated:\nRelated problems\nAlon-Saks-Seymour Conjecture\n\nDiscussion links:\n- Wikipedia: http://en.wikipedia.org/wiki/Erdős–Faber–Lovász_conjecture\n- Doug West's Webpage: http://www.math.uiuc.edu/%7Ewest/regs/efl.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Erdős–Faber–Lovász conjecture\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3251, "problem_number": "OPG-56230", "title": "2-colouring a graph without a monochromatic maximum clique", "statement": "Conjecture If $G$ is a non-empty graph containing no induced odd cycle of length at least $5$, then there is a $2$-vertex colouring of $G$ in which no maximum clique is monochromatic.", "background": "Source: Open Problem Garden. Original node ID: 56230. URL: http://www.openproblemgarden.org/op/2_colouring_a_graph_without_a_monochromatic_maximum_clique.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/2_colouring_a_graph_without_a_monochromatic_maximum_clique\n- Author(s): Hoang, Chinh T.; McDiarmid, Colin\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: maximum clique; Partitioning\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 25th, 2013 by Jon Noel\n\nProblem-page discussion:\nA $2$-division of a graph $G$ is a partitioning of $G$ into two subgraphs, neither of which contains a maximum clique. It is known that every perfect graph admits a $2$-division. Thus, by the Strong Perfect Graph Theorem [CRS], a graph which does not contain an induced copy of an odd cycle of length at least $5$ or its complement has a $2$-division. Hoàng and McDiarmid [HMcD] also prove that a claw-free graph admits a 2-division if and only if it does not contain an induced odd cycle of length at least $5$. The conjecture says that this holds for all graphs.\n\nThis problem was featured as unsolved problem #49 in Bondy and Murty's book \"Graph Theory\" [BM].\n\nSee also a posting on the American Institute of Mathematics website, contributed by Bruce Reed.\n\nBibliography:\n[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: The strong perfect graph theorem, Ann. of Math. (2) 164 (2006), no. 1, 51--229. MathSciNet\n\n[HMcD] C.T. Hoàng, C. McDiarmid, On the divisibility of graphs, Discrete Math. 242 (1–3) (2002) 145–156.\n\n[BM] J. A. Bondy and U. S. R. Murty. Graph theory, volume 244 of Graduate Texts in Mathematics. Springer, New York, 2008.\n\nDiscussion links:\n- posting: http://www.aimath.org/WWN/perfectgraph/articles/html/19a/\n\nBibliography links:\n- The strong perfect graph theorem: http://www.arxiv.org/abs/math.CO/0212070\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2233847\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"2-colouring a graph without a monochromatic maximum clique\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3252, "problem_number": "OPG-59927", "title": "List Colourings of Complete Multipartite Graphs with 2 Big Parts", "statement": "Question Given $a,b\\geq2$, what is the smallest integer $t\\geq0$ such that $\\chi_\\ell(K_{a,b}+K_t)= \\chi(K_{a,b}+K_t)$?", "background": "Source: Open Problem Garden. Original node ID: 59927. URL: http://www.openproblemgarden.org/op/list_colourings_of_complete_multipartite_graphs_with_2_big_parts.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/list_colourings_of_complete_multipartite_graphs_with_2_big_parts\n- Author(s): Allagan, Julian\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: complete bipartite graph; complete multipartite graph; list coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: April 12th, 2014 by Jon Noel\n\nProblem-page discussion:\nThe list chromatic number of a graph $G$, denoted $\\chi_\\ell(G)$, is the minimum $k$ such that for every assignment of lists of size $k$ to the vertices of $G$ there is a proper colouring in which every vertex is mapped to a colour in its own list. For more background on the list chromatic number, see [3].\n\nGiven graphs $G$ and $H$, the join of $G$ and $H$, denoted $G+H$, is obtained by taking disjoint copies of $G$ and $H$ and adding all edges between them. Ohba [1] proved that for every graph $G$ there exists $t\\geq0$ such that $\\chi_\\ell(G+K_t)= \\chi(G+K_t)$. The question above asks to determine the minimum value of $t$ in the case that $G$ is a complete bipartite graph. It seems that it was first studied in [4], although this is unclear; for the time being, we have chosen to attribute this problem to J. Allagan.\n\nDefine $\\phi(a,b)$ to be the minimum $t$ such that $\\chi_\\ell(K_{a,b}+K_t)= \\chi(K_{a,b}+K_t)$. Note that, if $G$ is a complete multipartite graph with at most one non-singleton part, then we see that $\\chi_\\ell(G)=\\chi(G)$ by colouring the vertices of the non-singleton part last. Thus, if $a$ or $b$ is equal to 1, then $\\phi(a,b)=0$. As it turns out, $\\phi(2,2)=\\phi(2,3)=0$ and $\\phi(3,3)=\\phi(2,4)=1$. This can be deduced from the following result of [2] and the fact that $\\chi_\\ell(K_{3,3})=\\chi_\\ell(K_{4,2})=3$:\n\nTheorem (Noel, Reed, Wu (2012)) If $|V(G)|\\leq 2\\chi(G)+1$, then $\\chi_\\ell(G)=\\chi(G)$.\n\nThe above result of [2] implies that if $a+b\\geq 5$, then $\\phi(a,b)\\leq a+b-5$. However it seems that, for most values of $a,b$, this bound is far from tight.\n\nA simple observation is that, since $\\chi_\\ell(K_{a,b}+K_t)\\geq \\chi_\\ell(K_{a,b})$ for all $t$, we must have\n$$\n\\phi(a,b)\\geq \\chi_\\ell(K_{a,b}) - \\chi(K_{a,b}) = \\chi_\\ell(K_{a,b}) -2.\n$$\n\nThe following is a result of Allagan [4]:\n\nTheorem (Allagan (2009)) If $a\\geq5$, then\n$$\n\\lfloor \\sqrt{a}\\rfloor - 1 \\leq \\phi(a,2)\\leq \\left\\lceil\\frac{-7+\\sqrt{8a+17}}{2}\\right\\rceil.\n$$\n\nThis implies that $\\phi(a,2)=1$ for $4\\leq a\\leq 8$ and that $\\phi(a,2)=2$ for $9\\leq a\\leq 13$.\n\nBibliography:\n[1] K. Ohba. On chromatic-choosable graphs, J. Graph Theory. 40 (2002) 130--135. MathSciNet.\n\n[2] J. A. Noel, B. A. Reed, H. Wu. A Proof of a Conjecture of Ohba. Submitted. pdf.\n\n[3] J. A. Noel. Choosability of Graphs with Bounded Order: Ohba's Conjecture and Beyond. Master's Thesis, McGill University. pdf.\n\n[4] J. A. D. Allagan. Choice Numbers, Ohba Numbers and Hall Numbers of some complete $k$-partite graphs. PhD Thesis. Auburn University. 2009.\n\nRelated:\nRelated problems\nOhba's Conjecture\nChoice Number of k-Chromatic Graphs of Bounded Order\nChoice number of complete multipartite graphs with parts of size 4\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1899118\n- pdf: http://www.openproblemgarden.org/people.maths.ox.ac.uk/noel/Ohba Paper.pdf\n- pdf: http://www.openproblemgarden.org/people.maths.ox.ac.uk/noel/Masters.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 30.\n\nAttempt notes:\nTarget:\nMake progress on \"List Colourings of Complete Multipartite Graphs with 2 Big Parts\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3253, "problem_number": "OPG-59943", "title": "List Hadwiger Conjecture", "statement": "Conjecture Every $K_t$-minor-free graph is $c t$-list-colourable for some constant $c\\geq1$.", "background": "Source: Open Problem Garden. Original node ID: 59943. URL: http://www.openproblemgarden.org/op/list_hadwiger_conjecture.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/list_hadwiger_conjecture\n- Author(s): Kawarabayashi, Ken-ichi; Mohar, Bojan\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: Hadwiger conjecture; list colouring; minors\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 7th, 2014 by David Wood\n\nProblem-page discussion:\nHadwiger's conjecture asserts that every $K_t$-minor-free graph is $(t − 1)$-colourable. Robertson, Seymour and Thomas [RST] proved Hadwiger's conjecture for $t \\leq 6$. It remains open for $t \\geq 7$. In fact, it is open whether every $K_t$-minor-free graph is $ct$-colourable for some constant $c\\geq 1$. It is natural to consider analogous problems for list colourings.\n\nFirst, consider planar graphs. While every planar graph is 4-colourable, Erdös, Rubin and Taylor. [ERT] conjectured that some planar graph is not 4-list-colourable, and that every planar graph is 5-list-colourable. The first conjecture was verified by Voigt [V] and the second by Thomassen [T].\n\nMore generally, Borowiecki [B] asked whether every $K_t$-minor-free graph is $(t − 1)$-list-colourable, which is true for $t \\leq 4$ but false for $t = 5$ by Voigt’s example. Kawarabayashi and Mohar [KM] proposed the stated conjecture, and suggested it might be true with $c=\\frac{3}{2}$. Barát, Joret and Wood [BJW] proved that $c\\geq\\frac{4}{3}$. In particular, they constructed a $K_{3t+2}$-minor-free graph that is not $4t$-list-colourable.\n\nBibliography:\n[B] Mieczyslaw Borowiecki. Research problem 172. Discrete Math., 121:235–236, 1993..\n\n[BJW] Janos Barát, Gwenael Joret, David R. Wood. Disproof of the List Hadwiger Conjecture, Electronic J. Combinatorics 18:P232, 2011.\n\n[ERT] Paul Erdo ̋s, Arthur L. Rubin, and Herbert Taylor. Choosability in graphs. In Proc. West Coast Conference on Combinatorics, Graph Theory and Computing, vol. XXVI of Congress. Numer., pp. 125–157. Utilitas Math., 1980. MathSciNet.\n\n*[KM] Ken-ichi Kawarabayashi and Bojan Mohar. A relaxed Hadwiger’s conjecture for list colorings. J. Combin. Theory Ser. B, 97(4):647–651, 2007. MathSciNet.\n\n[RST] Neil Robertson, Paul D. Seymour, and Robin Thomas. Hadwiger’s conjecture for $K_6$-free graphs. Combinatorica, 13(3):279–361, 1993. MathSciNet.\n\n[T] Carsten Thomassen. Every planar graph is 5-choosable. J. Combin. Theory Ser. B, 62(1):180–181, 1994. MathSciNet.\n\nDiscussion links:\n- minor: http://en.wikipedia.org/wiki/Graph_minor\n- list colourings: http://en.wikipedia.org/wiki/List_coloring\n\nBibliography links:\n- Research problem 172: http://dx.doi.org/10.1016/0012-365X(93)90557-A\n- Disproof of the List Hadwiger Conjecture: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p232\n- Choosability in graphs: http://www.renyi.hu/∼p erdos/1980-07.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=593902\n- A relaxed Hadwiger’s conjecture for list colorings: http://dx.doi.org/10.1016/j.jctb.2006.11.002\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2325803\n- Hadwiger’s conjecture for $K_6$-free graphs: http://dx.doi.org/10.1007/BF01202354\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1238823\n- Every planar graph is 5-choosable: http://dx.doi.org/10.1006/jctb.1994.1062\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1290638\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"List Hadwiger Conjecture\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3254, "problem_number": "OPG-60001", "title": "Cycles in Graphs of Large Chromatic Number", "statement": "Conjecture If $\\chi(G)>k$, then $G$ contains at least $\\frac{(k+1)(k-1)!}{2}$ cycles of length $0\\bmod k$.", "background": "Source: Open Problem Garden. Original node ID: 60001. URL: http://www.openproblemgarden.org/op/cycles_in_graphs_of_large_chromatic_number.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cycles_in_graphs_of_large_chromatic_number\n- Author(s): Brewster, Richard C.; McGuinness, Sean; Moore, Benjamin; Noel, Jonathan A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: chromatic number; cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 20th, 2015 by Jon Noel\n\nProblem-page discussion:\nChudnovsky, Plumettaz, Scott and Seymour [CPSS] proved that every graph with chromatic number at least $4$ contains a cycle of length $0\\bmod 3$. A simpler proof was found by Wrochna [W]. Wrochna's argument was generalised by Brewster, McGuinness, Moore and Noel [BMMN] to the following: if $\\chi(G)>k$, then $G$ contains at least TeX Embedding failed! cycles of length $0\\bmod k$.} The compete graph on $k+1$ vertices has exactly $\\frac{(k+1)(k-1)!}{2}$ cycles of length $0\\bmod k$ and so the conjecture above, if true, would be best possible.\n\nBibliography:\n[BMMN] R. C. Brewster, S. McGuinness, B. Moore, J. A. Noel, A Dichotomy Theorem for Circular Colouring Reconfiguration, submitted, arXiv:1508.05573v1.\n\n[CPSS] M. Chudnovsky, M. Plumettaz, A. Scott, and P. Seymour, The Structure of Graphs with no Cycles of Length Divisible by Three, in preparation.\n\n[Wro] M. Wrochna, unpublished.\n\nRelated:\nRelated problems\nBounding the chromatic number of graphs with no odd hole\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Cycles in Graphs of Large Chromatic Number\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3255, "problem_number": "OPG-169", "title": "The Two Color Conjecture", "statement": "Conjecture If $G$ is an orientation of a simple planar graph, then there is a partition of $V(G)$ into $\\{X_1,X_2\\}$ so that the graph induced by $X_i$ is acyclic for $i=1,2$.", "background": "Source: Open Problem Garden. Original node ID: 169. URL: http://www.openproblemgarden.org/op/partitioning_planar_digraphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partitioning_planar_digraphs\n- Author(s): Neumann-Lara, Victor\n- Subject(s): Graph Theory; Directed Graphs\n- Keywords: acyclic; digraph; planar\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 26th, 2007 by mdevos\n\nProblem-page discussion:\nThis is a type of coloring digraphs introduced by V. Neumann-Lara. More generally, if $G$ is a digraph, we wish to partition the vertex set of $G$ into as few parts as possible so that each induces an acyclic subgraph.\n\nBibliography:\n* [N] V. Neumann-Lara (1985). Vertex colourings in digraphs. Some problems. Technical report, University of Waterloo.\n\nComments:\n- May 31st, 2007 | Anonymous | Riste Skrekovski conjectured: Riste Skrekovski conjectured at a graph theory meeting in Budmerice, Slovakia, in 2005 that this conjecture is false.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"The Two Color Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3256, "problem_number": "OPG-179", "title": "Woodall's Conjecture", "statement": "Conjecture If $G$ is a directed graph with smallest directed cut of size $k$, then $G$ has $k$ disjoint dijoins.", "background": "Source: Open Problem Garden. Original node ID: 179. URL: http://www.openproblemgarden.org/op/woodalls_conjecture.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/woodalls_conjecture\n- Author(s): Woodall, Douglas R.\n- Subject(s): Graph Theory; Directed Graphs\n- Keywords: digraph; packing\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 5th, 2007 by mdevos\n\nProblem-page discussion:\nDefinitions: Let $G$ be a directed graph. A directed cut of $G$ is an edge-cut in which all edges are directed the same way. A dijoin is a set of edges which intersect every directed cut.\n\nThere is an important theorem of Lucchesi and Younger [LY] which asserts that the dual problem has an optimum integer packing. That is, for every digraph $G$ in which the smallest dijoin has size $k$, there exist $k$ pairwise disjoint directed cuts. This result implies more generally (in the terminology of Corneujols [C]) that the clutter of (minimal) directed cuts has the max-flow min-cut property (MFMC). It follows from this and Lehman's theorem [L] that the clutter of (minimal) dijoins is ideal. Therefore, if the smallest directed cut of $G$ has size $k$, then there exist nonnegative rational numbers $x_1,x_2,\\ldots,x_m$ summing to $k$ and dijoins $J_1,J_2,\\ldots,J_m$ so that if each dijoin $J_i$ is taken with weight $x_i$, the total weight of the dijoins containing any edge is at most 1. The above conjecture asserts that such a combination exists with $x_1,x_2,\\ldots,x_m$ integral.\n\nSchrijver [Sc80] has found a digraph in which the clutter of (minimal) dijoins does not have the MFMC property. Thus, the weighted version of Woodall's conjecture is not true in general. However, this clutter does have the MFMC property in a number of interesting special cases. One such example (due to Schrijver [Sc82] and Feofiloff, Younger [FY]) is when the digraph is acyclic and every source is joined to every sink by a directed path. Since every such digraph has the MFMC property, every such digraph must satisfy Woodall's conjecture.\n\nThere seem to be few positive results towards Woodall's conjecture for general digraphs. Although this was probably already known, Seymour and I (M. DeVos) observed that the conjecture is true for $k=2$. To see this, note that the underlying graph is 2-edge-connected, so it may be oriented to give a strongly connected digraph, call it $H$. Now partition the edges into two sets $\\{X,Y\\}$ where $X$ consists of those edges which have the same orientation in both $H$ and $G$, and $Y$ consists of those edges with different orientations in the two graphs. It is immediate that both $X$ and $Y$ meet every directed cut, so each is a dijoin. Extending this to $k=3$ appears difficult. Indeed, I believe the following weak version of this is still open.\n\nConjecture Prove that there exists a fixed integer $k$ so that every digraph with all directed cuts of size $\\ge k$ contains three pairwise disjoint dijoins.\n\nThe restriction of this conjecture to the special case of planar graphs is also open. Here we can use duality to restate the conjecture. Call a set of edges $A$ a feedback arc-set if $A$ intersects every directed cycle (or equivalently, $G-A$ is acyclic).\n\nConjecture If $G$ is a planar digraph with all directed cycles of length $\\ge k$, then $G$ contains $k$ pairwise disjoint feedback arc-sets.\n\nThe above conjecture is known to fail when the assumption of planarity is removed. On the other hand, it does hold for digraphs which are orientations of series-parallel graphs [LW]. It is also still open for $k=3$.\n\nBibliography:\n[C] G. Cornuejols, Combinatorial Optimization, Packing and Covering SIAM, Philadelphia (2001). MathSciNet\n\n[FY] P. Feofiloff and D. H. Younger, Directed cut transversal packing for source-sink connected graphs. Combinatorica 7 (1987), no. 3, 255--263. MathSciNet\n\n[LW] O. Lee, Y. Wakabayashi, Note on a min-max conjecture of Woodall. J. Graph Theory 38 (2001), no. 1, 36--41. MathSciNet\n\n[LY] C.L. Lucchesi and D. H. Younger A minimax theorem for directed graphs, Journal of the London Math. Soc. (2) 17 (1978) 369-374. MathSciNet\n\n[Sc80] A. Schrijver, A counterexample to a conjecture of Edmonds and Giles, Discrete Math. 32 (1980) 213-214. MathSciNet\n\n[Sc82] A. Schrijver, Min-max relations for directed graphs, Annals of Discrete Math. 16 (1982) 261-280. MathSciNet\n\n[W] D. R. Woodall, Menger and König systems. Theory and applications of graphs (Proc. Internat. Conf., Western Mich. Univ., Kalamazoo, Mich., 1976), pp. 620--635, Lecture Notes in Math., 642, Springer, Berlin, 1978. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1828452\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0918396\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1849557\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0500618\n- A counterexample to a conjecture of Edmonds and Giles: http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.1016/0012-365X(80)90057-6\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0592858\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0686312\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0499529\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Woodall's Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3257, "problem_number": "OPG-611", "title": "The Bermond-Thomassen Conjecture", "statement": "Conjecture For every positive integer $k$, every digraph with minimum out-degree at least $2k-1$ contains $k$ disjoint cycles.", "background": "Source: Open Problem Garden. Original node ID: 611. URL: http://www.openproblemgarden.org/op/the_bermond_thomassen_conjecture.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_bermond_thomassen_conjecture\n- Author(s): Bermond, Jean-Claude; Thomassen, Carsten\n- Subject(s): Graph Theory; Directed Graphs\n- Keywords: cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 1st, 2007 by JS\n\nProblem-page discussion:\nThis conjecture is a simple observation when $k=1$. It was proved by Thomassen~[Tho83] in 1983 when $k=2$, and more recently the case $k=3$ was settled~[LPS07].\n\nThe bound offered would be optimal — just consider a symmetric complete graph on $2k-1$ vertices. In 1996, Alon~[Alo96] proved that the statement is true with $2k-1$ replaced by $64k$. The conjecture was also verified for tournaments of minimum in-degree at least $2k-1$ ~[BLS07].\n\nBang-Jensen et al. [BBT] made a stronger conjecture for digraph with sufficiently large girth.\n\nConjecture For every integer $g >1$, every digraph $D$ with girth at least $g$ and with minimum out-degree at least $\\frac{g}{g-1}k$ contains $k$ disjoint cycles.\n\nThe constant $\\frac{g}{g-1}$ is best possible. Indeed, for every integers $p$ and $g$, consider the digraph $D(g,p)$ on $n = p(g − 1) + 1$ vertices with vertex set $\\{x_1, \\dots, x_n\\}$ and arc set $\\{x_ix_j: j − i \\mod n \\in \\{1,\\dots p\\}\\}$. It has girth $g$ and out-degree $p = \\left \\lfloor \\frac{g}{g−1} k \\right \\rfloor$. Moreover, for $n = 0 \\mod g$, the digraph $D(g,p)$ admits a partition into $k$ vertex disjoint 3-cycles and no more. For g = 3, the first case of this conjecture which differs from Bermond-Thomassen Conjecture and which is not already known corresponds to the following question:\n\nQuestion Does every digraph D without 2-cycles and out-degree at least 6 admit four vertex disjoint cycles?\n\nBibliography:\n[Alo96] N. Alon: Disjoint directed cycles, J. Combin. Theory Ser. B, 68(2):167--178, 1996. PDF\n\n[BBT] J. Bang-Jensen, S. Bessy and S. Thomassé, Disjoint 3-cycles in tournaments: a proof of the Bermond-Thomassen conjecture for tournaments, J. Graph Theory, to appear.\n\n*[BeTh81] J.-C. Bermond and C.~Thomassen: Cycles in digraphs---a survey, J. Graph Theory, 5(1):1--43, 1981. MathSciNet\n\n[BLS07] S.~Bessy, N.~Lichiardopol, and J.-S. Sereni: Two proofs of the {B}ermond-{T}homassen conjecture for tournaments with bounded minimum in-degree, Discrete Math., Special Issue dedicated to CS06, to appear.\n\n[LPS07] N.~Lichiardopol, A.~ P\\'or, and J.-S. Sereni: A step towards the Bermond-Thomassen conjecture about disjoint cycles in digraphs, Submitted, 2007.\n\n[Tho83] C.~Thomassen, Disjoint cycles in digraphs, Combinatorica, 3(3-4):393--396, 1983. MathSciNet\n\nBibliography links:\n- PDF: http://www.math.tau.ac.il/%7Enogaa/PDFS/dicycles3.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR604304\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR729792\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"The Bermond-Thomassen Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3258, "problem_number": "OPG-646", "title": "Seymour's Second Neighbourhood Conjecture", "statement": "Conjecture Any oriented graph has a vertex whose outdegree is at most its second outdegree.", "background": "Source: Open Problem Garden. Original node ID: 646. URL: http://www.openproblemgarden.org/op/seymours_second_neighbourhood_conjecture.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/seymours_second_neighbourhood_conjecture\n- Author(s): Seymour, Paul D.\n- Subject(s): Graph Theory; Directed Graphs\n- Keywords: Caccetta-Häggkvist; neighbourhood; second; Seymour\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: October 9th, 2007 by nkorppi\n\nProblem-page discussion:\nBy the $n$ th outdegree of $v$, we mean the number of vertices for which the minimal outward-directed path from $v$ to them is of length $n$.\n\nChen, Shen, and Yuster [CSY] proved that in any oriented graph there is a vertex whose second outdegree is at least $\\gamma$ times its outdegree, where $\\gamma=0.657298...$ is the unique real root of $2x^3+x^2 -1=0$.\n\nThis conjecture implies a special case of the \\Oprefnum[Caccetta-Häggkvist Conjecture]{46385}.\n\nBibliography:\n[ASY] Chen, G.; Shen, J.; Yuster, R. Second neighborhood via first neighborhood in digraphs, Annals of Combinatorics, 7 (2003), 15--20.\n\n[F] Fisher, David C. Squaring a tournament: a proof of Dean's conjecture. J. Graph Theory 23 (1996), no. 1, 43--48.\n\n[KL] Kaneko, Yoshihiro; Locke, Stephen C. The minimum degree approach for Paul Seymour's distance 2 conjecture. Proceedings of the Thirty-second Southeastern International Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, LA, 2001). Congr. Numer. 148 (2001), 201--206.\n\nRelated:\nRelated problems\nCaccetta-Häggkvist Conjecture\n\nComments:\n- December 12th, 2009 | Anonymous | It is proved: This conjecture has been proved. You can find the proof here.\n- December 12th, 2009 | Robert Samal | Re: It is proved: Thanks for the reference. The proof, however, seems flawed. The basic outline of the proof (if I understood it correctly) is as follows:\n\n- If $G$ has a sink, then this sink satisfies the conditions.\n- An oriented graph without directed cycles has a sink.\n- Then one proves directly that a graph containing a directed cycle has a vertex (even on that cycle) that satisfies the conditions.\n\nThe last part (Lemma 2 of the manuscript), is not true -- take a directed cycle $C$, add many independent points $X$, and add all the arcs from $C$ to $X$. Now the conjecture is true for this graph (one can take any of the sinks -- vertices in $X$ ), but it is not true that one can choose a vertex of $C$.\n\nThe omission in the proof of Lemma 2 is that it's tacitly assumed, that adjacent vertices of the cycle have no common out-neighbours.\n- March 14th, 2010 | Anonymous | So is this proof valid or: So is this proof valid or not? I am currently working on a proof and am wondering whether or not I should be bothering\n- August 8th, 2010 | rs | Re: So is this proof valid or: I believe the mentioned prof is not valid. Is your proof working? And sorry for the late reply, our system falsely recognized your comment as a spam.\n- March 18th, 2011 | sjcjoosten | Counterexample for the proof: As was indicated, Lemma 2 is false. Here is a counterexample to lemma 2. Note that it is not a counterexample to the conjecture, since it has two sinks.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Seymour's Second Neighbourhood Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3259, "problem_number": "OPG-1793", "title": "Non-edges vs. feedback edge sets in digraphs", "statement": "For any simple digraph $G$, we let $\\gamma(G)$ be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and $\\beta(G)$ be the size of the smallest feedback edge set.\n\nConjecture If $G$ is a simple digraph without directed cycles of length $\\le 3$, then $\\beta(G) \\le \\frac{1}{2} \\gamma(G)$.", "background": "Source: Open Problem Garden. Original node ID: 1793. URL: http://www.openproblemgarden.org/op/non_edges_vs_feedback_edge_sets_in_digraphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/non_edges_vs_feedback_edge_sets_in_digraphs\n- Author(s): Chudnovsky, Maria; Seymour, Paul D.; Sullivan, Blair\n- Subject(s): Graph Theory; Directed Graphs\n- Keywords: acyclic; digraph; feedback edge set; triangle free\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 8th, 2008 by mdevos\n\nProblem-page discussion:\nIf $G$ satisfies $\\gamma(G) = 0$, then $G$ is a tournament, and it is easy to check that $G$ will have a directed cycle of length three unless it is acyclic, in which case $\\beta(G) = 0$. So in this case, the conjecture holds. More generally, it is natural to suspect that a digraph with few non-edges and no directed triangles should be close to acyclic. Indeed, this conjecture asserts a precise relationship of this form.\n\nIf true, the above conjecture is essentially tight for a number of examples. We noted above that it is tight for transitive tournaments. Here is another basic class: let $G_k$ be the circulant digraph obtained by placing $3k+1$ vertices in a circle, and adding an edge directed from $u$ to $v$ whenever $v$ is distance $\\le k$ from $u$ in the clockwise order. Such examples may be nested to obtain new ones.\n\nChudnovsky, Seymour, and Sullivan [CSS] utilized a clever double counting argument to prove that $\\beta(G) \\le \\gamma(G)$ always holds. They also proved their conjecture in the case when $V(G)$ is the union of two cliques, and when $G$ is a circular interval digraph.\n\nBibliography:\n*[CSS] M. Chudnovsky, P.D. Seymour, and B. Sullivan, Cycles in dense digraphs.\n\nSource links:\n- feedback edge set: http://en.wikipedia.org/wiki/feedback arc set\n\nBibliography links:\n- Cycles in dense digraphs: http://www.math.princeton.edu/%7Epds/papers/triangles/paper.ps\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Non-edges vs. feedback edge sets in digraphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3260, "problem_number": "OPG-46167", "title": "Oriented trees in n-chromatic digraphs", "statement": "Conjecture Every digraph with chromatic number at least $2k-2$ contains every oriented tree of order $k$ as a subdigraph.", "background": "Source: Open Problem Garden. Original node ID: 46167. URL: http://www.openproblemgarden.org/op/oriented_trees_in_n_chromatic_digraphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/oriented_trees_in_n_chromatic_digraphs\n- Author(s): Burr, S. A.\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 25th, 2013 by fhavet\n\nProblem-page discussion:\nThe conjectured bound is best possible, because a regular tournament of order $2k-3$ does not contain the oriented tree consisting of a vertex dominating $k-1$ leaves.\n\nLet $f$ be the function $f$ such that every oriented tree of order $k$ is $f(k)$-universal, that is contained in every digraph with chromatic number at least $f(k)$. Burr proved that $f(k) \\leq (k-1)^2$. This was slightly improved by Addario-Berry et al. [AHL+] who proved $f(k)\\leq k^2/2-k/2+1$.\n\nBurr's conjecture has been proved only in few particular cases of digraphs: tournaments, and acyclic digraphs. Kühn, Mycroft, and Osthus [KMS] showed that every oriented tree of order $k$ is contained in every tournament of order $2k-2$ for all sufficiently large $k$ (so proving a Conjecture of Sumner); Addario-Berry et al. [AHL+] proved that every acyclic digraph with chromatic number $k$ contains every oriented tree of order $k$.\n\nBurr's conjecture or some approximation have been also proved for special classes of trees. Gallai-Roy's celebrated theorem states that every directed path of order $k$ is $k$-universal; El-Sahili [E] proved that every oriented path of order $4$ is $4$-universal and that the antidirected path of order $5$ is $5$-universal; Addario-Berry, Havet, and Thomassé [AHT] showed that every oriented path of order $k$ whose vertex set can be partioned into two directed paths is $k$-universal; Addario-Berry et al. [AHL+] showed that antidirected trees (oriented trees in which every vertex has in-degree $0$ or out-degree $0$ ) are $5k$-universal.\n\nHavet, generalizing a conjecture of Havet and Thomassé (see [H]) on tournaments, conjectured that the following could also be true.\n\nConjecture Every digraph with chromatic number at least $k+\\ell+1$ contains every oriented tree of order $k$ with $k$ leaves.\n\nBibliography:\n[AHL+] L. Addario-Berry, F. Havet, C. Linhares Sales, B. Reed, and S. Thomassé. Oriented trees in digraphs. Discrete Mathematics, 313(8):967-974, 2013.\n\n[AHT] L. Addario-Berry, F. Havet, and S. Thomassé, Paths with two blocks in $n$-chromatic digraphs, J. of Combinatorial Theory Ser. B, 97 (2007), 620--626.\n\n* [B] A. Burr, Subtrees of directed graphs and hypergraphs, Proceedings of the Eleventh Southeastern Conference on Combinatorics, Graph Theory and Computing, Boca Raton, Congr. Numer., 28 (1980), 227--239.\n\n[H] F. Havet, Trees in tournaments. Discrete Mathematics 243 (2002), no. 1-3, 121--134.\n\n[KOM] D. Kühn, D. Osthus, and R. Mycroft, A proof of Sumner's universal tournament conjecture for large tournaments, Proceedings of the London Mathematical Society 102 (2011), 731--766.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Oriented trees in n-chromatic digraphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3261, "problem_number": "OPG-46279", "title": "Antidirected trees in digraphs", "statement": "An antidirected tree is an orientation of a tree in which every vertex has either indegree 0 or outdergree 0.\n\nConjecture Let $D$ be a digraph. If $|A(D)| > (k-2) |V(D)|$, then $D$ contains every antidirected tree of order $k$.", "background": "Source: Open Problem Garden. Original node ID: 46279. URL: http://www.openproblemgarden.org/op/antidirected_trees_in_digraphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/antidirected_trees_in_digraphs\n- Author(s): Addario-Berry, Louigi; Havet, Frédéric; Linhares Sales, Claudia; Reed, Bruce A.; Thomassé, Stéphan\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 26th, 2013 by fhavet\n\nProblem-page discussion:\nThe value $k-2$ would be best possible, since the oriented tree consisting of a vertex dominating $k-1$ other vertices is not contained in any digraph in which every vertex has outdegree $k-2$. The condition on the trees be antidirected cannot be suppressed. In a bipartite digraph $D$ with bipartition $(A,B)$ such that all arcs are directed from $A$ to $B$, all the trees contained in $D$ are antidirected.\n\nThis conjecture for symmetric digraphs is equivalent to the celebrated Erdös-Sos conjecture for undirected graphs. (see [E]).\n\nConjecture Let $G$ be a graph. If $|E(G)| > \\frac{1}{2} (k-2) |V(G)|$, then $G$ contains every tree of order $k$.\n\nAddario-Berry et al. Conjecture also implies Burr's conjecture (see Oriented trees in n-chromatic digraphs) for antidirected trees, since every digraph with chromatic number $2k-2$ contains a colour-critical digraph has minimum degree at least $2k-3$, and so whose number of vertices is at least $\\frac{2k-3}{2}|V(D)|$, which exceeds $(k-2) |V(D)|$.\n\nThis conjecture has only been proved [AHL+] for antidirected trees of diameter at most $3$.\n\nBibliography:\n*[AHL+] L. Addario-Berry, F. Havet, C. Linhares Sales, B. Reed, and S. Thomassé. Oriented trees in digraphs. Discrete Mathematics, 313(8):967-974, 2013.\n\n[E] P. Erdös, Some problems in graph theory, Theory of Graphs and Its Applications, M. Fielder, Editor, Academic Press, New York, 1965, pp. 29--36.\n\nRelated:\nRelated problems\nOriented trees in n-chromatic digraphs\n\nDiscussion links:\n- Oriented trees in n-chromatic digraphs: http://www.openproblemgarden.org/?q=node/46167]\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Antidirected trees in digraphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3262, "problem_number": "OPG-46359", "title": "Directed path of length twice the minimum outdegree", "statement": "Conjecture Every oriented graph with minimum outdegree $k$ contains a directed path of length $2k$.", "background": "Source: Open Problem Garden. Original node ID: 46359. URL: http://www.openproblemgarden.org/op/directed_cycle_of_length_twice_the_minimum_outdegree.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/directed_cycle_of_length_twice_the_minimum_outdegree\n- Author(s): Thomassé, Stéphan\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 28th, 2013 by fhavet\n\nProblem-page discussion:\nIn fact, Thomassé made the following stronger conjecture which implies the celebrated Cacetta-Häggkvist Conjecture.\n\nConjecture Every digraph with minimum outdegree $k$ and directed girth $g$, contains a directed path of length $(g-1)k$.\n\nThis conjecture holds easily when $g=2$. For $g=3$ it is the above conjecture which is still open.\n\nBibliography:\n*[S] Blair D. Sullivan: A Summary of Problems and Results related to the Caccetta-Haggkvist Conjecture\n\nRelated:\nRelated problems\nCaccetta-Häggkvist Conjecture\n\nDiscussion links:\n- Cacetta-Häggkvist Conjecture: http://www.openproblemgarden.org/?q=node/46385\n\nBibliography links:\n- A Summary of Problems and Results related to the Caccetta-Haggkvist Conjecture: http://www.arxiv.org/abs/math.CO/0605646\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Directed path of length twice the minimum outdegree\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3263, "problem_number": "OPG-46385", "title": "Caccetta-Häggkvist Conjecture", "statement": "Conjecture Every simple digraph of order $n$ with minimum outdegree at least $r$ has a cycle with length at most $\\lceil n/r\\rceil$", "background": "Source: Open Problem Garden. Original node ID: 46385. URL: http://www.openproblemgarden.org/op/caccetta_haggkvist_conjecture.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/caccetta_haggkvist_conjecture\n- Author(s): Caccetta, L.; Häggkvist, Roland\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 28th, 2013 by fhavet\n\nProblem-page discussion:\nIt is one of the most famous conjectures in graph theory. It has many alternative formulations and lots of work have been done around it. Many interesting conjectures are related to it. See [Sul]. It is in particular implied by a conjecture of Thomassé and Hoàng-Reed Conjecture.\n\nThe Caccetta-Häggkvist Conjecture is a generalization of an earlier conjecture of Behzad, Chartrand, and Wall, who conjectured it only for diregular digraphs. Caccetta-H äggkvist Conjecture has been proved for $r\\leq \\sqrt{n/2}$ by Shen [She1]. For $r\\geq n/2$ it is trivial. But already for $r=n/3$, it is still open as well as Behzad-Chartrand-Wall Conjecture\n\nConjecture Every simple $n$-vertex digraph with minimum outdegree at least $r/3$ and minimum indegree at least $r/3$ has a cycle with length at most $3$.\n\nThis conjecture would be implied by Seymour's Second Neighbourhood Conjecure.\n\nShen [She2] also proved the following approximate version.\n\nTheorem Every simple digraph of order $n$ with minimum outdegree at least $r$ has a cycle with length at most $n/r + 73$.\n\nBollobás and Scott [BS] proposed a weighted version of the Caccetta-Häggkvist Conjecture.\n\nConjecture Let $w:E(D) \\rightarrow [0,1]$ be a weight function on the arcs of a digraph $D$. If $\\sum_{u\\in N^-(v)} w(uv) \\geq 1$ and $\\sum_{u\\in N^+(v)} w(vu) \\geq 1$ for all $v\\in V(D)$, then there is a directed cycle in $D$ of total weight at least 1.\n\nThey gave a nice proof that there is a directed path of total weight at least 1.\n\nBibliography:\n[BCW] M. Behzad, G. Chartrand, and C. Wall. On minimal regular digraphs with given girth. Fundamenta Mathematicae, 69:227–231, 1970.\n\n[BS] B. Bollobás and A. D. Scott, A proof of a conjecture of {B}ondy concerning paths in weighted digraphs. J. Combin. Theory Ser. B, 66:283-292, 1996.\n\n*[CH] L. Caccetta and R. Häggkvist. On minimal digraphs with given girth. Congressus Numerantium, XXI, 1978\n\n[She1J. Shen. On the girth of digraphs. Discrete Math, 211(1-3):167–181, 2000.\n\n[She2] J. Shen. On the Caccetta-Häggkvist conjecture. Graphs and Combinatorics, 18(3):645–654, 2002.\n\n[Sul] Blair D. Sullivan: A Summary of Problems and Results related to the Caccetta-Häggkvist Conjecture\n\nRelated:\nRelated problems\nSeymour's Second Neighbourhood Conjecture\nDirected path of length twice the minimum outdegree\nHoàng-Reed Conjecture\n\nDiscussion links:\n- conjecture of Thomassé: http://www.openproblemgarden.org/?q=node/46359\n- Hoàng-Reed Conjecture: http://www.openproblemgarden.org/?q=node/47282\n- Seymour's Second Neighbourhood Conjecure: http://www.openproblemgarden.org/?q=node/646\n\nBibliography links:\n- A Summary of Problems and Results related to the Caccetta-Häggkvist Conjecture: http://www.arxiv.org/abs/math.CO/0605646\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Caccetta-Häggkvist Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3264, "problem_number": "OPG-46432", "title": "Ádám's Conjecture", "statement": "Conjecture Every digraph with at least one directed cycle has an arc whose reversal reduces the number of directed cycles.", "background": "Source: Open Problem Garden. Original node ID: 46432. URL: http://www.openproblemgarden.org/op/adams_conjecture.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/adams_conjecture\n- Author(s): Ádám, András\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 1st, 2013 by fhavet\n\nProblem-page discussion:\nThe conjecture fails for multigraphs (multiple arcs are allowed). Counterexamples for multidigraphs have been given by Grinberg [G], Jirásek [J] and Thomassen [T].\n\nSurprisingly, the conjecture remains open for tournaments.\n\nConjecture Every tournament that is not transitive has an arc whose reversal reduces the number of directed cycles.\n\nBibliography:\n*[A] A. Ádám, Problem 2. In Theory of Graphs and its Applications (M. Fiedler, ed.), 234. (1964) Publishing House of the Czechoslovak Academy of Sciences, Prague.\n\n[G] E.Y. Grinberg, Examples of non-Ádám multigraphs (in Russian) Latv. Mat. Ezhegodnik, 31 (1988), pp. 128–138\n\n[J] J. Jirásek, On a certain class of multidigraphs, for which reversal of no arc decreases the number of their cycles, Comment. Math. Univ. Carolinae, 28 (1987), pp. 185–189.\n\n[T] C. Thomassen, Counterexamples to Ádám's conjecture on arc reversals in directed graphs, J. Combin. Theory Ser. B, 42 (1987), pp. 128–130.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Ádám's Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3265, "problem_number": "OPG-46456", "title": "Splitting a digraph with minimum outdegree constraints", "statement": "Problem Is there a minimum integer $f(d)$ such that the vertices of any digraph with minimum outdegree $d$ can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least $d$?", "background": "Source: Open Problem Garden. Original node ID: 46456. URL: http://www.openproblemgarden.org/op/splitting_a_digraph_with_minimum_outdegree_constraints.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/splitting_a_digraph_with_minimum_outdegree_constraints\n- Author(s): Alon, Noga\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 1st, 2013 by fhavet\n\nProblem-page discussion:\nThomassen [T] proved the conjecture when $d=1$ and showed $f(1)=3$. In fact, this case is equivalent to the case $k=2$ of the Bermond-Thomassen Conjecture.\n\nThe existence of the corresponding function $f$ for the undirected analogue is easy and has been observed by many authors. Stiebitz [S] even proved the following tight result: if the minimum degree of an undirected graph $G$ is $d_1+d_2+ \\cdots + d_k$, where each $d_i$ is a non-negative integer, then the vertex set of $G$ can be partitioned into $k$ pairwise disjoint sets $V_1,\\dots, V_k$, so that for all $i$, the induced subgraph on $V_i$ has minimum degree at least $d_i$. This is clearly tight, as shown by an appropriate complete graph.\n\nBibliography:\n*[A] Noga Alon, Disjoint Directed Cycles, Journal of Combinatorial Theory, Series B, 68 (1996), no. 2, 167-178.\n\n[S] M. Stiebitz, Decomposing graphs under degree constraints, J. Graph Theory 23 (1996), 31-324.\n\n[T] C. Thomassen, Disjoint cycles in digraphs, Combinatorica 3 (1983), 393 - 396.\n\nRelated:\nRelated problems\nThe Bermond-Thomassen Conjecture\n\nDiscussion links:\n- Bermond-Thomassen Conjecture: http://www.openproblemgarden.org/?q=node/611\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Splitting a digraph with minimum outdegree constraints\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3266, "problem_number": "OPG-46460", "title": "Long directed cycles in diregular digraphs", "statement": "Conjecture Every strong oriented graph in which each vertex has indegree and outdegree at least $d$ contains a directed cycle of length at least $2d+1$.", "background": "Source: Open Problem Garden. Original node ID: 46460. URL: http://www.openproblemgarden.org/op/long_directed_cycles_in_digraph_with_minimum_in_and_out_degree.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/long_directed_cycles_in_digraph_with_minimum_in_and_out_degree\n- Author(s): Jackson, Bill\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 1st, 2013 by fhavet\n\nProblem-page discussion:\nThe disjoint union of two regular tournaments on $2d+1$ vertices shows that this would be best possible.\n\nIf the oriented graph has order at most $4d+1$, Jackson conjecture the existence of a longer cycle, namely a Hamilton cycle\n\nBibliography:\n*[J] B. Jackson. Long paths and cycles in oriented graphs. J. Graph Theory 5 (1981), 145--157.\n\nRelated:\nRelated problems\nDirected path of length twice the minimum outdegree\nHamilton cycle in small d-diregular graphs\n\nDiscussion links:\n- Hamilton cycle: http://www.openproblemgarden.org/?q=node/47028\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Long directed cycles in diregular digraphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3267, "problem_number": "OPG-46495", "title": "Arc-disjoint out-branching and in-branching", "statement": "Conjecture There exists an integer $k$ such that every $k$-arc-strong digraph $D$ with specified vertices $u$ and $v$ contains an out-branching rooted at $u$ and an in-branching rooted at $v$ which are arc-disjoint.", "background": "Source: Open Problem Garden. Original node ID: 46495. URL: http://www.openproblemgarden.org/op/arc_disjoint_out_branching_and_in_branching.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/arc_disjoint_out_branching_and_in_branching\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 2nd, 2013 by fhavet\n\nProblem-page discussion:\nThomassen [T] showed that, given a digraph $D$ and two vertices $u$ and $v$, deciding whether there are an out-branching rooted at $u$ and an in-branching rooted at $v$ which are arc-disjoint is NP-complete.\n\nIn contrast, one can decide in polynomial time whether there are $k$ arc-disjoint out-branchings with specified roots $s_1, \\dots, s_k$ (some of which may be identical). This is a consequence of Edmonds’ well known branching theorem [E] states that a digraph $D$ has $k$ arc-disjoint out-branchings rooted at some fixed vertex $s$ if and only if there are $k$ arc-disjoint paths from $s$ to every other vertex of $D$.\n\nBang-Jensen [B] proved this conjecture for tournaments.\n\nA similar question can be asked about arc-disjoint strongly connected spanning subdigraphs. Several related problems are mentioned in the survey of Bang-Jensen and Kriesell [BK].\n\nBibliography:\n[B] J. Bang-Jensen, Edge-disjoint in- and out-branching in tournaments and related path problems. J. Combin. Theory Ser. B 51 (1991), 1-23.\n\n[BK] J. Bang-Jensen, M. Kriesell, Disjoint sub(di)graphs in digraphs, Electronic Notes in Discrete Mathematics 34 (2009), 179-183.\n\n[E] J. Edmonds, Edge-disjoint branchings. In Combinatorial Algorithms, B. Rustin, ed., Acad. Press, New York (1973), 91-96.\n\n*[T] C. Thomassen, Configurations in Graphs, Annals of The New York Acad. Sci. 555 (1989), 402-412.\n\nRelated:\nRelated problems\nArc-disjoint strongly connected spanning subdigraphs\n\nDiscussion links:\n- arc-disjoint strongly connected spanning subdigraphs: http://www.openproblemgarden.org/?q=node/46496\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Arc-disjoint out-branching and in-branching\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3268, "problem_number": "OPG-46680", "title": "Subdivision of a transitive tournament in digraphs with large outdegree.", "statement": "Conjecture For all $k$ there is an integer \u000e $f(k)$ such that every digraph of minimum outdegree at least \u000e $f(k)$ contains a subdivision of a transitive tournament of order $k$.", "background": "Source: Open Problem Garden. Original node ID: 46680. URL: http://www.openproblemgarden.org/op/subdivision_of_a_transitive_tournament_in_digraphs_with_large_outdegree.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/subdivision_of_a_transitive_tournament_in_digraphs_with_large_outdegree\n- Author(s): Mader, W.\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nA fundamental result of Mader [M1] states that for every integer $k$ there is a smallest $g(k)$ so that every graph of average degree at least $g(k)$ contains a subdivision of a complete graph on $k$ vertices. Bollobás and Thomason [BT] as well as Komlós and Szemerédi [KS] showed that $g$ is quadratic in $k$.\n\nThe above conjecture is a digraph analogue of this result. However one cannot replace the minimum outdegree in this conjecture by the average degree as in Mader's analogue for graphs: consider the complete bipartite graph $K_{n,n}$ and orient all edges from the first to the second class. The resulting digraph has average degree $n$ but not even a transitive tournament on 3 vertices.\n\nOne might be tempted to conjecture that large minimum outdegree would even force the existence of a subdivision of a large complete digraph. However, for all $n$ Thomassen [T] constructed a digraph on $n$ vertices whose minimum outdegree is at least $\\frac{1}{2} \\log_2 n$ but which does not contain an even directed cycle (and thus no complete digraph on 3 vertices). A simpler construction was found by DeVos et al. [DMMS].\n\nIt is easy to see that \u000e $f(1)=0$ and $f(2)=1$. Mader [M3] showed that $f(4) = 3$. Even the existence of \u000e $f(5)$ is not known.\n\nBibliography:\n[BT] B. Bollobás and A. Thomason, Proof of a conjecture of Mader, Erdös and Hajnal on topological complete subgraphs, European Journal of Combinatorics 19 (1998), 883–887.\n\n[DMMS] M. DeVos, J. McDonald, B. Mohar, and D. Scheide, Immersing complete digraphs, European Journal of Combinatorics, 33 (2012), no 6, 1294-1302.\n\n[KS] J. Komlós and E. Szemerédi, Topological Cliques in Graphs II, Combinatorics, Probability and Computing 5 (1996), 70–90.\n\n[M1] W. Mader, Homomorphieeigenschaften und mittlere Kantendichte von Graphen, Math. Annalen 174 (1967), 265–268.\n\n* [M2] W. Mader, Degree and Local Connectivity in Digraphs, Combinatorica 5 (1985), 161–165.\n\n[M3] W. Mader, On Topological Tournaments of order 4 in Digraphs of Outdegree 3, Journal of Graph Theory 21 (1996), 371–376.\n\n[T] C. Thomassen, Even Cycles in Directed Graphs, European Journal of Combinatorics 6 (1985), 85–89.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Subdivision of a transitive tournament in digraphs with large outdegree.\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3269, "problem_number": "OPG-47028", "title": "Hamilton cycle in small d-diregular graphs", "statement": "An directed graph is $k$-diregular if every vertex has indegree and outdegree at least $k$.\n\nConjecture For $d >2$, every $d$-diregular oriented graph on at most $4d+1$ vertices has a Hamilton cycle.", "background": "Source: Open Problem Garden. Original node ID: 47028. URL: http://www.openproblemgarden.org/op/hamilton_cycle_in_small_d_diregular_graphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hamilton_cycle_in_small_d_diregular_graphs\n- Author(s): Jackson, Bill\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 8th, 2013 by fhavet\n\nProblem-page discussion:\nThe disjoint union of two regular tournaments on $2d+1$ vertices shows that this would be best possible. For $d$-diregular oriented graphs with an arbitrary order of vertices, Jackson conjectured the existence of a long cycle.\n\nKühn and Osthus [KO] conjectured that it may actually be possible to increase the size of the graph even further if we assume that the graph is strongly 2-connected.\n\nProblem Is it true that for each $d >2$, every $d$-regular strongly $2$-connected oriented graph $G$ on at most $6d$ vertices has a Hamilton cycle?\n\nBibliography:\n*[J] B. Jackson. Long paths and cycles in oriented graphs, J. Graph Theory 5 (1981), 145-157.\n\n[KO] D. Osthus and D. Kühn, A survey on Hamilton cycles in directed graphs, European J. Combinatorics 33 (2012), 750-766.\n\nRelated:\nRelated problems\nLong directed cycles in diregular digraphs\n\nDiscussion links:\n- existence of a long cycle: http://www.openproblemgarden.org/?q=node/46460\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamilton cycle in small d-diregular graphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3270, "problem_number": "OPG-47282", "title": "Hoàng-Reed Conjecture", "statement": "Conjecture Every digraph in which each vertex has outdegree at least $k$ contains $k$ directed cycles $C_1, \\ldots, C_k$ such that $C_j$ meets $\\cup_{i=1}^{j-1}C_i$ in at most one vertex, $2 \\leq j \\leq k$.", "background": "Source: Open Problem Garden. Original node ID: 47282. URL: http://www.openproblemgarden.org/op/hoand_reed_conjecture.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hoand_reed_conjecture\n- Author(s): Hoang, Chinh T.; Reed, Bruce A.\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 11th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture is not even known to be true for $k=3$. In the case $k=2$, Thomassen proved [T] that every digraph with minimum outdegree 2 has two directed cycles intersecting on a vertex.\n\nThis conjecture would imply the Caccetta-Häggkvist Conjecture.\n\nBibliography:\n*[HR] C.T. Hoàng and B. Reed, A note on short cycles in digraphs, Discrete Math., 66 (1987), 103-107.\n\n[T] C. Thomassen, The 2-linkage problem for acyclic digraphs, Discrete Math., 55 (1985), 73-87.\n\nRelated:\nRelated problems\nCaccetta-Häggkvist Conjecture\n\nDiscussion links:\n- Caccetta-Häggkvist Conjecture: http://www.openproblemgarden.org/?q=node/46385\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Hoàng-Reed Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3271, "problem_number": "OPG-49573", "title": "Arc-disjoint directed cycles in regular directed graphs", "statement": "Conjecture If $G$ is a $k$-regular directed graph with no parallel arcs, then $G$ contains a collection of ${k+1 \\choose 2}$ arc-disjoint directed cycles.", "background": "Source: Open Problem Garden. Original node ID: 49573. URL: http://www.openproblemgarden.org/op/arc_disjoint_directed_cycles_in_regular_directed_graphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/arc_disjoint_directed_cycles_in_regular_directed_graphs\n- Author(s): Alon, Noga; McDiarmid, Colin; Molloy, Michael\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 17th, 2013 by fhavet\n\nProblem-page discussion:\nIf true, ${k+1 \\choose 2}$ would be best possible as shown by the complete symmetric digraph.\n\nAlon et al. [AMM] showed that a $k$-regular directed graph with no parallel arcs contains at least $\\frac{3}{2^{19}}k^2$ arc-disjoint directed cycles. It was then improved by Alon [A] who showed that every directed graph with minimum outdegree at least $k$ contains at least $\\frac{1}{128}k^2$ arc-disjoint directed cycles.\n\nBibliography:\n[A} N. Alon, Disjoint directed cycles, J. Combinatorial Theory, Ser. B, 68 (1996), 167-178.\n\n*[AMM] N. Alon, C. McDiarmid and M. Molloy, Edge-disjoint cycles in regular directed graphs, J. Graph Theory, 22 (1996), no. 3, 231-237.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Arc-disjoint directed cycles in regular directed graphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3272, "problem_number": "OPG-50631", "title": "Cyclic spanning subdigraph with small cyclomatic number", "statement": "Conjecture Let $D$ be a digraph all of whose strong components are nontrivial. Then $D$ contains a cyclic spanning subdigraph with cyclomatic number at most $\\alpha(D)$.", "background": "Source: Open Problem Garden. Original node ID: 50631. URL: http://www.openproblemgarden.org/op/cyclic_spanning_subdigraph_with_small_cyclomatic_number.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cyclic_spanning_subdigraph_with_small_cyclomatic_number\n- Author(s): Bondy, J. Adrian\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 2nd, 2013 by fhavet\n\nProblem-page discussion:\nThe {\\it cyclomatic number} of a digraph $D=(V,A)$ is $|A|-|V|+1$. For a strong digraph, it correspond to the minimum of directed ears in a directed ears decomposition. (See Chapter 5 of [BM]).\n\n$\\alpha(D)$ denotes the {\\it stability number} of the digraph $D$, that is the maximum number of pairwise non-adjacent vertices.\n\nBessy and Thomassé [BT04] showed that any nontrivial strong digraph $D$ has a spanning subdigraph which is the union of $\\alpha$ directed cycles. However, the structure of this subdigraph might be rather complicated. This leads one to ask whether there always exists a spanning subdigraph whose structure is relatively simple, one which is easily seen to be the union of $\\alpha$ directed cycles. A natural candidate would be a spanning subdigraph built from a directed cycle by adding $\\alpha(D)-1$ directed ears. But or any $\\alpha \\geq 2$, there exists a digraph $D$ with stability number $\\alpha$ requiring at least $2\\alpha -2$ directed ears. See Chapter 19 of [BM08].\n\nA possible way around this problem is to allow spanning subdigraphs which are disjoint union of strong digraphs. Such digraph are called cyclic (because each arc lies on a directed cycle). The conjecture was formulated by Bondy[B], based on a remark of Chen and Manalastas [CM].\n\nThe Conjecture holds for $\\alpha(D)=1$ by Camion's Theorem [C] and also for $\\alpha(D)=2$ and $\\alpha(D)=3$ by theorems of Chen and Manalastas [CM] and S. Thomassé (unpublished), respectively.\n\nThe conjecture implies not only the above-mentioned Bessy--Thomassé Theorem, but also a result of Thomassé [Thom01], that the vertex set of any strong digraph $D$ with $\\alpha(D) \\geq 2$ can be partitioned into $\\alpha(D)-1$ directed paths, as well as another theorem of Bessy and Thomassé [BT03], that every strong digraph $D$ has a strong spanning subdigraph with at most $n+2\\alpha(D)-2$ arcs.\n\nBibliography:\n[BT03] S. Bessy and S. Thomassé, Every strong digraph has a spanning strong subgraph with at most $n+2\\alpha-2$ arcs. J. Combin. Theory Ser. B 87 (2003), 289--299.\n\n[BT04] S. Bessy and S. Thomassé, Three min-max theorems concerning cyclic orders of strong digraphs. In Integer Programming and Combinatorial Optimization, 132--138. Lecture Notes in Comput. Sci., Vol. 3064, Springer, Berlin.\n\n*[B95] J.A. Bondy, Basic graph theory: paths and circuits. In Handbook of Combinatorics, Vol. 1, 3--110. Elsevier, Amsterdam.\n\n[BM] J.A. Bondy and U.S.R. Murty, Graph Theory, volume 244 of Graduate Texts in Mathematics. Springer, 2008.\n\n[C] P. Camion, Chemins et circuits hamiltoniens des graphes complets. C. R. Acad. Sci. Paris 249 (1959), 2151--2152.\n\n[CM] C.C. Chen C.C. and Jr. P. Manalastas, Every finite strongly connected digraph of stability 2 has a Hamiltonian path. Discrete Math. 44 (1983), 243--250.\n\n[T] S. Thomassé, Covering a strong digraph by $\\alpha-1$ disjoint paths: a proof of Las Vergnas' conjecture. J. Combin. Theory Ser. B 83 (2001), 331--333.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Cyclic spanning subdigraph with small cyclomatic number\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3273, "problem_number": "OPG-52197", "title": "Large acyclic induced subdigraph in a planar oriented graph.", "statement": "Conjecture Every planar oriented graph $D$ has an acyclic induced subdigraph of order at least $\\frac{3}{5} |V(D)|$.", "background": "Source: Open Problem Garden. Original node ID: 52197. URL: http://www.openproblemgarden.org/op/large_acyclic_induced_subdigraph_in_a_planar_oriented_graph.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/large_acyclic_induced_subdigraph_in_a_planar_oriented_graph\n- Author(s): Harutyunyan, Ararat\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 25th, 2013 by fhavet\n\nProblem-page discussion:\nBorodin's 5-Colour Theorem states that every planar graph has an acyclic 5-colouring This implies that every planar oriented graph $D$ has an acyclic induced subdigraph of order at least $\\frac{2}{5} |V(D)|$.\n\nAlready improving this bound to $\\frac{1}{2} |V(D)|$ would be interesting: it is a relaxtion of both a Conjecture of Albertson and Berman stating that every planar graph $G$ has an induced forest of order $\\frac{1}{2} |V(G)|$ and a Conjecture of Neumann-Lara stating that every planar oriented graph can be split into two acyclic subdigraphs.\n\nIf true, this conjecture would be best possible.\n\nRelated:\nRelated problems\nLarge induced forest in a planar graph.\nThe Two Color Conjecture\n\nDiscussion links:\n- Conjecture of Albertson and Berman: http://www.openproblemgarden.org/?q=node/46634\n- Conjecture of Neumann-Lara: http://www.openproblemgarden.org/?q=node/169\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Large acyclic induced subdigraph in a planar oriented graph.\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3274, "problem_number": "OPG-52200", "title": "Erdős-Posa property for long directed cycles", "statement": "Conjecture Let $\\ell \\geq 2$ be an integer. For every integer $n\\geq 0$, there exists an integer $t_n=t_n(\\ell)$ such that for every digraph $D$, either $D$ has a $n$ pairwise-disjoint directed cycles of length at least $\\ell$, or there exists a set $T$ of at most $t_n$ vertices such that $D-T$ has no directed cycles of length at least $\\ell$.", "background": "Source: Open Problem Garden. Original node ID: 52200. URL: http://www.openproblemgarden.org/op/erdos_posa_property_for_long_directed_cycles.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/erdos_posa_property_for_long_directed_cycles\n- Author(s): Havet, Frédéric; Maia, Ana Karolinna\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 25th, 2013 by fhavet\n\nProblem-page discussion:\nThe case $\\ell=2$ has been proved by Reed et al. [RRST], hence solving a conjecture of Gallai [G] and Younger [Y]. The case $\\ell=2$ and $n=2$ has previously been solved by McCuaig [M], who proved that $t_2(2)=3$. Havet and Maia [HM] proved the case $\\ell=3$.\n\nThe analogous statement for undirected graph has been proved by Birmelé, Bondy and Reed [BBR], hence generalizing Erdős-Posa [EP] result for $\\ell =3$.\n\nBibliography:\n[BBR] E. Birmelé, J.A. Bondy, and B.A. Reed. The Erdos-Posa property for long circuits, Combinatorica, 27(2), 135–145, 2007.\n\n[EP] P. Erdős and L. Pósa. On the independent circuits contained in a graph. Canad. J. Math., 17, 347--352, 1965.\n\n[G] T. Gallai. Problem 6, in Theory of Graphs, Proc. Colloq. Tihany 1966 (New York), Academic Press, p.362, 1968.\n\n*[HM] F. Havet and A. K. Maia. On disjoint directed cycles with prescribed minimum lengths. INRIA Research Report, RR-8286, 2013.\n\n[M] W. McCuaig, Intercyclic digraphs. Graph Structure Theory, (Neil Robertson and Paul Seymour, eds.), AMS Contemporary Math., 147:203--245, 1993.\n\n[RRST] B. Reed, N. Robertson, P.D. Seymour, and R. Thomas. Packing directed circuits. Combinatorica, 16(4):535--554, 1996.\n\n[Y] D. H. Younger. Graphs with interlinked directed circuits. Proceedings of the Midwest Symposium on Circuit Theory, 2:XVI 2.1 - XVI 2.7, 1973.\n\nBibliography links:\n- On disjoint directed cycles with prescribed minimum lengths: http://hal.inria.fr/hal-00816135/en\n- Packing directed circuits: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.89.5838&rep=rep1&type=pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Erdős-Posa property for long directed cycles\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3275, "problem_number": "OPG-60028", "title": "Monochromatic reachability in arc-colored digraphs", "statement": "Conjecture For every $k$, there exists an integer $f(k)$ such that if $D$ is a digraph whose arcs are colored with $k$ colors, then $D$ has a $S$ set which is the union of $f(k)$ stables sets so that every vertex has a monochromatic path to some vertex in $S$.", "background": "Source: Open Problem Garden. Original node ID: 60028. URL: http://www.openproblemgarden.org/op/monochromatoc_reachability_in_arc_colored_digraphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/monochromatoc_reachability_in_arc_colored_digraphs\n- Author(s): Sands, Bill; Sauer, Norbert W.; Woodrow, Robert E.\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 4th, 2017 by fhavet\n\nProblem-page discussion:\nIn the particular case of tournaments (and more generally when the stabilty number of $D$ is bounded), it has been proved by Bousquet, Lochet, and Thomassé [BLT].\n\nBibliography:\n[BLT] Nicolas Bousquet, William Lochet, Stéphan Thomassé: A proof of the Erdős-Sands-Sauer-Woodrow conjecture,\n\n[SSW] B. Sands, N. Sauer and R. Woodrow, On monochromatic paths in edge-coloured digraphs. Journal of Combinatorial Theory, Series B, 33, (1982), 271--275.\n\nRelated:\nRelated problems\nMonochromatic reachability in edge-colored tournaments\nMonochromatic reachability or rainbow triangles\n\nBibliography links:\n- A proof of the Erdős-Sands-Sauer-Woodrow conjecture: http://www.arxiv.org/abs/math.CO/1703.08123\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Monochromatic reachability in arc-colored digraphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3276, "problem_number": "OPG-1808", "title": "Monochromatic reachability or rainbow triangles", "statement": "In an edge-colored digraph, we say that a subgraph is rainbow if all its edges have distinct colors, and monochromatic if all its edges have the same color.\n\nProblem Let $G$ be a tournament with edges colored from a set of three colors. Is it true that $G$ must have either a rainbow directed cycle of length three or a vertex $v$ so that every other vertex can be reached from $v$ by a monochromatic (directed) path?", "background": "Source: Open Problem Garden. Original node ID: 1808. URL: http://www.openproblemgarden.org/op/monochromatic_reachability_vs_rainbow_triangles.\n\nSource subject path: Graph Theory > Directed Graphs > Tournaments.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/monochromatic_reachability_vs_rainbow_triangles\n- Author(s): Sands, Bill; Sauer, Norbert W.; Woodrow, Robert E.\n- Subject(s): Graph Theory; Directed Graphs; Tournaments\n- Keywords: digraph; edge-coloring; tournament\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 15th, 2008 by mdevos\n\nProblem-page discussion:\nThis problem was raised in a paper by Sands, Sauer, and Woodrow [SSW] where they prove that every tournament with 2-colored edges has a vertex $v$ so that every other vertex can be reached from $v$ by a monochromatic path.\n\nGaleana-Sanchez and Rojas-Monroy found a tournament on 6 vertices with 4-colored edges which has no rainbow triangle and does not have a vertex $v$ which has monochromatic paths to all remaining vertices. However, the following generalization of the above conjecture looks plausible.\n\nProblem Does every edge-colored tournament have either a rainbow directed cycle or a vertex $v$ so that every other vertex can be reached from $v$ by a monochromatic path?\n\nBibliography:\n*[SSW] B. Sands, N. Sauer, R. Woodrow, On monochromatic paths in edge-coloured digraphs. J. Combin. Theory Ser. B 33 (1982), no. 3, 271--275. MathSciNet.\n\nRelated:\nRelated problems\nMonochromatic reachability in edge-colored tournaments\n\nBibliography links:\n- On monochromatic paths in edge-coloured digraphs: http://www.sciencedirect.com/science/article/pii/0095895682900478\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0693367\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Monochromatic reachability or rainbow triangles\" in Graph Theory; Directed Graphs; Tournaments, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3277, "problem_number": "OPG-46237", "title": "Decomposing an even tournament in directed paths.", "statement": "Conjecture Every tournament $D$ on an even number of vertices can be decomposed into $\\sum_{v\\in V}\\max\\{0,d^+(v)-d^-(v)\\}$ directed paths.", "background": "Source: Open Problem Garden. Original node ID: 46237. URL: http://www.openproblemgarden.org/op/decomposing_an_even_tournament_in_directed_paths.\n\nSource subject path: Graph Theory > Directed Graphs > Tournaments.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_an_even_tournament_in_directed_paths\n- Author(s): Alspach, Brian; Mason, David W.; Pullman, Norman J.\n- Subject(s): Graph Theory; Directed Graphs; Tournaments\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 26th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture is clearly tight, because in a decomposition of a directed graph in directed paths, at least $\\max \\{0,d^+(v)-d^-(v)\\}$ directed paths must start at vertex $v$.\n\nObserve that the analogue is trivially false for odd tournament: in regular tournament $d^+(v)=d^-(v)$ for every vertex $v$, so $\\sum_{v\\in V}\\max\\{0,d^+(v)-d^-(v)\\}=0$. For a tournament of even order $n$, $\\sum_{v\\in V}\\max\\{0,d^+(v)-d^-(v)\\}\\geq n/2$. Since a directed path may have up to $n-1$ arcs, it might be possible to cover the $n(n-1)/2$ arcs of the tournament if $n$ is even. If the tournament is almost regular (i.e. $|d^+(v)-d^-(v)|=1$ for all vertex $v$ ), the conjecture asserts that it can be decomposed into directed Hamilton paths.\n\nThis conjecture for almost regular tournaments would imply the following one due to Kelly.\n\nConjecture Every regular tournament of order $n$ can be decomposed into $(n-1)/2$ Hamilton directed cycles.\n\nTo see this, consider a regular tournament $T$ and a vertex $v$ of $T$. The tournament $T-v$ has even order, and in $T-v$, $\\max \\{0,d^+(v)-d^-(v)\\}=0$ unless $v$ is an outneighbour of $v$ in $T$ in which case $\\max \\{0,d^+(v)-d^-(v)\\}=0$. Hence $\\sum_{v\\in V}\\max\\{0,d^+(v)-d^-(v)\\}=(n-1)/2$. Now if Alspach-Mason-Pulman conjecture holds, $T-v$ can be decomposed into $(n-1)/2$ directed paths. These paths must start at distinct outneighbours of $v$ in $T$ and ends at distinct inneighbours of $v$ in $T$. Hence, we can complete each directed path in a Hamilton directed cycle in $T$ to obtain a decomposition of $T$ into $(n-1)/2$ Hamilton cycles.\n\nKelly's conjecture has been proved for tournaments of sufficiently large order by Kühn and Osthus [KO].\n\nBibliography:\n*[AMP] Brian Alspach, David W. Mason, Norman J. Pullman, Path numbers of tournaments, Journal of Combinatorial Theory, Series B, 20 (1976), no. 3, June 1976, 222–228\n\n[KO] Daniela Kühn and Deryk Osthus, Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments, Advances in Mathematics 237 (2013), 62-146.\n\nRelated:\nRelated problems\nEdge-disjoint Hamilton cycles in highly strongly connected tournaments.\n\nBibliography links:\n- Path numbers of tournaments: http://www.sciencedirect.com/science/article/pii/0095895676900137\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposing an even tournament in directed paths.\" in Graph Theory; Directed Graphs; Tournaments, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3278, "problem_number": "OPG-47031", "title": "Edge-disjoint Hamilton cycles in highly strongly connected tournaments.", "statement": "Conjecture For every $k\\geq 2$, there is an integer $f(k)$ so that every strongly $f(k)$-connected tournament has $k$ edge-disjoint Hamilton cycles.", "background": "Source: Open Problem Garden. Original node ID: 47031. URL: http://www.openproblemgarden.org/op/edge_disjoint_hamilton_cycles.\n\nSource subject path: Graph Theory > Directed Graphs > Tournaments.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/edge_disjoint_hamilton_cycles\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Directed Graphs; Tournaments\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 8th, 2013 by fhavet\n\nProblem-page discussion:\nKelly made the following conjecture which replaces the assumption of high connectivity by regularity.\n\nConjecture Every regular tournament of order $n$ can be decomposed into $(n-1)/2$ Hamilton directed cycles.\n\nKelly's conjecture has been proved for tournaments of sufficiently large order by Kühn and Osthus [KO].\n\nBibliography:\n[KO] Daniela Kühn and Deryk Osthus, Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments, Advances in Mathematics 237 (2013), 62-146.\n\n*[T] C. Thomassen, Edge-disjoint Hamiltonian paths and cycles in tournaments, Proc. London Math. Soc. 45 (1982), 151-168.\n\nRelated:\nRelated problems\nDecomposing an even tournament in directed paths.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Edge-disjoint Hamilton cycles in highly strongly connected tournaments.\" in Graph Theory; Directed Graphs; Tournaments, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3279, "problem_number": "OPG-47643", "title": "Partitionning a tournament into k-strongly connected subtournaments.", "statement": "Problem Let $k_1, \\dots, k_p$ be positve integer Does there exists an integer $g(k_1, \\dots, k_p)$ such that every $g(k_1, \\dots, k_p)$-strong tournament $T$ admits a partition $(V_1\\dots, V_p)$ of its vertex set such that the subtournament induced by $V_i$ is a non-trivial $k_i$-strong for all $1\\leq i\\leq p$.", "background": "Source: Open Problem Garden. Original node ID: 47643. URL: http://www.openproblemgarden.org/op/partitionning_a_tournament_into_k_strongly_connected_subtournaments.\n\nSource subject path: Graph Theory > Directed Graphs > Tournaments.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partitionning_a_tournament_into_k_strongly_connected_subtournaments\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Directed Graphs; Tournaments\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 15th, 2013 by fhavet\n\nProblem-page discussion:\nIf $k_i=1$ for $2\\leq k_i\\leq k_p$, then $g(k_1, \\dots, k_p)$ exists and is at most $k_1+3p-3$. This follows by an easy induction on $p$, by taking $V_p$ to be a set inducing a directed $3$-cycle.\n\nThe following example shows that if it exists $g(k_1, \\dots, k_p)\\geq k_1+\\cdots + k_p$. Set $s=k_1 + \\cdots + k_p -1$. For $n\\geq 3s$, let $R_s(n)$ be a tournament on $n$ vertices having a set $R$ of $s$ vertices such that $T-R$ a transitive tournament of order $n-s$ with hamiltonian path $(v_1,\\dots, v_{n-s})$, and $R$ dominates $\\{v_1, \\dots, v_{s}\\}$ and is dominated by $\\{v_{n-2s+1}, \\dots, v_{n-s}\\}$. It easy to check that $R_s(n)$ is $s$-strongly connected. However, every (non-trivial) $k$-strong tournament of $R_s(n)$ must contain at least $k$ vertices of $R$. Hence $R_s(n)$ does not have a partition $(V_1\\dots, V_p)$ of its vertex set such that the subtournament induced by $V_i$ is a non-trivial $k_i$-strong for all $1\\leq i\\leq p$.\n\nSome small examples give better lower bound. For example, the Paley tournament on 7 vertices which is 3-strong cannot be partionned into two strong subtournaments. However, there are only finitely many known such tournaments. Chen, Gould, and Li [CGL] showed that every $k$-strongly connected tournament with at least $8k$ vertices has a partition into $k$ strongly connected tournaments.\n\nThe existence of $g(2,2)$ is still open.\n\nBibliography:\n[CGL] G. Chen, R.J. Gould, and H. Li, Partitioning vertices of a tournament into independent cycles, J. combin. Theory Ser B, Vol 83, no. 2 (2001) 213-220.\n\n*[R] K.B. Reid, Three problems on tournaments, Graph Theory and Its Applications, East. and West. Ann. New York Acad. Sci. 576 (1989), 466-473.\n\nDiscussion links:\n- Paley tournament: http://en.wikipedia.org/wiki/Paley graph\n\nBibliography links:\n- Partitioning vertices of a tournament into independent cycles: http://www.sciencedirect.com/science/article/pii/S0095895601920489\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 28.\n\nAttempt notes:\nTarget:\nMake progress on \"Partitionning a tournament into k-strongly connected subtournaments.\" in Graph Theory; Directed Graphs; Tournaments, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3280, "problem_number": "OPG-47651", "title": "Decomposing k-arc-strong tournament into k spanning strong digraphs", "statement": "Conjecture Every k-arc-strong tournament decomposes into k spanning strong digraphs.", "background": "Source: Open Problem Garden. Original node ID: 47651. URL: http://www.openproblemgarden.org/op/decomposing_k_arc_strong_tournament_into_k_spanning_strong_digraphs.\n\nSource subject path: Graph Theory > Directed Graphs > Tournaments.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_k_arc_strong_tournament_into_k_spanning_strong_digraphs\n- Author(s): Bang-Jensen, Joergen; Yeo, Anders\n- Subject(s): Graph Theory; Directed Graphs; Tournaments\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 15th, 2013 by fhavet\n\nProblem-page discussion:\nConjecture 8 implies Kelly's conjecture (Every regular tournament of order $n$ can be decomposed into $(n-1)/2$ Hamilton directed cycles.) which has been proved for tournaments of sufficiently large order by Kühn and Osthus [KO].\n\nBang-Jensen and Yeo [BY] gave several results supporting this conjecture. For example they proved it for $k$-arc-strong tournaments with minimum in- and out-degree at least $37k$.\n\nBibliography:\n*[BY] J. Bang-Jensen, A. Yeo, Decomposing k-arc-strong tournaments into strong spanning subdigraphs, Combinatorica 24 (2004) 331–349.\n\n[KO] Daniela Kühn and Deryk Osthus, Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments, Advances in Mathematics 237 (2013), 62-146.\n\nRelated:\nRelated problems\nArc-disjoint strongly connected spanning subdigraphs\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposing k-arc-strong tournament into k spanning strong digraphs\" in Graph Theory; Directed Graphs; Tournaments, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3281, "problem_number": "OPG-162", "title": "The Erdös-Hajnal Conjecture", "statement": "Conjecture For every fixed graph $H$, there exists a constant $\\delta(H)$, so that every graph $G$ without an induced subgraph isomorphic to $H$ contains either a clique or an independent set of size $|V(G)|^{\\delta(H)}$.", "background": "Source: Open Problem Garden. Original node ID: 162. URL: http://www.openproblemgarden.org/op/the_erdos_hajnal_conjecture.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_erdos_hajnal_conjecture\n- Author(s): Erdos, Paul; Hajnal, Andras\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Keywords: induced subgraph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 18th, 2007 by mdevos\n\nProblem-page discussion:\nThere are numerous interesting classes of graphs which are based upon forbidding one or more induced subgraphs. For instance: chordal graphs, split graphs, and claw-free graphs. Numerous other natural classes of graphs have been proved to have such characterizations, most famously perfect graphs, but also line graphs and comparability graphs. All of these classes are very well structured (far from random) and their members all either have large cliques or independent sets. On the flip side of this are random graphs. It is well known that a random graph on $n$ vertices has both clique and independence number highly concentrated around $2 \\log_2 n$. The Erdos-Hajnal conjecture suggests a fundamental separation between these two worlds in terms of independence/clique sizes.\n\nErdös and Hajnal proved that this conjecture is true for the recursive class of graphs ${\\mathcal C}$ defined as follows. The one vertex graph is in ${\\mathcal C}$, and if $G_1$ and $G_2$ lie in ${\\mathcal C}$, then the disjoint union of $G_1$ and $G_2$ lies in ${\\mathcal C}$, as does the graph obtained from the disjoint union by adding an edge between $v_1$ and $v_2$ for every $v_1 \\in V(G_1)$ and $v_2 \\in V(G_2)$. More generally, Alon, Pach, and Solymosi proved that if $F$ is a graph with $V(F) = \\{v_1,v_2,\\ldots,v_n\\}$ for which the Erdös-Hajnal conjecture holds, and $H_1,\\ldots,H_n$ are graphs for which the Erdos-Hajnal conjecture holds, then the graph obtained from $F$ by blowing up each vertex $v_i$ with a copy of $H_i$ (more precisely, starting from the disjoint union of $H_1,H_2,\\ldots,H_n$, we add all possible edges between the vertices of $V(H_i)$ and $V(H_j)$ if $ij \\in E(F)$ ) also satisfies the Erdos-Hajnal conjecture.\n\nThe Erdös-Hajnal property is known to hold for a number of small graphs (and using the above result this may be easily bootstrapped). For instance, the conjecture is known to hold when $H$ is a path of three edges, and recently M. Chudnovsky and S. Safra have announced a proof when $H$ is a bull (a triangle plus two pendant edges). However, our knowledge here is still quite limited. In particular, Lovasz has suggested the following very special case which remains open.\n\nQuestion Is the Erdös-Hajnal conjecture true when $H \\cong C_5$?\n\nBibliography:\n[APS] N. Alon, J. Pach, and J. Solymosi, Ramsey-type theorems with forbidden subgraphs, Combinatorica 21 (2001), 155-170.\n\n[EH] P. Erdös and A. Hajnal, Ramsey-type theorems, Discrete Appl. Math. 25 (1989), 37-52 MathSciNet\n\nDiscussion links:\n- chordal graphs: http://en.wikipedia.org/wiki/chordal graph\n- split graphs: http://en.wikipedia.org/wiki/split graph\n- perfect graphs: http://en.wikipedia.org/wiki/perfect graph\n- line graphs: http://en.wikipedia.org/wiki/line graph\n- comparability graphs: http://en.wikipedia.org/wiki/comparability graph\n- random graphs: http://en.wikipedia.org/wiki/random graph\n\nBibliography links:\n- Ramsey-type theorems with forbidden subgraphs: http://www.math.tau.ac.il/%7Enogaa/PDFS/aps4.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1031262\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 23.\n\nAttempt notes:\nTarget:\nMake progress on \"The Erdös-Hajnal Conjecture\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3282, "problem_number": "OPG-567", "title": "What is the smallest number of disjoint spanning trees made a graph Hamiltonian", "statement": "We are given a complete simple undirected weighted graph $G_1=(V,E)$ and its first arbitrary shortest spanning tree $T_1=(V,E_1)$. We define the next graph $G_2=(V,E\\setminus E_1)$ and find on $G_2$ the second arbitrary shortest spanning tree $T_2=(V,E_2)$. We continue similarly by finding $T_3=(V,E_3)$ on $G_3=(V,E\\setminus \\cup_{i=1}^{2}E_i)$, etc. Let k be the smallest number of disjoint shortest spanning trees as defined above and let $T^{k}=(V,\\cup_{i=1}^{k}E_i)$ be the graph obtained as union of all $k$ disjoint trees.\n\nQuestion 1. What is the smallest number of disjoint spanning trees creates a graph $T^{k}$ containing a Hamiltonian path.\n\nQuestion 2. What is the smallest number of disjoint spanning trees creates a graph $T^{k}$ containing a shortest Hamiltonian path?\n\nQuestions 3 and 4. Replace in questions 1 and 2 a shortest spanning tree by a 1-tree. What is the smallest number of disjoint 1-trees creates a Hamiltonian graph? What is the smallest number of disjoint 1-trees creates a graph containing a shortest Hamiltonian cycle?", "background": "Source: Open Problem Garden. Original node ID: 567. URL: http://www.openproblemgarden.org/op/what_is_the_smallest_number_of_disjoint_spanning_trees_made_a_graph_hamiltonian.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/what_is_the_smallest_number_of_disjoint_spanning_trees_made_a_graph_hamiltonian\n- Author(s): Goldengorin\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Keywords: 1-trees; cycle; Hamitonian path; spanning trees\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 10th, 2007 by boris\n\nProblem-page discussion:\nThese questions are induced by the following paper Chrobak and Poljak. On common edges in optimal solutions to travelling salesman and other optimization problems, Discrete Applied Mathematics 20 (1988) 101-111.\n\nBibliography:\nM. Chrobak and S. Poljak. On common edges in optimal solutions to travelling salesman and other optimization problems, Discrete Applied Mathematics 20 (1988) 101-111.\n\nComments:\n- September 10th, 2007 | mdevos | Is this statement correct?: Unless I am mistaken, it appears that there does not exist any $k$ for which any of the above problems has a positive solution. For instance, let $G$ be the complete graph with vertex set $V = \\{1,\\ldots,6k\\}$, and define $C$ to be the edge-cut of $G$ consisting of all edges between $\\{1,2,\\ldots,2k\\}$ and $\\{2k+1,2k+2,\\ldots,6k\\}$. Now define a weighting of $G$ by assigning each edge in $C$ weight 1 and every other edge weight 2. So, the subgraph $(V,C)$ (consisting of all edges of weight 1) is isomorphic to $K_{2k,4k}$- and since $K_{2k,4k}$ has $k$ edge-disjoint spanning trees (by the Nash-Williams theorem, say), our procedure may well choose trees $T_1,T_2,\\ldots,T_k$ so that all of the edges in all of these graphs are in $C$. But then $T^k$ will not have a Hamiltonian path since it is a subgraph of $(V,C) \\cong K_{2k,4k}$.\n\nOf course, we may modify the edge weights here so that the procedure is forced to choose $T_1,\\ldots,T_k$ so that all of these trees have their edges in $C$.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"What is the smallest number of disjoint spanning trees made a graph Hamiltonian\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3283, "problem_number": "OPG-37305", "title": "Extremal problem on the number of tree endomorphism", "statement": "Conjecture An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the $n$ vertices' trees, the star with $n$ vertices has the most endomorphisms, while the path with $n$ vertices has the least endomorphisms.", "background": "Source: Open Problem Garden. Original node ID: 37305. URL: http://www.openproblemgarden.org/op/extremal_problem_on_the_number_of_tree_endomorphism.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/extremal_problem_on_the_number_of_tree_endomorphism\n- Author(s): Zhicong Lin\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: March 1st, 2011 by shudeshijie\n\nBibliography:\n[BT] Bela Bollobas and Mykhaylo Tyomkyn, Walks and paths in trees, http://arxiv.org/abs/1002.2768.\n\nComments:\n- November 25th, 2020 | Anonymous | it is not open: This problem is not open. Look at this: https://mathscinet.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=3284058&loc=fromrevtext\n- March 23rd, 2012 | Anonymous | the upper bound is proved: the upper bound is proved recently.\n- March 20th, 2011 | leshabirukov | counterexample: Asymmetric tree (http://en.wikipedia.org/wiki/Asymmetric_graph, http://upload.wikimedia.org/wikipedia/commons/a/ad/Asymmetric_tree.svg) has single, trivial endomorphism.\n\nUpdate: Sorry, I have confused endomorphism with automorphism.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Extremal problem on the number of tree endomorphism\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3284, "problem_number": "OPG-46738", "title": "Complexity of the H-factor problem.", "statement": "An $H$-factor in a graph $G$ is a set of vertex-disjoint copies of $H$ covering all vertices of $G$.\n\nProblem Let $c$ be a fixed positive real number and $H$ a fixed graph. Is it NP-hard to determine whether a graph $G$ on $n$ vertices and minimum degree $cn$ contains and $H$-factor?", "background": "Source: Open Problem Garden. Original node ID: 46738. URL: http://www.openproblemgarden.org/op/complexity_of_the_h_factor_problem.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/complexity_of_the_h_factor_problem\n- Author(s): Kühn, Daniella; Osthus, Deryk\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 5th, 2013 by fhavet\n\nProblem-page discussion:\nThe answer is positive for cliques and a few other graphs [KO06].\n\nIf we remove the minimum degree condition, the problem is NP-complete if and only if $H$ has a component which contains at least 3 vertices, as shown by Hell and Kirkpatrick [HK].\n\nBibliography:\n[HK] P. Hell and D.G. Kirkpatrick, On the complexity of general graph factor problems, SIAM J. Computing 12 (1983), 601-609.\n\n[KO06] D. Kühn and D. Osthus, Critical chromatic number and the complexity of perfect packings in graphs, Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2006.\n\n*[KO09] D. Kühn and D. Osthus, The minimum degree threshold for perfect graph packings, Combinatorica 29 (2009), 65-107.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Complexity of the H-factor problem.\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3285, "problem_number": "OPG-46824", "title": "Odd-cycle transversal in triangle-free graphs", "statement": "Conjecture If $G$ is a simple triangle-free graph, then there is a set of at most $n^2/25$ edges whose deletion destroys every odd cycle.", "background": "Source: Open Problem Garden. Original node ID: 46824. URL: http://www.openproblemgarden.org/op/odd_cycle_transversal_in_triangle_free_graphs.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/odd_cycle_transversal_in_triangle_free_graphs\n- Author(s): Erdos, Paul; Faudree, Ralph; Pach, János; Spencer, Joel\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 6th, 2013 by fhavet\n\nBibliography:\n*[EFPS] P. Erdös, R. Faudree, J. Pach and J. Spencer, How to make a graph bipartite. J. Combin. Theory Ser. B 45 (1988), 86--98.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Odd-cycle transversal in triangle-free graphs\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3286, "problem_number": "OPG-46837", "title": "Triangle-packing vs triangle edge-transversal.", "statement": "Conjecture If $G$ has at most $k$ edge-disjoint triangles, then there is a set of $2k$ edges whose deletion destroys every triangle.", "background": "Source: Open Problem Garden. Original node ID: 46837. URL: http://www.openproblemgarden.org/op/triangle_packing_vs_triangle_edge_transversal.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/triangle_packing_vs_triangle_edge_transversal\n- Author(s): Tuza, Zsolt\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 6th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture may be rephrased in terms of packing and edge-transversal. A triangle packing is a set of pairwise edge-disjoint triangles. A triangle edge-tranversal is a set of edges meeting all triangles. Denote the maximum size of a triangle packing in $G$ by $\\nu(G)$ and the minimum size of a triangle edge-transversal of $G$ by $\\tau(G)$. Clearly $\\nu(G) \\leq \\tau(G)$. The conjecture translates in $\\tau(G)\\leq 2\\nu(G)$.\n\nThis conjecture, if true, is best possible as can be seen by taking, say $G=K_4$ or $G=K_5$. Trivially, $\\tau(G)\\leq 3\\nu(G)$, since the set of edges of a maximum triangle packing is a triangle edge-transversal. Haxell [H] proved that $\\tau(G) \\leq (3-\\frac{3}{23})\\nu(G)$ edges whose deletion destroys every triangle.\n\nAs usual, one can define fractional packing and fractional transversal. Let ${\\cal T}$ be the set of triangles of $G$. A fractional triangle packing is a function $f:{\\cal T}\\rightarrow \\mathbb{R}^+$ such that $\\sum_{T\\ni e} \\leq 1$ for every edge $e$. A fractional triangle edge-transversal is a function $g:E\\rightarrow \\mathbb{R}^+$ such that $\\sum_{e\\in T} g(e)\\geq 1$ for every triangle $T\\in {\\cal T}$. We denote by $\\nu^*(G)$ the maximum of $\\sum_{T\\in {\\cal T}} f(T)$ over all fractional triangle packing and by $\\tau^*(G)$ the minimum of $\\sum_{e\\in E(G)} g(e)$ over all fractional edge-transversals. By duality of linear programming $\\tau^*(G) = \\nu^*(G)$. Krivelevich [K] proved two fractional versions of the conjecture:\n\n$\\tau(G) \\leq 2\\nu^*(G)$ and $\\tau^*(G)\\leq 2\\nu(G)$.\n\nBibliography:\n[H] P.Haxell, Packing and covering triangles in graphs, Discrete Mathematics 195 (1999), no. 1–3, 251–254.\n\n[K] M. Krivelevich, On a conjecture of Tuza about packing and covering of triangles Discrete Mathematics 142 (1995), 281-286.\n\n*[T] Z. Tuza, A conjecture on triangles of graphs. Graphs Combin. 6 (1990), 373-380.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 25.\n\nAttempt notes:\nTarget:\nMake progress on \"Triangle-packing vs triangle edge-transversal.\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3287, "problem_number": "OPG-48232", "title": "The Bollobás-Eldridge-Catlin Conjecture on graph packing", "statement": "Conjecture (BEC-conjecture) If $G_1$ and $G_2$ are $n$-vertex graphs and $(\\Delta(G_1) + 1) (\\Delta(G_2) + 1) < n + 1$, then $G_1$ and $G_2$ pack.", "background": "Source: Open Problem Garden. Original node ID: 48232. URL: http://www.openproblemgarden.org/op/the_bollobas_eldridge_catlin_conjecture_on_graph_packing.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_bollobas_eldridge_catlin_conjecture_on_graph_packing\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Keywords: graph packing\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 23rd, 2013 by asp\n\nProblem-page discussion:\nA pair of $n$-vertex graphs $G_1$ and $G_2$ are said to ${\\it pack}$ if they are edge-disjoint subgraphs of the complete graph on $n$ vertices.\n\nThe main conjecture in the area of graph packing is the abovementioned conjecture by Bollobás, Eldridge [BE] and Catlin [C].\n\nIn support of the BEC-conjecture, Sauer and Spencer [SS] proved that if $G_1$ and $G_2$ are $n$-vertex graphs and $2 \\Delta(G_1) \\Delta(G_2) < n$ then $G_1$ and $G_2$ pack.\n\nGiven a graph $G$, $L(G)$ denotes the line graph of $G$ and $\\Theta(G)$ denotes the number $\\Delta(L(G)) + 2$. Kostochka and Yu [KY1] proved that if $G_1$ and $G_2$ are two $n$-vertex graphs with $\\Theta(G_1) \\Delta(G_2) \\leq n$, then $G_1$ and $G_2$ pack with the following exceptions: (1) $G_1$ is a perfect matching and $G_2$ is either $K_{n/2,n/2}$ with $n/2$ odd or contains $K_{n/2 + 1}$ or (2) $G_2$ is a perfect matching and $G_1$ is $K_{r,n-r}$ with $r$ odd or contains $K_{n/2 + 1}$.\n\nKostachka and Yu [KY2] conjectured that if $G_1$ and $G_2$ are $n$-vertex graphs with $\\Theta(G_1) \\Theta(G_2) < 2n$ then $G_1$ and $G_2$ pack.\n\nBibliography:\n*[BE] B. Bollabás and S. E. Eldridge, Maximal matchings in graphs with given maximal and minimal degrees, Congr. Numer. XV (1976), 165--168.\n\n*[C] P. A. Catlin, Embedding subgraphs and coloring graphs under extremal degree conditions, Ph. D. Thesis, Ohio State Univ., Columbus (1976).\n\n[KY1] A. V. Kostochka and G. Yu, An Ore-type analogue of the Sauer-Spencer Theorem, Graphs Combin. 23 (2007), 419--424.\n\n[KY2] A. V. Kostochka and G. Yu, An Ore-type graph packing problems, Combin. Probab. Comput. 16 (2007), 167--169.\n\n[SS] N. Sauer and J. Spencer, Edge disjoint placement of graphs, J. Combin. Theory Ser. B 25 (1978), 295--302.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 17.\n\nAttempt notes:\nTarget:\nMake progress on \"The Bollobás-Eldridge-Catlin Conjecture on graph packing\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3288, "problem_number": "OPG-60013", "title": "Weak saturation of the cube in the clique", "statement": "Problem\n\nDetermine $\\text{wsat}(K_n,Q_3)$.", "background": "Source: Open Problem Garden. Original node ID: 60013. URL: http://www.openproblemgarden.org/op/weak_saturation_of_the_cube_in_the_clique.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/weak_saturation_of_the_cube_in_the_clique\n- Author(s): Morrison, Natasha; Noel, Jonathan A.\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Keywords: bootstrap percolation; hypercube; Weak saturation\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: April 6th, 2016 by Jon Noel\n\nProblem-page discussion:\nGiven graphs $G$ and $H$, let $\\text{wsat}(G,H)$ denote the minimum number of edges in a subgraph $F$ of $G$ such that the edges of $E(G)\\setminus E(F)$ can be added to $F$, one edge at a time, so that each edge completes a copy of $H$ when it is added.\n\nOf course, if one can solve the problem above, then a natural next step is to determine $\\text{wsat}(K_n,Q_m)$ for all $n$ and $m$.\n\nMorrison, Noel and Scott [MNS] solved the related problem of determining $\\text{wsat}(Q_d,Q_m)$ for all $d$ and $m$.\n\nBibliography:\n[MNS] N. Morrison, J. A. Noel, A. Scott. Saturation in the hypercube and bootstrap percolation. To appear in Combin. Probab. Comput.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Weak saturation of the cube in the clique\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3289, "problem_number": "OPG-60042", "title": "Multicolour Erdős--Hajnal Conjecture", "statement": "Conjecture For every fixed $k\\geq2$ and fixed colouring $\\chi$ of $E(K_k)$ with $m$ colours, there exists $\\varepsilon>0$ such that every colouring of the edges of $K_n$ contains either $k$ vertices whose edges are coloured according to $\\chi$ or $n^\\varepsilon$ vertices whose edges are coloured with at most $m-1$ colours.", "background": "Source: Open Problem Garden. Original node ID: 60042. URL: http://www.openproblemgarden.org/op/multicolour_erdos_hajnal_conjecture.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/multicolour_erdos_hajnal_conjecture\n- Author(s): Erdos, Paul; Hajnal, Andras\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Keywords: ramsey theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 10th, 2019 by Jon Noel\n\nProblem-page discussion:\nSee [FGP].\n\nBibliography:\n[FGP] Jacob Fox, Andrey Grinshpun and János Pach: The Erdős–Hajnal conjecture for rainbow triangles, J. Combin. Theory, Series B. 111 (2016), 75--125.\n\nRelated:\nRelated problems\nThe Erdös-Hajnal Conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Multicolour Erdős--Hajnal Conjecture\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3290, "problem_number": "OPG-37079", "title": "A gold-grabbing game", "statement": "Setup Fix a tree $T$ and for every vertex $v \\in V(T)$ a non-negative integer $g(v)$ which we think of as the amount of gold at $v$.\n\n2-Player game Players alternate turns. On each turn, a player chooses a leaf vertex $v$ of the tree, takes the gold at this vertex, and then deletes $v$. The game ends when the tree is empty, and the winner is the player who has accumulated the most gold.\n\nProblem Find optimal strategies for the players.", "background": "Source: Open Problem Garden. Original node ID: 37079. URL: http://www.openproblemgarden.org/op/a_gold_grabbing_game.\n\nSource subject path: Graph Theory > Graph Algorithms.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_gold_grabbing_game\n- Author(s): Rosenfeld, Moshe\n- Subject(s): Graph Theory; Graph Algorithms\n- Keywords: game; tree\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 2nd, 2009 by mdevos\n\nProblem-page discussion:\nIn the special case when $T$ is a path of even length, the first player can ensure that she chooses either all of the even vertices, or all of the odd vertices. Thus, player 1 should never finish with less than player 2, and whenever the total gold on the odd vertices and the total gold on the even vertices are not equal, there is a winning strategy for player 1.\n\nComments:\n- October 4th, 2009 | porton | Not an open problem in the strict sense: There exists an obvious algorithm which just enumerates all variants.\n\nThe problem seems to mean to find a more efficient algorithm. This is not a strict formulation because it is not strictly defined what is \"more efficient\".\n\nI suggest to rip this problem, such as to put it into Second tier problems.\n\n--\n\nVictor Porton - http://www.mathematics21.org\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"A gold-grabbing game\" in Graph Theory; Graph Algorithms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3291, "problem_number": "OPG-47646", "title": "PTAS for feedback arc set in tournaments", "statement": "Question Is there a polynomial time approximation scheme for the feedback arc set problem for the class of tournaments?", "background": "Source: Open Problem Garden. Original node ID: 47646. URL: http://www.openproblemgarden.org/op/ptas_for_feedback_arc_set_in_tournaments.\n\nSource subject path: Graph Theory > Graph Algorithms.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/ptas_for_feedback_arc_set_in_tournaments\n- Author(s): Ailon, Nir; Alon, Noga\n- Subject(s): Graph Theory; Graph Algorithms\n- Keywords: feedback arc set; PTAS; tournament\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 15th, 2013 by fhavet\n\nProblem-page discussion:\nA tournament is an orientation of a complete graph. A feedback arc set is a set of arcs in a digraph whose removal leave the digraph acyclic. The feedback arc set problem consists in finding a feedback arc set of minimum size. A polynomial time approximation scheme is an algorithm which takes an instance of an optimization problem and a parameter $\\epsilon > 0$ and, in polynomial time, produces a solution that is within a factor $1+\\epsilon$ of being optimal.\n\nThe feedback arc set problem has been proved NP-hard. See [ACM, A, CTY, C]. It was shown in [RS] that the feedback arc set problem is fixed parameter tractable for tournaments.\n\nBibliography:\n*[AA] N. Ailon, N. Alon, link, Inform. and Comput. 205 (8) (2007) 1117–1129.\n\n[ACM] N. Alion, M. Charikar, A. Newman, Aggregating inconsistent information: Ranking and clustering, in: Proceedings of the 37th Symposium on the Theory of Computing, STOC, ACM Press, 2005, pp. 684–693.\n\n[A] N. Alon, Ranking tournaments, SIAM J. Discrete Math. 20 (2006) 137–142.\n\n[CTY] P. Charbit, P. Thomassé, A. Yeo, The minimum Feedback arc set problem is NP-hard for tournaments, Combin. Probab. Comput. 16 (1) (2007) 1–4.\n\n[C] V. Conitzer, Computing Slater rankings using similarities among candidates, in: Proceedings, The Twenty-First National Conference on Artificial Intelligence and the Eighteenth Innovative Applications of Artificial Intelligence Conference, July 16–20, AAAI Press, Boston, Massachusetts, USA, 2006.\n\n[RS] V. Raman, S. Saurabh, Parameterized complexity of directed feedback arc set problems in tournaments, in: Algorithms and Data Structures, in: Lecture Notes in Computer Science, vol. 2748, Springer, Berlin, 2003, pp. 484–492.\n\nDiscussion links:\n- tournament: http://en.wikipedia.org/wiki/Tournament (graph theory)\n- feedback arc set: http://en.wikipedia.org/wiki/Feedback arc set\n- polynomial time approximation scheme: http://en.wikipedia.org/wiki/Polynomial-time approximation scheme\n- fixed parameter tractable: http://en.wikipedia.org/wiki/FPT\n\nBibliography links:\n- link: http://www.openproblemgarden.org/Hardness of fully dense problems]{http:/www.tau.ac.il/%7Enogaa/PDFS/dense8.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"PTAS for feedback arc set in tournaments\" in Graph Theory; Graph Algorithms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3292, "problem_number": "OPG-165", "title": "Ryser's conjecture", "statement": "Conjecture Let $H$ be an $r$-uniform $r$-partite hypergraph. If $\\nu$ is the maximum number of pairwise disjoint edges in $H$, and $\\tau$ is the size of the smallest set of vertices which meets every edge, then $\\tau \\le (r-1) \\nu$.", "background": "Source: Open Problem Garden. Original node ID: 165. URL: http://www.openproblemgarden.org/op/rysers_conjecture.\n\nSource subject path: Graph Theory > Hypergraphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/rysers_conjecture\n- Author(s): Ryser, Herbert J.\n- Subject(s): Graph Theory; Hypergraphs\n- Keywords: hypergraph; matching; packing\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 19th, 2007 by mdevos\n\nProblem-page discussion:\nDefinitions: A (vertex) cover is a set of vertices which meets (has nonempty intersection with) every edge, and we let $\\tau(H)$ denote the size of the smallest vertex cover of $H$. A matching is a collection of pairwise disjoint edges, and we let $\\nu(H)$ denote the size of the largest matching in $H$. When the hypergraph is clear from context, we just write $\\tau$ or $\\nu$.\n\nIt is immediate that $\\nu \\le \\tau$, since every cover must contain at least one point from each edge in any matching. For $r$-uniform hypergraphs, $\\tau \\le r \\nu$, since the union of the edges from any maximal matching is a set of at most $r \\nu$ vertices that which meets every edge. Ryser's conjecture is that this second bound can be improved if $H$ is $r$-uniform and $r$-partite (the vertices may be partitioned into $r$ sets $V_1,V_2,\\ldots,V_r$ so that every edge contains exactly one element of each $V_i$ ).\n\nIn the special case when $r=2$ our trivial inequality yields $\\nu \\le \\tau$ and the conjecture implies $\\tau \\le \\nu$, so we should have $\\nu = \\tau$. In fact this is true, it is König's theorem on bipartite graphs [K]. Indeed, Ryser's conjecture is probably easiest to view as a high dimensional generalization of this early result of König. Recently, Aharoni [A] has applied the \"Hall's theorem for hypergraphs\" result of Aharoni and Haxell [AH] to prove this conjecture for $r=3$. However the case $r=4$ is still wide open.\n\nSome other interesting work on this problem concerns fractional covers and fractional matchings. A fractional cover of $H = (V,E)$ is a weighting $a: V \\rightarrow {\\mathbb R}^+$ so that $\\sum_{x \\in S} a(x) \\ge 1$ for every $S \\in E$, and the weight of this cover is $\\sum_{x \\in V} a(x)$. The fractional cover number, denoted $\\tau^*$ is the infimum of the set of weights of covers. Similarly, a fractional matching is an edge-weighting $b: E \\rightarrow {\\mathbb R}^+$ so that $\\sum_{S \\ni x} b(S) \\le 1$ for every $x \\in V$, and the weight of this matching is $\\sum_{S \\in E} b(S)$. The fractional matching number, denoted $\\nu^*$ is the supremum of the set of weights of fractional matchings. Fractional covers and matchings are the usual fractional relaxations, and by LP-duality, they satisfy $\\nu^* = \\tau^*$ for every hypergraph. For $r$-regular $r$-partite hypergraphs, Füredi [F] has proved that $\\tau^* \\le (r-1)\\nu$ and Lovasz [L] has shown $\\tau \\le \\frac{1}{2} r \\nu^*$.\n\nBibliography:\n[A] R. Aharoni, Ryser's conjecture for tripartite 3-graphs. Combinatorica 21 (2001), no. 1, 1--4. MathSciNet\n\n[AH] R. Aharoni and P. Haxell, Hall's theorem for hypergraphs. J. Graph Theory 35 (2000), no. 2, 83--88. MathSciNet\n\n[F] Z. Füredi, Maximum degree and fractional matchings in uniform hypergraphs, Combinatorica 1 (1981), 155--162. MathSciNet\n\n[K] D. König, Theorie der endlichen und unendlichen Graphen, Leipzig, 1936.\n\n[L] L. Lovász, On minimax theorems of combinatorics, Ph.D thesis, Matemathikai Lapok 26 (1975), 209--264 (in Hungarian). MathSciNet\n\nSource links:\n- uniform: http://en.wikipedia.org/wiki/hypergraph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1805710\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1781189\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0625548\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0510823\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 31.\n\nAttempt notes:\nTarget:\nMake progress on \"Ryser's conjecture\" in Graph Theory; Hypergraphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3293, "problem_number": "OPG-547", "title": "¿Are critical k-forests tight?", "statement": "Conjecture\n\nLet $H$ be a $k$-uniform hypergraph. If $H$ is a critical $k$-forest, then it is a $k$-tree.", "background": "Source: Open Problem Garden. Original node ID: 547. URL: http://www.openproblemgarden.org/op/are_critical_k_forests_tight.\n\nSource subject path: Graph Theory > Hypergraphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_critical_k_forests_tight\n- Author(s): Strausz, Ricardo\n- Subject(s): Graph Theory; Hypergraphs\n- Keywords: heterochromatic number\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 1st, 2007 by Dino\n\nProblem-page discussion:\nWe say that a hypergraph $H=(V,E)$ is a $k$-graph if it is $k$-uniform, and denote its order by $n=|V|$ and its size by $m=|E|$.\n\nLaszlo Lovasz introduced the following concept: a $k$-graph $H=(V,E)$ is said to be a $k$-forest if for every edge $e\\in E$ there exists a $k$-colouing $\\varsigma\\colon V\\to[k]$ such that $\\varsigma(e')=[k]\\Leftrightarrow e'=e$; that is, such that only the edge $e$ receives the $k$ colours in its vertices. Clearly a $2$-forest is simply a forest in the usual sense (i.e., an acyclic graph). Lovasz proved that\n\nTheorem A $k$-forest has size at most $m\\leq{n-1\\choose k-1}$.\n\nOn the other hand, Victor Neumann-Lara introduced the following invariant: the heterochromatic number of a $k$-graph $H=(V,G)$ is the minimum number of colours $c$ such that, in every colouring $\\varsigma\\colon V\\to[c]$ there is an edge wich receives different colours in each of its vertices; that is, there exists $e\\in E$ such that $|\\varsigma(e)|=k$. If the heterochromatic number and the rank are equal, the hypergraph is said to be tight. Clearly a $2$-graph is tight if and only if it is connected. A tight $k$-forest is called a $k$-tree.\n\nI can prove the following\n\nTheorem If a $k$-forest has size $m={n-1\\choose k-1}$ then it is tight — and therefore a $k$-tree.\n\nFinally, we say that a $k$-forest is critical if no edge can be added to it without loosing the property of being a $k$-forest; it is maximal (in size) with such a property. Observe that there are critical $k$-forests of size $m<{n-1\\choose k-1}$, whenever $k>2$.\n\nSo, the conjecture is to motivate the question: ¿are critical $k$-forests tight?\n\nSource links:\n- uniform: http://en.wikipedia.org/wiki/hypergraph\n\nDiscussion links:\n- acyclic: http://en.wikipedia.org/wiki/Tree (graph theory)\n- connected: http://en.wikipedia.org/wiki/Connectivity (graph theory)\n\nComments:\n- May 5th, 2014 | Anonymous | This has been solved.: See http://arxiv.org/abs/1109.3390\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"¿Are critical k-forests tight?\" in Graph Theory; Hypergraphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3294, "problem_number": "OPG-2108", "title": "Frankl's union-closed sets conjecture", "statement": "Conjecture Let $F$ be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists $x$ such that $x$ is an element of at least half the members of $F$.", "background": "Source: Open Problem Garden. Original node ID: 2108. URL: http://www.openproblemgarden.org/op/frankls_union_closed_sets_conjecture.\n\nSource subject path: Graph Theory > Hypergraphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/frankls_union_closed_sets_conjecture\n- Author(s): Frankl, Peter\n- Subject(s): Graph Theory; Hypergraphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 25th, 2008 by tchow\n\nProblem-page discussion:\nThis conjecture is notoriously difficult, even though (or should we say `because'?) it involves almost no mathematical structure whatsoever. It was posed by Frankl in the late 1970's. The recent paper of Morris [M] provides a good illustration of the kind of partial results known: Morris extends earlier work to show that the conjecture holds for families containing three 3-subsets of a 5-set, four 3-subsets of a 6-set, or eight 4-subsets of a 6-set. In a different direction, Czédli [C] has proved the conjecture in the case when $|F| \\ge 2^n - 2^{n/2}$ where $n = |\\bigcup F| \\ge 3$.\n\nBibliography:\n[C] G. Czédli, On averaging Frankl's conjecture for large union-closed-sets, J. Combin. Theory Ser. A, to appear.\n\n[M] R. Morris, FC-families and improved bounds for Frankl's conjecture, European J. Combin. 27 (2006), no. 2, 269–282.\n\n[P] B. Poonen, Union-closed families, J. Combin. Theory Ser. A 59 (1992), no. 2, 253–268.\n\n[V] T. P. Vaughan, Three-sets in a union-closed family, J. Combin. Math. Combin. Comput. 49 (2004), 73–84.\n\n[W] P. Wójcik, Union-closed families of sets, Discrete Math. 199 (1999), no. 1–3, 173–182.\n\nComments:\n- March 7th, 2009 | Anonymous | Frankl's conjecture: For mor einformation and many references see also West's account in: http://www.math.uiuc.edu/~west/openp/unionclos.html\n- December 17th, 2009 | Anonymous | question: Does anyone know if there is any linear bound known instead of 1/2?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Frankl's union-closed sets conjecture\" in Graph Theory; Hypergraphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3295, "problem_number": "OPG-46817", "title": "Simultaneous partition of hypergraphs", "statement": "Problem Let $H_1$ and $H_2$ be two $r$-uniform hypergraph on the same vertex set $V$. Does there always exist a partition of $V$ into $r$ classes $V_1, \\dots, V_r$ such that for both $i=1,2$, at least $r!m_i/r^r -o(m_i)$ hyperedges of $H_i$ meet each of the classes $V_1, \\dots, V_r$?", "background": "Source: Open Problem Garden. Original node ID: 46817. URL: http://www.openproblemgarden.org/op/simultaneous_partition_of_hypergraphs.\n\nSource subject path: Graph Theory > Hypergraphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/simultaneous_partition_of_hypergraphs\n- Author(s): Kühn, Daniella; Osthus, Deryk\n- Subject(s): Graph Theory; Hypergraphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 6th, 2013 by fhavet\n\nProblem-page discussion:\nThe bound on the number of hyperedges is what one would expect for a random partition. For graphs, the question was answered in the a\u000effirmative in [KO]. Keevash and Sudakov observed that the answer is negative if we consider many hypergraphs instead of just 2 (see [KO] for the example).\n\nBibliography:\n*[KO] D. Kühn and D. Osthus, Maximizing several cuts simultaneously, Combinatorics, Probability and Computing 16 (2007), 277-283.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Simultaneous partition of hypergraphs\" in Graph Theory; Hypergraphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3296, "problem_number": "OPG-47343", "title": "Turán's problem for hypergraphs", "statement": "Conjecture Every simple $3$-uniform hypergraph on $3n$ vertices which contains no complete $3$-uniform hypergraph on four vertices has at most $\\frac12 n^2(5n-3)$ hyperedges.\n\nConjecture Every simple $3$-uniform hypergraph on $2n$ vertices which contains no complete $3$-uniform hypergraph on five vertices has at most $n^2(n-1)$ hyperedges.", "background": "Source: Open Problem Garden. Original node ID: 47343. URL: http://www.openproblemgarden.org/op/turans_problem_for_hypergraphs.\n\nSource subject path: Graph Theory > Hypergraphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/turans_problem_for_hypergraphs\n- Author(s): Turan, Paul\n- Subject(s): Graph Theory; Hypergraphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 12th, 2013 by fhavet\n\nProblem-page discussion:\nLet $V$ be an $n$-set. A $k$-uniform hypergraph $(V,{\\cal F})$ is complete if ${\\cal F}={V \\choose k}$, the set of all ${n\\choose{k}}$ $k$-subsets of $V$.\n\nLet $\\{X,Y,Z\\}$ be a partition of $V$ into three sets which are as nearly equal in size as possible, and let ${\\cal F}$ be the union of $\\{\\{x,y,z\\}:x\\in X, y\\in Y, z\\in Z\\}$, $\\{\\{x_1,x_2,y\\}:x_1\\in X, x_2\\in X, y\\in Y\\}$, $\\{\\{y_1,y_2,z\\}:y_1\\in Y, y_2\\in Y, z\\in Z\\}$, and $\\{\\{z_1,z_2,x\\}:z_1\\in Z, z_2\\in Z, x\\in X\\}$. This $3$-uniform hypergraph has $\\frac12 n^2(5n-3)$ hyperedges and contains no complete $3$-uniform hypergraph on four vertices. Hence the first conjecture asserts that this hypergraph is extremal with this prpoerty.\n\nLet $\\{X,Y\\}$ be a partition of $V$ into two sets which are as nearly equal in size as possible, and let ${\\cal F}$ be the set of all $3$-subsets of $V$ which intersect both $X$ and $Y$. This $3$-uniform hypergraph has $n^2(n-1)$ hyperedges and contains no complete $3$-uniform hypergraph on five vertices. Hence the second conjecture asserts that this hypergraph is extremal with this property.\n\nBibliography:\n*[T] P. Turán, Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48 (1941), 436--452.\n\nDiscussion links:\n- hypergraph: http://en.wikipedia.org/wiki/hypergraph\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 20.\n\nAttempt notes:\nTarget:\nMake progress on \"Turán's problem for hypergraphs\" in Graph Theory; Hypergraphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3297, "problem_number": "OPG-333", "title": "Seymour's self-minor conjecture", "statement": "Conjecture Every infinite graph is a proper minor of itself.", "background": "Source: Open Problem Garden. Original node ID: 333. URL: http://www.openproblemgarden.org/op/seymours_self_minor_conjecture.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/seymours_self_minor_conjecture\n- Author(s): Seymour, Paul D.\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: infinite graph; minor\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 22nd, 2007 by mdevos\n\nProblem-page discussion:\nRobertson and Seymour famously proved that the set of all finite graphs is well-quasi-ordered by the minor relation. More precisely, if we let $G_1,G_2,\\ldots$ be an infinite sequence of graphs, then there exist $i < j$ so that $G_i$ is a minor of $G_j$. Their theory also gives us a rough structure theorem for the family of graphs without a fixed minor - a powerful tool for studying minors in finite graphs.\n\nOn the other hand, there are still large gaps in our understanding of minors of infinite graphs. For instance, while it is known that Wagner's conjecture does not hold in general for infinite graphs [T], it is possible that countably infinite graphs are well quasi-ordered.\n\nThe conjecture highlighted above is especially interesting, because (if true) it would imply the well quasi-ordering of finite graphs. Indeed, the well quasi-ordering of finite graphs is equivalent to the statement that every infinite set $\\Omega$ of finite graphs contains two distinct members, one of which is a minor of the other. This latter statement follows from the above conjecture, since we may form a single infinite graph $G$ from the disjoint union of the graphs in $\\Omega$, and the proper self-minor of $G$ gives us a pair of graphs in $\\Omega$ with one a minor of the other.\n\nBibliography:\n[T] R. Thomas, A counterexample to \"Wagner's conjecture\" for infinite graphs. Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 1, 55--57. MathSciNet\n\nSource links:\n- minor: http://en.wikipedia.org/wiki/minor (graph theory)\n\nDiscussion links:\n- well-quasi-ordered: http://en.wikipedia.org/wiki/well quasi-order\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0913450\n\nComments:\n- April 4th, 2010 | Anonymous | Counterexample to Seymour's self-minor conjecture: This conjecture has been shown to be false:\n\nB. Oporowski, 'A counterexample to Seymour's self-minor conjecture.' Journal of Graph Theory, (14) 5 (521--524)\n- April 15th, 2010 | Anonymous | Real Problem: The real problem, of course, is to prove Seymour's Conjectre for countable graphs (which suffices for the Graph Minor Theorem). Note that Oporowski's example involves uncountable many vertices.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Seymour's self-minor conjecture\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3298, "problem_number": "OPG-349", "title": "Unions of triangle free graphs", "statement": "Problem Does there exist a graph with no subgraph isomorphic to $K_4$ which cannot be expressed as a union of $\\aleph_0$ triangle free graphs?", "background": "Source: Open Problem Garden. Original node ID: 349. URL: http://www.openproblemgarden.org/op/unions_of_triangle_free_graphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/unions_of_triangle_free_graphs\n- Author(s): Erdos, Paul; Hajnal, Andras\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: forbidden subgraph; infinite graph; triangle free\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 4th, 2007 by mdevos\n\nProblem-page discussion:\nShelah [S] has proved that the existence of such a graph is consistent with ZFC.\n\nBibliography:\n*[EH] P. Erdos and A. Hajnal, On decomposition of graphs, Acta Math. Acad. Sci. Hungar. 18 (1967), 359–377.\n\n[S] S. Shelah, Consistency of positive partition theorems for graphs and models, in Set Theory and Applications, Springer Lecture Notes 1401, (Toronto, ON, 1987), 167–193.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Unions of triangle free graphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3299, "problem_number": "OPG-484", "title": "Infinite uniquely hamiltonian graphs", "statement": "Problem Are there any uniquely hamiltonian locally finite 1-ended graphs which are regular of degree $r > 2$?", "background": "Source: Open Problem Garden. Original node ID: 484. URL: http://www.openproblemgarden.org/op/infinite_uniquely_hamiltonian_graphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/infinite_uniquely_hamiltonian_graphs\n- Author(s): Mohar, Bojan\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: hamiltonian; infinite graph; uniquely hamiltonian\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 24th, 2007 by Robert Samal\n\nProblem-page discussion:\n(Originally appeared as [M].)\n\nLet $G$ be a locally finite infinite graph and let $I(G)$ be the set of ends of~ $G$. The Freudenthal compactification of $G$ is the topological space $|G|$ which is obtained from the usual topological space of the graph, when viewed as a 1-dimensional cell complex, by adding all points of $I(G)$ and setting, for each end $t \\in I(G)$, the basic set of neighborhoods of $t$ to consist of sets of the form $C(S, t) \\cup I(S,t) \\cup E'(S,t)$, where $S$ ranges over the finite subsets of $V(G)$, $C(S, t)$ is the component of $G - S$ containing all rays in $t$, the set $I(S,t)$ contains all ends in $I(G)$ having rays in $C(S, t)$, and $E'(S,t)$ is the union of half-edges $(z,y]$, one for every edge $xy$ joining $S$ and $C(S,t)$. We define a hamilton circle in $|G|$ as a homeomorphic image $C$ of the unit circle $S^1$ into $|G|$ such that every vertex (and hence every end) of $G$ appears in $C$. More details about these notions can be found in [D].\n\nA graph $G$ (finite or infinite) is said to be uniquely hamiltonian if it contains precisely one hamilton circle.\n\nFor finite graphs, the celebrated Sheehan's conjecture states that there are no $r$-regular uniquely hamiltonian graphs for $r>2$; this is known for all odd $r$ and even $r > 23$. For infinite graphs this is false even for odd $r$ (e.g. for the two-way infinite ladder), but each of the known counterexamples has at least 2 ends, leading to the problem stated.\n\nAnother way to extend Sheehan's conjecture to infinite graphs is to define degree of an end $t \\in I(G)$ to be the maximal number of disjoint rays in $t$ and ask the following:\n\nProblem Are there any uniquely hamiltonian locally finite graphs where every vertex and every end has the same degree $r > 2$?\n\nBibliography:\n[D] R. Diestel, Graph Theory, Third Edition, Springer, 2005.\n\n*[M] Bojan Mohar, Problem of the Month\n\nRelated:\nRelated problems\nr-regular graphs are not uniquely hamiltonian.\n\nDiscussion links:\n- Freudenthal compactification: http://en.wikipedia.org/wiki/End_(topology)\n- Sheehan's conjecture: http://www.openproblemgarden.org/?q=node/480\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P0703_HamiltonicityInfinite.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"Infinite uniquely hamiltonian graphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3300, "problem_number": "OPG-488", "title": "Hamiltonian cycles in line graphs of infinite graphs", "statement": "Conjecture\n\n- If $G$ is a 4-edge-connected locally finite graph, then its line graph is hamiltonian.\n- If the line graph $L(G)$ of a locally finite graph $G$ is 4-connected, then $L(G)$ is hamiltonian.", "background": "Source: Open Problem Garden. Original node ID: 488. URL: http://www.openproblemgarden.org/op/hamiltonian_cycles_in_line_graphs_of_infinite_graphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hamiltonian_cycles_in_line_graphs_of_infinite_graphs\n- Author(s): Georgakopoulos, Agelos\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: hamiltonian; infinite graph; line graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 24th, 2007 by Robert Samal\n\nProblem-page discussion:\n(Reproduced from [M].)\n\nA locally finite graph is hamiltonian, if its Freudenthal compactification (also called the end compactification, see [D]) contains a hamilton circle, i.e. a homeomorphic copy of $S^1$ containing all vertices.\n\nThe first part is known for finite graphs. The proof uses the existence of two edge-disjoint spanning trees in 4-edge-connected graphs. In the infinite case, it would be enough to prove that a 4-edge-connected locally finite graph $G$ has two edge-disjoint topological spanning trees (see [D]), one of which is connected as a subgraph of $G$. The problem is open even for the 1-ended case (where hamilton circles correspond to 2-way-infinite paths).\n\nThe second part is widely open even in the finite case, where it was proposed by Thomassen [T].\n\nBibliography:\n[D] Reinhard Diestel, Graph Theory, Third Edition, Springer, 2005.\n\n*[G] A. Georgakopoulos, Oberwolfach reports, 2007.\n\n[M] Bojan Mohar, Problem of the Month\n\n[T] Carsten Thomassen, Reflections on graph theory, J. Graph Theory 10 (1986) 309-324, MathSciNet\n\nRelated:\nRelated problems\nHamiltonian cycles in line graphs\n\nSource links:\n- line graph: http://en.wikipedia.org/wiki/line graph\n\nDiscussion links:\n- Thomassen: http://www.openproblemgarden.org/?q=node/485\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P0703_HamiltonicityInfinite.html\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0856118\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamiltonian cycles in line graphs of infinite graphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3301, "problem_number": "OPG-490", "title": "Hamiltonian cycles in powers of infinite graphs", "statement": "Conjecture\n\n- If $G$ is a countable connected graph then its third power is hamiltonian.\n- If $G$ is a 2-connected countable graph then its square is hamiltonian.", "background": "Source: Open Problem Garden. Original node ID: 490. URL: http://www.openproblemgarden.org/op/hamiltonian_cycles_in_powers_of_infinite_graphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hamiltonian_cycles_in_powers_of_infinite_graphs\n- Author(s): Georgakopoulos, Agelos\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: hamiltonian; infinite graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 24th, 2007 by Robert Samal\n\nProblem-page discussion:\n(Reproduced from [M].)\n\nBoth results are known to be true for finite graphs (the second part is the celebrated result of Fleischner) and also for locally finite graphs [G].\n\nBibliography:\n[G] A. Georgakopoulos, Oberwolfach reports, 2007.\n\n[M] Bojan Mohar, Problem of the Month\n\nRelated:\nRelated problems\nHamiltonian cycles in line graphs of infinite graphs\n\nSource links:\n- power: http://en.wikipedia.org/wiki/Glossary_of_graph_theory#Distance\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P0703_HamiltonicityInfinite.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamiltonian cycles in powers of infinite graphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3302, "problem_number": "OPG-677", "title": "Universal highly arc transitive digraphs", "statement": "An alternating walk in a digraph is a walk $v_0,e_1,v_1,\\ldots,v_m$ so that the vertex $v_i$ is either the head of both $e_i$ and $e_{i+1}$ or the tail of both $e_i$ and $e_{i+1}$ for every $1 \\le i \\le m-1$. A digraph is universal if for every pair of edges $e,f$, there is an alternating walk containing both $e$ and $f$\n\nQuestion Does there exist a locally finite highly arc transitive digraph which is universal?", "background": "Source: Open Problem Garden. Original node ID: 677. URL: http://www.openproblemgarden.org/op/universal_highly_arc_transitive_digraphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/universal_highly_arc_transitive_digraphs\n- Author(s): Cameron, Peter J.; Praeger, Cheryl E.; Wormald, Nicholas C.\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: arc transitive; digraph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 21st, 2007 by mdevos\n\nProblem-page discussion:\nLet $D$ be a digraph. For a nonnegative integer $s$, a $s$-arc in $D$ is a sequence $(x_0,x_1,\\ldots,x_s)$ of vertices so that $(x_i,x_{i+1})$ is an edge for every $0 \\le i \\le s-1$ and $x_{i-1} \\neq x_{i+1}$ for every $1 \\le i \\le s-1$. We say that $D$ is $s$-arc transitive if its automorphism group acts transitively on the set of $s$ arcs, and we say that $D$ is highly arc transitive if it is $s$-arc transitive for every $s$. Note that the condition $0$-arc transitive is precisely equivalent to vertex transitive.\n\nIt is an easy exercise to show that the only finite digraphs which are highly arc transitive are directed cycles. Since such graphs have only trivial alternating walks (only one edge can be used), they are not universal. Thus, any graph satisfying the criteria of the conjecture must be infinite.\n\nLet $P$ be a two way infinite directed path (i.e. the Cayley graph on ${\\mathbb Z}$ with generating set $\\{1\\}$ ). The digraph $P$ is not universal, but moreover, any digraph with a homomorphism onto $P$ cannot be universal. In the same article where the above question was posed, the authors asked wether there exist infinite highly transitive digraphs with no homomorphism onto $P$. This question has since been resolved in the affirmative: Evans [E] constructed such a digraph with infinite indegree, and Malnic et. al. [MMSZ] have constructed a locally finite one.\n\nIn a vertex transitive digraph, every vertex must have the same indegree and the same outdegree, and we shall denote these by $d^-$ and $d^+$ respectively. A theorem of Praeger [P] shows that every locally finite highly transitive digraph for which $d^- \\neq d^+$ has a homomorphism onto $P$ and thus is not universal. More recently, Malnic et. al. [MMMSTZ] have established a condition on edge stabilizers in arc transitive digraphs which implies that any such digraph with $d^- = d^+$ a prime is not universal. It follows that any digraph satisfying the conditions of the highlighted question must have $d^+ = d^-$ a composite number.\n\nBibliography:\n*[CPW] P. J. Cameron, C. E. Praeger, and N. C. Wormald, Infinite highly arc transitive digraphs and universal covering digraphs. Combinatorica 13 (1993), no. 4, 377--396. MathSciNet.\n\n[E] D. M. Evans, An infinite highly arc-transitive digraph, European J. Combin., 18 (1997) 281--286. MathSciNet.\n\n[MMMSTZ] A. Malnic, D. Marusic, R. G. Moller, N. Seifter, V. Trofimov, and B. Zgrablic, Highly arc transitive digraphs: reachability, topological groups. European J. Combin. 26 (2005), no. 1, 19--28. MathSciNet.\n\n[MMSZ] A. Malnic, D. Marusic, N. Seifter, and B. Zgrablic, Highly arc-transitive digraphs with no homomorphism onto Z. Combinatorica 22 (2002), no. 3, 435--443. MathSciNet\n\n[P] C. E. Praeger, On homomorphic images of edge transitive directed graphs, Australas. J. Combin., 3 (1991), 207--210. MathSciNet.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1262915\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1437003\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2101032\n- Highly arc-transitive digraphs with no homomorphism onto Z: http://www.ijp.si/ftp/pub/preprints/ps/99/pp659.ps\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1932063\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1122225\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 24.\n\nAttempt notes:\nTarget:\nMake progress on \"Universal highly arc transitive digraphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3303, "problem_number": "OPG-687", "title": "Unfriendly partitions", "statement": "If $G$ is a graph, we say that a partition of $V(G)$ is unfriendly if every vertex has at least as many neighbors in the other classes as in its own.\n\nProblem Does every countably infinite graph have an unfriendly partition into two sets?", "background": "Source: Open Problem Garden. Original node ID: 687. URL: http://www.openproblemgarden.org/op/unfriendly_partitions.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/unfriendly_partitions\n- Author(s): Cowan, Robert H.; Emerson, William R.\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: coloring; infinite graph; partition\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 22nd, 2007 by mdevos\n\nProblem-page discussion:\nIt is a simple property that every finite graph $G$ has an unfriendly partition into two sets - just choose a partition of $V(G)$ into two sets so that the number of edges with one end in each is maximum. Cowan and Emerson [CE] conjectured that the same property should hold true of infinite graphs. A counterexample to this was constructed by Milner and Shelah [MS], but their construction uses uncountably many vertices, leaving the countable case (highlighted above) still open. In the same article by Milner and Shelah [MS], they show that every graph does have an unfriendly partition into three sets.\n\nCuriously, it is quite easy to see that the answer to the above question is yes in the case when all vertices have finite degree, and also in the case when all vertices have infinite degree. The former follows from the unfriendly partition property for finite graphs together with a standard compactness argument. The latter can be achieved with a \"back and forth\" construction. Thus, the difficult case is the mixed one. Aharoni, Milner, and Prikry [AMP] showed that every graph with only finitely many vertices of infinite degree has an unfriendly partition into two sets, but this seems the extent of our knowledge.\n\nIt does not appear that there is any consensus among experts as to whether this conjecture should be true or false.\n\nBibliography:\n*[CE] R. Cowan and W. Emerson, Proportional colorings of graphs, unpublished.\n\n[MS] E. C. Milner and S. Shelah, Graphs with no unfriendly partitions. A tribute to Paul Erdös, 373--384, Cambridge Univ. Press, Cambridge, 1990. MathSciNet.\n\n[AMP] R. Aharoni, E. C. Milner, K. Prikry, Unfriendly partitions of a graph. J. Combin. Theory Ser. B 50 (1990), no. 1, 1--10. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1117030\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1070461\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Unfriendly partitions\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3304, "problem_number": "OPG-690", "title": "Strong matchings and covers", "statement": "Let $H$ be a hypergraph. A strongly maximal matching is a matching $F \\subseteq E(H)$ so that $|F' \\setminus F| \\le |F \\setminus F'|$ for every matching $F'$. A strongly minimal cover is a (vertex) cover $X \\subseteq V(H)$ so that $|X' \\setminus X| \\ge |X \\setminus X'|$ for every cover $X'$.\n\nConjecture If $H$ is a (possibly infinite) hypergraph in which all edges have size $\\le k$ for some integer $k$, then $H$ has a strongly maximal matching and a strongly minimal cover.", "background": "Source: Open Problem Garden. Original node ID: 690. URL: http://www.openproblemgarden.org/op/strong_matchings_and_covers.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/strong_matchings_and_covers\n- Author(s): Aharoni, Ron\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: cover; infinite graph; matching\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 23rd, 2007 by mdevos\n\nProblem-page discussion:\nThe theory of matching in finite graphs is quite well understood. Now, thanks to the work of Aharoni and others, much of this theory has been extended to infinite graphs. On the other hand, matching in hypergraphs - both finite and infinite - is a subject where our knowledge apears to be lacking. The above conjecture asserts a rather basic property of hypergraphs which would be nice to verify.\n\nThis conjecture is (of course) trivial for finite hypergraphs, but it looks very difficult for infinite ones. It has been proved by Aharoni [A2] for the case when $k=2$, that is, for infinite graphs. Here the key tool is an infinite version of the Tutte-Edmonds-Gallai decomposition theorem [A1].\n\nNext we offer another interesting conjecture of Aharoni on minimal covers.\n\nConjecture If $G$ is a (possibly infinite) graph and $H$ is the hypergraph of independent sets in $G$, then $H$ has a strongly minimal cover.\n\nBibliography:\n[A1] R. Aharoni, Matchings in infinite graphs. J. Combin. Theory Ser. B 44 (1988), no. 1, 87--125. MathSciNet.\n\n*[A2] R. Aharoni, Infinite matching theory. Directions in infinite graph theory and combinatorics (Cambridge, 1989). Discrete Math. 95 (1991), no. 1-3, 5--22. MathSciNet.\n\nSource links:\n- hypergraph: http://en.wikipedia.org/wiki/hypergraph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0923268\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1141929\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Strong matchings and covers\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3305, "problem_number": "OPG-691", "title": "Highly arc transitive two ended digraphs", "statement": "Conjecture If $G$ is a highly arc transitive digraph with two ends, then every tile of $G$ is a disjoint union of complete bipartite graphs.", "background": "Source: Open Problem Garden. Original node ID: 691. URL: http://www.openproblemgarden.org/op/highly_arc_transitive_two_ended_digraphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/highly_arc_transitive_two_ended_digraphs\n- Author(s): Cameron, Peter J.; Praeger, Cheryl E.; Wormald, Nicholas C.\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: arc transitive; digraph; infinite graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 29th, 2007 by mdevos\n\nProblem-page discussion:\nIt follows from a theorem of Dunwoody [D] that every vertex transitive graph $G$ with two ends has a system of imprimitivity $\\{ X_i: i \\in {\\mathbb Z} \\}$ with finite blocks so that the cyclic order $\\ldots X_{-2}, X_{-1},X_0,X_1,X_2,\\ldots$ is preserved by the automorphism group (of $G$ ). If $G$ is edge-transitive, then every edge of $G$ must have its ends in two consecutive blocks, so in this case $G$ is an edge-disjoint union of the (isomorphic) bipartite graphs $G[X_i,X_{i+1}]$ for $i \\in {\\mathbb Z}$- which we shall call tiles. Note that the tiles are edge-transitive.\n\nThis gives us a good description of edge-transitive graphs with two ends; each is made up by gluing together copies of a tile in a linear order. If $G$ is a 2-arc transitive digraph with two ends, then all edges in each tile must be oriented consistently, so by possibly reordering, we may assume that every edge in $G[X_i,X_{i+1}]$ is oriented from $X_i$ to $X_{i+1}$. The above conjecture asserts that under the added symmetry condition of high arc transitivity, each tile has a simple structure - namely it is a union of (consistently oriented) complete bipartite graphs.\n\nIt is easy to construct a highly arc transitive two ended graph by simply using the complete bipartite graph $K_{n,n}$ (with all edges oriented consistently) as a tile. Mckay and Praeger found the following pretty construction of a highly arc transitive digraph with tiles isomorphic to a disjoint union of complete bipartite graphs: Let $S$ be a finite set, let $n$ be a positive integer, and define $G$ to be the digraph with vertex set ${\\mathbb Z} \\times S^n$ and an edge from $(i, \\mathbf{x}, y)$ to $(i+1, z, \\mathbf{x})$ if $i \\in {\\mathbb Z}$, $\\mathbf{x} \\in S^{n-1}$, and $y,z \\in S$. A generalized (twisted) version of this construction was introduced by Cameron, Praeger, and Wormald [CPW], but again, every tile in this construction is a disjoint unions of bipartite graphs, and it looks hard to do anything else.\n\nBibliography:\n*[CPW] P. J. Cameron, C. E. Praeger, and N. C. Wormald, Infinite highly arc transitive digraphs and universal covering digraphs. Combinatorica 13 (1993), no. 4, 377--396. MathSciNet.\n\n[D] M. J. Dunwoody, Cutting up graphs. Combinatorica 2 (1982), no. 1, 15--23. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1262915\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0671142\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Highly arc transitive two ended digraphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3306, "problem_number": "OPG-735", "title": "End-Devouring Rays", "statement": "Problem Let $G$ be a graph, $\\omega$ a countable end of $G$, and $K$ an infinite set of pairwise disjoint $\\omega$-rays in $G$. Prove that there is a set $K'$ of pairwise disjoint $\\omega$-rays that devours $\\omega$ such that the set of starting vertices of rays in $K'$ equals the set of starting vertices of rays in $K$.", "background": "Source: Open Problem Garden. Original node ID: 735. URL: http://www.openproblemgarden.org/op/end_devouring_rays.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/end_devouring_rays\n- Author(s): Georgakopoulos, Agelos\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: end; ray\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: February 3rd, 2008 by Agelos\n\nProblem-page discussion:\nWe say that a set of rays $K$ devours the end $\\omega$ if every ray in $\\omega$ meets some ray in $K$. An end is countable if there is a countable set of rays devouring it.\n\nIf $K$ is a finite set of rays then it is not hard to prove (see [G]) that this problem has a positive answer:\n\nTheorem For every graph $G$ and every countable end $\\omega$ of $G$, if $G$ has a set $K$ of $k\\in \\mathcal N$ pairwise disjoint $\\omega$-rays, then it also has a set $K'$ of $k$ pairwise disjoint $\\omega$-rays that devours $\\omega$. Moreover, $K'$ can be chosen so that its rays have the same starting vertices as the rays in~ $K$.\n\nBibliography:\n*[G] A. Georgakopoulos, Infinite Hamilton Cycles in Squares of Locally Finite Graphs, Preprint.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"End-Devouring Rays\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3307, "problem_number": "OPG-1750", "title": "Characterizing (aleph_0,aleph_1)-graphs", "statement": "Call a graph an $(\\aleph_0,\\aleph_1)$-graph if it has a bipartition $(A,B)$ so that every vertex in $A$ has degree $\\aleph_0$ and every vertex in $B$ has degree $\\aleph_1$.\n\nProblem Characterize the $(\\aleph_0,\\aleph_1)$-graphs.", "background": "Source: Open Problem Garden. Original node ID: 1750. URL: http://www.openproblemgarden.org/op/characterizing_aleph_0_aleph_1_graphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/characterizing_aleph_0_aleph_1_graphs\n- Author(s): Diestel, Reinhard; Leader, Imre\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: binary tree; infinite graph; normal spanning tree; set theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 26th, 2008 by mdevos\n\nProblem-page discussion:\nThe motivation for this problem comes from a lovely paper of Diestel and Leader [DL] where they prove that an infinite graph has a normal spanning tree (the natural infinite analogue of a depth-first search tree) if and only if it has no minor isomorphic to either an $(\\aleph_0,\\aleph_1)$-graph or an Aronszajn tree. (An earlier conjecture of Halin asserted that only the first of these excluded minors was needed.) So, $(\\aleph_0,\\aleph_1)$-graphs appear as a forbidden minor obstruction to the existence of a kind of depth-first search tree for infinite graphs.\n\nThe obvious example of an $(\\aleph_0,\\aleph_1)$-graph is $K_{\\aleph_0,\\aleph_1}$, but there are other natural families of such graphs. For instance, let $T$ be an infinite binary tree with root $r$, and let $X$ be the set of all rays (one way infinite paths) with endpoint $r$. Now, form a bipartite graph with vertex bipartition $(X,V(T))$ and adjacency given by the rule that $v \\in V(T)$ adjacent to $x \\in X$ if and only if $v$ lies on the ray $x$ (in $G$ ). Any $(\\aleph_0,\\aleph_1)$-graph which is isomorphic to a subgraph of this graph is said to be of binary type.\n\nSay that a $(\\aleph_0,\\aleph_1)$-graph is divisible if there exist disjoint subsets $A',A\" \\subseteq A$ and disjoint subsets $B',B\" \\subseteq B$ so that the graphs induced by both $A' \\cup B'$ and $A\" \\cup B\"$ are $(\\aleph_0,\\aleph_1)$-graphs. It is not difficult to show that every binary type graph is divisible. Curiously, the existence of non-divisible $(\\aleph_0,\\aleph_1)$-graphs depends on the Continuum Hypothesis (see [DL]).\n\nAlthough it is not clear wether or not there is a nice characterization of $(\\aleph_0,\\aleph_1)$-graphs, it would certainly be interesting to find more natural families of these graphs. The following rather more concrete question is posed by Diestel and Leader who suspect the answer is 'no'.\n\nProblem Does every $(\\aleph_0,\\aleph_1)$-graph have an $(\\aleph_0,\\aleph_1)$-graph as a minor which is either indivisible or of binary type?\n\nBibliography:\n[DL] R. Diestel and I. Leader, Normal spanning trees, Aronszajn trees and excluded minors, J. London Math. Soc. 63 (2001), 16-32;\n\nBibliography links:\n- Normal spanning trees, Aronszajn trees and excluded minors: http://www.math.uni-hamburg.de/home/diestel/papers/Aronszajn.pdf\n\nComments:\n- November 21st, 2011 | Anonymous | problem claimed to be solved by scott streit texas professor: Has his solution ever been viewed/reviewed? He claims to have solved it with his students\n- November 24th, 2011 | Robert Samal | Re: problem claimed to be solved by scott streit texas professor: Please be more specific: who claims it a where? If I googled the right Scott Streit, he doesn't seem to be the kind of person interested in this kind of problems.\n\nBest wishes, Robert\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 20.\n\nAttempt notes:\nTarget:\nMake progress on \"Characterizing (aleph_0,aleph_1)-graphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3308, "problem_number": "OPG-827", "title": "Coloring random subgraphs", "statement": "If $G$ is a graph and $p \\in [0,1]$, we let $G_p$ denote a subgraph of $G$ where each edge of $G$ appears in $G_p$ with independently with probability $p$.\n\nProblem Does there exist a constant $c$ so that ${\\mathbb E}(\\chi(G_{1/2})) > c \\frac{\\chi(G)}{\\log \\chi(G)}$?", "background": "Source: Open Problem Garden. Original node ID: 827. URL: http://www.openproblemgarden.org/op/coloring_random_subgraphs.\n\nSource subject path: Graph Theory > Probabilistic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/coloring_random_subgraphs\n- Author(s): Bukh, Boris\n- Subject(s): Graph Theory; Probabilistic Graph Theory\n- Keywords: coloring; random graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 18th, 2008 by mdevos\n\nProblem-page discussion:\nIt is a classical result that the above problem has a positive answer when $G$ is the complete graph. More generally, the lower bound ${\\mathbb E}(\\chi(G_{1/2})) \\ge c \\frac{\\chi(G)}{\\log |V(G)|}$ is known.\n\nIt is easy to obtain the bound ${\\mathbb E}(\\chi(G_{1/2})) \\ge (\\chi(G))^{1/2}$, since we may imagine forming two random subgraphs $H,H'$ of $G$ by putting each edge of $G$ in either $H$ or $H'$ independently with probability $1/2$. Then $\\chi(H) \\chi(H') \\ge \\chi(G)$ and this gives the desired bound. A similar argument with three subgraphs shows that ${\\mathbb E}(\\chi(G_{1/3})) \\ge (\\chi(G))^{1/3}$, however these arguments all seem to require integer multiples, so the best known lower bound on ${\\mathbb E}(\\chi(G_{49/100}))$ of this form is $(\\chi(G))^{1/3}$.\n\nBibliography:\n*[B] Boris Bukh's problem page.\n\nBibliography links:\n- Boris Bukh's problem page: http://www.math.princeton.edu/%7Ebbukh/problems.html\n\nComments:\n- June 4th, 2009 | chrisrudy502 | No?: I haven't worked through the details yet, but what if $G$ is the union of $K_n$ and a sufficiently huge bipartite graph $H$? Then $\\chi(G) = n$, and by taking $H$ huge enough, you can get $\\mathbb{E}(\\chi(G_{1/2}))$ as close to 2 as you like, forcing $c$ as small as you like.\n\nEDIT: Whoo boy. Nevermind.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 19.\n\nAttempt notes:\nTarget:\nMake progress on \"Coloring random subgraphs\" in Graph Theory; Probabilistic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3309, "problem_number": "OPG-1757", "title": "Negative association in uniform forests", "statement": "Conjecture Let $G$ be a finite graph, let $e,f \\in E(G)$, and let $F$ be the edge set of a forest chosen uniformly at random from all forests of $G$. Then\n$$\n{\\mathbb P}(e \\in F \\mid f \\in F}) \\le {\\mathbb P}(e \\in F)\n$$", "background": "Source: Open Problem Garden. Original node ID: 1757. URL: http://www.openproblemgarden.org/op/negative_association_in_uniform_forests.\n\nSource subject path: Graph Theory > Probabilistic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/negative_association_in_uniform_forests\n- Author(s): Pemantle, Robin\n- Subject(s): Graph Theory; Probabilistic Graph Theory\n- Keywords: forest; negative association\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 30th, 2008 by mdevos\n\nProblem-page discussion:\nThe FKG inequality is the cornerstone of a respectable theory of positive association; If a natural lattice condition holds, we can use it to deduce positive association. On the other hand, the theory of negative associations is still lacking good techniques. See Pemantle's lovely paper [P] for an excellent description of this situation. The conjecture highlighted above seems to be almost obviously true, but we have no tools to prove it.\n\nModifying the conjecture by replacing \"forest\" by \"spanning tree\" gives a true statement which was proved by Feder and Mihail [FM]. Actually, they prove that this holds more generally for uniform bases of balanced matroids. Perhaps surprisingly, this is false for general matroids, see [SW].\n\nBibliography:\n[FM] T. Feder and M. Mihail, Balanced Matroids. Proc 24th Annual STOC 26 - 38 (1992).\n\n*[P] R. Pemantle, Towards a theory of negative dependence, Journal of Mathematical Physics 41 (2000), 1371–1390.\n\n[SW] P. D. Seymour and D. J. A. Welsh, Combinatorial applications of an inequality from statistical mechanics. Math. Proc. Camb. Phil. Soc. 77 485 - 495 (1975).\n\nBibliography links:\n- Towards a theory of negative dependence: http://arxiv.org/abs/math/0404095\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Negative association in uniform forests\" in Graph Theory; Probabilistic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3310, "problem_number": "OPG-37656", "title": "Chromatic number of random lifts of complete graphs", "statement": "Question Is the chromatic number of a random lift of $K_5$ concentrated on a single value?", "background": "Source: Open Problem Garden. Original node ID: 37656. URL: http://www.openproblemgarden.org/op/chromatic_number_of_random_lifts_of_complete_graphs.\n\nSource subject path: Graph Theory > Probabilistic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/chromatic_number_of_random_lifts_of_complete_graphs\n- Author(s): Amit, Linial, Matousek\n- Subject(s): Graph Theory; Probabilistic Graph Theory\n- Keywords: random lifts, coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 3rd, 2012 by DOT\n\nProblem-page discussion:\nLet $G$ be a graph with vertex set $V$ and edge set $E$. An $h$-lift $H$ is a graph with vertex set $V\\times\\{1,\\dots,h\\}$, such that $(u,k)$ and $(v,\\ell)$ may only be adjacent in $H$ if $uv \\in E$, and for each $uv\\in E$, the edges between $\\{u\\}\\times\\{1,\\dots,h\\}$ and $\\{v\\}\\times\\{1,\\dots,h\\}$ form a perfect matching.\n\nA random $h$-lift of $G$ is a graph drawn uniformly at random from the set of all $h$-lifts of $G$. This amounts to choosing, independently at random, a perfect matching for each edge of $G$. One is generally interested in properties of random $h$-lifts when $h\\to\\infty$.\n\nAmit, Linial, and Matousek [ALM02] have studied the chromatic number of random lifts. They ask whether a the chromatic number of a random $h$-lift of $K_5$ is asymptotically almost surely a single number.\n\nIt is easy to see that this number may be either 3 or 4. Farzad and Theis [FT12] have shown that random lifts of $K_5\\setminus e$ are asymptotically almost surely 3-colorable.\n\nA more general question is this.\n\nQuestion Is the chromatic number of a random lift of $K_n$ concentrated on a single value?\n\nAmit, Linial, and Matousek [ALM02] have shown that the chromatic number of a random lift of $K_n$ is in $\\Theta(n/\\log n)$.\n\nBibliography:\n*[ALM02] Random Lifts of Graphs III: Independence and Chromatic Number, A. Amit, N. Linial and J. Matousek, Random Structures and Algorithms, 20(2002) 1-22.\n\n[FT12] Random lifts of $K_5\\setminus e$ are 3-colourable. B. Farzad and D.O. Theis. SIAM J. Discrete Math. 26:1 (2012), 169–179.\n\nDiscussion links:\n- $h$-lift: http://en.wikipedia.org/wiki/Covering_graph\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 17.\n\nAttempt notes:\nTarget:\nMake progress on \"Chromatic number of random lifts of complete graphs\" in Graph Theory; Probabilistic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3311, "problem_number": "OPG-36922", "title": "Domination in plane triangulations", "statement": "Conjecture Every sufficiently large plane triangulation $G$ has a dominating set of size $\\le \\frac{1}{4} |V(G)|$.", "background": "Source: Open Problem Garden. Original node ID: 36922. URL: http://www.openproblemgarden.org/op/domination_in_plane_triangulations.\n\nSource subject path: Graph Theory > Topological Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/domination_in_plane_triangulations\n- Author(s): Matheson, Lesley R.; Tarjan, Robert E.\n- Subject(s): Graph Theory; Topological Graph Theory\n- Keywords: coloring; domination; multigrid; planar graph; triangulation\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 4th, 2009 by mdevos\n\nProblem-page discussion:\nMotivated by some problems in multigrid computations, Matheson and Tarjan [MT] considered the problem of finding small dominating sets in plane triangulations. They proved that every such graph $G$ has a dominating set of size $\\le \\frac{1}{3} |V(G)|$ and posed the above question.\n\nThe Octahedron is a triangulation with 6 vertices for which every dominating set has size $\\ge 2$, so no constant better than $\\frac{1}{3}$ can be achieved in general. However, it appears that one can do better for larger graphs. The most extreme examples here (also from [MT]) are constructed as follows: Start with $n$ disjoint copies of $K_4$ embedded in the plane, and then add edges to complete this graph to a triangulation (with $4n$ vertices). Now each of the original copies of $K_4$ has an inner vertex which has degree 3 in the final graph, and in order to cover it, one must take at least one vertex from this $K_4$. It follows that every dominating set has size $\\ge n$.\n\nSince the Matheson-Tarjan proof is short and instructive, we sketch it here. In fact, we shall prove (as they did) the stronger statement that every near-triangulation (a graph embedded in the plane with all finite faces of size three) has a (possibly improper) 3-coloring so that each color class is a dominating set and so that the subgraph induced by those vertices incident with the infinite face is properly colored. This stronger fact we prove by induction. If the infinite face is not bounded by a cycle or the infinite face is bounded by a cycle which has a chord, then the graph may be written as the union of two near-triangulations $G_1,G_2$ where $G_1$ and $G_2$ either share one vertex or two adjacent vertices and one edge. In either case, the result follows by applying induction to $G_1$ and $G_2$. Otherwise, choose a vertex $v$ on the infinite face, delete $v$ and apply induction. Since the neighbors of $v$ are all on the infinite face, and do not form an independent set, there are at least two colors, say $1$ and $2$, which appear on the neighbors of $v$. Now giving $v$ the color $3$ gives a solution.\n\nBibliography:\n[MT] L. R. Matheson, R. E. Tarjan, Dominating sets in planar graphs. European J. Combin. 17 (1996), no. 6, 565--568. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1401911\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Domination in plane triangulations\" in Graph Theory; Topological Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3312, "problem_number": "OPG-46634", "title": "Large induced forest in a planar graph.", "statement": "Conjecture Every planar graph on $n$ verices has an induced forest with at least $n/2$ vertices.", "background": "Source: Open Problem Garden. Original node ID: 46634. URL: http://www.openproblemgarden.org/op/large_induced_forest_in_a_planar_graph.\n\nSource subject path: Graph Theory > Topological Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/large_induced_forest_in_a_planar_graph\n- Author(s): Abertson, Michael O.; Berman, David M.\n- Subject(s): Graph Theory; Topological Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture is best possible. (See [AW]). It follows from Borodin's theorem stating that every planar graph has an acyclic $5$-colouring that every planar graph on $n$ verices has an induced forest with at least $2n/5$ vertices. The conjecture holds for planar graph with girth at least $5$, because they can be partitionned into a stable set and a forest [BG] (see also [KT]).\n\nAkiyama-Watanabe [AW] conjectured an even larger induced forest for bipartite planar graphs.\n\nConjecture Every bipartite planar graph on $n$ verices has an induced forest with at least $5n/8$ vertices.\n\nThis conjecture is also best possible. (See [AW]).\n\nBibliography:\n*[AB] M. O. Albertson and D. M. Berman. A conjecture on planar graphs. Graph Theory and Related Topics (J. A. Bondy and U. S. R. Murty, eds.), (Academic Press, 1979), 357.\n\n[AW] J. Akiyama and M. Watanabe. Maximum induced forests of planar graphs. Graphs and Combinatorics 3 (1987), 201--202.\n\n[B] O. V. Borodin. A proof of B. Grünbaum's conjecture on the acyclic 5-colorability of planar graphs. (Russian) Dokl. Akad. Nauk SSSR 231 (1976), no. 1, 18--20.\n\n[BG] O. V. Borodin and A. N. Glebov. On the partition of a planar graph of girth 5 into an empty graph and an acyclic subgraph. Diskretn. Anal. Issled. Oper. Ser. 1 8:34–53, 2001\n\n[KT] K. Kawarabayashi and C. Thomassen. Decomposing a planar graph of girth 5 into an independent set and a forest. Journal of Combinatorial Theory, Series B 99(4):674–684, 2009.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Large induced forest in a planar graph.\" in Graph Theory; Topological Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3313, "problem_number": "OPG-46943", "title": "Every 4-connected toroidal graph has a Hamilton cycle", "statement": "Conjecture Every 4-connected toroidal graph has a Hamilton cycle.", "background": "Source: Open Problem Garden. Original node ID: 46943. URL: http://www.openproblemgarden.org/op/every_4_connected_toroidal_graph_has_a_hamilton_cycle.\n\nSource subject path: Graph Theory > Topological Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/every_4_connected_toroidal_graph_has_a_hamilton_cycle\n- Author(s): Grunbaum, Branko; Nash-Williams, Crispin, St. J. A.\n- Subject(s): Graph Theory; Topological Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2013 by fhavet\n\nProblem-page discussion:\nTutte [Tu] proved that every 4-connected planar graph has a Hamilton cycle. (See also [Th]). Thomas and Yu [TY] proved that every 4-connected projective-planar graph has a Hamilton cycle.\n\nThomas and Yu [TY] also proved that every 5-connected toroidal graph has a Hamilton cycle. In fact, they show something stronger: every edge in a 5-connected toroidal graph is contained in a Hamilton cycle. This stronger result cannot be extended to 4-connected toroidal graphs: Thomassen [Th] showed 4-connected toroidal graphs in which certain edges are not contained in any Hamilton cycle.\n\nBibliography:\n*[G] B. Grünbaum, Polytopes, graphs, and complexes. Bull. Amer. Math. Soc. 76 (1970) 1131-1201.\n\n*[N] C. St. J. A. Nash-Williams, Unexplored and semi-explored territories in graph theory. New Directions in Graph Theory, Academic Press, New York (1973) 169-176.\n\n[TY] R. Thomas and X. Yu, 5-connected toroidal graphs are hamiltonian. J. Combinat. Theory Ser. B 69 (1997), no.1, 79-96.\n\n[Th] C. Thomassen, A theorem on paths in planar graphs. J. Graph Theory 7 (1983) 169-176.\n\n[Tu] W. T. Tutte, A theorem on planar graphs. Trans. Amer. Math. Soc. 82 (1956) 99-116.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Every 4-connected toroidal graph has a Hamilton cycle\" in Graph Theory; Topological Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3314, "problem_number": "OPG-177", "title": "Grunbaum's Conjecture", "statement": "Conjecture If $G$ is a simple loopless triangulation of an orientable surface, then the dual of $G$ is 3-edge-colorable.", "background": "Source: Open Problem Garden. Original node ID: 177. URL: http://www.openproblemgarden.org/op/grunbaums_conjecture.\n\nSource subject path: Graph Theory > Topological Graph Theory > Coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/grunbaums_conjecture\n- Author(s): Grunbaum, Branko\n- Subject(s): Graph Theory; Topological Graph Theory; Coloring\n- Keywords: coloring; surface\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 4th, 2007 by mdevos\n\nProblem-page discussion:\nThe Four Color Theorem is equivalent to the statement that every cubic planar graph with no bridge is 3-edge-colorable. This is precisely equivalent to Grunbaum's conjecture restricted to the plane. Thus, Grunbaum's conjecture, if true, would imply the Four Color Theorem. Indeed, this conjecture suggests a deep generalization of the 4-color theorem.\n\nDefinition: A cubic graph $G$ is a snark if $G$ is internally 4-edge-connected and $G$ is not 3-edge-colorable.\n\nGrunbaum's conjecture states that no snark is the dual of a simple loopless triangulation of an orientable surface. In this light, the conjecture looks to be almost obviously false. To find a counterexample, it suffices to embed a snark in an orientable surface so that the dual has no loops or parallel edges. Of course, the difficulty is in satisfying this last constraint. All known embeddings of snarks in orientable surfaces give rise to either loops or parallel edges in the dual. It is striking to compare this conjecture with the Orientable cycle double cover conjecture. Both conjectures may be stated in terms of embedding snarks in orientable surfaces as follows:\n\nConjecture (Grunbaum's conjecture (version 2)) Every embedding of a snark in an orientable surface has a cycle of length 1 or 2 (a loop or parallel edges) in the dual.\n\nConjecture (Orientable cycle double cover conjecture) Every snark may be embedded in an orientable surface so that the dual graph has no cycle of length 1 (no loop).\n\nIn this light it may look unlikely that both Grunbaum's conjecture and the orientable cycle double cover conjecture are true. I (M. DeVos) don't have a strong sense for or against either of these conjectures, and I don't believe there is a strong consensus among experts.\n\nMohar and Robertson have suggested the following weak version of Grunbaum's conjecture: There exists a fixed constant $k$ so that the dual of every loopless triangulation of an orientable surface of face-width $>k$ is 3-edge-colorable. Robertson has suggested that this may still hold true even for nonorientable surfaces. The following conjecture is a further weakening of Grunbaum's conjecture which would allow the parameter $k$ to depend on the surface. This is probably the weakest open problem in this vein.\n\nConjecture (Weak Grunbaum conjecture) For every orientable surface $\\Sigma$, there is a fixed constant $k$ so that the dual of every loopless triangulation of $\\Sigma$ with face-width $>k$ is 3-edge-colorable.\n\nBibliography:\n[G] B. Grunbaum, Conjecture 6. In Recent progress in combinatorics, (W.T. Tutte Ed.), Academic Press (1969) 343.\n\nSource links:\n- triangulation: http://en.wikipedia.org/wiki/triangulation (topology)\n- orientable surface: http://en.wikipedia.org/wiki/orientable surface\n\nDiscussion links:\n- Four Color Theorem: http://en.wikipedia.org/wiki/four color theorem\n- cubic: http://en.wikipedia.org/wiki/cubic graph\n- planar: http://en.wikipedia.org/wiki/planar graph\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n- edge-colorable: http://en.wikipedia.org/wiki/edge coloring\n- snark: http://en.wikipedia.org/wiki/snark (graph theory)\n- Orientable cycle double cover conjecture: http://www.openproblemgarden.org/?q=op/cycle_double_cover_conjecture\n\nComments:\n- October 4th, 2007 | Anonymous | Grunbaum's conjecture is false!: Martin Kochol and Bojan Mohar announced a counterexample to Grunbaum's conjecture at the PIMS Workshop on the Cycle Double Cover Conjecture (Vancouver, 2007). By using Kochol's \"superposition\" operation on several copies of Petersen's graph, they constructed a snark which embeds on the orientable surface of genus 9, and whose dual contains no loops or parallel edges.\n\nOf course Grunbaum's Conjecture may still hold true for lower-genus surfaces, in particular, the torus.\n\nRef: Kochol, M; Mohar, B; preprint 2007.\n- March 23rd, 2009 | Anonymous | Solution: The solution of the Grunbaum's conjecture is published in (see also http://www.mat.savba.sk/~kochol):\n\nM. Kochol, 3-Regular non 3-edge-colorable graphs with polyhedral embeddings in orientable surfaces, in: Graph Drawing 2008, Editors: I.G. Tollis, M. Patrignani, Lecture Notes in Computer Science, Vol. 5417, Springer-Verlag, Berlin, 2009, pp. 319-323\n\nM. Kochol, Polyhedral embeddings of snarks in orientable surfaces, Proceedings of the American Mathematical Society vol. 137 (2009), pp. 1613-1619.\n- November 25th, 2007 | Anonymous | Wrong information. Martin: Wrong information. Martin Kochol\n- December 17th, 2007 | Anonymous | so?: Sorry I don't understand, is the conjecture false or still opened?\n\nNR\n- December 19th, 2007 | Anonymous | False in general: A counter example was found on the nine holed torus, and I have heard there is now one on the five holed torus as well so the conjecture is false in general. However what is still not known is for which n does the conjecture hold for the n-holed torus, and in particular the one holed torus is always of interest and remains open.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Grunbaum's Conjecture\" in Graph Theory; Topological Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3315, "problem_number": "OPG-411", "title": "5-local-tensions", "statement": "Conjecture There exists a fixed constant $c$ (probably $c=4$ suffices) so that every embedded (loopless) graph with edge-width $\\ge c$ has a 5-local-tension.", "background": "Source: Open Problem Garden. Original node ID: 411. URL: http://www.openproblemgarden.org/op/5_local_tensions.\n\nSource subject path: Graph Theory > Topological Graph Theory > Coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/5_local_tensions\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Topological Graph Theory; Coloring\n- Keywords: coloring; surface; tension\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 22nd, 2007 by mdevos\n\nProblem-page discussion:\nThe edge-width of an embedded graph is the length of the shortest non-contractible cycle.\n\nDefinition Let $G$ be a directed graph, let $\\Gamma$ be an abelian group, and let $\\phi: E(G) \\rightarrow \\Gamma$. Define the height of a walk $W$ to be the sum of $\\phi$ on the forward edges of $W$ minus the sum of $\\phi$ on the backward edges of $W$ (edges are counted according to multiplicity). We call $\\phi$ a tension if the height of every closed walk is zero, and if $G$ is an embedded graph, we call $\\phi$ a local-tension if the height of every closed walk which forms a contractible curve is zero. If in addition, $\\Gamma = {\\mathbb Z}$ and $0 < \\phi(e) < k$ for some $k \\in {\\mathbb Z}$, we say that $\\phi$ is a $k$-tension or a $k$-local-tension. If we reverse an edge $e$ and replace $\\phi(e)$ by $-\\phi(e)$, this preserves the properties of tension or local-tension. Accordingly, we say that an undirected graph (embedded graph) $G$ has a $k$-tension ( $k$-local-tension) if some and thus every orientation of it admits such a map.\n\nProposition A graph has a $k$-tension if and only if it is $k$-colorable.\n\nProof To see the \"if\" direction, let $f: V(G) \\rightarrow \\{0,\\ldots,k-1\\}$ be a coloring, orient the edges of $G$ arbitrarily, and defining $\\phi: E(G) \\rightarrow {\\mathbb Z}$ by the rule $\\phi(uv) = f(v) - f(u)$. It is straightforward to check that $\\phi$ is a $k$-tension. For the \"only if\" direction, let $\\phi: E(G) \\rightarrow {\\mathbb Z}$ be a $k$-tension. Now choose a point $u \\in V(G)$ and define the map $f: V(G) \\rightarrow {\\mathbb Z}_k$ by the rule that $f(v)$ is the height of some (and thus every) walk from $u$ to $v$ modulo $k$. Again, it is straightforward to check that this defines a proper $k$-coloring.\n\nFor graphs on orientable surfaces, local-tensions are dual to flows. More precisely, if $G$ and $G^*$ are dual graphs embedded in an orientable surface, then $G$ has a $k$-local-tension if and only if $G^*$ has a nowhere-zero $k$-flow. On non-orientable surfaces, there is a duality between $k$-local-tensions in $G$ and nowhere-zero $k$-flows in a bidirected $G^*$. Based on this duality we have a couple of conjectures. The first follows from Tutte's 5-flow conjecture, the second from Bouchet's 6-flow conjecture.\n\nConjecture (Tutte) Every loopless graph embedded in an orientable surface has a 5-local-tension.\n\nConjecture (Bouchet) Every loopless graph embedded in any surface has a 6-local-tension.\n\nSo although, graphs on surfaces may have high chromatic number, thanks to some partial results toward the above conjectures, we know that they always have small local-tensions. For orientable surfaces, there is a famous Conjecture of Grunbaum which is equivalent to the following.\n\nConjecture (Grunbaum) If $G$ is a simple loopless graph embedded in an orientable surface with edge-width $\\ge 3$, then $G$ has a 4-local-tension.\n\nOn non-orientable surfaces, it is known that there are graphs of arbitrarily high edge-width which do not admit 4-local-tensions (see [DGMVZ]). However, it remains open whether sufficiently high edge-width forces the existence of a 5-local-tension. Indeed, as suggested by the conjecture at the start of this page, it may be that edge-width at least 4 is enough. Edge-width 3 does not suffice since the embedding of $K_6$ in the projective plane does not admit a 5-local-tension.\n\nBibliography:\n*[DGMVZ] M. DeVos, L. Goddyn, B. Mohar, D. Vertigan, and X. Zhu, Coloring-flow duality of embedded graphs. Trans. Amer. Math. Soc. 357 (2005), no. 10 MathSciNet\n\nRelated:\nRelated problems\nBouchet's 6-flow conjecture\nGrunbaum's Conjecture\n5-flow conjecture\n\nDiscussion links:\n- Tutte's 5-flow conjecture: http://www.openproblemgarden.org/?q=node/126\n- Bouchet's 6-flow conjecture: http://www.openproblemgarden.org/?q=node/131\n- Conjecture of Grunbaum: http://www.openproblemgarden.org/?q=node/177\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2159697\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 26.\n\nAttempt notes:\nTarget:\nMake progress on \"5-local-tensions\" in Graph Theory; Topological Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3316, "problem_number": "OPG-798", "title": "Degenerate colorings of planar graphs", "statement": "A graph $G$ is $k$-degenerate if every subgraph of $G$ has a vertex of degree $\\le k$.\n\nConjecture Every simple planar graph has a 5-coloring so that for $1 \\le k \\le 4$, the union of any $k$ color classes induces a $(k-1)$-degenerate graph.", "background": "Source: Open Problem Garden. Original node ID: 798. URL: http://www.openproblemgarden.org/op/degenerate_colorings_of_planar_graphs.\n\nSource subject path: Graph Theory > Topological Graph Theory > Coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/degenerate_colorings_of_planar_graphs\n- Author(s): Borodin, Oleg V.\n- Subject(s): Graph Theory; Topological Graph Theory; Coloring\n- Keywords: coloring; degenerate; planar\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 21st, 2008 by mdevos\n\nProblem-page discussion:\nAn acyclic coloring of a graph $G$ is a proper coloring with the added property that the union of any two color classes induces a forest. Grunbaum famously conjectured that every simple planar graph has an acyclic 5-coloring. Following a sequence of partial results, Borodin [B] resolved this conjecture with an impressive and detailed argument. In the same paper, Borodin made the above conjecture, which, if true, would give a stronger result (as forests are precisely the 1-degenerate graphs).\n\nA degenerate coloring of a graph $G$ is a proper coloring with the added property that the union of any $k$ color classes induces a $(k-1)$-degenerate graph. A planar graph of minimum degree 5 cannot have a degenerate 5-coloring, but if the above conjecture holds, something just short of this is true. Rautenbach [R] proved that every planar graph has a degenerate 18-coloring, and recently, Mohar, Spacepan, and Zhu showed that every planar graph has a degenerate 9-coloring.\n\nBibliography:\n*[B] O. V. Borodin, A proof of B. Grünbaum's conjecture on the acyclic $5$-colorability of planar graphs. Dokl. Akad. Nauk SSSR 231 (1976), no. 1, 18--20. MathSciNet\n\n[R] D. Rautenbach, A conjecture of Borodin and a coloring of Grünbaum. Fifth Cracow Conference on Graph Theory USTRON '06, 187--194 Electron. Notes Discrete Math., 24, Elsevier, Amsterdam, 2006.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0447031\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Degenerate colorings of planar graphs\" in Graph Theory; Topological Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3317, "problem_number": "OPG-34915", "title": "3-Colourability of Arrangements of Great Circles", "statement": "Consider a set $S$ of great circles on a sphere with no three circles meeting at a point. The arrangement graph of $S$ has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.\n\nConjecture Every arrangement graph of a set of great circles is $3$-colourable.", "background": "Source: Open Problem Garden. Original node ID: 34915. URL: http://www.openproblemgarden.org/op/3_colourability_of_arrangements_of_great_circles.\n\nSource subject path: Graph Theory > Topological Graph Theory > Coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/3_colourability_of_arrangements_of_great_circles\n- Author(s): Felsner, Stefan; Hurtado, Ferran; Noy, Marc; Streinu, Ileana\n- Subject(s): Graph Theory; Topological Graph Theory; Coloring\n- Keywords: arrangement graph; graph coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: January 19th, 2009 by David Wood\n\nProblem-page discussion:\nIt is NP-complete to test 3-colourability of planar 4-regular graphs in general [D80].\n\nArrangement graphs of general circles on the sphere can require four colors [K90].\n\nA stronger conjecture states that the arrangement graph of every set of great circles is $3$-choosable. A natural approach is to use the machinery of [AT92].\n\nPreviously appeared here.\n\nBibliography:\n[AT92] Noga Alon and Michael Tarsi. Colourings and orientations of graphs. Combinatorica 12:125--134, 1992.\n\n*[FHNS00] Stefan Felsner, Ferran Hurtado, Marc Noy, and Ileana Streinu. Hamiltonicity and colorings of arrangement graphs. In Proc. 11th Annual ACM-SIAM Symp. Discrete Algorithms (SODA), pages 155--164, January 2000.\n\n[FHNS06] Felsner, Stefan; Hurtado, Ferran; Noy, Marc; Streinu, Ileana. Hamiltonicity and colorings of arrangement graphs. Discrete Appl. Math. 154 (2006), no. 17, 2470--2483.\n\n[K90] G. Koester. 4-critical, 4-valent planar graphs constructed with crowns. Math. Scand., 67:15--22, 1990.\n\n[D80] D. P. Dailey. Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete, Discrete Math. 30:289--193, 2980.\n\nDiscussion links:\n- $3$-choosable: http://en.wikipedia.org/wiki/List_coloring\n- here: http://maven.smith.edu/%7Eorourke/TOPP/P44.html#Problem.44\n\nComments:\n- September 8th, 2014 | Anonymous | Answer to the conjecture: This problem has been solved. See here: http://arxiv.org/abs/math/0408363. - Anthony Hernandez\n- October 22nd, 2020 | Anonymous | Conjecture still open: The conjecture is still open. I'm not sure what Cahit's arxiv preprint (http://arxiv.org/abs/math/0408363) contains, but it certainly does not contain a proof. - Manfred Scheucher\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"3-Colourability of Arrangements of Great Circles\" in Graph Theory; Topological Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3318, "problem_number": "OPG-307", "title": "The Crossing Number of the Complete Graph", "statement": "The crossing number $cr(G)$ of $G$ is the minimum number of crossings in all drawings of $G$ in the plane.\n\nConjecture $\\displaystyle cr(K_n) = \\frac 14 \\floor{\\frac n2} \\floor{\\frac{n-1}2} \\floor{\\frac{n-2}2} \\floor{\\frac{n-3}2}$", "background": "Source: Open Problem Garden. Original node ID: 307. URL: http://www.openproblemgarden.org/op/the_crossing_number_of_the_complete_graph.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_crossing_number_of_the_complete_graph\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: complete graph; crossing number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 11th, 2007 by Robert Samal\n\nProblem-page discussion:\n(This discussion appears as [M].)\n\nA drawing of a graph $G$ in the plane has the vertices represented by distinct points and the edges represented by polygonal lines joining their endpoints such that:\n\n- no edge contains a vertex other than its endpoints,\n- no two adjacent edges share a point other than their common endpoint,\n- two nonadjacent edges share at most one point at which they cross transversally, and\n- no three edges cross at the same point.\n\nThe conjectured value for the crossing number of $K_n$ is known to be an upper bound. This is shown by exhibiting a drawing with that number of crossings. If $n = 2m$, place $m$ vertices regularly spaced along two circles of radii 1 and 2, respectively. Two vertices on the inner circle are connected by a straight line; two vertices on the outer circle are connected by a polygonal line outside the circle. A vertex on the inner circle is connected to one on the outer circle with a polygonal line segment of minimum possible positive winding angle around the cylinder. A simple count shows that the number of crossings in such a drawing achieves the conjectured minimum. For $n = 2m-1$ we delete one vertex from the drawing described and achieve the conjectured minimum.\n\nThe conjecture is known to be true for $n$ at most 10 [G]. If the conjecture is true for $n = 2m$, then it is also true for $n-1$. This follows from an argument counting the number of crossings in drawings of all $K_{n-1}$ 's contained in an optimal drawing of $K_n$.\n\nIt would also be interesting to prove that the conjectured upper bound is asymptotically correct, that is, that $\\lim \\frac{cr(K_n)}{\\binom{n}4} = \\frac38$.\n\nThe best known lower bound is due to Kleitman [K], who showed that this limit is at least $3/10$.\n\nBibliography:\n[G] R. Guy, The decline and fall of Zarankiewicz's theorem, in Proof Techniques in Graph Theory (F. Harary Ed.), Academic Press, New York (1969) 63-69.\n\n[K] D. Kleitman, The crossing number of $K_{5,n}$, J. Combin. Theory 9 (1970) 315-323.\n\n[M] B. Mohar, Problem of the Month\n\nRelated:\nRelated problems\nThe Crossing Number of the Complete Bipartite Graph\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P1CrossingNumberKn.html\n\nComments:\n- July 28th, 2008 | Anonymous | On lower bound.: It has been shown that $\\lim_{n\\rightarrow\\infty}\\frac{cr(K_n)}{Z(n)}\\geq 0.83$, where $Z(n)$ is the conjectured value. For the proof, see de Klerk, E.; Maharry, J.; Pasechnik, D. V.; Richter, R. B.; Salazar, G. Improved bounds for the crossing numbers of $K\\sb {m,n}$ and $K\\sb n$. (2007).\n- July 28th, 2008 | Anonymous | This is not the best known bound.: The same article below, that proves that $cr(K_{11})=100$ and $cr(K_{12})=150$, states that $0.8594 \\cdot Z(n)\\leq cr(K_n) \\leq Z(n)$.\n- July 10th, 2007 | Robert Samal | true upto n=12: The conjecture was recently verified for n=11 and 12 (The Crossing Number of $K_{11}$ Is 100 by Shengjun Pan and R. Bruce Richter).\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"The Crossing Number of the Complete Graph\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3319, "problem_number": "OPG-310", "title": "The Crossing Number of the Complete Bipartite Graph", "statement": "The crossing number $cr(G)$ of $G$ is the minimum number of crossings in all drawings of $G$ in the plane.\n\nConjecture $\\displaystyle cr(K_{m,n}) = \\floor{\\frac m2} \\floor{\\frac {m-1}2} \\floor{\\frac n2} \\floor{\\frac {n-1}2}$", "background": "Source: Open Problem Garden. Original node ID: 310. URL: http://www.openproblemgarden.org/op/the_crossing_number_of_the_complete_bipartite_graph.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_crossing_number_of_the_complete_bipartite_graph\n- Author(s): Turan, Paul\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: complete bipartite graph; crossing number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 11th, 2007 by Robert Samal\n\nProblem-page discussion:\n(This discussion appears as [M].)\n\nA drawing of a graph $G$ in the plane has the vertices represented by distinct points and the edges represented by polygonal lines joining their endpoints such that:\n\n- no edge contains a vertex other than its endpoints,\n- no two adjacent edges share a point other than their common endpoint,\n- two nonadjacent edges share at most one point at which they cross transversally, and\n- no three edges cross at the same point.\n\nThis problem is also known as Turan's Brickyard Problem (since it was formulated by Turan when he was working at a brickyard - the edges of the drawing would correspond to train tracks connecting different shipping depots, and fewer crossings would mean smaller chance for collision of little trains and smaller chance for their derailing).\n\nThis conjectured value for the crossing number of $K_{m,n}$ can be realized by the following drawing. Place $\\ceil{n/2}$ vertices on the positive $x$-axis and $\\floor{n/2}$ vertices on the negative $x$-axis. Similarly, place $\\ceil{m/2}$ and $\\floor{m/2}$ along the positive and negative $y$-axis. Now connect each pair of vertices on different axes with straight line segments.\n\nBibliography:\n[G] R. Guy, The decline and fall of Zarankiewicz's theorem, in Proof Techniques in Graph Theory (F. Harary Ed.), Academic Press, New York (1969) 63-69.\n\n[K] D. Kleitman, The crossing number of K_{5,n}, J. Combin. Theory 9 (1970) 315-32\n\n[M] B. Mohar, Problem of the Month\n\nRelated:\nRelated problems\nThe Crossing Number of the Complete Graph\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P2CrossingNumberKmn.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"The Crossing Number of the Complete Bipartite Graph\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3320, "problem_number": "OPG-313", "title": "The Crossing Number of the Hypercube", "statement": "The crossing number $cr(G)$ of $G$ is the minimum number of crossings in all drawings of $G$ in the plane.\n\nThe $d$-dimensional (hyper)cube $Q_d$ is the graph whose vertices are all binary sequences of length $d$, and two of the sequences are adjacent in $Q_d$ if they differ in precisely one coordinate.\n\nConjecture $\\displaystyle \\lim \\frac{cr(Q_d)}{4^d} = \\frac{5}{32}$", "background": "Source: Open Problem Garden. Original node ID: 313. URL: http://www.openproblemgarden.org/op/the_crossing_number_of_the_hypercube.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_crossing_number_of_the_hypercube\n- Author(s): Erdos, Paul; Guy, Richard K.\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: crossing number; hypercube\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 11th, 2007 by Robert Samal\n\nProblem-page discussion:\nIt is known that $cr(Q_d) = 0$ for $d = 1,2,3$ and that $cr(Q_4) = 8$. No other exact values are known. Madej [M] proved that $cr(Q_d) \\le 4^d/6 + o(4^d/6)$. Faria and de Figueiredo [FF] improved the upper bound to $(165/1024) 4^d$. Sykora and Vrto [SV] proved that $4^d/20 + o(4^d/20)$ is a lower bound on $cr(Q_d)$.\n\nBibliography:\n*[EG] P. Erdős and R.K. Guy, Crossing number problems, Amer. Math. Monthly 80 (1973) 52-58.\n\n[FF] L. Faria, C.M.H. de Figueiredo, On Eggleton and Guy's conjectured upper bound for the crossing number of the $n$-cube, Math. Slovaca 50 (2000) 271-287.\n\n[M] T. Madej, Bounds for the crossing number of the $n$-cube, J. Graph Theory 15 (1991) 81-97.\n\n[SV] O. Sykora and I. Vrto, On crossing numbers of hypercubes and cube connected cycles, BIT 33 (1993) 232-237.\n\nRelated:\nRelated problems\nThe Crossing Number of the Complete Graph\nThe Crossing Number of the Complete Bipartite Graph\n\nComments:\n- July 18th, 2008 | Robert Samal | Improved upper bound: I came accross a paper Faria, Herrera de Figueiredo, Sykora, Vrto: An improved upper bound on the crossing number of the hypercube that proves half of this, getting the correct upper bound.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"The Crossing Number of the Hypercube\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3321, "problem_number": "OPG-322", "title": "Drawing disconnected graphs on surfaces", "statement": "Conjecture Let $G$ be the disjoint union of the graphs $G_1$ and $G_2$ and let $\\Sigma$ be a surface. Is it true that every optimal drawing of $G$ on $\\Sigma$ has the property that $G_1$ and $G_2$ are disjoint?", "background": "Source: Open Problem Garden. Original node ID: 322. URL: http://www.openproblemgarden.org/op/drawing_disconnected_graphs_on_surfaces.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/drawing_disconnected_graphs_on_surfaces\n- Author(s): DeVos, Matt; Mohar, Bojan; Samal, Robert\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: crossing number; surface\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 12th, 2007 by mdevos\n\nProblem-page discussion:\nWe insist on the usual restrictions for drawings (as in The Crossing Number of the Complete Graph).\n\nAlthough both crossing numbers and embeddings of graphs on general surfaces are rich and well-studied subjects, their common generalization - drawing graphs on general surfaces has received very little attention. The question highlighted here appears to be quite basic in nature, but due to the combined difficulties of crossings and general surfaces, it may be quite difficult to resolve.\n\nThis conjecture is trivially true when $\\Sigma$ is the plane, and DeVos, Mohar, and Samal have proved that it also holds when $\\Sigma$ is the projective plane. It is open for all other surfaces to the best of my (M. DeVos) knowledge.\n\nDiscussion links:\n- The Crossing Number of the Complete Graph: http://www.openproblemgarden.org/?q=op/the_crossing_number_of_the_complete_graph\n\nComments:\n- September 30th, 2007 | lbeaudou | Drawing disconnected graphs on surfaces: any reference?: Hello,\n\nYou don't mention reference in this problem, though it is said that some work has been made. Would it be possible to know how the projective plane has been shown to verify this conjecture?\n\nThanks in advance for any piece of information.\n\nLaurent Beaudou\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Drawing disconnected graphs on surfaces\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3322, "problem_number": "OPG-1812", "title": "Crossing sequences", "statement": "Conjecture Let $(a_0,a_1,a_2,\\ldots,0)$ be a sequence of nonnegative integers which strictly decreases until $0$.\n\nThen there exists a graph that be drawn on a surface with orientable (nonorientable, resp.) genus $i$ with $a_i$ crossings, but not with less crossings.", "background": "Source: Open Problem Garden. Original node ID: 1812. URL: http://www.openproblemgarden.org/op/crossing_sequences.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/crossing_sequences\n- Author(s): Archdeacon, Dan; Bonnington, C. Paul; Siran, Jozef\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: crossing number; crossing sequence\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 30th, 2008 by Robert Samal\n\nProblem-page discussion:\nThis actually are two conjectures, one for the orientable case and another for nonorientable one. For sequences $(a_0,a_1,0)$ the nonorientable case was resolved in [ABS] and the orientable one in [DMS].\n\nThe conclusion also holds (for the orientable case) whenever the sequence $(a_i)$ is convex [S], that is whenever $a_i - a_{i-1}$ is nonincreasing. It might seem that this condition is also necessary: For the most extreme sequence $(N,N-1,0)$ (suggested by Salazar) one needs to construct a graph for which adding one handle saves just one crossing, while adding another saves many -- but then why not add the second handle first? Somewhat surprisingly, graphs with this counterintuitive property exist, at least for sequences $(a_0,a_1,0)$.\n\nAn interesting open case is to consider sequences for which $$a_0 - a_s < \\varepsilon (a_s - a_{s+1})$$for some$s$and small$\\varepsilon$.\n\nBibliography:\n*[ABS] Dan Archdeacon, C. Paul Bonnington, and Jozef Siran, Trading crossings for handles and crosscaps, J.Graph Theory 38 (2001), 230--243.\n\n[DMS] Matt DeVos, Bojan Mohar, Robert Samal, Unexpected behaviour of crossing sequences, in preparation\n\n[S] Jozef Siran, The crossing function of a graph, Abh. Math. Sem. Univ. Hamburg 53 (1983), 131--133.\n\nRelated:\nRelated problems\nDrawing disconnected graphs on surfaces\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"Crossing sequences\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3323, "problem_number": "OPG-37068", "title": "Crossing numbers and coloring", "statement": "We let $cr(G)$ denote the crossing number of a graph $G$.\n\nConjecture Every graph $G$ with $\\chi(G) \\ge t$ satisfies $cr(G) \\ge cr(K_t)$.", "background": "Source: Open Problem Garden. Original node ID: 37068. URL: http://www.openproblemgarden.org/op/crossing_numbers_and_coloring.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/crossing_numbers_and_coloring\n- Author(s): Albertson, Michael O.\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: coloring; complete graph; crossing number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 4th, 2009 by mdevos\n\nProblem-page discussion:\nThis conjecture is an interesting weakening of the disproved Hajos Conjecture which asserted that $\\chi(G) \\ge t$ implies that $G$ contains a subdivision of $K_t$.\n\nA minimal counterexample to Albertson's conjecture is critical, with minimum degree $\\ge t$. Using this and the crossing lemma, Albertson, Cranston and Fox showed that a minimum counterexample has at most $4t$ vertices. They then analyzed small cases to show that the conjecture holds for $t \\le 12$. More recently, Barat and Toth [BT] sharpened these arguments to show that the conjecture holds for $t \\le 16$.\n\nBibliography:\n[BT] J. Barat and G. Toth, Towards the Albertson Conjecture\n\nSource links:\n- crossing number: http://en.wikipedia.org/wiki/crossing number (graph theory)\n\nBibliography links:\n- Towards the Albertson Conjecture: http://arxiv1.library.cornell.edu/abs/0909.0413\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Crossing numbers and coloring\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3324, "problem_number": "OPG-37117", "title": "Are different notions of the crossing number the same?", "statement": "Problem Does the following equality hold for every graph $G$?\n$$\n\\text{pair-cr}(G) = \\text{cr}(G)\n$$\n\nThe crossing number $\\text{cr}(G)$ of a graph $G$ is the minimum number of edge crossings in any drawing of $G$ in the plane. In the pairwise crossing number $\\text{pair-cr}(G)$, we minimize the number of pairs of edges that cross.", "background": "Source: Open Problem Garden. Original node ID: 37117. URL: http://www.openproblemgarden.org/op/are_different_notions_of_the_crossing_number_the_same.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_different_notions_of_the_crossing_number_the_same\n- Author(s): Pach, János; Tóth, Géza\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: crossing number; pair-crossing number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 3rd, 2009 by cibulka\n\nProblem-page discussion:\nObviously we have $\\text{pair-cr}(G) \\leq \\text{cr}(G)$.\n\nThe problem was first posed by Pach and Tóth in~[PT], who first spotted the possibility that the pairwise crossing number might be different from the crossing number. They proved $\\text{cr}(G) \\leq 2k^2$ for graphs with pairwise crossing number $k$, which was later improved by Valtr~[V05] to $O(k^2/ \\log(k))$ and by Tóth~[T08] to $O(k^2/ \\log^2(k))$.\n\nBibliography:\n*[PT] János Pach, Géza Tóth, Which crossing number is it anyway?, Journal of Combinatorial Theory Series B 80 (2000), no. 2, 225--246. MathSciNet\n\n[V05] Pavel Valtr, On the pair-crossing number, Combinatorial and computational geometry, 52 (2005), 569--575. MathSciNet\n\n[T08] Géza Tóth, Note on the pair-crossing number and the odd-crossing number, Discrete Comput. Geom., 39 (2008), no. 4, 791--799. MathSciNet\n\nSource links:\n- crossing number: http://en.wikipedia.org/wiki/Crossing number (graph theory)\n\nBibliography links:\n- Which crossing number is it anyway?: http://www.cs.bme.hu/%7Egeza/whichcrossing.ps\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1794693\n- On the pair-crossing number: http://www.msri.org/communications/books/Book52/files/31valtr.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2178340\n- Note on the pair-crossing number and the odd-crossing number: http://www.cs.bme.hu/%7Egeza/pair-cr.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2413161\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Are different notions of the crossing number the same?\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3325, "problem_number": "OPG-326", "title": "Universal point sets for planar graphs", "statement": "We say that a set $P \\subseteq {\\mathbb R}^2$ is $n$-universal if every $n$ vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in $P$, and all edges are (non-intersecting) straight line segments.\n\nQuestion Does there exist an $n$-universal set of size $O(n)$?", "background": "Source: Open Problem Garden. Original node ID: 326. URL: http://www.openproblemgarden.org/op/small_universal_point_sets_for_planar_graphs.\n\nSource subject path: Graph Theory > Topological Graph Theory > Drawings.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/small_universal_point_sets_for_planar_graphs\n- Author(s): Mohar, Bojan\n- Subject(s): Graph Theory; Topological Graph Theory; Drawings\n- Keywords: geometric graph; planar graph; universal set\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 22nd, 2007 by mdevos\n\nProblem-page discussion:\nMore generally, if we let $f(n)$ denote the size of the smallest $n$-universal set, we are interested in the behaviour of $f$. The best known upper bound is $f(n) = O(n^2)$. Indeed, every $n$-vertex planar graph can be drawn as required in the $n \\times n$ grid [dFPP], [S]. On the flip side, it is known that $f(n) \\ge 1.098n$ for sufficiently large $n$ [CH].\n\nBibliography:\n[CH] M. Chrobak and H.Karloff. A lower bound on the size of universal sets for planar graphs. SIGACT News, 20:83-86, 1989.\n\n[dFPP] H. de Fraysseix, J. Pach, and R. Pollack. How to draw a planar graph on a grid. Combinatorica, 10(1):41-51, 1990. MathSciNet\n\n[S] W. Schnyder. Embedding planar graphs on the grid. In Proc. 1st ACM-SIAM Sympos. Discrete Algorithms, pages 138-148, 1990.\n\nBibliography links:\n- How to draw a planar graph on a grid: http://www.springerlink.com/content/b471830661k10534/\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1075065\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Universal point sets for planar graphs\" in Graph Theory; Topological Graph Theory; Drawings, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3326, "problem_number": "OPG-596", "title": "Linear Hypergraphs with Dimension 3", "statement": "Conjecture Any linear hypergraph with incidence poset of dimension at most 3 is the intersection hypergraph of a family of triangles and segments in the plane.", "background": "Source: Open Problem Garden. Original node ID: 596. URL: http://www.openproblemgarden.org/op/linear_hypergraphs_with_dimension_3.\n\nSource subject path: Graph Theory > Topological Graph Theory > Drawings.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/linear_hypergraphs_with_dimension_3\n- Author(s): de Fraysseix, Hubert; Ossona de Mendez, Patrice; Rosenstiehl, Pierre\n- Subject(s): Graph Theory; Topological Graph Theory; Drawings\n- Keywords: Hypergraphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 26th, 2007 by taxipom\n\nProblem-page discussion:\nA hypergraph is linear if any two edges may share at most one vertex. The incidence poset of a hypergraph is the vertex-edge inclusion poset. The dimension of a poset $P$ is the minimum number of linear extentions of $P$, whose intersection is $P$ [DM].\n\nSchnyder proved that the incidence poset of a graph $G$ has dimension at most $3$ if and only if $G$ is planar [S89]. Fraysseix, Rosenstiehl and Ossona de Mendez proved that every planar graph has a representation by contacts of triangles [FOR94] and Scheinerman conjectured that every planar graph has a representation by intersection of segments [S84] (claimed to be proved by Gonçalves et al.).\n\nA hypergraph is planar if its vertex-edge incidence graph is planar [W]. Fraysseix, Rosenstiehl and Ossona de Mendez proved that every planar linear hypergraph has a representation by contacts of triangles [FOR07] and it has been conjectured that they have a representation by intersection of straight line segments [FO07] (cf Straight line representation of planar linear hypergraphs).\n\nAlthough the incidence poset of a simple planar hypergraph has dimension at most $3$ (what follows from [BT]), the converse is false: The linear hypergraph with vertices $1,\\dots,5$ and edge set $\\{\\{1,2\\},\\{2,3\\},\\{3,4\\},\\{1,4\\},\\{1,3,5\\},\\{2,4,5\\}\\}$ has incidence dimension $3$ but is not planar (its vertex-edge incidence graph is a subdivision of $K_{3,3}$ ). It follows from [O] that the vertices of simple hypergraphs with incidence posets of dimensions $d$ can be represented by convex sets of the Euclidean space of dimension $d-1$, in such a way that the edges of the hypergraph are exactly the maximal subsets of vertices, such that the corresponding subset of convexes has a non-empty intersection.\n\nBibliography:\n[BT] G.~Brightwell and W.T. Trotter, The order dimension of planar maps, SIAM journal on Discrete Mathematics 10 (1997), no.~4, 515--528.\n\n[DM] B.~Dushnik and E.W. Miller, Partially ordered sets, Amer. J. Math. 63 (1941), 600--610.\n\n[FO07] Hubert de Fraysseix, Patrice Ossona de Mendez: Stretching of Jordan arc contact systems, Discrete Applied Mathematics 155 (2007), no. 9, 1079--1095.\n\n[FOR94] H.~de Fraysseix, P.~Ossona~de Mendez, and P.~Rosenstiehl, On triangle contact graphs, Combinatorics, Probability and Computing 3 (1994), 233--246.\n\n*[FOR07] H.~de Fraysseix, P.~Ossona~de Mendez, and P.~Rosenstiehl, Representation of Planar Hypergraphs by Contacts of Triangles, Proc. of Graph Drawing '07, to appear.\n\n[O] P.~Ossona~de Mendez, Realization of posets, Journal of Graph Algorithms and Applications 6 (2002), no.~1, 149--153.\n\n[S84] E.R. Scheinerman, Intersection classes and multiple intersection parameters of graphs, Ph.D. thesis, Princeton University, 1984.\n\n[S89] W.~Schnyder, Planar graphs and poset dimension, Order 5 (1989), 323--343.\n\n[W] T.R.S. Walsh, Hypermaps versus bipartite maps, J. Combinatorial Theory 18(B) (1975), 155--163.\n\nRelated:\nRelated problems\nStraight line representation of planar linear hypergraphs\n\nDiscussion links:\n- Straight line representation of planar linear hypergraphs: http://www.openproblemgarden.org/?q=node/554\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Linear Hypergraphs with Dimension 3\" in Graph Theory; Topological Graph Theory; Drawings, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3327, "problem_number": "OPG-172", "title": "Consecutive non-orientable embedding obstructions", "statement": "Conjecture Is there a graph $G$ that is a minor-minimal obstruction for two non-orientable surfaces?", "background": "Source: Open Problem Garden. Original node ID: 172. URL: http://www.openproblemgarden.org/op/consecutive_non_orientable_embedding_obstructions.\n\nSource subject path: Graph Theory > Topological Graph Theory > Genus.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/consecutive_non_orientable_embedding_obstructions\n- Subject(s): Graph Theory; Topological Graph Theory; Genus\n- Keywords: minor; surface\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 27th, 2007 by Bruce Richter\n\nComments:\n- September 15th, 2010 | Anonymous | Minor-Minimal Obstruction: Is a minor-minimal obstruction the same as a forbidden minor?\n- September 15th, 2010 | mdevos | yes: yes\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Consecutive non-orientable embedding obstructions\" in Graph Theory; Topological Graph Theory; Genus, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3328, "problem_number": "OPG-157", "title": "What is the largest graph of positive curvature?", "statement": "Problem What is the largest connected planar graph of minimum degree 3 which has everywhere positive combinatorial curvature, but is not a prism or antiprism?", "background": "Source: Open Problem Garden. Original node ID: 157. URL: http://www.openproblemgarden.org/op/what_is_the_largest_graph_of_positive_curvature.\n\nSource subject path: Graph Theory > Topological Graph Theory > Planar graphs.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/what_is_the_largest_graph_of_positive_curvature\n- Author(s): DeVos, Matt; Mohar, Bojan\n- Subject(s): Graph Theory; Topological Graph Theory; Planar graphs\n- Keywords: curvature; planar graph\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: March 10th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: For a graph $G$ embedded in the sphere, the combinatorial curvature of a vertex $v$ is defined to be $1 - \\frac{ {\\mathit deg}(v)}{2} + \\sum_{f \\sim v} \\frac{1}{ {\\mathit size}(f) }$ (here the summation is over all faces $f$ incident with $v$ ).\n\nLet $G$ be a graph embedded in the sphere, and consider the polygonal surface formed by treating each face of size $n$ as a regular $n$-gon of side length $1$. The gaussian curvature at a vertex $v$ is defined to be $2 \\pi$ minus the sum of the angles incident with $v$. So, our vertex $v$ has positive curvature if the sum of the incident angles is less than $2 \\pi$. In fact, the combinatorial curvature at $v$ is exactly $2 \\pi$ times the gaussian curvature, so these quantities will always have the same sign.\n\nLet us call a convex polyhedron regular-faced if each face is a regular polygon. Based on the previous discussion, we know that every convex regular-faced polyhedron gives us a graph with everywhere positive combinatorial curvature. Indeed, we may view planar graphs with everywhere positive curvature as a kind of generalization of these polyhedra. The polyhedra in this class have been studied and classified. The Platonic solids and Archimedean solids are all convex and regular faced, and there are two infinite families: prisms and antiprisms. In addition to this, there are 92 other exceptional ones, known as Johnson Solids.\n\nEuler's formula tells us that the sum of the combinatorial curvatures over all of the vertices is equal to 2. Indeed, the combinatorial curvature is exactly what we get when we assign $1$ to each vertex and face, $-1$ to each edge, and then \"discharge\" evenly onto the vertices. So, if we wish to construct large planar graphs where every vertex has positive curvature, we will need to make the curvature arbitrarily small. This can be achieved with prisms and antiprisms, but apart from these two families, all other graphs with everywhere positive curvature have a bounded number of vertices. Improving upon [DM], Zhang [Z] has shown this upper bound to be at most 580. The great rhombicosidodecahedron has 120 vertices and everywhere positive curvature (this is the largest regular-faced convex polyhedron which is not a prism or antiprism). Reti, Bitay, and Kosztolanyi [RBK] have improved upon this lower bound by constructing a graph with everywhere positive curvature and 138 vertices. These are the best bounds I (M. DeVos) know of.\n\nBibliography:\n[DM] M. DeVos and B. Mohar, An analogue of the Descarte-Euler formula for infinite graphs and Higuchi's conjecture preprint.\n\n[H] Y. Higuchi, Combinatorial curvature for planar graph, J. Graph Theory, Vol 38 (2001), no. 4, 220-229. MathSciNet\n\n[RBK] T. Reti, E. Bitay, and Zs. Kosztolanyi, On the polyhedral graphs with positive combinatorial curvature, Acta Polytechnica Hungarica Vol. 2, No. 2 (2005) 19-37.\n\n[Z] L. Zhang, A result on combinatorial curvature for embedded graphs on a surface, Discrete Math (2007) in press\n\nSource links:\n- planar graph: http://en.wikipedia.org/wiki/planar graph\n- prism: http://en.wikipedia.org/wiki/prism (geometry)\n- antiprism: http://en.wikipedia.org/wiki/antiprism\n\nDiscussion links:\n- Platonic solids: http://en.wikipedia.org/wiki/Platonic solids\n- Archimedean solids: http://en.wikipedia.org/wiki/Archimedean solids\n- prisms: http://en.wikipedia.org/wiki/prism (geometry)\n- antiprisms: http://en.wikipedia.org/wiki/antiprism\n- Johnson Solids: http://en.wikipedia.org/wiki/johnson solid\n- great rhombicosidodecahedron: http://en.wikipedia.org/wiki/great rhombicosidodecahedron\n\nBibliography links:\n- An analogue of the Descarte-Euler formula for infinite graphs and Higuchi's conjecture: http://www.ijp.si/ftp/pub/preprints/ps/2004/pp930.ps\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1864922\n- On the polyhedral graphs with positive combinatorial curvature: http://www.bmf.hu/journal/Reti_4.pdf\n\nComments:\n- August 30th, 2011 | Anonymous | new 'largest' PCC graph: We have a new largest PCC graph with 208 vertices, improving the lower bound.\n\nIt will be published soon in the New Zealand Journal of Mathematics. \"New graphs with thinly spread positive combinatorial curvature.\" j.sneddon@auckland.ac.nz\n- August 3rd, 2010 | Anonymous | project underway: This is an active project for an honours student at the University of Auckland. We'd be interested in hearing from anyone else working on the problem: contact j.sneddon@auckland.ac.nz\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"What is the largest graph of positive curvature?\" in Graph Theory; Topological Graph Theory; Planar graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3329, "problem_number": "OPG-702", "title": "Growth of finitely presented groups", "statement": "Problem Does there exist a finitely presented group of intermediate growth?", "background": "Source: Open Problem Garden. Original node ID: 702. URL: http://www.openproblemgarden.org/op/growth_of_finitely_presented_groups.\n\nSource subject path: Group Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/growth_of_finitely_presented_groups\n- Author(s): Adyan, Sergeui I.\n- Subject(s): Group Theory\n- Keywords: finitely presented; growth\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 12th, 2007 by mdevos\n\nProblem-page discussion:\nSee Wikipedia's growth of groups for definitions of the basics reguarding growth rate in groups (in particular polynomial and exponential growth rates). A finitely generated group has intermediate growth if its growth rate (for every finite generating set) is subexponential but superpolynomial.\n\nMost naturally occuring groups have either polynomial growth (such as ${\\mathbb Z}^n$ ) or exponential growth (such as a free group with rank $n > 1$ ). Milnor [M] famously asked if there exists a finitely generated group with intermediate growth, and this problem was resolved in the affirmative by Grigorchuk [G]. The groups constructed by Grigorchuk are not finitely presented, thus leaving the above problem.\n\nExpert opinion seems to be that there are no finitely presented groups of intermediate growth.\n\nBibliography:\n[G] R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means., Izv. Akad. Nauk SSSR Ser. Mat. 48:5 (1984), 939-985.\n\n[M] J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1-7.\n\nDiscussion links:\n- Wikipedia's growth of groups: http://en.wikipedia.org/wiki/growth rate (group theory)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Growth of finitely presented groups\" in Group Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 17, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3330, "problem_number": "OPG-732", "title": "Subgroup formed by elements of order dividing n", "statement": "Conjecture\n\nSuppose $G$ is a finite group, and $n$ is a positive integer dividing $|G|$. Suppose that $G$ has exactly $n$ solutions to $x^{n} = 1$. Does it follow that these solutions form a subgroup of $G$?", "background": "Source: Open Problem Garden. Original node ID: 732. URL: http://www.openproblemgarden.org/op/subgroup_formed_by_elements_of_order_dividing_n.\n\nSource subject path: Group Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/subgroup_formed_by_elements_of_order_dividing_n\n- Author(s): Frobenius, Ferdinand G.\n- Subject(s): Group Theory\n- Keywords: order, dividing\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: January 28th, 2008 by dlh12\n\nProblem-page discussion:\nIf these solutions form a subgroup, they form a characteristic (and therefore normal) subgroup of $G$. This easily follows from the First Sylow Theorem if $n$ is the highest power of a prime $p$ dividing $|G|$.\n\nIn a 1980 article, Feit commented that the case where $(n, \\frac{|G|}{n}) = 1$ (i.e., $n$ 'exactly divides' $|G|$ ) had been reduced to considering $G$ simple. Thus it should be resolvable using the classification of finite simple groups.\n\nIt is known that if $n$ divides $|G|$, the number of solutions of $x^{n} = 1$ in $G$ is a multiple of $n$. A generalization of this theorem, replacing $x^{n} = 1$ by $x^{n} \\in C$ for a conjugacy class $C$ of $G$, can be found in Marshall Hall Jr.'s book.\n\nObservation This conjecture implies the (known) theorem of Frobenius:\n\nTheorem If $G$ is a finite transitive permutation group in which only the identity has more than one fixed point, then the derangements of $G$, together with the identity, form a subgroup of $G$.\n\nTo prove Frobenius' Theorem from this (say $G$ is such a permutation group on $k$ points), use the standard elementary counting arguments to show that the solutions to $x^{k} = 1$ are only the derangements and the identity and that there are exactly $k$ of these. This theorem of Frobenius has not yet been proven in general without the use of group characters.\n\nBibliography:\nMarshall Hall Jr., Theory of Groups, Macmillan (1959)\n\nWalter Feit, On a Conjecture of Frobenius, Proceedings of the American Mathematical Society, Vol.7, No. 2 (Apr. 1956), 177-187.\n\nComments:\n- March 30th, 2012 | Anonymous | Solved... again: it can be proved by a isomorphism between the solutions of x^n=1 in G and the solutions of the same eq in C (complex numbers).\n- June 26th, 2009 | Anonymous | Solved: This conjecture has been proven.\n- June 29th, 2009 | Anonymous | Re: Solved: Could you add a reference please?\n- December 1st, 2010 | Anonymous | Reference: Iiyori, Nobuo; Yamaki, Hiroyoshi On a conjecture of Frobenius. Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 413--416.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Subgroup formed by elements of order dividing n\" in Group Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 17, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3331, "problem_number": "OPG-760", "title": "Burnside problem", "statement": "Conjecture If a group has $r$ generators and exponent $n$, is it necessarily finite?", "background": "Source: Open Problem Garden. Original node ID: 760. URL: http://www.openproblemgarden.org/op/burnside_problem.\n\nSource subject path: Group Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/burnside_problem\n- Author(s): Burnside, William\n- Subject(s): Group Theory\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 11th, 2008 by dlh12\n\nProblem-page discussion:\nIt is possible to define the $free Burnside group$ $B(r,n)$ to be the group generated by $x_{1}, \\ldots, x_{r}$ with relations $w^{n}=1$ where $w$ ranges over every word in the generators. There is a universality property: Any homomorphism $\\phi: G \\to H$ where $H$ has r generators and exponent dividing $n$ can be written as a composition of a homomorphism $\\psi: G \\to B(r,n)$ with a homomorphism $\\pi: B(r,n) \\to H$. Some cases of this are known: $B(1,n)$ is a cyclic group of order $n$, for any positive integer $n$. $B(r,1)$ is trivial for any positive integer $r$. $B(r,2)$ is isomorphic to the Cartesian product of $r$ cyclic groups of order $2$, for any positive integer $r$. This is because the relations make it easy to prove that the generators commute. $B(r,3)$ is a finite group, and its order is\n$$\n3^{r + \\binom{r}{2} + \\binom{r}{3}}.\n$$\n $B(r,6)$ is a finite group, and its order is\n$$\n2^{1 + (r-1)3^{r + \\binom{r}{2} + \\binom{r}{3}} }3^{1 + (r-1)2^{r + \\binom{r}{2} + \\binom{r}{3}} }.\n$$\n $B(r,4)$ is a finite group for any positive integer $r$. The order is known for $r$ up to $5$:\n$$\n|B(1,4)| = 2^{2}\n$$\n\n$$\n|B(2,4)| = 2^{12}\n$$\n\n$$\n|B(3,4)| = 2^{69}\n$$\n\n$$\n|B(4,4)| = 2^{422}\n$$\n\n$$\n|B(5,4)| = 2^{2728}\n$$\n $B(r,n)$ is known to be infinite for sufficiently large $r$ and odd $n \\geq 665$, as well as $r > 1$ and $n \\geq 10^{48}$ divisible by $2^{9}$.\n\nBurnside_problem\n$$\n\\\\\n$$\n Burnside Problem -- from Wolfram MathWorld\n\nDiscussion links:\n- Burnside_problem: http://en.wikipedia.org/wiki/Burnside_problem\n- Burnside Problem -- from Wolfram MathWorld: http://mathworld.wolfram.com/BurnsideProblem.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 31.\n\nAttempt notes:\nTarget:\nMake progress on \"Burnside problem\" in Group Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 17, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3332, "problem_number": "OPG-3572", "title": "Inverse Galois Problem", "statement": "Conjecture Every finite group is the Galois group of some finite algebraic extension of $\\mathbb Q$.", "background": "Source: Open Problem Garden. Original node ID: 3572. URL: http://www.openproblemgarden.org/op/inverse_galois_problem.\n\nSource subject path: Group Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/inverse_galois_problem\n- Author(s): Hilbert, David\n- Subject(s): Group Theory\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 13th, 2008 by tchow\n\nProblem-page discussion:\nThis problem is one of the greatest open problems in group theory. Hilbert was the first to study it in earnest. His irreducibility theorem established a connection between Galois groups over $\\mathbb Q$ and Galois groups over ${\\mathbb Q}(x)$; the latter could be attacked by geometric methods, and in this way, Hilbert showed that the symmetric and alternating groups are Galois realizable over $\\mathbb Q$. In the 1950's, Shafarevich showed using number-theoretic methods that all finite solvable groups are Galois realizable over $\\mathbb Q$. Another spectacular result was John Thompson's realization of the Monster group as a Galois group over $\\mathbb Q$. One of Thompson's main tools was a concept he called \"rigidity\", a concept discovered independently by several people that continues to be important to this day. It is now known that 25 of the 26 sporadic simple groups are Galois realizable over $\\mathbb Q$ (the sole exception being the Mathieu group $M_{23}$ ).\n\nBibliography:\n[MM] Gunter Malle and B. Heinrich Matzat, Inverse Galois Theory, Springer, 1999.\n\n[V] Helmut Völklein, Groups as Galois Groups: An introduction, Cambridge University Press, 1996.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Inverse Galois Problem\" in Group Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 17, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3333, "problem_number": "OPG-37302", "title": "Which lattices occur as intervals in subgroup lattices of finite groups?", "statement": "Conjecture\n\nThere exists a finite lattice that is not an interval in the subgroup lattice of a finite group.", "background": "Source: Open Problem Garden. Original node ID: 37302. URL: http://www.openproblemgarden.org/op/which_lattices_occur_as_intervals_in_finite_groups.\n\nSource subject path: Group Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/which_lattices_occur_as_intervals_in_finite_groups\n- Subject(s): Group Theory\n- Keywords: congruence lattice; finite groups\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 22nd, 2011 by williamdemeo\n\nProblem-page discussion:\nThis would settle the Finite Lattice Representation Problem.\n\nBibliography:\n[P5] Palfy and Pudlak. Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis, 11 (1980), 22–27.\n\nRelated:\nRelated problems\nFinite Lattice Representation Problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Which lattices occur as intervals in subgroup lattices of finite groups?\" in Group Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 17, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3334, "problem_number": "OPG-660", "title": "F_d versus F_{d+1}", "statement": "Problem Find a constant $k$ such that for any $d$ there is a sequence of tautologies of depth $k$ that have polynomial (or quasi-polynomial) size proofs in depth $d+1$ Frege system $F_{d+1}$ but requires exponential size $F_d$ proofs.", "background": "Source: Open Problem Garden. Original node ID: 660. URL: http://www.openproblemgarden.org/op/f_d_versus_f_d_1.\n\nSource subject path: Logic.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/f_d_versus_f_d_1\n- Author(s): Krajicek, Jan\n- Subject(s): Logic\n- Keywords: Frege system; short proof\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 18th, 2007 by zitterbewegung\n\nProblem-page discussion:\nProblem statement from link.\n\nSuch tautologies are known if $k$ may depend on $d$ (for $k:= d$ ). This problem is also closely related to conservativity relations among bounded arithmetic theories.\n\nBibliography:\n*[K1] J.Krajicek: \"Lower Bounds to the Size of Constant-Depth Propositional Proofs\", J. of Symbolic Logic, 59(1), (1994), pp.73-86\n\n[K2] J. Krajicek: \"Bounded arithmetic, propositional logic, and complexity theory\", Encyclopedia of Mathematics and Its Applications, Vol.60, Cambridge University Press, Cambridge - New York - Melbourne, (1995), 343 p. (on page 243)\n\nDiscussion links:\n- link: http://www.math.cas.cz/%7Ekrajicek/problemy.html#pl\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"F_d versus F_{d+1}\" in Logic, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3335, "problem_number": "OPG-1790", "title": "Tarski's exponential function problem", "statement": "Conjecture Is the theory of the real numbers with the exponential function decidable?", "background": "Source: Open Problem Garden. Original node ID: 1790. URL: http://www.openproblemgarden.org/op/tarskis_exponential_function_problem.\n\nSource subject path: Logic.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/tarskis_exponential_function_problem\n- Author(s): Tarski, Alfred\n- Subject(s): Logic\n- Keywords: Decidability\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 8th, 2008 by Charles\n\nProblem-page discussion:\nSee Tarski's exponential function problem. Tarski proved that the theory of the real numbers without the exponential is decidable before asking this.\n\nRelated:\nRelated problems\nAlgebraic independence of pi and e\nSchanuel's Conjecture\n\nDiscussion links:\n- Tarski's exponential function problem: http://en.wikipedia.org/wiki/Tarski's exponential function problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Tarski's exponential function problem\" in Logic, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3336, "problem_number": "OPG-2379", "title": "Termination of the sixth Goodstein Sequence", "statement": "Question How many steps does it take the sixth Goodstein sequence to terminate?", "background": "Source: Open Problem Garden. Original node ID: 2379. URL: http://www.openproblemgarden.org/op/termination_of_the_sixth_goodstein_sequence.\n\nSource subject path: Logic.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/termination_of_the_sixth_goodstein_sequence\n- Author(s): Graham, Ronald L.\n- Subject(s): Logic\n- Keywords: Goodstein Sequence\n- Importance: Low ✭\n- Recommended for undergraduates: no\n- Posted: October 7th, 2008 by mdevos\n\nProblem-page discussion:\nFor a positive integer $n$, the $n^{th}$ Goodstein Sequence is defined as follows. The first term of the sequence in $n$. To obtain the $k^{th}$ term, write the $(k-1)^{st}$ term in hereditary base k notation, change all $k$ 's to $(k+1)$ 's and then subtract 1. If the sequence hits 0, then it terminates. So, the first terms of the sixth Goodstein Sequence are as follows:\n\n$$\n\\begin{array}{lll} \\mbox{term} & \\mbox{value} \\\\ 1 & 2^2 + 2 = 6 \\\\ 2 & 3^3 + 2 = 29 \\\\ 3 & 4^4 + 1 = 257 \\\\ 4 & 5^5 = 3125 \\\\ 5 & 5 \\cdot 6^5 + 5 \\cdot 6^5 + 5 \\cdot 6^4 + 5 \\cdot 6^3 + 5 \\cdot 6^2 + 5 \\cdot 6 + 5 = 46655 \\end{array}\n$$\n\nSurprisingly, despite the fact that Goodstein Sequences grow quite quickly at the start, all such sequences do eventually hit 0 and terminate. This result, first discovered by Goodstein, is of interest in logic since it cannot be proved in Peano arithmetic.\n\nAlthough determining particular properties of a specific Goodstein Sequence are of limited mathematical value, this problem is an interesting computational challenge.\n\nDiscussion links:\n- Goodstein Sequence: http://en.wikipedia.org/wiki/Goodstein Sequence\n- hereditary base k notation: http://en.wikipedia.org/wiki/Goodstein Sequence\n\nComments:\n- September 17th, 2010 | Deedlit | approximate value: So how much is $F_6(6) - 2$ in terms of, say, Knuth arrows? we have $$F_6(6) = F_5^6(6) = F_5^5(F_5(6)) = F_5^5(F_4^6(6)) =... = F_5^5(F_4^5(F_3^5(F_2^6(6)))) \\\\ F_2(6) = 2^6*6 = 384 \\\\ F_2^2(6) = 2^384 * 384 > 2^{392} \\\\ F_2^6(6) > 2^{2^{2^{2^{2^{392}}}}} \\\\ F_5^5(F_4^5(F_3^5(F_2^6(6)))) > (2\\uparrow\\uparrow\\uparrow\\uparrow)^5 (2\\uparrow\\uparrow\\uparrow)^5 (2\\uparrow\\uparrow)^5 (2^{2^{2^{2^{2^{392}}}}})$$ That's about as close an approximation as you can get.\n- September 17th, 2010 | Deedlit | The actual value is much higher: You've underestimated the true value by quite bit.\n\nTo get the value of the Goodstein function at n, you take n, write it in hereditary base 2, then replace every appearance to with the infinite ordinal $\\omega$. Call the result R(n). The value of G(n) is then $$H_{R(n)}(3) - 2$$where$H_a(x)$ is the Hardy hierarchy, defined by\n\n$H_0(x) = x$\n\n$H_{a+1} (x) = H_a (x+1)$\n\n$H_a (x) = H_{a[x]} (x)$ for limit ordinals a\n\nSo to find G(6), we write $6 = 2^2 + 2$, so $R(6) = \\omega^\\omega + \\omega$. Hence, $$G(6) = H_{\\omega^\\omega + \\omega} (3) - 2 = H_{\\omega^\\omega}( H_{\\omega} (3)) - 2 = F_{\\omega} (F_1 (3)) - 2 = F_{\\omega} (6) - 2 = F_6 (6) - 2$$\n\nwhere $F_a(x)$ is the fast-growing hierarchy, defined by\n\n$F_0(x) = x+1$\n\n$F_{a+1} (x) = F_a^x (x)$\n\n$F_a (x) = F_{a[x]} (x)$ for limit ordinals a\n\n(or you could just leave the answer in terms of the Hardy hierarchy, I just changed to the fast-growing hierarchy because the answer is a little simpler.)\n- June 4th, 2010 | kagidab | Solution: $k(a,n)=$ amount of steps to reduce $(a-1)*a^n+(a-1)a^{n-1}+...(a-1)a^0$ to -1\n\n$(a-1)a^1+a-1\\rightarrow(a-1)(a+a)^1-1\\rightarrow(a-2)(2a)^1-(2a-1)$\n\nIf you do this a - 1 times: $(a - 1) a ^ 1 + a - 1 -> a * 2^{a - 1} - 1$\n\nWhich means it is reduced to a 0th power and will take $a*2^{a-1}$ steps to finish.\n\nSo total steps $=a(2^{0}+2^{1}+2^{2}+...+2^{a-2})+a*2^{a-1}=a*(2^{a}-1)=k(a, 1)+1$\n\nFor $(a-1)*a^n+(a-1)a^(n-1)+... +(a-1) a^0$:\n\n$k(a,n) = k(k...k(a,n-1)...)) - 2$ [k a times]\n\n$f(n)=g(h(n)-2,h(n), h(n))-2$\n\nWhere $g(o) = g(0, n, o) = o * 2^o$, $g(m, n, o)=g(m - 1, n, g(g(...g(o)))...)$ [n copies of g] and $h(n)$ is the first number in the sequence in the form $(a - 1) * a^(a-1)+(a - 1) a ^ (a - 2) +... (a - 1)$ $h([3,4,5,6,7])=[2, 3, 4, 6, 8]$\n\nE.g.\n\n$f(4) = g(1, 3, 3) - 2 = g(0, 3, g(g(g(3)))) - 2 = 3 * 2^{402653211} - 2$\n\nUsing $g(n)\\sim2^{n}$:\n\n$f(6) = g(4,6,6)-2\\sim g(3, 6, 2\\uparrow\\uparrow 7})\\sim2\\uparrow\\uparrow25$\n- August 17th, 2010 | Anonymous | Has this solution been: Has this solution been verified by the author? Just curious.\n- August 21st, 2010 | kagidab | Nah, I just posted it here: Nah, I just posted it here as my attempt at a solution. It probably needs to be done a bit better, but I believe it works until n = 8 or so, where you have to do a bit extra. I might try make it a bit clearer and rigorous at some point. Plus a better approximation would be useful.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 42.\n\nAttempt notes:\nTarget:\nMake progress on \"Termination of the sixth Goodstein Sequence\" in Logic, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3337, "problem_number": "OPG-37424", "title": "Fixed-point logic with counting", "statement": "Question Can either of the following be expressed in fixed-point logic plus counting:\n\n- Given a graph, does it have a perfect matching, i.e., a set $M$ of edges such that every vertex is incident to exactly one edge from $M$?\n- Given a square matrix over a finite field (regarded as a structure in the natural way, as described in [BGS02]), what is its determinant?", "background": "Source: Open Problem Garden. Original node ID: 37424. URL: http://www.openproblemgarden.org/op/fixed_point_logic_with_counting.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/fixed_point_logic_with_counting\n- Author(s): Blass, Andreas\n- Subject(s): Logic; Finite Model Theory\n- Keywords: Capturing PTime; counting quantifiers; Fixed-point logic; FMT03-Bedlewo\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 18th, 2012 by dberwanger\n\nProblem-page discussion:\nIt is known that (1) is expressible if restricted to bipartite graphs and that (2) is expressible if the field has only two elements. Both these results are in [BGS02].\n\nBibliography:\n[BGS02] A. Blass, Y. Gurevich, and S. Shelah, On polynomial time computation over unordered structures, J. Symbolic Logic 67 (2002) 1093--1125.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Fixed-point logic with counting\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3338, "problem_number": "OPG-37429", "title": "Order-invariant queries", "statement": "Question\n\n- Does ${<}\\text{-invariant\\:FO} = \\text{FO}$ hold over graphs of bounded tree-width?\n- Is ${<}\\text{-invariant\\:FO}$ included in $\\text{MSO}$ over graphs?\n- Does ${<}\\text{-invariant\\:FO}$ have a 0-1 law?\n- Are properties of ${<}\\text{-invariant\\:FO}$ Hanf-local?\n- Is there a logic (with an effective syntax) that captures ${<}\\text{-invariant\\:FO}$?", "background": "Source: Open Problem Garden. Original node ID: 37429. URL: http://www.openproblemgarden.org/op/order_invariant_queries.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/order_invariant_queries\n- Author(s): Segoufin, Luc\n- Subject(s): Logic; Finite Model Theory\n- Keywords: Effective syntax; FMT12-LesHouches; Locality; MSO; Order invariance\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 18th, 2012 by dberwanger\n\nProblem-page discussion:\nWe describe the problem over finite vertex-colored graphs which we call graphs in the sequel. An ordered graph is a graph together with a linear order on its vertices. A property $p$ of ordered graphs is said to be order-invariant if it is independent of the linear order. I.e. $G,<_1 \\models p$ iff $G,<_2 \\models p$ for all graphs $G$ and all linear orders $<_1,<_2$ on $G$. Therefore, we now view an order-invariant property as a property over (unordered) graphs.\n\nWe denote by ${<}\\text{-invariant\\:FO}$, the set of order-invariant first-order definable properties over graphs, where the first-order signature contains the vocabulary for graphs but also a linear order predicate. Note that it is undecidable whether a first-order query is order-invariant. In terms of expressive power, Gurevich showed that ${<}\\text{-invariant\\:FO}$ is strictly more expressive than $\\text{FO}$. However it is known that queries expressible in ${<}\\text{-invariant\\:FO}$ are Gaifman-local~[GS00]. It is also known that ${<}\\text{-invariant\\:FO} = \\text{FO}$ over finite trees~[BS09] (see also~[N05]).\n\nA glimpse beyond\n\nOne can view the linear order on top of the graph as a bijection between the vertices of the graph and an ordered prefix of the positive natural numbers. With this point of view, being order-invariant corresponds to being independent from the choice of the bijection. We could imagine allowing more predicates on the numerical side, and not just the linear order. Typically addition and multiplication. When both these predicates are present we denote by $(+,*)\\text{-invariant\\:FO}$ the properties definable in first-order independently of the bijection. It was shown in~[AMSS10] that those properties are Gaifman local but with a polylog radius for the neighborhoods, and this polylog is tight. However the case when only addition is present is unclear. On top of all the questions above we could add:\n\nQuestion Is $+\\text{-invariant\\:FO}$ Gaifman-local?\n\nThe question of the inclusion of $+\\text{-invariant\\:FO}$ in $\\text{MSO}$ is already relevant over words:\n\nQuestion Can $+\\text{-invariant\\:FO}$ define non-regular languages over words?\n\nSee~[SS10] for more background about this question.\n\nBibliography:\n[AMSS10] Matthew Anderson, Dieter van Melkebeek, Nicole Schweikardt, and Luc Segoufin, Locality of queries definable in invariant first-order logic with arbitrary built-in predicates. In ICALP'11.\n\n[BS09] Michael Benedikt and Luc Segoufin, Towards a characterization of order-invariant queries over tame graphs. J. Symb. Log. 74(1), 2009. Pages 168-186.\n\n[GS00] Martin Grohe and Thomas Schwentick, Locality of order-invariant first-order formulas. ACM Trans. Comput. Log. 1(1), 2000. Pages 112-130.\n\n[N05] Hannu Niemistö, On Locality and Uniform Reduction, LICS'05.\n\n[SS10] Nicole Schweikardt and Luc Segoufin, Addition-Invariant FO and regularity, LICS'10.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Order-invariant queries\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3339, "problem_number": "OPG-37440", "title": "Monadic second-order logic with cardinality predicates", "statement": "The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the following atomic formulas:\n\n- $\\text{\"}\\,\\mathrm{Card}(X) = \\mathrm{Card}(Y)\\,\\text{\"}$\n- $\\text{\"}\\,\\mathrm{Card}(X) \\text{ belongs to } A\\,\\text{\"}$\n\nwhere $A$ is a fixed recursive set of integers.\n\nLet us fix $k$ and a closed formula $F$ in this language.\n\nConjecture Is it true that the validity of $F$ for a graph $G$ of tree-width at most $k$ can be tested in polynomial time in the size of $G$?", "background": "Source: Open Problem Garden. Original node ID: 37440. URL: http://www.openproblemgarden.org/op/monadic_second_order_logic_with_cardinality_predicates.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/monadic_second_order_logic_with_cardinality_predicates\n- Author(s): Courcelle, Bruno\n- Subject(s): Logic; Finite Model Theory\n- Keywords: bounded tree width; cardinality predicates; FMT03-Bedlewo; MSO\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 18th, 2012 by dberwanger\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Monadic second-order logic with cardinality predicates\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3340, "problem_number": "OPG-37444", "title": "Blatter-Specker Theorem for ternary relations", "statement": "Let $C$ be a class of finite relational structures. We denote by $f_C(n)$ the number of structures in $C$ over the labeled set $\\{0, \\dots, n-1 \\}$. For any class $C$ definable in monadic second-order logic with unary and binary relation symbols, Specker and Blatter showed that, for every $m \\in \\mathbb{N}$, the function $f_C(n)$ is ultimately periodic modulo $m$.\n\nQuestion Does the Blatter-Specker Theorem hold for ternary relations.", "background": "Source: Open Problem Garden. Original node ID: 37444. URL: http://www.openproblemgarden.org/op/blatter_specker_theorem_for_ternary_relations.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/blatter_specker_theorem_for_ternary_relations\n- Author(s): Makowsky, Janos A.\n- Subject(s): Logic; Finite Model Theory\n- Keywords: Blatter-Specker Theorem; FMT00-Luminy\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 18th, 2012 by dberwanger\n\nProblem-page discussion:\nOur exposition follows closely [BS84].\n\nCounting labeled structures modulo $m$\n\nLet $C$ be a class of finite structures for one binary relation symbol $R$. We define for $A = \\{ 1, \\ldots, n \\}$ $$F_C(n) = \\mid \\{ R^A \\subseteq A^2: \\langle A, R^A \\rangle \\in C \\} \\mid$$\n\nExamples:\n\n- If $C=U$ consists of all $R$-structures, $f_U(n)= 2^{n^2}$.\n- If $C=B$ consists of bijections, $f_B(n)= n!$\n- If $C= G$ is the class of all (undirected, simple) graphs, $f_G(n)= 2^{\\binom{n}{2}}$.\n- If $C=E$ is the class of all equivalence relations, then $f_E(n)= B_n$, the {\\em Bell Numbers}.\n- If $C=E_2$ is the class of all equivalence relations with two classes only, of the same size, $f_{E_2}(2n)= \\frac{1}{2} \\cdot {\\binom{2n}{n}}$. Clearly, $f_{E_2}(2n+1)= 0$.\n- If $C=T$ is the class of all trees, $f_T(n)= n^{n-2}$, {\\em Caley}.\n\nWe observe the following:\n\n$$f_C(n)= 2^{n^2} = (-1)^{n^2} \\pmod{3}$$\n\n$$f_C(n)= n! = 0 \\pmod{m} \\mbox{ for } n \\geq m$$\n\nAnd for each $m$ the functions, $f_G(n)= 2^{\\binom{n}{2}}$, $f_E(n)= B_n$, $f_T(n)= n^{n-2}$ are ultimately periodic $\\pmod{m}$.\n\nHowever, $f_{E_2}(2n)= \\frac{1}{2} \\cdot {\\binom{2n}{n}} = 1 \\pmod{2}$ iff $n = 2^{2k}$, hence is not periodic $\\pmod{2}$.\n\nMonadic second-order logic definable classes\n\nThe first four examples (all relations, all bijections, all graphs, all equivalence relations) are definable in First Order Logic $\\text{FO}$. The trees are definable in Monadic Second Order Logic $\\text{MSO}$..\n\n$E_2$ is definable in Second Order Logic $\\text{SO}$, but it is not $\\text{MSO}$-definable. If we expand $E_2$ to have the bijection between the classes we get structures with two binary relations. The class is now $\\text{FO}$-definable. Let us denote the corresponding counting function $F_{E_2}(2n)$. We have $$f_{E_2}(2n) \\cdot n! = F_{E_2}(n) = 0 \\pmod{m}$$for$n$ large enough.\n\nPeriodicity and linear recurrence relations\n\nThe periodicity of $f_C(n)$ $\\pmod{m}$ is usually established by exhibiting a linear recurrence relation:\n\nThere exists $1 \\leq k \\in \\mathbb{N}$ and integers $a_1, \\ldots, a_k$ such that for all $n$ $$f_C(n) = \\sum_{j=1}^{k} a_j \\cdot f_C(n-j) \\pmod{m}$$\n\nExamples.\n\n- In the case of $f_C(n) = 2^{n^2}$ we have $$f_C(n) = f_C(n-2) + 2 \\cdot f_C(n-1) \\pmod{3}$$\n- In the case of$f_C(n) = n!$we have for all$m$$$f_C(n) = 0 \\cdot f_C(n-1) \\pmod{m}$$In this case we say that$f_C$ trivializes.\n\nThe Blatter-Specker Theorem\n\nTheorem (BS84) Let $\\tau$ be a binary vocabulary, i.e. all relation symbols are at most binary. If $C$ is a class of finite $\\tau$-structures which is $\\text{MSO}$-definable, then for all $m \\in \\mathbb{N}$ $f_C(n)$ is ultimately periodic $\\pmod{m}$.\n\nMoreover, there exists $1 \\leq k \\in \\mathbb{N}$ and integers $a_1, \\ldots, a_k$ such that for all $n$ $$f_C(n) = \\sum_{j=1}^{k} a_j \\cdot f_C(n-j) \\pmod{m}$$ i.e we have a linear recurrence relation.\n\nBibliography:\n[BS84]* C. Blatter and E. Specker, Recurrence relations for the number of labeled structures on a finite set, Logic and Machines: Decision Problems and Complexity, E. Börger, G. Hasenjaeger and D. Rödding, eds, LNCS 171 (1984) pp. 43-61.\n\n[F03] E. Fischer, The Specker-Blatter theorem does not hold for quaternary relations, Journal of Combinatorial Theory Series A 103(2003), 121-136.\n\n[FM06] E. Fischer and J. A. Makowsky, The Specker-Blatter Theorem revisited: Generating functions for definable classes of stuctures. In Computing and Combinatorics (COCOON 2003) Proc., LNCS vol. 2697 (2003), 90-101.\n\n[S88] E. Specker, Application of Logic and Combinatorics to Enumeration Problems, Trends in Theoretical Computer Science, E. Börger ed., Computer Science Press, 1988, pp. 141-169. Reprinted in: Ernst Specker, Selecta, Birkhäuser 1990, pp. 324-350.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 52.\n\nAttempt notes:\nTarget:\nMake progress on \"Blatter-Specker Theorem for ternary relations\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3341, "problem_number": "OPG-37448", "title": "MSO alternation hierarchy over pictures", "statement": "Question Is the MSO-alternation hierarchy strict for pictures that are balanced, in the sense that the width and the length are polynomially (or linearly) related.", "background": "Source: Open Problem Garden. Original node ID: 37448. URL: http://www.openproblemgarden.org/op/mso_alternation_hierarchy_over_pictures.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/mso_alternation_hierarchy_over_pictures\n- Author(s): Grandjean, Etienne\n- Subject(s): Logic; Finite Model Theory\n- Keywords: FMT12-LesHouches; MSO, alternation hierarchy; picture languages\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 18th, 2012 by dberwanger\n\nProblem-page discussion:\nIn [MST02], Matz, Schweikardt, and Thomas, proved that the MSO-alternation hierarchy is strict over the class of 2-dimensional rectangular pictures (and, as a consequence, is also strict over the class of finite graphs).\n\nThe proof of this hierarchy strictness is essentially based on the fact that, for any positive integer $k$, there is a function $f_k: \\mathbb{N}\\to \\mathbb{N}$ (defined as a fixed height tower of exponentials) such that the set of rectangular grids of format $n\\times f_k(n)$ (i.e, of width $n$ and length $f_k(n)$ ) can be defined by some $\\Sigma_k$ MSO sentence but cannot be defined by some $\\Sigma_{k-1}$ MSO sentence.\n\nSo, the hierarchy result essentially rests on the (more than exponential) imbalance between the two dimensions of the rectangular grid.\n\nIn view of this result a natural question is as follows.\n\nQuestion Is the MSO-alternation hierarchy strict for more well-balanced pictures, for example, if it is required that the width and the length of the pictures are polynomially (resp. linearly) related?\n\nFor example, for square picture languages (or equivalently, rectangular picture languages for which the width and the length of the pictures are linearly related), the only thing we know is that EMSO (that is Existential or $\\Sigma_1$ MSO) over square pictures is not closed under complement.\n\nOliver Matz (personal communication) thinks it is possible that any MSO sentence over square pictures be equivalent to a Boolean combination of existntial MSO sentences.\n\nBibliography:\n[MST02] O. Matz, N. Schweikardt and W. Thomas, The Monadic Quantifier Alternation Hierarchy over Grids and Graphs, Information and Computation 179(2002), 356-383.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"MSO alternation hierarchy over pictures\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3342, "problem_number": "OPG-37863", "title": "Finite entailment of Positive Horn logic", "statement": "Question Positive Horn logic (pH) is the fragment of FO involving exactly $\\exists, \\forall, \\wedge, =$. Does the fragment $pH \\wedge \\neg pH$ have the finite model property?", "background": "Source: Open Problem Garden. Original node ID: 37863. URL: http://www.openproblemgarden.org/op/finite_satisfiability_of_positive_horn_logic_entailment.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/finite_satisfiability_of_positive_horn_logic_entailment\n- Author(s): Martin, Barnaby\n- Subject(s): Logic; Finite Model Theory\n- Keywords: entailment; finite satisfiability; horn logic\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 21st, 2012 by LucSegoufin\n\nProblem-page discussion:\nIt doesn't really matter whether or not equality is allowed, as it may mostly be propagated out by substitution. The question is whether there an infinity axiom of the form $\\phi \\wedge \\neg \\psi$, for $\\phi, \\psi$ in pH?\n\nIn [CMM08] it is proved that entailment of pH sentences is decidable. I.e. input, $\\phi, \\psi$ in pH and return yes if $\\phi \\rightarrow \\psi$ is true on all models. The question is whether this is the same as asking if entailment of pH is equivalent to finite entailment, i.e. if $\\phi \\rightarrow \\psi$ is true on all models iff it is true on all finite models.\n\nFor the positive equality-free fragment of FO (bigger than pH), finite entailment and general entailment do not coincide, and the latter problem is undecidable. For existential positive logic, (smaller than pH), finite entailment and general entailment do coincide, and of course both are decidable. At present, the question as to whether finite entailment of pH is decidable is also open.\n\nBibliography:\n[CMM08] Hubie Chen, Florent R. Madelaine, Barnaby Martin: Quantified Constraints and Containment Problems. LICS 2008: 317-328\n\nComments:\n- June 15th, 2020 | Anonymous | Solved.: Solved in LICS 2017 \"Herbrand Property, Finite Quasi-Herbrand Models, and a Chandra-Merlin Theorem for Quantified Conjunctive Queries\"\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Finite entailment of Positive Horn logic\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3343, "problem_number": "OPG-38188", "title": "Vertex Cover Integrality Gap", "statement": "Conjecture For every $\\varepsilon > 0$ there is $\\delta > 0$ such that, for every large $n$, there are $n$-vertex graphs $G$ and $H$ such that $G \\equiv_{\\delta n}^{\\mathrm{C}} H$ and $\\mathrm{vc}(G) \\ge (2 - \\varepsilon) \\cdot \\mathrm{vc}(H)$.", "background": "Source: Open Problem Garden. Original node ID: 38188. URL: http://www.openproblemgarden.org/op/vertex_cover_integrality_gap.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/vertex_cover_integrality_gap\n- Author(s): Atserias, Albert\n- Subject(s): Logic; Finite Model Theory\n- Keywords: counting quantifiers; FMT12-LesHouches\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 2nd, 2012 by dberwanger\n\nProblem-page discussion:\nHere $\\equiv^{\\mathrm{C}}_{k}$ denotes indistinguishability in $k$-variable first-order logic with counting quantifiers, and $\\mathrm{vc}(G)$ denotes the cardinality of the minimum vertex-cover of $G$. By~[1], $G \\equiv_{3}^{\\mathrm{C}} H$ implies $\\mathrm{vc}(G) \\leq 2 \\cdot \\mathrm{vc}(H)$. Also by~[1] a positive answer would imply that an integrality gap of $2-\\varepsilon$ resists $\\delta n$ levels of Sherali-Adams linear programming relaxations of vertex-cover, on $n$-vertex graphs. It is known that such a gap resists $n^{\\delta}$ levels~[2]. What we ask would let us replace $n^{\\delta}$ by $\\delta n$. If improving over $n^{\\delta}$ were not possible, then we could approximate vertex-cover by a factor better than~ $2$ in subexponential time (i.e. $2^{n^{o(1)}}$ ). Approximating vertex-cover by a factor better than~1.36 is NP-hard~[3], and approximating vertex-cover by factor better than~2 is UG-hard~[4], where UG stands for Unique Games (from the Unique Games Conjecture); but note that UG-hardness does not rule out subexponential-time algorithms because UG itself is solvable in subexponential time~[5]\n\nBibliography:\n[1] A. Atserias and E. Maneva. Sherali-Adams Relaxations and Indistinguishability in Counting Logics, in Proc. 3rd ACM ITCS, pp. 367-379, 2012.\n\n[2] M. Charikar, K. Makarychev and Y. Makarychev. Integrality Gaps for Sherali-Adams Relaxations, in Proc. 41st ACM STOC, pp. 283-292, 2009.\n\n[3] I. Dinur and S. Safra. On the Hardness of Approximating Minimum Vertex-Cover, Annals of Mathematics, 162(1):439-485, 2005.\n\n[4] S. Khot and O. Regev. Vertex cover might be hard to approximate to within 2-epsilon, J. Comput. Syst. Sci. 74(3):335-349, 2008.\n\n[5] S. Arora, B. Barak, and D. Steurer. Subexponential Algorithms for Unique Games and Related problems, in Proc. 51th IEEE FOCS, pp. 563-572, 2010.}\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 17.\n\nAttempt notes:\nTarget:\nMake progress on \"Vertex Cover Integrality Gap\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3344, "problem_number": "OPG-416", "title": "Lonely runner conjecture", "statement": "Conjecture Suppose $k$ runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for any given runner, there is a time at which that runner is distance at least $\\frac{1}{k}$ (along the track) away from every other runner.", "background": "Source: Open Problem Garden. Original node ID: 416. URL: http://www.openproblemgarden.org/op/lonely_runner_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/lonely_runner_conjecture\n- Author(s): Cusick, Thomas W.; Wills, Jorg M.\n- Subject(s): Number Theory\n- Keywords: diophantine approximation; view obstruction\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 24th, 2007 by mdevos\n\nProblem-page discussion:\nThis conjecture was independently introduced in two very different contexts. Wills [W] introduced it as a problem in diophantine approximation, and Cusick [C1] discovered it as a geometric view obstruction problem. The poetic name is due to Goddyn.\n\nThere are a number of different proofs of this conjecture for small values of $k$ (as a warning, there are different formulations of this conjecture, and what appears here as the problem for $k$ runners is sometimes considered to be the problem for $k-1$ runners). The cases with $k \\le 3$ runners are easy to check. The $k=4$ case was proved independently by Betke and Wills [BW] and by Cusick. The $k=5$ case was first established by Cusick and Pomerance [CP] with the help of some computer checking, and this argument was later simplified by Bienia et al. [BGGS] who also found applications of this theorem to the study of flows on graphs. The $k=6$ case was first proved by Bohman et al. [BHK] and this was later simplified by Renault [R]. Recently, the $k=7$ case was proved by Barajas and Serra [BS].\n\nBibliography:\n[BS] J. Barajas and O. Serra, The lonely runner problem with seven runners.\n\n[BW] U. Betke and J. M. Wills, Untere Schranken fur zwei diophantische Approximations-Funktionen, Monatsch. Math. 76 (1972), 214-217.\n\n[BGST] W. Bienia, L. Goddyn, P. Gvozdjak, A. Sebo, Flows, View Obstructions, and the Lonely Runner, J. Combinatorial Theory Ser. B 72 (1998) 1-9.\n\n[BHK] T. Bohman, R. Holzman, and D. Kleitman, Six lonely runners, Electron. J. Combin. 8 (2001), no. 2\n\n[CC] Y.G. Chen, T.W. Cusick, The View-Obstruction Problem for n-Dimensional Cubes, J. Number Theory 74, no. 1 (1999) 126-133.\n\n*[C1] T.W. Cusick, View-Obstruction Problems in n-Dimensional Geometry, J. Combinatorial Theory Ser. A 16 (1974) 1-11.\n\n[C2] T.W. Cusick, View-Obstruction Problems II, Proc. Amer. Math. Soc. 84 (1982) 25-28.\n\n[C3] T.W. Cusick, The view-obstruction problem for $5$-dimensional cubes Monatsh. Math. 127 (1999), no. 3, 183--187.\n\n[CP] T.W. Cusick and C. Pomerance, View-Obstruction Problems III, J. Number Theory 19 (1984) 131-139.\n\n[R] J. Renault, View-obstruction: a shorter proof for 6 lonely runners. Discrete Math. 287 (2004), no. 1-3, 93-101.\n\n*[W] J.M. Wills, Zwei Satze uber Inhomogene Diophantische Approximation von Irrationalzahlen, Monatsch. Math. 71 (1967) 263-269.\n\nBibliography links:\n- The lonely runner problem with seven runners: http://www.mac.cie.uva.es/%7Erevilla/vjmda/files/044.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Lonely runner conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3345, "problem_number": "OPG-671", "title": "MacEachen Conjecture", "statement": "Conjecture Every odd prime number must either be adjacent to, or a prime distance away from a primorial or primorial product.", "background": "Source: Open Problem Garden. Original node ID: 671. URL: http://www.openproblemgarden.org/op/maceachen_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/maceachen_conjecture\n- Author(s): McEachen, Bill R.\n- Subject(s): Number Theory\n- Keywords: primality; prime distribution\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: October 19th, 2007 by billymac00\n\nProblem-page discussion:\nThis conjecture speaks to the distribution of all prime numbers, relating them to primorials. Recall that the primorials are simply the consecutive product of prime numbers eg 2,2*3,2*3*5, etc. this is OEIS A002110 {2,6,30,210,30030,...}. A new sequence A129912, related to the conjecture, ie A129912 and combines A002110 with the products of unique primorials eg 2*6,2*30,6*30, etc. The PariGP code to generate the terms is on the author's wiki site. By unique, the products mentioned would not use an entry of A002110 more than once. A numerical example is candidate number N=189239. This cannot be prime unless it is an absolute prime distance from a sequence term covering the range 0 thru 2*N. of course, it is 9059 away from sequence entry 180180, and so it may be prime (and it is). Note that the conjecture treats a required but not sufficient condition for primality. As it works for offset distance less than the candidate it could be used in a primality method if that somehow could be applied effectively (seriously doubtful). However, the insight provided into the distribution of the primes is the worthwhile part. It leaves no doubt about the non random, concrete structure of prime interdependency.\n\nNote that the conjecture implies as others have suspected, the existence of Twin primes to be found adjacent to sequence terms. Note that the conjecture was independently confirmed through the first 50 million primes. Also, the author has attempted a strict proof, that is conditional upon Goldbach's Conjecture being true. Realizing that the author is merely an amateur hobbiest, the proof may be flawed but it can be supplied, it is quite elementary, less than a page.\n\nI do note the independent work of several others found online, all of whom I do not personally know. One is John Sokol, who in 2002 made the following conjecture (incorrect is as far as the distance being < the candidate, think it breaks down around 331). I independently started with this same conjecture until I quickly realized the correct one. I also note the work of Bob Potter and his primorial conjecture at the link shown lower. A third person is Hank Harrell, who has done work in an area similar to Potter's.\n\nThe author plotted a normalized minimum offset seen at the primes, with the resulting scatter plot clearly indicating an asymptotic trend towards the relevant sequence entry being one greater than the prime candidate.\n\nThe author has also speculated that the A129912 sequence is useful in locating Twin Primes (not his particular interest). An earlier conjecture by the author in 2006 (A117825) concerning highly composite numbers basically parallels Fortune's conjecture (A005325).\n\nThe links mentioned above are now listed: OEIS 2110 OEIS 129912 author's wiki Wikipedia PlanetMath Harrell WikiCommons\n\nSokol's conjecture (sic): A primes can only exists + or - a prime from a primorial. Where 1 is considered a prime and 2 is not.\n\nSokol\n\nPotter conjecture:\n\n\"All prime numbers in the vicinity of a primorial (or primorial multiple) will combine to make a Goldbach pair for the primorial. The length of sequence for which this effect holds increases as the value of the primorial or primorial multiple increases.\"\n\nPotter\n\nMany more online resources were accessed for this work, some of them are:\n\nPari-GP program Chris Caldwell's well respected site Dario Alpern's online ECM calculator\n\nI am sure there are others I have forgotten.\n\nSource links:\n- primorial: http://en.wikipedia.org/wiki/primorial\n\nDiscussion links:\n- OEIS 2110: http://www.research.att.com/%7Enjas/sequences/?q=A2110&language=english&go=Search\n- OEIS 129912: http://www.research.att.com/%7Enjas/sequences/?q=A129912&sort=0&fmt=0&language=english&go=Search\n- author's wiki: http://billymac00.pbwiki.com/main\n- Wikipedia: http://en.wikipedia.org/wiki/Primorials\n- PlanetMath: http://planetmath.org/encyclopedia/FortunesConjecture.html\n- Harrell: http://www.prime-equations.com/index.html\n- WikiCommons: https://commons.wikimedia.org/wiki/File:OEIS_A129912_spin1.svg\n- Sokol: http://www.dnull.com/%7Esokol/prime/conjecture1.html\n- Potter: http://primorialconjecture.wordpress.com/2012/07/\n- Pari-GP program: http://pari.math.u-bordeaux.fr/] \\href [WIMS]{http://wims.unice.fr/wims/en_tool~algebra%7Efactor.html\n- Chris Caldwell's well respected site: http://primes.utm.edu/\n- Dario Alpern's online ECM calculator: http://www.alpertron.com.ar/ECM.HTM\n\nComments:\n- July 7th, 2012 | chongchingbak | is this already solved?: is this already solved?\n- March 24th, 2008 | Anonymous | wrong reference: Hi, Hope I'm not being too picky but the reference to my web site is wrong. www.primorialconjecture.org - maybe if you're updating this sometime you would tweak it.\n\nThanks! Bob Potter\n- January 22nd, 2009 | billymac00 | link correction: sorry, I just saw this today, I fixed the link in the 2 places I saw Bob.\n- February 26th, 2008 | billymac00 | proof progress: due to an email xchange with a fella Simon Horvat Feb 24 2008, who I am otherwise unfamiliar with, I believe he has come upon a fairly straightforward mathematical proof of my conjecture. I defer to him for the time being but it is very promising...\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"MacEachen Conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3346, "problem_number": "OPG-739", "title": "Chowla's cosine problem", "statement": "Problem Let $A \\subseteq {\\mathbb N}$ be a set of $n$ positive integers and set\n$$\nm(A) = - \\min_x \\sum_{a \\in A} \\cos(ax).\n$$\n What is $m(n) = \\min_A m(A)$?", "background": "Source: Open Problem Garden. Original node ID: 739. URL: http://www.openproblemgarden.org/op/chowlas_cosine_problem.\n\nSource subject path: Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/chowlas_cosine_problem\n- Author(s): Chowla, Sarvadaman\n- Subject(s): Number Theory\n- Keywords: circle; cosine polynomial\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 22nd, 2008 by mdevos\n\nProblem-page discussion:\nIt is easy to see that $m(A) > 0$, since the average value of the sum of the cosines is zero. Bourgain [B] proved that $m(n) > e^{(\\log n)^c}$ for some $c>0$ and $n$ sufficiently large. Recently, Ruzsa [R] tightened this argument, proving that $m(n) > c_1 e^{c_2 \\sqrt{ \\log n}}$ where $c_2 = \\sqrt{ (\\log 2)/ 8}$. The proof utilizes a clever manipulation of norms to reveal a (somewhat surprising) additive structure to the problem.\n\nIt seems the only known upper bound is $m(n) \\ll \\sqrt{n}$.\n\nBibliography:\n[B] J. Bourgain, Sur le minimum d'une somme de cosinus, Acta Arith. 45 (1986), 381--389. MathSciNet\n\n*[C] S. Chowla, Some applications of a method of A. Selberg. J. Reine Angew. Math. 217 (1965) 128--132. MathSciNet\n\n[R] I.Z. Ruzsa, Negative values of cosine sums. Acta Arith. 111 (2004), no. 2, 179--186. MathSciNet\n\nRelated:\nRelated problems\nLonely runner conjecture\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0847298\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0172853\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2039421\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Chowla's cosine problem\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3347, "problem_number": "OPG-791", "title": "Quartic rationally derived polynomials", "statement": "Call a polynomial $p \\in {\\mathbb Q}[x]$ rationally derived if all roots of $p$ and the nonzero derivatives of $p$ are rational.\n\nConjecture There does not exist a quartic rationally derived polynomial $p \\in {\\mathbb Q}[x]$ with four distinct roots.", "background": "Source: Open Problem Garden. Original node ID: 791. URL: http://www.openproblemgarden.org/op/quartic_rationally_derived_polynomials.\n\nSource subject path: Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/quartic_rationally_derived_polynomials\n- Author(s): Buchholz, Ralph H.; MacDougall, James A.\n- Subject(s): Number Theory\n- Keywords: derivative; diophantine; elliptic; polynomial\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 17th, 2008 by mdevos\n\nProblem-page discussion:\nProbably anyone who has ever designed simple problems for calculus students has looked for polynomials $p$ with the property that both $p$ and some small derivatives of it are easy to factor. Perhaps inspired by this, Buchholz and MacDougall attempted to classify all univariate polynomials defined over a domain $k$ with the property that they and all their nonzero derivatives have all their roots in $k$. This problem can be split into cases dependent upon the multiplicity of the roots, and Buchholz and MacDougall solved many of the small ones for $k={\\mahtbb Q}$. Based on their results and a theorem of Flynn [F], an affirmative solution to the above conjecture would complete this classification problem for $k={\\mathbb Q}$.\n\nBibliography:\n*[BM] R. Buchholz, and J. MacDougall, When Newton met Diophantus: a study of rational-derived polynomials and their extension to quadratic fields. J. Number Theory 81 (2000), no. 2, 210--233. MathSciNet\n\n[F] E. V. Flynn, On Q-derived polynomials. Proc. Edinb. Math. Soc. (2) 44 (2001), no. 1, 103--110. MathSciNet\n\nBibliography links:\n- When Newton met Diophantus: a study of rational-derived polynomials and their extension to quadratic fields: http://www.geocities.com/teufel_pi/papers/rdp.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1752251\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1879212\n\nComments:\n- February 16th, 2011 | Comet | Bibliography: Hyperlink in the bibliography is no longer valid, but the article can be found at: http://web.archive.org/web/20011127182208/http://www.geocities.com/teufel_pi/papers/rdp.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Quartic rationally derived polynomials\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3348, "problem_number": "OPG-819", "title": "A discrete iteration related to Pierce expansions", "statement": "Conjecture Let $a > b > 0$ be integers. Set $b_1 = b$ and $b_{i+1} = {a \\bmod {b_i}}$ for $i \\geq 0$. Eventually we have $b_{n+1} = 0$; put $P(a,b) = n$.\n\nExample: $P(35, 22) = 7$, since $b_1 = 22$, $b_2 = 13$, $b_3 = 9$, $b_4 = 8$, $b_5 = 3$, $b_6 = 2$, $b_7 = 1$, $b_8 = 0$.\n\nProve or disprove: $P(a,b) = O((\\log a)^2)$.", "background": "Source: Open Problem Garden. Original node ID: 819. URL: http://www.openproblemgarden.org/op/a_discrete_iteration_related_to_pierce_expansions.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_discrete_iteration_related_to_pierce_expansions\n- Author(s): Shallit, Jeffrey O.\n- Subject(s): Number Theory\n- Keywords: Pierce expansions\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: June 11th, 2008 by shallit\n\nProblem-page discussion:\nThe best upper bound is currently $P(a,b) = O(a^{1/3})$. For more information, see [ES].\n\nBibliography:\n[ES] P. Erd\\\"os and J. Shallit, \"New bounds on the length of finite Pierce and Engel series\", S\\'eminaire de Th\\'eorie des Nombres de Bordeaux 3 (1991), 43--53.\n\nComments:\n- June 11th, 2008 | Porges | A different upper bound: This paper shows an upper bound of $O(\\sqrt[3]a \\sqrt[3]{\\log(a)})$.\n\nEdit: But looking at the title page of the paper, I see you already knew that;)\n- May 30th, 2020 | Anonymous | bound: That's because the best currently known bound is not the one in the paper. It is in a technical report by Vlado Keselj.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"A discrete iteration related to Pierce expansions\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3349, "problem_number": "OPG-1786", "title": "Algebraic independence of pi and e", "statement": "Conjecture $\\pi$ and $e$ are algebraically independent", "background": "Source: Open Problem Garden. Original node ID: 1786. URL: http://www.openproblemgarden.org/op/algebraic_independence_of_pi_and_e.\n\nSource subject path: Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/algebraic_independence_of_pi_and_e\n- Subject(s): Number Theory\n- Keywords: algebraic independence\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 8th, 2008 by porton\n\nSource links:\n- algebraically independent: http://en.wikipedia.org/wiki/Algebraic_independence\n\nComments:\n- April 29th, 2012 | warut | Schanuel's conjecture: Assuming Schanuel's conjecture, one can show that $\\pi$ and $e$ are algebraically independent over $\\mathbb Q$.\n- February 16th, 2011 | Comet | in which subfield K of which field L?: After all, e to the pi i = -1, so this shows that pi and e are not always algebraically independent.\n- July 16th, 2011 | Anonymous | By definition?: I think any two distinct transcendental numbers must be algebraically independent, almost by definition. Since e and pi are transcendental, they must be a. i. No? - David Spector\n- August 5th, 2011 | cubola zaruka | not all transcedentials are algebraically independant: pi and 4-pi are both transcedential and sum to 4, so are not algebraically independant.\n- July 21st, 2011 | Anonymous | two transcendentals are not necessarily algebraically independen: e and e^2 are both transcendental but (e,e^2) makes the two-variable polynomial f(x,y)=x^2-y equal to zero\n- May 18th, 2011 | Jon Noel | ?: but that's not a polynomial.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Algebraic independence of pi and e\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3350, "problem_number": "OPG-2147", "title": "Odd perfect numbers", "statement": "Conjecture There is no odd perfect number.", "background": "Source: Open Problem Garden. Original node ID: 2147. URL: http://www.openproblemgarden.org/op/odd_perfect_number.\n\nSource subject path: Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/odd_perfect_number\n- Author(s): Ancient/folklore\n- Subject(s): Number Theory\n- Keywords: perfect number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: September 27th, 2008 by azi\n\nProblem-page discussion:\nThere is substantial literature on the problem. Most proceeds from a study of the multiplicative function $\\sigma_{-1}(n)=\\sigma(n)/n$ where the conjecture can be stated: $\\sigma_{-1}(n)=2$ implies that $n$ is even.\n\nSource links:\n- perfect number: http://en.wikipedia.org/wiki/perfect number\n\nComments:\n- December 7th, 2011 | Anonymous | limiting divisors: My idea is to assume that the OPN is divisible by a prime number (e.x. 3) then use the properties of perfect numbers to figure out other numbers the OPN is divisible by.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Odd perfect numbers\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3351, "problem_number": "OPG-16555", "title": "Diophantine quintuple conjecture", "statement": "Definition A set of m positive integers $\\{a_1, a_2, \\dots, a_m\\}$ is called a Diophantine $m$-tuple if $a_i\\cdot a_j + 1$ is a perfect square for all $1 \\leq i < j \\leq m$.\n\nConjecture (1) Diophantine quintuple does not exist.\n\nIt would follow from the following stronger conjecture [Da]:\n\nConjecture (2) If $\\{a, b, c, d\\}$ is a Diophantine quadruple and $d > \\max \\{a, b, c\\}$, then $d = a + b + c + 2bc + 2\\sqrt{(ab+1)(ac+1)(bc+1)}.$", "background": "Source: Open Problem Garden. Original node ID: 16555. URL: http://www.openproblemgarden.org/op/diophantine_quintuple_does_not_exist.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/diophantine_quintuple_does_not_exist\n- Subject(s): Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: November 22nd, 2008 by maxal\n\nProblem-page discussion:\nIt was proved in [Db] that there are only finitely many Diophantine quintuples and no Diophantine sextuples.\n\nConjecture (2) is motivated by an observation of [AHS] that every Diophantine triple $\\{a,b,c\\}$ can be extended to a Diophantine quadruple $\\{a,b,c,a + b + c + 2bc + 2\\sqrt{(ab+1)(ac+1)(bc+1)}\\}.$\n\nBibliography:\n[Da] A. Dujella Diophantine $m$-tuples, a survey of the main problems and results concerning Diophantine m-tuples.\n\n[Db] A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183-214.\n\n[AHS] J. Arkin, V. E. Hoggatt and E. G. Strauss, On Euler's solution of a problem of Diophantus, Fibonacci Quart. 17 (1979), 333-339.\n\nBibliography links:\n- Diophantine $m$-tuples: http://web.math.hr/%7Eduje/dtuples.html\n- There are only finitely many Diophantine quintuples: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.58.8571\n\nComments:\n- February 25th, 2020 | Anonymous | This result has been proven: in a paper announced in 2016 and published in 2019, He, Togbé and Ziegler [350] gave the proof of the Diophantine quintuple conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Diophantine quintuple conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3352, "problem_number": "OPG-36952", "title": "Twin prime conjecture", "statement": "Conjecture There exist infinitely many positive integers $n$ so that both $n$ and $n+2$ are prime.", "background": "Source: Open Problem Garden. Original node ID: 36952. URL: http://www.openproblemgarden.org/op/twin_prime_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/twin_prime_conjecture\n- Subject(s): Number Theory\n- Keywords: prime; twin prime\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 4th, 2009 by kaushiks.nitt\n\nComments:\n- November 22nd, 2011 | Kermit1941 | Twin Prime Conjecture: Hello.\n\nI see a reference to a proof called Tyte's proof but saw no details on it. When I click on the link, it only gives me the same page that I'm already on.\n\nAlso, I could not get any additional details about the following post:\n\nOn January 10th, 2011 Hugh Barker says: I've added a thread here, linking to an attempted proof of Twin Primes and the Polignac conjecture in general.\n\nhttp://garden.irmacs.sfu.ca/?q=op/twin_primes_and_polignacs_conjecture\n\nWill receive any input, debunking etc gratefully.\n\nI am most interested in searching for any attempted proofs of the twin prime conjecture. I believe that I have a proof, but anticipate clearly filling in some details.\n\nThe core idea in our proof is that we specify exactly a lower bound for the number of twin primes less than a given integer, N, and that this lower bound goes to infinity as N goes to infinity.\n\nKermit\n- November 25th, 2009 | ducafelipe | Regarding the Tyte's proof I: Regarding the Tyte's proof I have received three enthusiastic comments-contributions, pointing out that while the averaging step made by Alan is questionable, maybe this approach shows where to look for in the solution of this Conjecture. These first three comments are from Chris Nash, Fabrice Marchant and Leadhyena Inrandomtan:\n\n\"About Alan Tyte's proof: all the beginning up to \"Lemma 5\" is right but there are 2 errors in the end of the proof, after each \"Hence, on the average:\" because we do not know the way our beloved Ds are spanned: no reason to be sure they are put at the same rate between x and x^2 than between a whole pattern. However, I think the idea of the proof with As, Bs... is great and I'll try to work in the way of Alan.\" (F. Marchant)\n- August 18th, 2009 | Anonymous | Where is the conjecture?: The conjecture is not stated. It's just the definition.\n- August 18th, 2009 | mdevos | thanks: I know little about the status of this conjecture, but I did correct the definition. We would welcome anyone with more knowledge to update it further.\n- January 10th, 2011 | Hugh Barker | I've added a thread here,: I've added a thread here, linking to an attempted proof of Twin Primes and the Polignac conjecture in general.\n\nhttp://garden.irmacs.sfu.ca/?q=op/twin_primes_and_polignacs_conjecture\n\nWill receive any input, debunking etc gratefully.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Twin prime conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3353, "problem_number": "OPG-37289", "title": "Polignac's Conjecture", "statement": "Conjecture Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.\n\nIn particular, this implies:\n\nConjecture Twin Prime Conjecture: There are an infinite number of twin primes.", "background": "Source: Open Problem Garden. Original node ID: 37289. URL: http://www.openproblemgarden.org/op/twin_primes_and_polignacs_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/twin_primes_and_polignacs_conjecture\n- Author(s): de Polignac, Alphonse\n- Subject(s): Number Theory\n- Keywords: prime; prime gap\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: January 10th, 2011 by Hugh Barker\n\nBibliography:\n*[P] A. de Polignac, Six propositions arithmologiques déduites de crible d'Ératosthène. Nouv. Ann. Math. 8 (1849), pp. 423--429.\n\nRelated:\nRelated problems\nTwin prime conjecture\n\nComments:\n- June 18th, 2013 | Charles R Great... | Link: I removed this link and its description from the problem, since it is now known to be incorrect. For future reference here it is: http://barkerhugh.blogspot.com/2011/01/twin-primes-and-polignac-conjecture.html\n- January 13th, 2011 | Hugh Barker | Flaw: OK, someone has spotted the inevitable flaw in the logic and pointed it out, so not worth looking after all (though feel free if you want to play \"spot the error\"...\n- January 11th, 2011 | Anonymous | Compressed version: There's a slightly compressed version of this proof here:\n\nhttp://barkerhugh.blogspot.com/2011/01/twin-prime-proof-compressed-version.html\n\nProbably better to refer to this one as it is more focused.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Polignac's Conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3354, "problem_number": "OPG-37300", "title": "Special Primes", "statement": "Conjecture Let $p$ be a prime natural number. Find all primes $q\\equiv1\\left(\\mathrm{mod}\\: p\\right)$, such that $2^{\\frac{\\left(q-1\\right)}{p}}\\equiv1\\left(\\mathrm{mod}\\: q\\right)$.", "background": "Source: Open Problem Garden. Original node ID: 37300. URL: http://www.openproblemgarden.org/op/special_primes.\n\nSource subject path: Number Theory.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/special_primes\n- Author(s): George BALAN\n- Subject(s): Number Theory\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: February 18th, 2011 by maththebalans\n\nComments:\n- February 7th, 2013 | Anonymous | All primes are: All primes are p=(q-1)/(order of 2 mod q)\n- February 17th, 2012 | Anonymous | paul newell: q divides 2^((q-1)/P))-1 iff p divides (q-1)/( Order of2 mod q )\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Special Primes\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3355, "problem_number": "OPG-37318", "title": "Primitive pythagorean n-tuple tree", "statement": "Conjecture Find linear transformation construction of primitive pythagorean n-tuple tree!", "background": "Source: Open Problem Garden. Original node ID: 37318. URL: http://www.openproblemgarden.org/op/primitive_pythagorean_n_tuple_tree.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/primitive_pythagorean_n_tuple_tree\n- Subject(s): Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 24th, 2011 by tsihonglau\n\nProblem-page discussion:\nPrimitive pythagorean n-tuple is a n-tuple $(a_{1},a_{2},a_{3},...,a_{n})$ such that $a_{1}^2 + a_{2}^2 + a_{3}^2 +... + a_{n-1}^2 = a_{n}^2$\n\nand the greatest common divisor of $(a_{1},a_{2},a_{3},...,a_{n})$ is 1.\n\nThere are at least two known linear transformation construction of primitive pythagorean triple tree!\n\nWikipedia\n\nIs there any other linear transformation construction of primitive pythagorean triple tree?\n\nMoreover, find linear transformation construction of primitive pythagorean n-tuple tree!\n\nDiscussion links:\n- There are at least two known linear transformation construction of primitive pythagorean triple tree!: http://home.educities.edu.tw/tsihonglau/senior/primitive_pythagorean_triple_ternary_tree.html\n- Wikipedia: http://en.wikipedia.org/wiki/Pythagorean_triple#Parent.2Fchild_relationships\n\nComments:\n- June 16th, 2021 | Anonymous | Linear transformation of primitive pythagorean n-tuple: Is it true that for primitive pythagorean n-tuple (a_1, a_2...... a_n), if there are (n-2) even terms and 1 odd term in the summation side of the equation and a_n is odd, a linear transformation can be made? And can we find any such primitive pythagorean n-tuple where a_n is even?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Primitive pythagorean n-tuple tree\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3356, "problem_number": "OPG-37396", "title": "3 is a primitive root modulo primes of the form 16 q^4 + 1, where q>3 is prime", "statement": "Conjecture $3~$ is a primitive root modulo $~p$ for all primes $~p=16\\cdot q^4+1$, where $~q>3$ is prime.", "background": "Source: Open Problem Garden. Original node ID: 37396. URL: http://www.openproblemgarden.org/op/primes_p_such_that_3_is_a_primitive_root_modulo_p.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/primes_p_such_that_3_is_a_primitive_root_modulo_p\n- Subject(s): Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 25th, 2012 by princeps\n\nSource links:\n- primitive root: http://en.wikipedia.org/wiki/Primitive_root_modulo_n\n\nComments:\n- August 24th, 2012 | Anonymous | group theory answer: Using group theory, the multiplicative group of order p=16q^4+1 has order p-1=16q^4. Using lagrange's theorem, the order of any element divides the order of the group. Therefore, any element is either a primitive root, a quadratic residue, or a qth power residue mod 16q^4+1. Using the laws of quadratic reciprocity, 3 is a quadratic residue modulo a prime if and only if the prime is congruent to plus or minus 1 mod 12. Since q>3 is a prime and therefore not divisible by 3, 16q^4=1(mod 3), so 16q^4+1=2(mod 3). That means that 16q^4+1=5(mod 12), and therefore 3 is not a quadratic residue mod p. Therefore the only thing left to prove is that 3 is not a qth power residue.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"3 is a primitive root modulo primes of the form 16 q^4 + 1, where q>3 is prime\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3357, "problem_number": "OPG-37397", "title": "Erdős–Straus conjecture", "statement": "Conjecture\n\nFor all $n > 2$, there exist positive integers $x$, $y$, $z$ such that $$1/x + 1/y + 1/z = 4/n$$.", "background": "Source: Open Problem Garden. Original node ID: 37397. URL: http://www.openproblemgarden.org/op/erdos_straus_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/erdos_straus_conjecture\n- Author(s): Erdos, Paul; Straus, Ernst G.\n- Subject(s): Number Theory\n- Keywords: Egyptian fraction\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: February 29th, 2012 by ACW\n\nProblem-page discussion:\nSee Erdős–Straus conjecture for more details.\n\nDiscussion links:\n- Erdős–Straus conjecture: http://en.wikipedia.org/wiki/Erdős–Straus conjecture\n\nComments:\n- July 14th, 2014 | Anonymous | Formula Individa: It was necessary to write the solution in a more General form: $$\\frac{t}{q}=\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}$$$t,q$- integers. Decomposing on the factors as follows:$p^2-s^2=(p-s)(p+s)=2qL$The solutions have the form:$$x=\\frac{p(p-s)}{tL-q}$$$$y=\\frac{p(p+s)}{tL-q}$$$$z=L$$Decomposing on the factors as follows:$p^2-s^2=(p-s)(p+s)=qL$The solutions have the form:$$x=\\frac{2p(p-s)}{tL-q}$$$$y=\\frac{2p(p+s)}{tL-q}$$$$z=L$$\n- July 14th, 2014 | Anonymous | Solution: For the equation: $$\\frac{4}{q}=\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}$$The solution can be written using the factorization, as follows.$$p^2-s^2=(p-s)(p+s)=2qL$$Then the solutions have the form:$$x=\\frac{p(p-s)}{4L-q}$$$$y=\\frac{p(p+s)}{4L-q}$$$$z=L$$I usually choose the number$L$such that the difference:$(4L-q)$was equal to:$1,2,3,4$Although your desire you can choose other. You can write a little differently. If unfold like this:$$p^2-s^2=(p-s)(p+s)=qL$$The solutions have the form:$$x=\\frac{2p(p-s)}{4L-q}$$$$y=\\frac{2p(p+s)}{4L-q}$$$$z=L$$\n- July 14th, 2013 | cpbm | Further restriction: I think you need to specify that $x$, $y$ and $z$ be positive for this to be challenging (and open).\n- July 15th, 2013 | ACW | Restriction: Done. Thank you.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 41.\n\nAttempt notes:\nTarget:\nMake progress on \"Erdős–Straus conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3358, "problem_number": "OPG-37402", "title": "Lucas Numbers Modulo m", "statement": "Conjecture The sequence {L(n) mod m}, where L(n) are the Lucas numbers, contains a complete residue system modulo m if and only if m is one of the following: 2, 4, 6, 7, 14, 3^k, k >=1.", "background": "Source: Open Problem Garden. Original node ID: 37402. URL: http://www.openproblemgarden.org/op/lucas_numbers_modulo_m.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/lucas_numbers_modulo_m\n- Subject(s): Number Theory\n- Keywords: Lucas numbers\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: March 19th, 2012 by Martin Erickson\n\nProblem-page discussion:\nThe Lucas numbers are defined by L(0)=2, L(1)=1, and L(n)=L(n-1)+L(n-2), for n >=2. Thus the sequence is 2, 1, 3, 4, 7, 11, 18, 29, 47,....\n\nExample: If m = 5, then we have the sequence 2, 1, 3, 4, 2, 1,..., and since the sequence repeats we never obtain 0 mod 5.\n\nExample: If m = 6, then we have 2, 1, 3, 4, 1, 5, 0,..., and we obtain a complete residue system mod 6.\n\nThe corresponding problem for the Fibonacci sequence was solved by S. A. Burr. The sequence {F(n) mod m} contains a complete residue system mod m if and only if m is one of the following: 5^k, 2.5^k, 4.5^k, 3^j.5^k, 6.5^k, 7.5^k, 14.5^k.\n\nBibliography:\nS. A. Burr, \"On Moduli for Which the Fibonacci Sequence Contains a Complete System of Residue\", Fibonacci Quarterly, December 1971, pp. 497-504.\n\nComments:\n- July 1st, 2014 | Anonymous | Solved: This problem has been solved: http://mathacadabra.com/Items2013/LucasCompleteResidue.aspx\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Lucas Numbers Modulo m\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3359, "problem_number": "OPG-37404", "title": "Sum of prime and semiprime conjecture", "statement": "Conjecture Every even number greater than $10$ can be represented as the sum of an odd prime number and an odd semiprime.", "background": "Source: Open Problem Garden. Original node ID: 37404. URL: http://www.openproblemgarden.org/op/sum_of_prime_and_semiprime_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/sum_of_prime_and_semiprime_conjecture\n- Author(s): Geoffrey Marnell\n- Subject(s): Number Theory\n- Keywords: prime; semiprime\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 23rd, 2012 by princeps\n\nBibliography:\n*[M] Geoffrey R. Marnell, \"Ten Prime Conjectures\", Journal of Recreational Mathematics 33:3 (2004-2005), pp. 193--196.\n\nRelated:\nRelated problems\nGoldbach conjecture\n\nSource links:\n- semiprime: http://en.wikipedia.org/wiki/semiprime\n\nComments:\n- July 1st, 2012 | Anonymous | surely that's Chen's theorem: Every sufficiently large even number is the sum of either 2 primes or a prime and a semiprime.\n- July 31st, 2016 | Charles R Great... | Yes, apart from the: Yes, apart from the \"sufficiently large\" and allowing prime + prime as well as prime + semiprime. The parity problem makes the latter hard, but some progress has been made, see arXiv:math/0609615 and arXiv:0803.2636. (The key progress is their use of E2 = semiprimes rather than P2 = primes or semiprimes.)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Sum of prime and semiprime conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3360, "problem_number": "OPG-37411", "title": "Giuga's Conjecture on Primality", "statement": "Conjecture $p$ is a prime iff $~\\displaystyle \\sum_{i=1}^{p-1} i^{p-1} \\equiv -1 \\pmod p$", "background": "Source: Open Problem Garden. Original node ID: 37411. URL: http://www.openproblemgarden.org/op/giugas_conjecture_on_primality.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/giugas_conjecture_on_primality\n- Author(s): Giuseppe Giuga\n- Subject(s): Number Theory\n- Keywords: primality\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 27th, 2012 by princeps\n\nBibliography:\n[BBBG] Borwein, D.; Borwein, J. M., Borwein, P. B., and Girgensohn, R. \"Giuga's Conjecture on Primality\", American Mathematical Monthly, 103, 40–50, (1996)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Giuga's Conjecture on Primality\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3361, "problem_number": "OPG-37413", "title": "Alexa's Conjecture on Primality", "statement": "Definition Let $r_i$ be the unique integer (with respect to a fixed $p\\in\\mathbb{N}$ ) such that\n\n$$(2i+1)^{p-1} \\equiv r_i \\pmod p ~~\\text{ and } ~ 0 \\le r_i < p.$$\n\nConjecture A natural number $p \\ge 8$ is a prime iff $$\\displaystyle \\sum_{i=1}^{\\left \\lfloor \\frac{\\sqrt[3]p}{2} \\right \\rfloor} r_i = \\left \\lfloor \\frac{\\sqrt[3]p}{2} \\right \\rfloor$$", "background": "Source: Open Problem Garden. Original node ID: 37413. URL: http://www.openproblemgarden.org/op/alexas_conjecture_on_primality.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/alexas_conjecture_on_primality\n- Author(s): Alexa\n- Subject(s): Number Theory\n- Keywords: primality\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 28th, 2012 by princeps\n\nProblem-page discussion:\nThe conjecture is obviously true when $p$ is prime, so it suffices to check when $p$ is composite.\n\nRelated:\nRelated problems\nGiuga's Conjecture on Primality\n\nComments:\n- March 28th, 2012 | Anonymous | counter-example for p=66: formula works for p from 8 through 100, except for p=66.\n- March 30th, 2012 | princeps | counter-example for p=66: Thanks, I have corrected statement.\n- March 30th, 2012 | Anonymous | don't work either: your new statement is ambiguous (which r_i should one choose inside the sum?). I'm assuming you're just trying to move the \"mod p\" to apply to the sum only (and not to the RHS). If that's what you're doing, it still doesn't work. Same counter-examples at p=66, 102, 246 and 492 for p from 8 to 500.\n- March 30th, 2012 | princeps | donit work either: Yes it works.I have checked statement up to 10^6,there is no counterexample...\n- March 30th, 2012 | Anonymous | still ambiguous: then please re-word your conjecture, because as it stands, it's ambiguous and not true. It's ambiguous, because the way you defined r_i, one could have chosen r_i, r_i + p, r_i + 2p etc., but when you plug these into the sum, you get a different sum and the equality doesn't make sense.\n- April 2nd, 2012 | Anonymous | still not good: your modified version now reduces back to putting mod p on the LHS of the equation, which as I've pointed out above, doesn't work (see counter-examples I gave). Where did you come up with this conjecture? Is there any published reference for it?\n- June 14th, 2013 | Charles R Great... | Fixed: I've corrected the statement on Alexa's behalf. This version holds up to at least 100 million.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Alexa's Conjecture on Primality\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3362, "problem_number": "OPG-37423", "title": "Birch & Swinnerton-Dyer conjecture", "statement": "Conjecture Let $E/K$ be an elliptic curve over a number field $K$. Then the order of the zeros of its $L$-function, $L(E, s)$, at $s = 1$ is the Mordell-Weil rank of $E(K)$.", "background": "Source: Open Problem Garden. Original node ID: 37423. URL: http://www.openproblemgarden.org/op/birch_swinnerton_dyer_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/birch_swinnerton_dyer_conjecture\n- Subject(s): Number Theory\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 12th, 2012 by eyoong\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Birch & Swinnerton-Dyer conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3363, "problem_number": "OPG-367", "title": "The Erdos-Turan conjecture on additive bases", "statement": "Let $B \\subseteq {\\mathbb N}$. The representation function $r_B: {\\mathbb N} \\rightarrow {\\mathbb N}$ for $B$ is given by the rule $r_B(k) = \\#\\{ (i,j) \\in B \\times B: i + j = k \\}$. We call $B$ an additive basis if $r_B$ is never $0$.\n\nConjecture If $B$ is an additive basis, then $r_B$ is unbounded.", "background": "Source: Open Problem Garden. Original node ID: 367. URL: http://www.openproblemgarden.org/op/the_erdos_turan_conjecture_on_additive_bases.\n\nSource subject path: Number Theory > Additive Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_erdos_turan_conjecture_on_additive_bases\n- Author(s): Erdos, Paul; Turan, Paul\n- Subject(s): Number Theory; Additive Number Theory\n- Keywords: additive basis; representation function\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 8th, 2007 by mdevos\n\nProblem-page discussion:\nThis famous conjecture seems intuitively likely, but to date, there has been relatively little progress on it despite considerable attention. Two positive results are a theorem of Dirac [D] which shows that $r_B$ cannot be constant from some point on, and a theorem of Borwein, Choi, and Chu [BCC] which shows that $r_B$ cannot be bounded above by $6$.\n\nOn the other hand, if we consider the related problem for subsets of integers instead of natural numbers, Nathanson [N] has shown that the conjecture does not hold.\n\nBibliography:\n[BCC] P. Borwein, S. Choi, and F. Chu, An old conjecture of Erdos-Turan on additive bases, Mathematics of Computation. Volume 75, Number 253, Pages 475–484.\n\n[D] G. A. Dirac, Note on a problem in additive number theory, J. London Math. Soc. 26 (1951), 312–313.\n\n[EG] P. Erdos and R. L. Graham, Old and new problems and results in combinatorial number theory: van der Waerden’s theorem and related topics, Enseign. Math. (2) 25 (1979), no. 3-4, 325–344 (1980). MathSciNet\n\n*[ET] P. Erdos and P. Turan, On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc. 16 (1941), 212–215. MathSciNet\n\n[N] Melvyn B. Nathanson, Unique representation bases for the integers, Acta Arith. 108 (2003), no. 1, 1–8. MathSciNet\n\nBibliography links:\n- An old conjecture of Erdos-Turan on additive bases: http://www.ams.org/mcom/2006-75-253/S0025-5718-05-01777-1/S0025-5718-05-01777-1.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0570317\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0006197\n- Unique representation bases for the integers: http://front.math.ucdavis.edu/math.NT/0302091\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1971077\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"The Erdos-Turan conjecture on additive bases\" in Number Theory; Additive Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3364, "problem_number": "OPG-706", "title": "Goldbach conjecture", "statement": "Conjecture Every even integer greater than 2 is the sum of two primes.", "background": "Source: Open Problem Garden. Original node ID: 706. URL: http://www.openproblemgarden.org/op/goldbach_conjecture.\n\nSource subject path: Number Theory > Additive Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/goldbach_conjecture\n- Author(s): Goldbach, Christian\n- Subject(s): Number Theory; Additive Number Theory\n- Keywords: additive basis; prime\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 20th, 2007 by Benschop\n\nProblem-page discussion:\nThis famous conjecture is one of the oldest unsolved problems in mathematics. It arose originally out of a correspondence (1742) between Goldbach and Euler. See Wikipedia's Goldbach's conjecture for more.\n\nDiscussion links:\n- Wikipedia's Goldbach's conjecture: http://en.wikipedia.org/wiki/goldbach's conjecture\n\nComments:\n- June 14th, 2013 | Charles R Great... | Weak conjecture now solved: Note that Harald Helfgott has proved the ternary version of Goldbach's conjecture, that every odd number greater than 7 is the sum of three odd primes. See http://arxiv.org/abs/1205.5252 (minor arc estimates) and http://arxiv.org/abs/1305.2897 (major arc estimates).\n- November 22nd, 2007 | Benschop | Goldbach conjecture: Those interested in a suggestion for a proof via semigroup theory and carry extension, see: \"Additive structure of Z(.) mod [\\prod first k primes], with carry extension to prime pair sums\". - http://home.iae.nl/users/benschop/ngb0203.pdf (10 pgs, submitted for publication)\n- December 12th, 2007 | Anonymous | Goldbach conjecture: In addition to the above link to the paper with Goldbach-Conj. proof, the abstract and links to an intro-article are at http://home.iae.nl/users/benschop/ng-abstr.htm\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Goldbach conjecture\" in Number Theory; Additive Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3365, "problem_number": "OPG-37192", "title": "Are there an infinite number of lucky primes?", "statement": "Conjecture If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc., an infinite number of the remaining integers are prime.", "background": "Source: Open Problem Garden. Original node ID: 37192. URL: http://www.openproblemgarden.org/op/are_there_an_infinite_number_of_lucky_primes.\n\nSource subject path: Number Theory > Additive Number Theory.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_there_an_infinite_number_of_lucky_primes\n- Author(s): Lazarus: Gardiner: Metropolis; Ulam, Stanislaw M.\n- Subject(s): Number Theory; Additive Number Theory\n- Keywords: lucky; prime; seive\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: March 24th, 2010 by cubola zaruka\n\nProblem-page discussion:\nThe difference between the seive for generating primes and the seive for gnerating lucky numbers is that the former removes every nth integer whereas the latter removes every nth remaining integer. There are known to be an infinite number of lucky numbers because removing every nth remaining number leaves numbers unremoved as n increases and eventually becomes larger than any given value. There is also the unproved lucky analogue of the Goldbach conjecture, that every even number is expressible as the sum of two lucky numbers.\n\nBibliography:\nhttp://en.wikipedia.org/wiki/Lucky_number\n\nRelated:\nRelated problems\nGoldbach conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Are there an infinite number of lucky primes?\" in Number Theory; Additive Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3366, "problem_number": "OPG-573", "title": "The Riemann Hypothesis", "statement": "The zeroes of the Riemann zeta function that are inside the Critical Strip (i.e. the vertical strip of the complex plane where the real part of the complex variable is in ]0;1[), are actually located on the Critical line ( the vertical line of the complex plane with real part equal to 1/2)", "background": "Source: Open Problem Garden. Original node ID: 573. URL: http://www.openproblemgarden.org/op/the_riemann_hypothesis.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_riemann_hypothesis\n- Author(s): Riemann, Bernhard\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: Millenium Problems; zeta\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 13th, 2007 by eric\n\nProblem-page discussion:\nThe Riemann zeta serie is the function of the complex variable $s$ defined by $\\zeta(s) = \\sum_{n=1}^\\infty \\frac{1}{n^s}$. It is defined only for a real part of $s$ greater than 1. It is an analytic function on this domain, and there exists a unique analytic function defined over the whole complex plane (except at 1) that coincides with zeta when $Re(s)>1$. This function is the analytic continuation of the Riemann zeta serie, and is called the Riemann zeta function, on which is based the Riemann Hypothesis. It was stated by Bernhard Riemann in 1859 and is still open. The zeta function and the Riemann Hypothesis are closely related to number theory and the distribution of prime numbers, as is well described in wikipedia. For that reason this item could also lie under the \"Number Theory\" category of this website.\n\nA lot of variants and extensions of the Riemann hypothesis have been raised till today. The location on the Critical line of the so-called \"non trivial zeroes\" of zeta (the ones in the Critical Strip, by opposition to the trivial ones that are negative even integers and are well known) is supposed to be also valid for the analytic continuation of Dirichlet L-series associated to a primitive Dirichlet character $\\chi$: $L(s) = \\sum_{n=1}^\\infty \\frac{\\chi(n)}{n^s}$. It is also believed to be valid for Dedekind zeta functions (generalization of zeta related to number fields, that is a finite dimensional extension of the field of rational numbers), also for Hecke L-functions associated to Hecke Grossencharacters (generalization to number fields of the Dirichlet L-functions), for Artin L-functions etc... The list of various generalizations is now long. Today, the largest class of functions that are expected to obey a Riemann Hypothesis are functions in the Selberg Class, even though zeta functions for motives over schemes are also candidates.\n\nThere exists a few variants of the Riemann Hypothesis for which the hypothesis is now solved: For the zeta functions of elliptic curves over finite fields, the problem was solved by André Weil (1950). For zeta functions associated to local fields it has been proved by Daniel Bump, Eugene Ng, Jeffrey Vaaler, Stephen Choi, Par Kurlberg [B] in the real Case (= the Mellin transform of the hermite functions behave like zeta), by Par Kurlberg in the non-archimedean case with odd residue characteristics and recently par Oloffson (2006) in the complex case even though it was previously believed it was wrong in that case. It is remarkable that the local (archimedean) results also apply to the Mellin transform of the laguerre functions, thanks to the properties of the second order differential equation fulfilled by the Laguerre functions. It supports (if necessary) the link between this problem and Harmonic Analysis (see also publications by Davidson, Olafson, Faraut [F] etc on representations of conformal groups underlying Jordan algebras on bounded symmetric domains). Polya [P] succeeded around 1926 to proove that some approximations of zeta do actually have their zeroes on the Critical Line, However his results cannot be directly generalized to zeta itself (see [T]). But there are actually a lot of ways to explore the Riemann Hypothesis, from pure Number Theory to Random Matrices or Non-Commutative Geometry, which makes this problem one of the most difficult mathematical problem today.\n\nThe zeta function is also amazing, since it is the first explicit function discovered to be \"Universal\" (in the sense that any analytic function that does not vanish in a small disk can be uniformly approximated by zeta up to a suitable translation of zeta in the complex plane). This result was proved in 1975 by Voronin, and Karatsuba generalized the result to a finite set of L-functions approximating a finite set of analytic functions. Before this discovery, the existence of a universal function was proven in the 50's by a construction requiring the use of the axiom of choice. This result has an impact in the context of the Riemann hypothesis, because it shows that any linear combination of some Dirichlet L-functions (with non-vanishing coefficients) do not follow the Riemann Hypothesis (such a linear combination actually vanishes infinitely many times in any vertical strip inside the Critical Strip). This result is even more important when specializing to a special kind of linear combinations of Dirichlet L-functions, the prototype of which is the Davenport Heilbronn L-function. This specific linear combination share a symmetry property with individual Dirichlet L-functions (a symmetry by the change of variable $s \\mapsto 1-s$ ) which is expressed by the so-called \"functional equation\" fulfilled by zeta as well as Dirichlet L-functions, Hecke L-functions etc.. This functional equation is very important in this problem since in the specific case of zeta it exactly characterizes the zeta function, and with additional conditions it characterizes also Dirichlet L-functions (these are the so-called \"Converse theorems\", the first of which was proven by Hans Hamburger [H]). The example of the Davenport Heilbronn L-functions shows that the exact form of the functional equation is essential in the context of the Riemann Hypothesis (the functional equation of the Davenport Heilbronn L-function is almost the same as the one of a single Dirichlet L-function but there is a slight difference). However, the functional equation $\\xi(1-s)=\\xi(s)$ where $\\xi(s) = s(1-s)\\pi^{s/2}\\Gamma(s/2)\\zeta(s)$ (and similarly for L-functions), is far from beeing sufficient to prove that the non trivial zeroes are located on the Critical Line, and it is widely agreed that the missing information will require to imagine new mathematical concepts.\n\nSee also the article of Peter Sarnak on the Clay Institute website, as well as the link about the Millenium Prize.\n\nE.C.\n\nBibliography:\n[R] Bernhard Riemann, \"Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse\", (1859) Monatsberichte der Berliner Akademie.\n\n[T] E.C. Titchmarsh, \"The theory of the Riemann zeta function\", Oxford. Univ Press\n\n[P] G. Polya, \"Bemerkung ueber die Integraldarstellung der Riemannsche zeta-Funktion\", Acta Math. 48 (1926), 305-317.\n\n[B] D. Bump, K.K. Choi, P. Kurlberg, J. Vaaler, \"A Local Riemann Hypothesis\", Math. Zeit. 233 (2000) p1-19.\n\n[H] H. Hamburger, \"Ueber die Riemannsche Funktionalgleichung der zeta-Funktion\", Math. Zeit. 10 (1921), 240-254.\n\n[V] A. Karatsuba, Voronin S., \"The Riemann Zeta function\", De Gruyter Exposition of Mathematics, Transl. Neil Koeblitz (1975) p212.\n\n[F] J. Faraut, A. Koranyi, \"Function spaces and reproducing kernels on bounded symmetric domains\", J. Funct. An. 88 (1990) p64-89.\n\nDiscussion links:\n- wikipedia: http://en.wikipedia.org/wiki/Riemann_hypothesis\n- Selberg Class: http://en.wikipedia.org/wiki/Selberg_class\n- Peter Sarnak: http://www.claymath.org/news/sarnak.php\n- Millenium Prize: http://www.claymath.org/millennium/Riemann_Hypothesis/\n\nComments:\n- February 16th, 2011 | Comet | Proposed (dis)proof of RH: A collection of (dis)proofs of the Riemann Hypothesis can be found at http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/RHproofs.htm. Many of them have been refuted, but some of them show approaches that do not seem to have been answered. There reader interested in RH may find the aforementioned page and its pointers to papers quite interesting.\n\nP.S. My former e-mail address comet@bayvax.decus.org is obsolete.\n- July 10th, 2008 | eric | Post initialy placed in the Analysis Category: Being the author of this post, please note that this item was initialy placed under the Analysis category on this website, because as for me in its very first statement it falls under this category. The editor of the site decided to move it under \"Number Theory\" because a lot of mathematicians think that the problem is intrinsincally related to number theory. Even though they represent the majority, and numerous links between this problem and Number Theory are well established, this is not my opinion. There are lot of ways to approach this problem, sometimes completely unrelated to number theory, and if I were to locate this entry into a mathematical domain that is best suited for a direct proof of the Riemann Hypothesis, I would have located it under Group Theory\\Representation Theory. But it is only the opinion of an amateur in mathematics...\n\nRgds, eric\n- December 27th, 2012 | zeraoulia | A positive answer to the Riemann hypothesis: A new result predic: Dear Prof,\n\nI am Prof. Zeraoulia Elhadj from the university of Tébessa, Algeria. Please see this link http://vixra.org/pdf/1210.0176v7.pdf http://arxiv.org/pdf/1210.1517v10.pdf\n\nfor a Solution of the Riemann Hypothesis: A positive answer to the Riemann hypothesis: A new result predicting the location of zeros. I think that this is a fine solution. Please let me know about your opinion on it. I think that your opinion is the final decison to accept or reject this solution. Any furhter comments are welcome. With kind regards. Elhadj\n- January 27th, 2013 | eric | This supposed proof is incorrect: Dear Zeraoulia Elhadj,\n\nThe proof detailed in http://arxiv.org/pdf/1210.1517v10.pdf is wrong since eq. 9 (alpha=1/2) cannot be deduced at all from eq. (8), hence invalidating the whole proof.\n\nTo make a more general comment, I don't think this section should be used to propose attempts of a proof, but only a fully validated proof if there's one some day (and I hope so). There are indeed numerous invalid proofs every year for this mathematical problem, and regular forums are much more adapted to discuss on these attempts. This site is much more a collection of open mathematical problems with their current status and would be rapidly obfuscated by long standing discussions about various attempts of proof on each problem... Of course I'm not the webmaster of the site and he may confirm or contradict this personal opinion\n\nRegards, Eric Chopin\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"The Riemann Hypothesis\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3367, "problem_number": "OPG-1788", "title": "Schanuel's Conjecture", "statement": "Conjecture Given any $n$ complex numbers $z_1,...,z_n$ which are linearly independent over the rational numbers $\\mathbb{Q}$, then the extension field $\\mathbb{Q}(z_1,...,z_n,\\exp(z_1),...,\\exp(z_n))$ has transcendence degree of at least $n$ over $\\mathbb{Q}$.", "background": "Source: Open Problem Garden. Original node ID: 1788. URL: http://www.openproblemgarden.org/op/schanuels_conjecture.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/schanuels_conjecture\n- Author(s): Schanuel, Stephen\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: algebraic independence\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 8th, 2008 by Charles\n\nProblem-page discussion:\nSchanuel's Conjecture implies the algebraic independence of $\\pi$ and $e$, as well as a positive solution to Tarski's exponential function problem.\n\nRelated:\nRelated problems\nAlgebraic independence of pi and e\nTarski's exponential function problem\n\nDiscussion links:\n- Schanuel's Conjecture: http://en.wikipedia.org/wiki/Schanuel's Conjecture\n- Tarski's exponential function problem: http://www.openproblemgarden.org/?q=node/1790\n\nComments:\n- January 18th, 2010 | Anonymous | I must agree with the: I must agree with the previous comment. Schanuel's conjecture is likely the most important open problem in Transcendental Number Theory. I realize that this might not be as major a field as the study of \"mimic\" numbers, but.....\n- January 19th, 2010 | Robert Samal | Re: I must agree with the: Encouraged by the previous comments, I changed the rating of this problem and the \"mimic\" one. Thanks for the feedback.\n- December 27th, 2009 | Anonymous | from Gasses: I am just curious why 'importance' is given as 2 stars when (according to wikipedia) \"The conjecture, if proven, would subsume most known results in transcendental number theory.\" Some of these results include results on this page that have greater importance than 2 stars.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Schanuel's Conjecture\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3368, "problem_number": "OPG-36961", "title": "Distribution and upper bound of mimic numbers", "statement": "Problem\n\nLet the notation $a|b$ denote \" $a$ divides $b$ \". The mimic function in number theory is defined as follows [1].\n\nDefinition For any positive integer $\\mathcal{N} = \\sum_{i=0}^{n}\\mathcal{X}_{i}\\mathcal{M}^{i}$ divisible by $\\mathcal{D}$, the mimic function, $f(\\mathcal{D} | \\mathcal{N})$, is given by,\n\n$$f(\\mathcal{D} | \\mathcal{N}) = \\sum_{i=0}^{n}\\mathcal{X}_{i}(\\mathcal{M}-\\mathcal{D})^{i}$$\n\nBy using this definition of mimic function, the mimic number of any non-prime integer is defined as follows [1].\n\nDefinition The number $m$ is defined to be the mimic number of any positive integer $\\mathcal{N} = \\sum_{i=0}^{n}\\mathcal{X}_{i}\\mathcal{M}^{i}$, with respect to $\\mathcal{D}$, for the minimum value of which $f^{m}(\\mathcal{D} | \\mathcal{N}) = \\mathcal{D}$.\n\nGiven these two definitions and a positive integer $\\mathcal{D}$, find the distribution of mimic numbers of those numbers divisible by $\\mathcal{D}$.\n\nAgain, find whether there is an upper bound of mimic numbers for a set of numbers divisible by any fixed positive integer $\\mathcal{D}$.", "background": "Source: Open Problem Garden. Original node ID: 36961. URL: http://www.openproblemgarden.org/op/distribution_and_upper_bound_of_mimic_numbers.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/distribution_and_upper_bound_of_mimic_numbers\n- Author(s): Bhattacharyya, M.\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: Divisibility; mimic function; mimic number\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: June 20th, 2009 by facility_cttb@i...\n\nBibliography:\n*[1] Malay Bhattacharyya, Sanghamitra Bandyopadhyay and U Maulik, Non-primes are recursively divisible, Acta Universitatis Apulensis 19 (2009).\n\nBibliography links:\n- Non-primes are recursively divisible: http://www.emis.de/journals/AUA\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Distribution and upper bound of mimic numbers\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3369, "problem_number": "OPG-37255", "title": "Lindelöf hypothesis", "statement": "Conjecture For any $\\epsilon>0$ $$\\zeta\\left(\\frac12 + it\\right) \\mbox{ is }\\mathcal{O}(t^\\epsilon).$$\n\nSince $\\epsilon$ can be replaced by a smaller value, we can also write the conjecture as, for any positive $\\epsilon$, $$\\zeta\\left(\\frac12 + it\\right) \\mbox{ is }o(t^\\varepsilon).$$", "background": "Source: Open Problem Garden. Original node ID: 37255. URL: http://www.openproblemgarden.org/op/lindelof_hypothesis.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/lindelof_hypothesis\n- Author(s): Lindelöf, Ernst\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: Riemann Hypothesis; zeta\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 20th, 2010 by porton\n\nProblem-page discussion:\nLindelof hypothesis in Wikipedia.\n\nAccordingly Wikipedia this hypothesis is implied by Riemann hypothesis.\n\nRelated:\nRelated problems\nThe Riemann Hypothesis\n\nDiscussion links:\n- Lindelof hypothesis: http://en.wikipedia.org/wiki/Lindelof hypothesis\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Lindelöf hypothesis\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3370, "problem_number": "OPG-37329", "title": "Euler-Mascheroni constant", "statement": "Question Is Euler-Mascheroni constant an transcendental number?", "background": "Source: Open Problem Garden. Original node ID: 37329. URL: http://www.openproblemgarden.org/op/euler_mascheroni_constant.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/euler_mascheroni_constant\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: constant; Euler; irrational; Mascheroni; rational; transcendental\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 19th, 2011 by Juggernaut\n\nProblem-page discussion:\nLet $\\gamma:=\\lim_{n\\rightarrow\\infty}\\left(\\sum_{k=1}^n\\left(\\frac{1}{k}\\right)-\\ln(n)\\right)$. The number $\\gamma$ has not been proved algebraic or transcendental. In fact, it is not even known whether $\\gamma$ is irrational.\n\nComments:\n- January 21st, 2013 | Anonymous | Euler-Masqueroni contant: If n = 2^k then log(n) is irrational. But Sum{1/k} is ever rational. Then Sum{1/k} - Log(n) is ever irrational.. Ludovicus\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Euler-Mascheroni constant\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3371, "problem_number": "OPG-37366", "title": "Is Skewes' number e^e^e^79 an integer?", "statement": "Conjecture\n\nSkewes' number $e^{e^{e^{79}}}$ is not an integer.", "background": "Source: Open Problem Garden. Original node ID: 37366. URL: http://www.openproblemgarden.org/op/is_skewes_number_e_e_e_79_an_integer.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/is_skewes_number_e_e_e_79_an_integer\n- Subject(s): Number Theory; Analytic Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: December 13th, 2011 by VladimirReshetnikov\n\nBibliography:\nSkewes, S. (1933), \"On the difference $\\pi(x) − Li(x)$ \", Journal of the London Mathematical Society 8: 277–283\n\nRelated:\nRelated problems\nSchanuel's Conjecture\n\nComments:\n- April 29th, 2012 | warut | Schanuel's conjecture: Assuming Schanuel's conjecture, one can show that e^e^e^79 is transcendental.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Is Skewes' number e^e^e^79 an integer?\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3372, "problem_number": "OPG-55810", "title": "Are all Fermat Numbers square-free?", "statement": "Conjecture Are all Fermat Numbers\n$$\nF_n = 2^{2^{n } } + 1\n$$\n Square-Free?", "background": "Source: Open Problem Garden. Original node ID: 55810. URL: http://www.openproblemgarden.org/op/are_all_fermat_numbers_square_free.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_all_fermat_numbers_square_free\n- Subject(s): Number Theory; Analytic Number Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: August 21st, 2013 by kurtulmehtap\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Are all Fermat Numbers square-free?\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3373, "problem_number": "OPG-55812", "title": "Are there only finite Fermat Primes?", "statement": "Conjecture A Fermat prime is a Fermat number\n$$\nF_n = 2^{2^n } + 1\n$$\n that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257,F_4=65537 It is unknown if other fermat primes exist.", "background": "Source: Open Problem Garden. Original node ID: 55812. URL: http://www.openproblemgarden.org/op/are_only_finite_fermat_primes.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_only_finite_fermat_primes\n- Subject(s): Number Theory; Analytic Number Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: August 21st, 2013 by kurtulmehtap\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Are there only finite Fermat Primes?\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3374, "problem_number": "OPG-59976", "title": "Are all Mersenne Numbers with prime exponent square-free?", "statement": "Conjecture Are all Mersenne Numbers with prime exponent ${2^p-1}$ Square free?", "background": "Source: Open Problem Garden. Original node ID: 59976. URL: http://www.openproblemgarden.org/op/are_all_mersenne_numbers_with_prime_exponent_square_free.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_all_mersenne_numbers_with_prime_exponent_square_free\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: Mersenne number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 4th, 2015 by kurtulmehtap\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Are all Mersenne Numbers with prime exponent square-free?\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3375, "problem_number": "OPG-59977", "title": "Are there infinite number of Mersenne Primes?", "statement": "Conjecture A Mersenne prime is a Mersenne number\n$$\nM_n = 2^p - 1\n$$\n that is prime.\n\nAre there infinite number of Mersenne Primes?", "background": "Source: Open Problem Garden. Original node ID: 59977. URL: http://www.openproblemgarden.org/op/are_there_infinite_number_of_mersenne_primes.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_there_infinite_number_of_mersenne_primes\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: Mersenne number; Mersenne prime\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 4th, 2015 by kurtulmehtap\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Are there infinite number of Mersenne Primes?\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3376, "problem_number": "OPG-155", "title": "Olson's Conjecture", "statement": "Conjecture If $a_1,a_2,\\ldots,a_{2n-1}$ is a sequence of elements from a multiplicative group of order $n$, then there exist $1 \\le j_1 < j_2 \\ldots < j_n \\le 2n-1$ so that $\\prod_{i=1}^n a_{j_i} = 1$.", "background": "Source: Open Problem Garden. Original node ID: 155. URL: http://www.openproblemgarden.org/op/olsons_conjecture.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/olsons_conjecture\n- Author(s): Olson, John E.\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: zero sum\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 10th, 2007 by mdevos\n\nProblem-page discussion:\nA famous theorem of Erdos, Ginzburg, and Ziv asserts that every sequence of $2n-1$ elements from an additive abelian group has a subsequence of length $n$ which sums to $0$. This pretty result has lead to numerous generalizations. In particular, Olsen generalized this result by showing that every sequence of $2n-1$ elements from an arbitrary multiplicative group of order $n$ has a subsequence of length $n$ which has product equal to $1$ in some order. The above conjecture asserts that this reordering is not needed. Apart from Olson's result, there appears to be very little known about this problem. Next we highlight an obvious question which appears untouched.\n\nFor every finite multiplicative group $G$, let $z(G)$ denote the smallest integer $m$ so that every sequence of $m$ elements of $G$ has a subsequence of length $|G|$ with product equal to $1$ in the given order (so Olsen's conjecture is equivalent to $z(G) \\le 2|G|-1$ ). It is clear that $z(G) \\le |G|(|G|-1) + 1$, since any sequence of length $> |G|(|G|-1)$ must contain at least $|G|$ copies of the same element, and the product of these will be $1$. However, I (M. DeVos) don't know how to improve significantly on this upper bound, and it would appear to me that any significant progress in this direction would require a little something new.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Olson's Conjecture\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3377, "problem_number": "OPG-156", "title": "Few subsequence sums in Z_n x Z_n", "statement": "Conjecture For every $0 \\le t \\le n-1$, the sequence in ${\\mathbb Z}_n^2$ consisting of $n-1$ copes of $(1,0)$ and $t$ copies of $(0,1)$ has the fewest number of distinct subsequence sums over all zero-free sequences from ${\\mathbb Z}_n^2$ of length $n-1+t$.", "background": "Source: Open Problem Garden. Original node ID: 156. URL: http://www.openproblemgarden.org/op/limiting_subsequence_sums_in_z_n_x_z_n.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/limiting_subsequence_sums_in_z_n_x_z_n\n- Author(s): Bollobas, Bela; Leader, Imre\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: subsequence sum; zero sum\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 10th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: Given a sequence $\\bf a$ of elements from an additive abelian group, we call a subsequence sum any group element expressable as a sum of some nontrivial subsequence of $\\bf a$. We say that $\\bf a$ is zero-free if $0$ is not a subsequence sum.\n\nIt is easy to see that every sequence $a_1,\\ldots,a_n$ of elements from ${\\mathbb Z}_n$ has a nontrivial subsequence which sums to zero (actually this holds for every group of order $n$ ). Just consider the elements $a_1$, $a_1 + a_2$, $\\ldots$, $a_1 + \\ldots, a_n$. If these elements are distinct, we have a zero sum. Otherwise, we have $a_1 + \\ldots + a_j = a_1 + \\ldots + a_k$ for some $1 \\le j < k \\le n$, but then $a_{j+1} + a_{j+2} + \\ldots a_k = 0$. The same argument shows that whenever $0 \\le t \\le n-1$, every zero-free sequence of $t$ elements of ${\\mathbb Z}_n$ must have at least $t$ distinct subsequence sums. In other words, the sequence consisting of $t$ copies of $1$ has the fewest number of distinct subsequence sums over all zero-free sequences in ${\\mathbb Z}_n$ of length $t$.\n\nIn the group ${\\mathbb Z}_n^2$, a theorem of Olsen shows that every sequence of length $\\ge 2n-1$ has a nontrivial subsequence which sums to zero. However, we do not know what the minimum number of distinct subsequence sums is for a zero-free sequence of a given length. The above conjecture would appear to be the natural optimum.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"Few subsequence sums in Z_n x Z_n\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3378, "problem_number": "OPG-337", "title": "Gao's theorem for nonabelian groups", "statement": "For every finite multiplicative group $G$, let $s(G)$ ( $s'(G)$ ) denote the smallest integer $m$ so that every sequence of $m$ elements of $G$ has a subsequence of length $>0$ (length $|G|$ ) which has product equal to 1 in some order.\n\nConjecture $s'(G) = s(G) + |G| - 1$ for every finite group $G$.", "background": "Source: Open Problem Garden. Original node ID: 337. URL: http://www.openproblemgarden.org/op/gaos_theorem_for_nonabelian_groups.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/gaos_theorem_for_nonabelian_groups\n- Author(s): DeVos, Matt\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: subsequence sum; zero sum\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 23rd, 2007 by mdevos\n\nProblem-page discussion:\nA beautiful theorem of Gao (previously conjectured by Caro) shows that the above property holds for all abelian groups. Rather surprisingly, almost all of the proof for the abelian case seems to work as well for the general case - only one rather innocent looking bit does not carry through. Next we explore this curiosity in detail, beginning with an easy observation.\n\nObservation $s'(G) \\ge s(G) + |G| - 1$ for every (finite) group $G$.\n\nTo see this, choose a sequence of length $s(G) - 1$ of elements which has no nontrivial subsequence with product equal to 1 in any order. Now, append $|G| - 1$ copies of 1 to this sequence. The new sequence has length $s(G) + |G| - 2$ and has no subsequence of length $|G|$ with product 1 in any order.\n\nSo, the hard part of Gao's theorem is to prove $s'(G) \\le s(G) + |G| - 1$, and we now have multiple proofs of this fact. One of the nicest arguments uses a theorem of Kempermann-Scherck, and can be split into the following two parts.\n\nLemma Let $m = s(G) + |G| - 1$ and let ${\\bf a} = (a_1,\\ldots,a_m)$ be a sequence in an arbitrary finite multiplicative $G$ with the added property that 1 is the most frequently occurring in ${\\bf a}$. Then there is a subsequence of ${\\bf a}$ of length $|G|$ which has product equal to 1 in some order.\n\nObservation If ${\\bf a}$ is a sequence of elements in the finite abelian group $G$ and $g \\in G$, then replacing each element $a_i$ of ${\\bf a}$ by $ga_i$ has no effect on the products of length $|G|$ subsequences of ${\\bf a}$.\n\nThe lemma and observation now combine easily to show $s'(G) \\le s(G) + |G| - 1$ in abelian groups, since we may take any sequence ${\\bf a}$ of length $s(G) + |G| - 1$ and modify it by mutiplying each element by a fixed constant so that 1 is the most common element of ${\\bf a}$. The lemma shows that there is now a subsequence with product 1, and the observation shows that the corresponding subsequence has product 1 in the original. So, surprisingly, the Lemma - which includes all of the real difficutly - works just fine for general groups. The only place we required the assumption $G$ is abelian is for the observation.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 19.\n\nAttempt notes:\nTarget:\nMake progress on \"Gao's theorem for nonabelian groups\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3379, "problem_number": "OPG-414", "title": "Sets with distinct subset sums", "statement": "Say that a set $S \\subseteq {\\mathbb Z}$ has distinct subset sums if distinct subsets of $S$ have distinct sums.\n\nConjecture There exists a fixed constant $c$ so that $|S| \\le \\log_2(n) + c$ whenever $S \\subseteq \\{1,2,\\ldots,n\\}$ has distinct subset sums.", "background": "Source: Open Problem Garden. Original node ID: 414. URL: http://www.openproblemgarden.org/op/sets_with_distinct_subset_sums.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/sets_with_distinct_subset_sums\n- Author(s): Erdos, Paul\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: subset sum\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 24th, 2007 by mdevos\n\nProblem-page discussion:\nErdos valued this problem at $500, and I (M. DeVos) believe these prizes are now supported by Ron Graham.\n\nDefine the function $f: {\\mathbb N} \\rightarrow {\\mathbb N}$ by the rule\n$$\nf(n) = \\min \\{ \\max S: S \\subseteq {\\mathbb N} \\mbox{ has distinct subset sums and } |S| = n \\}\n$$\n\nThen Erdos' conjecture is equivalent to the assertion that $f(n) \\ge c 2^n$ for a fixed constant $c$, and more generally, we would like to understand the behavior of $f$.\n\nErdos and Moser established an upper bound on $f$, proving that $f(n) \\ge 2^n / 4 \\sqrt{n}$. This was later improved by a constant factor by Elkies [E].\n\nWe get an easy lower bound on $f$ by observing that the set $S$ consisting of the first $n$ powers of 2 has distinct subset sums, and has maximal element $2^{n-1}$. This shows that $f(n) \\le 2^{n-1}$. At first glance, it might appear that such sets are optimal, but these sets have too many small numbers, and it is possible to improve upon them. Conway and Guy [CG] found a construction of sets with distinct subset sum, now called the Conway-Guy sequence, which gives an interesting upper bound on $f$. This was this was later improved by Lunnan [L], and then by Bohman [B] to $f(n) \\le.22002 \\cdot 2^n$ (for $n$ sufficiently large).\n\nBibliography:\n[B] T. Bohman, A construction for sets of integers with distinct subset sums, The Electronic. Journal of Combinatorics 5 (1998) /#R3\n\n[CG] J. H. Conway and R. K. Guy, Sets of natural numbers with distinct subset sums, Notices, Amer. Math. Soc., 15 (1968) 345.\n\n[E] N. Elkies, An improved lower bound on the greatest element of a sum-distinct set of fixed order, J. Comb. Th. A, 41 (1986) 89-94.\n\n[G1] R. K. Guy, Sets of integers whose subsets have distinct sums, Ann. Discrete Math., 12 (1982) 141-154.\n\n[G2] R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1981.\n\n[L] W. F. Lunnon, Integers sets with distinct subset sums, Math. Compute, 50 (1988) 297-320.\n\nBibliography links:\n- A construction for sets of integers with distinct subset sums: http://www.combinatorics.org/Volume_5/PDF/v5i1r3.pdf\n\nComments:\n- April 7th, 2011 | Anonymous | Mistake: You are referring to the first upper bound as a lower bound, and the lower bound as an upper bound.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Sets with distinct subset sums\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3380, "problem_number": "OPG-432", "title": "The 3n+1 conjecture", "statement": "Conjecture Let $f(n) = 3n+1$ if $n$ is odd and $\\frac{n}{2}$ if $n$ is even. Let $f(1) = 1$. Assume we start with some number $n$ and repeatedly take the $f$ of the current number. Prove that no matter what the initial number is we eventually reach $1$.", "background": "Source: Open Problem Garden. Original node ID: 432. URL: http://www.openproblemgarden.org/op/strange_series.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/strange_series\n- Author(s): Collatz, Lothar\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: integer sequence\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 13th, 2007 by dododododo\n\nProblem-page discussion:\nThis problem is also called Collatz conjecture, Ulam conjecture, or the Syracuse problem. For a more extensive discussion, visit the wikipedia article or [L].\n\nBibliography:\n[L] Jeffrey C. Lagarias: The 3x+1 problem: An annotated bibliography (1963--2000)\n\nDiscussion links:\n- wikipedia article: http://en.wikipedia.org/wiki/Collatz_conjecture\n\nBibliography links:\n- The 3x+1 problem: An annotated bibliography (1963--2000): http://www.arxiv.org/abs/math/0309224\n\nComments:\n- July 11th, 2014 | Anonymous | Peter Schorer, again a possible proof: Just last month, July 2014, Schorer published another paper which uses similar methods as from his 2008 paper, claiming again to have proven the conjecture. This is early, but I am wanting to know what the general opinion of Schorer is in the world of serious mathematicians. All of my research has turned up mixed reviews and heated arguments.\n- April 9th, 2008 | porton | Bruckman proved 3x+1 problem: From PlanetMath's forums:\n\n> Bruckman has published his proof of the conjecture in the International Journal of Mathematical Education in Science and Technology,\n\n> Vol 39, Issue 3 April 2008.\n\n--\n\nVictor Porton - http://www.mathematics21.org\n- June 11th, 2008 | brenton | In The 3x+1 Problem: An: In The 3x+1 Problem: An Annotated Bibliography, II (2001-), Lagarias explains why this proof is incomplete.\n- January 14th, 2008 | porton | Somebody's efforts: At http://occampress.com/ somebody publishes something about 3n+1 problem. Read him and check whether he is true.\n\nVictor Porton - http://www.mathematics21.org\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"The 3n+1 conjecture\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3381, "problem_number": "OPG-491", "title": "Odd incongruent covering systems", "statement": "Conjecture There is no covering system whose moduli are odd, distinct, and greater than 1.", "background": "Source: Open Problem Garden. Original node ID: 491. URL: http://www.openproblemgarden.org/op/odd_incongruent_covering_systems.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/odd_incongruent_covering_systems\n- Author(s): Erdos, Paul; Selfridge, John L.\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: covering system\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: August 4th, 2007 by Robert Samal\n\nProblem-page discussion:\nLet $a(n)$ denote the residue class $\\{a+nt \\mid t \\in \\Z\\}$. A covering system (defined by Paul Erdos in early 1930's) is a finite collection $\\{a_1(n_1), \\dots, a_k(n_k) \\}$ of residue classes whose union covers all the integers. Such systems are easy to find if the moduli are allowed to repeat, or if we allow even numbers. The covering system is called incongruent if all the moduli are distinct.\n\nPartial results are known. Berger, Felzenbaum and Fraenkel ([BFF1], [BFF2]) show (among else) that if covering system with odd distinct moduli (greater than 1) exists, $N$ is the least common multiple of $n_1$, \\dots, $n_t$, and $p_1$, \\dots, $p_s$ are all distinct prime divisors of~ $N$, then $$\\prod_{i=1}^s \\frac{p_i -1}{p_i-2} - \\sum_{i=1}^s \\frac {1}{p_i-2} > 2 \\,.$$\n\nSimpson and Zeilberger [SZ] proved that if in addition $n_1$, \\dots, $n_k$ are square-free, then $N$ has at least 18 prime divisors.\n\nGuo and Sun [GS] recently improved this to show that if $N$ is square-free, then it has at least 22 prime divisors.\n\nBibliography:\n[BFF1] M. A. Berger, A. Felzenbaum and A. S. Fraenkel, Necessary condition for the existence of an incongruent covering system with odd moduli, Acta. Arith. 45 (1986), 375–379\n\n[BFF2] M. A. Berger, A. Felzenbaum and A. S. Fraenkel, Necessary condition for the existence of an incongruent covering system with odd moduli. II, Acta Arith. 48 (1987), 73–79.\n\n[GS] Song Guo and Zhi-Wei Sun: On odd covering systems with distinct moduli; Adv. Appl. Math. 35(2005), 182–187\n\n[SZ] R. J. Simpson and D. Zeilberger, Necessary conditions for distinct covering systems with square-free moduli, Acta. Arith. 59 (1991), 59–70.\n\nRelated:\nRelated problems\nCovering systems with big moduli\n\nSource links:\n- covering system: http://en.wikipedia.org/wiki/covering system\n\nBibliography links:\n- On odd covering systems with distinct moduli: http://www.arxiv.org/abs/math/0412217\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Odd incongruent covering systems\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3382, "problem_number": "OPG-493", "title": "Covering systems with big moduli", "statement": "Problem Does for every integer $N$ exist a covering system with all moduli distinct and at least equal to~ $N$?", "background": "Source: Open Problem Garden. Original node ID: 493. URL: http://www.openproblemgarden.org/op/covering_systems_with_big_moduli.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/covering_systems_with_big_moduli\n- Author(s): Erdos, Paul; Selfridge, John L.\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: covering system\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 4th, 2007 by Robert Samal\n\nProblem-page discussion:\nLet $a(n)$ denote the residue class $\\{a+nt \\mid t \\in \\Z\\}$. A covering system (defined by Paul Erdos in early 1930's) is a finite collection $\\{a_1(n_1), \\dots, a_k(n_k) \\}$ of residue classes whose union covers all the integers.\n\nSuch systems are easy to find if the moduli are allowed to repeat. They are known for many lower bounds $N$ on the size of moduli: e.g. $\\{0(2), 0(3), 1(4), 5(6), 7(12) \\}$ is such system for $N=2$. Choi proved that it is possible to give an example for N = 20.\n\nOn the other hand, recently it was shown [FFKPY] that if such systems exist for arbitrary large $N$, then $\\sum_{i=1}^k \\frac 1{n_i}$ is not bounded.\n\nBibliography:\n[FFKPY] Michael Filaseta, Kevin Ford, Sergei Konyagin, Carl Pomerance, Gang Yu: Sieving by large integers and covering systems of congruences, J. Amer. Math. Soc. 20 (2007), 495-517.\n\nRelated:\nRelated problems\nOdd incongruent covering systems\n\nSource links:\n- covering system: http://en.wikipedia.org/wiki/covering system\n\nBibliography links:\n- Sieving by large integers and covering systems of congruences: http://www.arxiv.org/abs/math.NT/0507374\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Covering systems with big moduli\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3383, "problem_number": "OPG-506", "title": "Divisibility of central binomial coefficients", "statement": "Problem (1) Prove that there exist infinitely many positive integers $n$ such that $$\\gcd({2n\\choose n}, 3\\cdot 5\\cdot 7) = 1.$$\n\nProblem (2) Prove that there exists only a finite number of positive integers $n$ such that $$\\gcd({2n\\choose n}, 3\\cdot 5\\cdot 7\\cdot 11) = 1.$$", "background": "Source: Open Problem Garden. Original node ID: 506. URL: http://www.openproblemgarden.org/op/divisibility_of_central_binomial_coefficients.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/divisibility_of_central_binomial_coefficients\n- Author(s): Graham, Ronald L.\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: August 5th, 2007 by maxal\n\nProblem-page discussion:\nThe binomial coefficient ${2n\\choose n}$ is not divisible by prime $p$ iff all the base- $p$ digits of $n$ are smaller than $\\frac{p}{2}.$\n\nIt has been conjectured that 1, 2, 10, 3159, and 3160 are the only positive numbers for which $\\gcd({2n\\choose n}, 3\\cdot 5\\cdot 7\\cdot 11) = 1$ holds.\n\nBibliography:\nP. Erdos, R.L. Graham, I.Z. Ruzsa and E.G. Straus \"On the Prime Factor of $2n \\choose n$.\" Math. Comp. 29 (1975), 83-92.\n\nSequence A030979: Numbers n such that C(2n,n) is not divisible by 3, 5 or 7.\n\nAndrew Granville. \"The Arithmetic Properties of Binomial Coefficients.\"\n\nBibliography links:\n- P. Erdos, R.L. Graham, I.Z. Ruzsa and E.G. Straus \"On the Prime Factor of $2n \\choose n$.\" Math. Comp. 29 (1975), 83-92.: http://www.math.ucsd.edu/%7Esbutler/ron/75_03_prime_factors.pdf\n- Sequence A030979: Numbers n such that C(2n,n) is not divisible by 3, 5 or 7.: https://oeis.org/A030979\n- Andrew Granville. \"The Arithmetic Properties of Binomial Coefficients.\": http://www.cecm.sfu.ca/organics/papers/granville/index.html\n\nComments:\n- June 29th, 2014 | Anonymous | It seems Problem (1) was: It seems Problem (1) was solved last year, see http://arxiv.org/abs/1010.3070\n\nHow about Problem 2?\n- January 18th, 2012 | Anonymous | 2, 10, and 3159 are not valid for problem (2): In the initial comment five values are stated to be not divisible by 3, 5, 7, and 11. That is true for 1 and 3160 but not for 2, 10, and 3159:\n\nThe last base-3-digit of 2 is 2. The last base-11-digit of 10 is 10. The last base-5-digit 3159 is 4.\n\nOr check these facts:\n\n${{2 \\times 2} \\choose 2} = 6 = 3 \\times 2$\n\n${{2 \\times 10} \\choose 10} = 184756 = 11 \\times 16796$\n\n2 and 3159 are not in the sequence A030979 (mentioned in the bibliography).\n- July 23rd, 2012 | Ng Yong Hao | Reference for problem (2): It appears that the right reference for question 2 should be A151750 instead.\n- March 14th, 2011 | tba | Problem 1 solved?: Looks like Problem 1 was solved by http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.3070v1.pdf\n- May 27th, 2011 | Anonymous | Does not look like it.: Does not look like it.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"Divisibility of central binomial coefficients\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3384, "problem_number": "OPG-563", "title": "Davenport's constant", "statement": "For a finite (additive) abelian group $G$, the Davenport constant of $G$, denoted $s(G)$, is the smallest integer $t$ so that every sequence of elements of $G$ with length $\\ge t$ has a nontrivial subsequence which sums to zero.\n\nConjecture $s( {\\mathbb Z}_n^d) = d(n-1) + 1$", "background": "Source: Open Problem Garden. Original node ID: 563. URL: http://www.openproblemgarden.org/op/davenports_constant.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/davenports_constant\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: Davenport constant; subsequence sum; zero sum\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 8th, 2007 by mdevos\n\nProblem-page discussion:\nDavenport's original motivation for introducing the constant $s(G)$ concerned prime ideal decompositions in algebraic number fields. However, determining this constant even for some very restricted families of groups has proved to be an interesting combinatorial problem. Indeed, the highlighted conjecture is considered to be one of the most important unsolved problems concerning finite abelian groups. I (M. DeVos) have reguarded this conjecture as folklore, but I await correction here.\n\nIt is easy to see that $s( {\\mathbb Z}_{n_1} \\times {\\mathbb Z}_{n_2} \\ldots \\times {\\mathbb Z}_{n_{\\ell}} ) \\ge 1 + \\sum_{i=1}^\\ell (n_i - 1)$ because the sequence constructed by taking $n_i - 1$ copies of the element with a $1$ in the $i^{th}$ position and $0$ 's elsewhere has no nontrivial subsequence which sums to zero. There is also an easy upper bound of $s(G) \\le |G|$. To see this, assume $|G| = n$, let $a_1,\\ldots,a_n$ be a sequence of elements from $G$, and consider the terms $a_1, a_1 + a_2, \\ldots, a_1 + a_2 + \\ldots a_n$. If these terms are distinct, then one must be 0 (giving us a zero sum subseqence). Otherwise two of them must be equal, so we have $a_1 + \\ldots a_i = a_1 + \\ldots a_j$ for some $i < j$, but then $a_{i+1} + a_{i+1} \\ldots + a_j = 0$.\n\nFor cyclic groups, our trivial upper and lower bound match, so we have $s({\\mathbb Z}_n) = n$. However, the situation gets much more difficult as soon as we go any further. The following theorem summarizes two classic results of Olson which remain state of the art.\n\nTheorem (Olson)\n\n- $s( {\\mathbb Z}_a \\times {\\mathbb Z}_b ) = a + b - 1$ if $a|b$.\n- if $p$ is prime, $s( {\\mathbb Z}_{p^{d_1}} \\times {\\mathbb Z}_{p^{d_2}} \\ldots \\times {\\mathbb Z}_{p^{d_{\\ell}}}) = 1 + \\sum_{i=1}^{\\ell} (p^{d_i} - 1)$\n\nAlthough there does not even exist a conjecture as to the value of $s(G)$ for a general $G$, recently a number of authors have proved theorems which give upper bounds on $s(G)$ under some structural assumptions. For instance, Caro has proved that $s(G) \\le \\frac{|G|}{3} + 1$ for every $G$ which is not cyclic and not of the form ${\\mathbb Z}_2 \\times {\\mathbb Z}_m$.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 24.\n\nAttempt notes:\nTarget:\nMake progress on \"Davenport's constant\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3385, "problem_number": "OPG-655", "title": "Snevily's conjecture", "statement": "Conjecture Let $G$ be an abelian group of odd order and let $A,B \\subseteq G$ satisfy $|A| = |B| = k$. Then the elements of $A$ and $B$ may be ordered $A = \\{a_1,\\ldots,a_k\\}$ and $B = \\{b_1,\\ldots,b_k\\}$ so that the sums $a_1+b_1, a_2+b_2 \\ldots, a_k + b_k$ are pairwise distinct.", "background": "Source: Open Problem Garden. Original node ID: 655. URL: http://www.openproblemgarden.org/op/snevilys_conjecture.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/snevilys_conjecture\n- Author(s): Snevily, Hunter S.\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: addition table; latin square; transversal\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: October 13th, 2007 by mdevos\n\nProblem-page discussion:\nThe motivation for this question comes from the study of latin squares. The addition table of every (additive) group forms a latin square, and this gives us a rich source of interesting squares. To explain further, we require a couple of definitions. A transversal of a $k \\times k$ matrix is a collection of $k$ cells, no two of which are in the same row or column, and we say that a transversal is latin if no two of its cells contain the same element. Latin transversals are nice structures to find in latin squares. In particular, note that the cells of a $k \\times k$ latin square $L$ may be partitioned into $k$ latin transversals if and only if there is a latin square orthogonal to $L$ (see this for a definition of orthogonal latin squares). The above conjecture is perhaps most naturally phrased in terms of latin transversals as follows.\n\nConjecture (Snevily's conjecture - version 2) Every $k \\times k$ submatrix of the addition table of every abelian group of odd order has a latin transversal.\n\nSnevily's conjecture was proved by Alon [A] for abelian groups of prime order using a fairly standard application of the Alon-Tarsi polynomial technique. Later, Dasgupta, Karolyi, Serra, and Szegedy [DKSSz] used a sneaky application of the same technique to prove the conjecture for cyclic groups of odd order (the key to their approach is the fact that for $n$ odd, ${\\mathbb Z}_n$ is a subgroup of the multiplicative group of the field of order $2^{\\phi(n)}$ where $\\phi$ is Euler's totient function). The conjecture is still open for non-cyclic groups.\n\nThe full addition table of ${\\mathbb Z}_{2n}$ does not have a latin transversal. To see this, note that the sum of the elements in this group is equal to $n$ (here we identify $\\{0,1,\\ldots,2n-1\\}$ with ${\\mathbb Z}_{2n}$ in the usual manner). So, if $a_1,\\ldots,a_{2n}$ and $b_1,\\ldots,b_{2n}$ are two orderings of ${\\mathbb Z}_{2n}$, then $\\sum_{i=1}^{2n} (a_i + b_i) = 0$, and therefore $a_1 + b_1,\\ldots,a_{2n} + b_{2n}$ cannot be an ordering of ${\\mathbb Z}_{2n}$. This parity problem is the only obstruction known, and the following conjecture asserts that apart from it, the above conjectures holds for cyclic groups of even order.\n\nConjecture (Snevily) Every $k \\times k$ submatrix of the addition table of ${\\mathbb Z}_{2n}$ has a latin transversal, unless it is a translate of a cyclic subgroup of ${\\mathbb Z}_{2n}$ of even order.\n\nIn fact, it appears that the above conjecture might hold with ${\\mathbb Z}_{2n}$ replaced by any abelian group.\n\nBibliography:\n[A] N. Alon, Additive Latin transversals. Israel J. Math. 117 (2000), 125--130. MathSciNet\n\n[DKSSz] S. Dasgupta, Gy. Károlyi, O. Serra, B. Szegedy, Transversals of additive Latin squares. Israel J. Math. 126 (2001), 17--28. MathSciNet\n\n*[S] H. S. Snevily, Unsolved Problems: The Cayley Addition Table of Z $\\sb n$. Amer. Math. Monthly 106 (1999), no. 6, 584--585. MathSciNet.\n\nDiscussion links:\n- latin squares: http://en.wikipedia.org/wiki/latin square\n- this: http://www.cut-the-knot.org/arithmetic/latin3.shtml\n\nBibliography links:\n- Additive Latin transversals: http://www.tau.ac.il/%7Enogaa/PDFS/alt2.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1760589\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1882032\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1543489\n\nComments:\n- May 27th, 2011 | G. Eric Moorhouse | Proof of Snevily's Conjecture: Snevily's Conjecture was in fact proved in 2009. See: Bodan Arsovski, 'A proof of Snevily's Conjecture', Israel Journal of Mathematics, vol. 182 (2011), pp. 505-508. See also Gergely Harcos, Gyula Károlyi and Géza Kós, 'Remarks to Arsovski's proof of Snevily's Conjecture', Annales Univ. Sci. Budapest., vol. 54 (2011), pp. 57-61.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 22.\n\nAttempt notes:\nTarget:\nMake progress on \"Snevily's conjecture\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3386, "problem_number": "OPG-17958", "title": "Frobenius number of four or more integers", "statement": "Problem Find an explicit formula for Frobenius number $g(a_1, a_2, \\dots, a_n)$ of co-prime positive integers $a_1, a_2, \\dots, a_n$ for $n\\geq 4$.", "background": "Source: Open Problem Garden. Original node ID: 17958. URL: http://www.openproblemgarden.org/op/frobenius_number_of_four_or_more_integers.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/frobenius_number_of_four_or_more_integers\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: November 26th, 2008 by maxal\n\nProblem-page discussion:\nFor $n=2$, the formula $g(a_1,a_2) = a_1 a_2 − a_1 − a_2$ was discovered by Sylvester discovered in 1884 [S]. For $n=3$, an explicit solution is also known [G,R,SB]. No explicit solution is known for $n\\geq 4$.\n\nBibliography:\n[G] Greenberg, H. \"Solution to a Linear Diophantine Equation for Nonnegative Integers.\" J. Algorithms 9, 343-353, 1988.\n\n[R] Rødseth, Ø. J. \"On a Linear Diophantine Problem of Frobenius.\" J. reine angew. Math. 301, 171-178, 1978.\n\n[SB] Selmer, E. S. and Beyer, Ö. \"On the Linear Diophantine Problem of Frobenius in Three Variables.\" J. reine angew. Math. 301, 161-170, 1978.\n\n[S] Sylvester, J. J. \"Question 7382.\" Mathematical Questions from the Educational Times 41, 21, 1884.\n\nSource links:\n- Frobenius number: http://en.wikipedia.org/wiki/Frobenius number\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Frobenius number of four or more integers\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3387, "problem_number": "OPG-60034", "title": "Singmaster's conjecture", "statement": "Conjecture There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number $1$.\n\nThe number $2$ appears once in Pascal's triangle, $3$ appears twice, $6$ appears three times, and $10$ appears $4$ times. There are infinite families of numbers known to appear $6$ times. The only number known to appear $8$ times is $3003$. It is not known whether any number appears more than $8$ times. The conjectured upper bound could be $8$; Singmaster thought it might be $10$ or $12$. See Singmaster's conjecture.", "background": "Source: Open Problem Garden. Original node ID: 60034. URL: http://www.openproblemgarden.org/op/singmasters_conjecture.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/singmasters_conjecture\n- Author(s): Singmaster, David\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: Pascal's triangle\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: March 15th, 2019 by Zach Teitler\n\nSource links:\n- Singmaster's conjecture: http://en.wikipedia.org/wiki/Singmaster's conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Singmaster's conjecture\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3388, "problem_number": "OPG-508", "title": "A sextic counterexample to Euler's sum of powers conjecture", "statement": "Problem Find six positive integers $x_1, x_2, \\dots, x_6$ such that $$x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = x_6^6$$ or prove that such integers do not exist.", "background": "Source: Open Problem Garden. Original node ID: 508. URL: http://www.openproblemgarden.org/op/a_sextic_counterexample_to_eulers_sum_of_powers_conjecture.\n\nSource subject path: Number Theory > Computational Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_sextic_counterexample_to_eulers_sum_of_powers_conjecture\n- Author(s): Euler, Leonhard P.\n- Subject(s): Number Theory; Computational Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: August 5th, 2007 by maxal\n\nProblem-page discussion:\nEuler's sum of powers conjecture states that for $k\\geq 3$ the Diophantine equation $\\sum_{i=1}^{n} a_i^k = b^k$ does not have solutions in positive integers as soon as $n Computational Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/counterexamples_to_the_baillie_psw_primality_test\n- Subject(s): Number Theory; Computational Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: August 7th, 2007 by maxal\n\nProblem-page discussion:\nSelfridge, Wagstaff, and Pomerance offered $500 +$100 + $20 for$n$satisfying Problem 2, and$20 + $100 +$500 for a proof that there is no such $n$ (R. Guy, 1994).\n\nBibliography:\nCarl Pomerance. \"Are There Counterexamples to the Baillie-PSW Primality Test?\"\n\nThomas R. Nicely. \" The Baillie-PSW primality test.\"\n\nR. K. Guy. \"Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes\". §A12 in \"Unsolved Problems in Number Theory\", 2nd ed. New York: Springer-Verlag, pp. 27-30, 1994.\n\nSource links:\n- Baillie-PSW primality test: http://en.wikipedia.org/wiki/Baillie-PSW primality test\n- Fermat pseudoprime: http://en.wikipedia.org/wiki/Fermat pseudoprime\n- Lucas pseudoprime: http://en.wikipedia.org/wiki/Lucas pseudoprime\n\nBibliography links:\n- Carl Pomerance. \"Are There Counterexamples to the Baillie-PSW Primality Test?\": http://www.pseudoprime.com/dopo.pdf\n- Thomas R. Nicely. \" The Baillie-PSW primality test.\": http://www.trnicely.net/misc/bpsw.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Counterexamples to the Baillie-PSW primality test\" in Number Theory; Computational Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3390, "problem_number": "OPG-822", "title": "Wall-Sun-Sun primes and Fibonacci divisibility", "statement": "Conjecture For any prime $p$, there exists a Fibonacci number divisible by $p$ exactly once.\n\nEquivalently:\n\nConjecture For any prime $p>5$, $p^2$ does not divide $F_{p-\\left(\\frac p5\\right)}$ where $\\left(\\frac mn\\right)$ is the Legendre symbol.", "background": "Source: Open Problem Garden. Original node ID: 822. URL: http://www.openproblemgarden.org/op/fibonacci_divisibility.\n\nSource subject path: Number Theory > Computational Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/fibonacci_divisibility\n- Subject(s): Number Theory; Computational Number Theory\n- Keywords: Fibonacci; prime\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 14th, 2008 by adudzik\n\nProblem-page discussion:\nLet $p$ be an odd prime, and let $\\nu_p(n)$ denote the $p$-adic valuation of $n$. Let $F_{k(p)}$ be the smallest Fibonacci number that is divisible by $p$ (which must exist by a simple counting argument). A well-known result says that $\\nu_p(F_n)=0$ unless $k(p)$ divides $n$, and $\\nu_p(F_{k(p)m}) = \\nu_p(F_{k(p)}) + \\nu_p(m)$. This conjecture asserts that $\\nu_p(F_{k(p)})=1$ for all $p$. This has been verified up to at least $p<10^{14}$. [EJ]\n\nThis conjecture is equivalent to non-existence of Wall-Sun-Sun primes.\n\nBibliography:\n[EJ] Andreas-Stephan Elsenhansand and Jörg Jahnel, The Fibonacci sequence modulo p^2\n\n[R] Marc Renault, Properties of the Fibonacci Sequence Under Various Moduli\n\n*[W] D. D. Wall, Fibonacci Series Modulo m, American Mathematical Monthly, 67 (1960), pp. 525-532.\n\nDiscussion links:\n- Wall-Sun-Sun primes: http://en.wikipedia.org/wiki/Wall-Sun-Sun_prime\n\nBibliography links:\n- The Fibonacci sequence modulo p^2: http://www.uni-math.gwdg.de/tschinkel/gauss/Fibon.pdf\n- Properties of the Fibonacci Sequence Under Various Moduli: http://www.math.temple.edu/%7Erenault/fibonacci/thesis.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Wall-Sun-Sun primes and Fibonacci divisibility\" in Number Theory; Computational Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3391, "problem_number": "OPG-16570", "title": "Magic square of squares", "statement": "Question Does there exist a $3\\times 3$ magic square composed of distinct perfect squares?", "background": "Source: Open Problem Garden. Original node ID: 16570. URL: http://www.openproblemgarden.org/op/magic_square_of_squares.\n\nSource subject path: Number Theory > Computational Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/magic_square_of_squares\n- Author(s): LaBar, Martin\n- Subject(s): Number Theory; Computational Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: November 23rd, 2008 by maxal\n\nProblem-page discussion:\nThis question was first asked in 1984 by Martin LaBar and popularized in 1996 by Martin Gardner, who offered $100 to the first person to construct such a square. In 2005 Christian Boyer offered €1,000 and a bottle of champagne for a solution to a somewhat easier problem [Bc]. For a review of the history of research, see [Ba, Bb, Bc]. For basic facts about the anticipated$3\\times 3$ magic square of squares, see [Br, Mo].\n\nBibliography:\n[Ba] Christian Boyer. Some notes on the magic squares of squares problem. The Mathematical Intelligencer 27 (2005), 2, 52-64.\n\n[Bb] Christian Boyer. Magic squares of squares, \"Multimagic Squares\" website.\n\n[Bc] Christian Boyer. Latest research on the \"3x3 magic square of squares\" problem, \"Multimagic Squares\" website.\n\n[Br] Kevin Brown. Magic Square of Squares, \"Math Pages\" website.\n\n[Mo] Lee Morgenstern. 3x3 Magic Square of Squares Formulations\n\nSource links:\n- magic square: http://en.wikipedia.org/wiki/magic square\n\nBibliography links:\n- Magic squares of squares: http://multimagie.com/English/SquaresOfSquares.htm\n- Latest research on the \"3x3 magic square of squares\" problem: http://multimagie.com/English/SquaresOfSquaresSearch.htm\n- Magic Square of Squares: http://www.mathpages.com/HOME/kmath417.htm\n- 3x3 Magic Square of Squares Formulations: http://home.earthlink.net/%7Emorgenstern/magic/sq3.htm\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Magic square of squares\" in Number Theory; Computational Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3392, "problem_number": "OPG-37221", "title": "Perfect cuboid", "statement": "Conjecture Does a perfect cuboid exist?", "background": "Source: Open Problem Garden. Original node ID: 37221. URL: http://www.openproblemgarden.org/op/perfect_cuboid.\n\nSource subject path: Number Theory > Computational Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/perfect_cuboid\n- Subject(s): Number Theory; Computational Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 4th, 2010 by tsihonglau\n\nProblem-page discussion:\nPerfect cuboid is a cuboid whose edges and face and body diagonals are all integers. In other words, is there any solution to the following system of Diophantine equations:\n\n$a^2 + b^2 = d^2$\n\n$b^2 + c^2 = e^2$\n\n$a^2 + c^2 = f^2$\n\n$a^2 + b^2 + c^2 = g^2$\n\nDiscussion links:\n- Perfect cuboid: http://en.wikipedia.org/wiki/Perfect cuboid\n\nComments:\n- August 24th, 2010 | Anonymous | Primitive perfect cuboid (Primitive perfect Euler brick): I think that I have a simple proof that there cannot be any primitive perfect cuboid (primitive perfect Euler brick). I am willing to provide it if anyone requests it. T Herndon\n- January 27th, 2012 | Anonymous | the proof: send to tomk@globalserve.net\n- December 28th, 2010 | Anonymous | well, I'll bite...: If you still think you have a proof, I'd love to take a look - my email is timro21@gmail.com, or you could have it published on the Unsolved Problems web site at http://www.unsolvedproblems.org/\n\nTim\n- May 4th, 2010 | tsihonglau | Is there any 4D Euler brick?: Perfect cuboid is related to Euler brick whose edges and face diagonals are all integers. It is know that there are infinite Euler bricks. But is there any 4D Euler brick? In other words, is there any solution to the following system of Diophantine equations:\n\n$a^2 + b^2 = e^2$\n\n$a^2 + c^2 = f^2$\n\n$b^2 + c^2 = g^2$\n\n$a^2 + d^2 = h^2$\n\n$b^2 + d^2 = i^2$\n\n$c^2 + d^2 = j^2$\n\nI computed a, b, c, d up to 1 million with brute force and found no solution. Any idea?\n- April 24th, 2011 | tsihonglau | Divisibility: I found the divisibility conditions of four sides a, b, c and d in a primitive 4d euler brick (if exists):\n\n1. One is divided by 64, another by 16, another by 4, another odd.\n\n2. One is divided by 27, another by 9, another by 3, another not by 3.\n\n3. Two is divided by 5.\n\n4. Two is divided by 11.\n\n5. One is divided by 13.\n\n6. One is divided by 19.\n- October 1st, 2010 | tsihonglau | Without loss of generality,: Without loss of generality, we can suppose a > b > c > d and remove a Diophantine equation from the system. I found some solutions, for example: without the last equation, the following quadruple is a solution. a=6325,b=5796,c=5520,d=528. They are so small, so i guess 4D euler bricks should exist.\n- May 19th, 2010 | Anonymous | 4d brick: Well, it's easy to show that for any primitive 4D brick (that is, one where a, b, c, and d have no common factor), then exactly one of a, b, c, and d must be odd, and the rest even....\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Perfect cuboid\" in Number Theory; Computational Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-05-13T14:45:21.186Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3393, "problem_number": "OPG-60052", "title": "KPZ Universality Conjecture", "statement": "Conjecture Formulate a central limit theorem for the KPZ universality class.", "background": "Source: Open Problem Garden. Original node ID: 60052. URL: http://www.openproblemgarden.org/op/kpz_universality_conjecture.\n\nSource subject path: Probability.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/kpz_universality_conjecture\n- Subject(s): Probability\n- Keywords: KPZ equation, central limit theorem\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 3rd, 2020 by Tomas Kojar\n\nProblem-page discussion:\nThe KPZ equation is given by\n\n$$\\partial_{t}h(x,t)=\\partial_{x}^{2}h(x,t)+\\lambda(\\partial_{x}h(x,t))^{2}+\\xi,$$\n\nwhere $\\xi$ denotes space-time white noise and $\\lambda\\in \\mathbb{R}$ is a parameter describing the strength of its \"asymmetry\". It has been conjectured (see [BPRS93, BG97, Cor 12,GJ14,HQ18] for a number of results in this direction) that the KPZ equation has a “universal” character in the sense that any one-dimensional model of surface growth should converge to it provided that it has the following features:\n\n• There is a microscopic smoothing mechanism. Pictorially this means that large valleys are quickly filled.\n\n• The system has microscopic fluctuations with short-range correlations. Pictorially this means that height function change depends only on neighboring heights.\n\n• The system has some “lateral growth” mechanism in the sense that the growth speed depends in a nontrivial way on the slope. The vertical effective growth rate depends non-linearly on local slope.\n\n• At the microscopic scale, the strengths of the growth and fluctuation mechanisms are well separated: either the growth mechanism dominates (intermediate disorder) or the fluctuations dominate (weak asymmetry). Growth is drive by noise which quickly decorrelates in space / time and is not heavy tailed.\n\nHere is a concrete surface growth mathematical model to give a sense of the above features. The random deposition model is one of the simplest (and least realistic) models for a randomly growing one-dimensional interface. Unit blocks fall independently and in parallel from the sky above each site of $\\mathbb{Z}$ according to exponentially distributed waiting times. Recall that a random variable X has exponential distribution of rate $\\lambda>0$ (or mean $1/\\lambda$ ) if $P(X > x) = e^{-\\lambda x}$. Such random variables are characterized by the memoryless property – conditioned on the event that $X > x$, $X - x$ still has the exponential distribution of the same rate. Consequently, the random deposition model is Markov – its future evolution only depends on the present state (and not on its history). The ballistic deposition (or sticky block) model was introduced by Vold [V59] in 1959 and, as one expects in real growing interfaces, displays spatial correlation. As before, blocks fall according to iid exponential waiting times, however, now a block will stick to the first edge against which it becomes incident. This creates overhangs and we define the height function h(t, x) as the maximal height above x which is occupied by a box.\n\nBibliography:\n[BG97] L. BERTINI and G. GIACOMIN. Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys. 183, no. 3, (1997), 571–607.\n\n[BPRS93] L. BERTINI, E. PRESUTTI, B. RUDIGER ¨, and E. SAADA. Dynamical fluctuations at the critical point: convergence to a nonlinear stochastic PDE. Teor. Veroyatnost. i Primenen. 38, no. 4, (1993), 689–741\n\n[Cor 12] I. Corwin, The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), 1130001, 76. MR 2930377. Zbl 1247.82040. http://dx.doi.org\n\n[GJ14] P. GONC¸ ALVES and M. JARA. Nonlinear fluctuations of weakly asymmetric interacting particle systems. Arch. Ration. Mech. Anal. 212, no. 2, (2014), 597–644\n\n[HQ18] HAIRER, M. and QUASTEL, J. (2018). A class of growth models rescaling to KPZ. Forum Math. Pi 6 e3\n\n[V59] M. J. Vold. A numerical approach to the problem of sediment volume. J. Colloid Sci., 14:168 (1959).\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"KPZ Universality Conjecture\" in Probability, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 19, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3394, "problem_number": "OPG-36887", "title": "Sums of independent random variables with unbounded variance", "statement": "Conjecture If $X_1, \\dotsc, X_n \\geq 0$ are independent random variables with $\\mathbb{E}[X_i] \\leq \\mu$, then $$\\mathrm{Pr} \\left( \\sum X_i - \\mathbb{E} \\left[ \\sum X_i \\right ] < \\delta \\mu \\right) \\geq \\min \\left ( (1 + \\delta)^{-1} \\delta, e^{-1} \\right).$$", "background": "Source: Open Problem Garden. Original node ID: 36887. URL: http://www.openproblemgarden.org/op/sums_of_independent_random_variables_with_unbounded_variance.\n\nSource subject path: Theoretical Computer Science.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/sums_of_independent_random_variables_with_unbounded_variance\n- Author(s): Feige, Uriel\n- Subject(s): Theoretical Computer Science\n- Keywords: Inequality; Probability Theory; randomness in TCS\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 2nd, 2009 by cwenner\n\nProblem-page discussion:\nIn comparison to most probabilistic inequalities (like Hoeffding's), Feige's inequality does not deteriorate as $n$ goes to infinity, something that is useful for computer scientists.\n\nLet $T = \\mathbb{E}\\left [ \\sum X_i \\right ] + \\delta$. Feige argued that to prove the conjecture, one only needs to prove it for the case when $\\mu = 1$ and each variable $X_i$ has the entire probability mass distributed on 0 and $t_i$ for some $\\mathbb{E}[X_i] \\leq t_i \\leq T$. He proved that $\\mathrm{Pr} \\left( \\sum X_i - \\mathbb{E} \\left[ \\sum X_i \\right ] < \\delta \\right) \\geq \\min \\left ( (1 + \\delta)^{-1} \\delta, 1/13 \\right),$ and conjectured that the constant 1/13 may be replaced with $e^{-1}$. It was further conjectured that \"the worst case\" would be one of\n\n- one variable has $1 + \\delta$ as maximum value and the remaining $n-1$ random variables are always 1 (hence the probability that the sum is less than $T$ is $(1 + \\delta)^{-1} \\delta$ ),\n- each variable has $T = n + \\delta$ as maximum (hence the probability that the sum is less than $T$ is $\\left(1 - \\frac{1}{T}\\right)^n \\stackrel{n \\rightarrow \\infty}{\\longrightarrow} e^{-1}$ ).\n\nOne way to initiate an attack on this problem is to assume $\\delta = \\mathbb{E}[X_i] = 1$ and argue that the case when each variable assumes $n + 1$ with probability $(n+1)^{-1}$ and otherwise 0 is indeed the worst.\n\nBibliography:\n*[F04] Uriel Feige: On sums of independent random variables with unbounded variance, and estimating the average degree in a graph, STOC '04: Proceedings of the thirty-sixth annual ACM symposium on Theory of computing (2004), pp. 594 - 603. ACM\n\n*[F05] Uriel Feige: On sums of independent random variables with unbounded variance, and estimating the average degree in a graph, Manuscript, 2005, [pdf]\n\nThe problem was also referenced at population algorithms, the blog.\n\nBibliography links:\n- ACM: http://doi.acm.org/10.1145/1007352.1007443\n- [pdf]: http://www.wisdom.weizmann.ac.il/%7Efeige/Others/newmarkov.pdf\n- population algorithms, the blog: http://petar.blog.lcs.mit.edu/?p=66\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"Sums of independent random variables with unbounded variance\" in Theoretical Computer Science, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3395, "problem_number": "OPG-661", "title": "P vs. NP", "statement": "Problem Is P = NP?", "background": "Source: Open Problem Garden. Original node ID: 661. URL: http://www.openproblemgarden.org/op/p_vs_np.\n\nSource subject path: Theoretical Computer Science > Algorithms.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/p_vs_np\n- Author(s): Cook, Stephen; Levin, Leonid A.\n- Subject(s): Theoretical Computer Science; Algorithms\n- Keywords: Complexity Class; Computational Complexity; Millenium Problems; NP; P; polynomial algorithm\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 18th, 2007 by zitterbewegung\n\nProblem-page discussion:\nThis problem is the central open problem in Theoretical Computer Science and is regarded as one of the most outstanding unsolved problems in mathematics. See Wikipedia's P versus NP problem and The Clay Mathematical Institute's P vs. NP problem for more.\n\nRelated:\nRelated problems\nOne-way functions exist\n\nDiscussion links:\n- P versus NP problem: http://en.wikipedia.org/wiki/P versus NP problem\n- P vs. NP problem: http://www.claymath.org/millenium-problems/p-vs-np-problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"P vs. NP\" in Theoretical Computer Science; Algorithms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3396, "problem_number": "OPG-36311", "title": "Exponential Algorithms for Knapsack", "statement": "Conjecture\n\nThe famous 0-1 Knapsack problem is: Given $a_{1},a_{2},\\dots,a_{n}$ and $b$ integers, determine whether or not there are $0-1$ values $x_{1},x_{2},\\dots,x_{n}$ so that $$\\sum_{i=1}^{n} a_{i}x_{i} = b.$$The best known worst-case algorithm runs in time$2^{n/2}$times a polynomial in$n$. Is there an algorithm that runs in time$2^{n/3}$?", "background": "Source: Open Problem Garden. Original node ID: 36311. URL: http://www.openproblemgarden.org/op/exponential_algorithms_for_knapsack.\n\nSource subject path: Theoretical Computer Science > Algorithms.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/exponential_algorithms_for_knapsack\n- Author(s): Lipton, Dick\n- Subject(s): Theoretical Computer Science; Algorithms\n- Keywords: Algorithm construction; Exponential-time algorithm; Knapsack\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: February 4th, 2009 by dick lipton\n\nBibliography:\n[SS] Richard Schroeppel and Adi Shamir. A T=O(2n/2), S=O(2n/4) algorithm for certain NP-complete problems, SIAM J. Comput. 10 (1981), no. 3, 456--464.\n\nComments:\n- August 6th, 2010 | rzwaan | References: Solving this problem in pseudopolynomial time with polynomial space: Daniel Lokshtanov and Jesper Nederlof: \"Saving Space by Algebraization\". To appear in the proceedings of ACM Symposium on Theory of Computing (STOC 2010).\n\nThe next reference solves this problem (even general knapsack, I believe). An 2^{0.311 n} Algorithm: Nick Howgrave-Graham and Antoine Joux: \"A new generic algorithm for hard knapsacks\". To appear in the proceedings of EUROCRYPT 2010.\n\nHowever, the problem might be easily adjusted to \"find an algorithm that takes O(2^{c n}) time, where c < 0.311\", etc etc.\n- August 3rd, 2010 | Anonymous | Updates: Publications from 2010 on this topic: Pseudopolynomial algorithm with only polynomial space: Daniel Lokshtanov and Jesper Nederlof: \"Saving Space by Algebraization\". To appear in the proceedings of ACM Symposium on Theory of Computing (STOC 2010).\n\nFast knapsack: Nick Howgrave-Graham and Antoine Joux: \"New Generic Algorithms for Hard Knapsacks\". To appear in the proceedings of EUROCRYPT 2010 The running time is O^*(2^0.311 n), which is faster than the question posted here.\n\nObviously, the question could become: find a lower constant than 0.311.\n- March 29th, 2010 | Stephen Le Guen | 0-1 knapsack: I do not know how to determine the run time but I have an algorithm based on dynamic programming which solves the problem more efficiently than a simple search but not in polynomial time!\n\nI have implemented the procedure as a Java program and tried to express it as a flow chart. Files for both of these are available at; www.cybase.co.uk/wlcs/Software.html\n\nalthough at the moment the website is down.\n\nI have contacted the ISP to try and restore it.\n- February 7th, 2010 | Anonymous | This appears to be answered: see paper at http://www.joux.biz/publications/Knapsacks.pdf and rjlipton's blog post about it at http://rjlipton.wordpress.com/2010/02/05/a-2010-algorithm-for-the-knapsack-problem/\n\nof course, this is an open ended question, there is still room for better algorithms.\n- January 13th, 2010 | Anonymous | A solution has been announced: A (randomized, probably heuristic) algorithm has been announced to solve this problem in time close to 2^(0.311 n), at the ESC 2010 seminar.\n\nFor details, see: https://cryptolux.org/ESC/Antoine_Joux\n- September 8th, 2009 | Anonymous | Subset sum or 0-1?: When your values - your a_n - can be negative, and your b (the goal) is zero, then it's called \"subset sum\". If the a_n are non-negative (i.e., some of the a's may be zero), the b is positive, and the choice is to either exclude (0) or include (1) one \"copy\" of each value, then it's a \"0-1 knapsack\" problem. Usually, but not always, a knapsack problem has components with multiple values, and the goal is a minimax problem: maximize the a's, while minimizing the c's.\n\nIt's always darkest, just after the lights go out.\n- April 4th, 2009 | cwenner | 0-1 Knapsack?: I know this problem as subset sum and not as 0-1 knapsack.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Exponential Algorithms for Knapsack\" in Theoretical Computer Science; Algorithms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3397, "problem_number": "OPG-445", "title": "The robustness of the tensor product", "statement": "Problem Given two codes $R,C$, their Tensor Product $R \\otimes C$ is the code that consists of the matrices whose rows are codewords of $R$ and whose columns are codewords of $C$. The product $R \\otimes C$ is said to be robust if whenever a matrix $M$ is far from $R \\otimes C$, the rows (columns) of $M$ are far from $R$ ( $C$, respectively).\n\nThe problem is to give a characterization of the pairs $R,C$ whose tensor product is robust.", "background": "Source: Open Problem Garden. Original node ID: 445. URL: http://www.openproblemgarden.org/op/the_robustness_of_the_tensor_product_0.\n\nSource subject path: Theoretical Computer Science > Coding Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_robustness_of_the_tensor_product_0\n- Author(s): Ben-Sasson, Eli; Sudan, Madhu\n- Subject(s): Theoretical Computer Science; Coding Theory\n- Keywords: codes; coding; locally testable; robustness\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 13th, 2007 by ormeir\n\nProblem-page discussion:\nThe question is studied in the context of Locally Testable Codes.\n\nBibliography:\n*[BS] Eli Ben-Sasson, Madhu Sudan, Robust locally testable codes and products of codes, APPROX-RANDOM 2004, pp. 286-297 (See ECCC TR04-046).\n\n[CR] D. Coppersmith and A. Rudra, On the robust testability of tensor products of codes, ECCC TR07-061.\n\n[DSW] Irit Dinur, Madhu Sudan and Avi Wigderson, Robust local testability of tensor products of LDPC codes, APPROX-RANDOM 2006, pp. 304-315 (See ECCC TR06-118).\n\n[GM] Oded Goldreich, Or Meir, The Tensor Product of Two Good Codes Is Not Necessarily Robustly Testable, ECCC TR07-062.\n\n[M] Or Meir, On the Rectangle Method in proofs of Robustness of Tensor Products, ECCC TR07-061.\n\n[V] Paul Valiant, The Tensor Product of Two Codes Is Not Necessarily Robustly Testable, APPROX-RANDOM 2005, pp. 472-481.\n\nBibliography links:\n- ECCC TR04-046: http://eccc.hpi-web.de/eccc-reports/2004/TR04-046/index.html\n- ECCC TR07-061: http://eccc.hpi-web.de/eccc-reports/2005/TR05-104/index.html\n- ECCC TR06-118: http://eccc.hpi-web.de/eccc-reports/2006/TR06-118/index.html\n- ECCC TR07-062: http://eccc.hpi-web.de/eccc-reports/2007/TR07-062/index.html\n- ECCC TR07-061: http://eccc.hpi-web.de/eccc-reports/2007/TR07-061/index.html\n\nComments:\n- February 17th, 2010 | Anonymous | Results in: Eli Ben-Sasson and Michael Viderman. \"Composition of semi-LTCs by two-wise Tensor Products\" (RANDOM 09)\n\nEli Ben-Sasson and Michael Viderman. \"Tensor Products of Weakly Smooth Codes are Robust\" (RANDOM 08)\n- July 16th, 2007 | ormeir | The formal definition of robustness, and of the problem: In all of the following definitions, the term \"distance\" refers to \"relative Hamming distance\".\n\nGiven a matrix $M$, let $\\delta_{R \\otimes C}(M)$ denote the distance from $M$ to the nearest codeword of $R \\otimes C$. Let $\\delta_{\\rm{row}}(M)$ denote the average distance of a row of $M$ to $R$, and let $\\delta_{\\rm{col}}(M)$ denote the average distance of a column of $M$ to $C$. Finally, let $\\rho(M)$ denote the average of $\\delta_{\\rm{row}}(M)$ and $\\delta_{\\rm{col}}(M)$.\n\nThe tensor product $R \\otimes C$ is said to be $\\alpha$-robust iff for every matrix $M$ we have that $\\rho(M) \\ge \\alpha \\cdot \\delta_{R \\otimes C}(M)$.\n\nThe question is, under what conditions the tensor product $R \\otimes C$ is $\\alpha$-robust for some constant $\\alpha$.\n- July 14th, 2007 | Robert Samal | To be more precise...: 1) When you say codes, do you mean linear codes?\n\n2) What distance you are using when you're saying \"far from\"?\n- July 16th, 2007 | ormeir | To be more precise: 1) The question is most interesting for linear codes, but it can also be defined for non-linear codes.\n\n2) The distance is (relative or absolute) Hamming Distance.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"The robustness of the tensor product\" in Theoretical Computer Science; Coding Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3398, "problem_number": "OPG-163", "title": "Subset-sums equality (pigeonhole version)", "statement": "Problem Let $a_1,a_2,\\ldots,a_n$ be natural numbers with $\\sum_{i=1}^n a_i < 2^n - 1$. It follows from the pigeon-hole principle that there exist distinct subsets $I,J \\subseteq \\{1,\\ldots,n\\}$ with $\\sum_{i \\in I} a_i = \\sum_{j \\in J} a_j$. Is it possible to find such a pair $I,J$ in polynomial time?", "background": "Source: Open Problem Garden. Original node ID: 163. URL: http://www.openproblemgarden.org/op/theoretical_computer_science/subset_sums_equality.\n\nSource subject path: Theoretical Computer Science > Complexity.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/theoretical_computer_science/subset_sums_equality\n- Subject(s): Theoretical Computer Science; Complexity\n- Keywords: polynomial algorithm; search problem\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 18th, 2007 by mdevos\n\nProblem-page discussion:\nThis is one of a class of search problems for which a positive solution is garaunteed (so the corresponding decision problem is trivial) based on a theoretical property of the problem. Another such problem is given a Hamiltonian cycle in a cubic graph, find a second Hamiltonian cycle (here a theorem of Smith guarantee's a positive solution). The above problem is particularly attractive, since the proof that a pair $I,J$ must exist is quite simple, but it gives no insight into how to find the pair $I,J$.\n\nIt seems to be consensus among the cryptography community that this problem is hard.\n\nComments:\n- July 13th, 2007 | Anonymous | The problems are in PPP and PPA, respectively: Both this problem and the Smith/2nd Hamiltonian Path problems were suggested by Papadimitriou in his 1991 paper 'On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence\". (See http://www.cs.berkeley.edu/~christos/papers/ for a copy of the journal version.)\n\nThese problems are in natural complexity classes related to though apparently somewhat larger than PPAD. The subset sum problem is in the class PPP and the Smith problem is in the class PPA. Neither is known to be complete for the respective complexity class as far as I know.\n- July 9th, 2007 | kvanetten | You're right: I finally located a copy of the paper on the author's web page at http://www.lri.fr/~santha/ (why didn't that occur to me before?), and it's much clearer to me now--you're absolutely right. I guess the one suggestion I would make is to adopt their terminology and include the term 'pigeonhole' in the name of the problem to distinguish this variation from the more general case.\n- July 9th, 2007 | mdevos | How's this?: I am quite certian I have seen this problem called simply \"subset-sums equality\" before, so I have just appended \"(pigeonhole version)\" to the title. I hope that this is increasing the clarity.\n- July 9th, 2007 | mdevos | I think it is still open: First a disclaimer: I am no expert on this problem.. I heard it awhile ago, liked it, wrote it down, and that's what you see here. However, I do believe it is still open. If my understanding is correct, the problem considered in the paper you mention (\"Efficient approximation algorithms for the subset-sums equality problem\" by Bazgan, Santha and Tuza, JCSS Vol.64 Issue 2, March 2002) is a relaxed version of the one stated here where the restriction $\\sum_{i=1}^n a_i < 2^n - 1$ is not present, and the goal is to find a pair of nonempty disjoint subsets $I,J \\subseteq \\{1,\\ldots,n\\}$ so that $\\sum_{i \\in I} a_i$ and $\\sum_{j \\in J} a_j$ are as close as possible. I believe the only hardness results they obtain are for this problem.\n\nThe thing still missing is what to do with the funny assumption $\\sum_{i=1}^n a_i < 2^n - 1$. Although the problem is a search problem which looks roughly like a knapsack-type problem (and thus should be NP-hard), the associated decision problem is trivially \"true\", yet there is no obvious way to use this information.\n\nPS: thanks for the post!\n- July 8th, 2007 | kvanetten | A little more: Just an afterthought to my previous comment... Even if this problem is known not to be computable in polynomial time, it would still be interesting to know if it is NP-hard or not.\n- July 15th, 2007 | Anonymous | Re: A little more: Problems with guaranteed-to-exist solutions (like this one, & others in PPA, PPAD, etc.) are not NP-complete unless NP = coNP and the Polynomial Hierarchy collapses. Now, we don't yet know that P!= NP, or even that P!= PH.\n- July 16th, 2007 | kvanetten | Re: A little more: Yes, I made that comment before I read the paper by Bazgan, Santha and Tuza. They mentioned in passing that the problem isn't NP-hard unless NP = coNP, and apparently felt that it was obvious enough to not require any further comment. I'm probably missing something obvious, but I can't quite follow this. With factorization (for example), a number's prime decomposition is unique and can serve as a certificate for both 'yes' and 'no' instances. For pigeonhole subset-sums equality, the solutions are not necessarily unique. It would depend on exactly how one crafted the corresponding decision problem, but it's not clear to me what the certificate would be for a 'no' instance.\n- July 8th, 2007 | kvanetten | Should this problem still be considered open?: I was doing a quick search for background on this question and came across this article: http://dx.doi.org/10.1006/jcss.2001.1784, \"Efficient approximation algorithms for the subset-sums equality problem\" by Bazgan, Santha and Tuza, JCSS Vol.64 Issue 2 (March 2002). I only have access to the abstract, so I might be misinterpreting this, but they say, \"... On the other hand, we show that in the case where the value of a solution is the positive difference between the two partial sums, the problem is not 2nk -approximable in polynomial time unless P = NP, for any constant k.\" I take this to mean that there are no exact polynomial time algorithms unless P=NP. If so, this problem is only open in the sense that any question reducible to P vs. NP can be considered open.\n- June 30th, 2007 | Anonymous | Problem formulation: It seems to me that there are some typos in the problem formulation. The sum should be less than $2^n - 1$. The subsets should be nonempty.\n\n- JS\n- July 2nd, 2007 | mdevos | Thanks!: Thanks for your comment, I will edit the problerm and correct it.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Subset-sums equality (pigeonhole version)\" in Theoretical Computer Science; Complexity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3399, "problem_number": "OPG-467", "title": "Complexity of square-root sum", "statement": "Question What is the complexity of the following problem?\n\nGiven $a_1,\\dots,a_n; k$, determine whether or not $\\sum_i \\sqrt{a_i} \\leq k.$", "background": "Source: Open Problem Garden. Original node ID: 467. URL: http://www.openproblemgarden.org/op/complexity_of_square_root_sum.\n\nSource subject path: Theoretical Computer Science > Complexity.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/complexity_of_square_root_sum\n- Author(s): Goemans, Michel (?)\n- Subject(s): Theoretical Computer Science; Complexity\n- Keywords: semi-definite programming\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 18th, 2007 by abie\n\nProblem-page discussion:\nAs of a 1998 survey, the complexity of this problem was unknown. I'm not sure if that's still the case. But I wanted to see how easy it was to make a page about it.\n\nThis is the key to determining if semi-definite programming is truly solvable in polynomial time (it can be approximated to within $\\varepsilon$ using the interior point method or the ellipsoid algorithm in time polynomial in the size of the instance and $\\log 1/\\varepsilon$.\n\nBibliography:\n[G] Michal Goemans, Semidefinite Programming and Combinatorial Optimization\n\nDiscussion links:\n- a 1998 survey: http://www.emis.ams.org/journals/DMJDMV/xvol-icm/17/Goemans.MAN.ps.gz\n\nBibliography links:\n- Semidefinite Programming and Combinatorial Optimization: http://www.emis.ams.org/journals/DMJDMV/xvol-icm/17/Goemans.MAN.ps.gz\n\nComments:\n- December 29th, 2009 | Anonymous | Complexity: The complexity of the problem as stated is worst case O(n^3) in the magnitude of the greatest value a[i]. The reason being that it is a simple sum, as stated, of square roots. The square root operation is O(n^2) in the number of bits, which has a 3.xxx:1 relation to the magnitude of each a[i]. Either the problem is mistated or it has not been significant enough for anyone to waste time on stating the obvious.\n- December 30th, 2009 | Anonymous | Filching 10:00(2): I think the problem is that you may need to compute square roots to high precision. If you need to know the first m digits in the decimal expansion of sqrt(5), it will probably take time linear in m to find them, even though 5 took only constantly many bits to encode.\n- June 23rd, 2008 | Anonymous | A slightly more general: A slightly more general problem (in which the square roots may be subtracted as well as added) is very important in the context of computational geometry, both for actual algorithms and for proving complexity-theoretic bounds on problems, as it encapsulates the difficulty of comparing lengths of polygonal chains; see http://maven.smith.edu/~orourke/TOPP/P33.html.\n\n—David Eppstein\n- July 19th, 2007 | Robert Samal | Some update: Quotation from Etessami and Yannakakis, 2007:\n\n- [Sqrt-sum problem] is known to be solvable in PSPACE but it has been a major open problem ([GareyGrahamJohnson’76]) whether it is solvable even in NP.\n- [Allender et. al.,’06] Showed that Sqrt-Sum reduces to a more general problem, which they showed lies in the 4th level of the Counting Hierarchy ( $P^{PP^{PP^{PP}}}$ ).\n- July 1st, 2011 | Anonymous | From Allender et al.: square-root sum is in CH: http://ftp.cs.rutgers.edu/pub/allender/slp.pdf\n\nCorollary 1.5 The Sum-of-square-roots problem and the Euclidean Traveling Salesman Problem are in CH.\n\n(CH = Counting hierarchy.)\n\nDo we also know that it is in P^(PP^(PP^PP))?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Complexity of square-root sum\" in Theoretical Computer Science; Complexity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3400, "problem_number": "OPG-474", "title": "Linear-size circuits for stable $0,1 < 2$ sorting?", "statement": "Problem Can $O(n)$-size circuits compute the function $f$ on $\\{0,1,2\\}^*$ defined inductively by $f(\\lambda) = \\lambda$, $f(0x) = 0f(x)$, $f(1x) = 1f(x)$, and $f(2x) = f(x)2$?", "background": "Source: Open Problem Garden. Original node ID: 474. URL: http://www.openproblemgarden.org/op/linear_size_circuits_for_stable_0_1_2_sorting.\n\nSource subject path: Theoretical Computer Science > Complexity.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/linear_size_circuits_for_stable_0_1_2_sorting\n- Author(s): Regan, Kenneth\n- Subject(s): Theoretical Computer Science; Complexity\n- Keywords: Circuits; sorting\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: July 19th, 2007 by KWRegan\n\nProblem-page discussion:\nThis function moves all 2s in $x$ flush-right, leaving the sequence of 0s and 1s the same, and represents stable topological sort of the partial order $0,1 < 2$. It is linear-time computable in any model that supports the operations of a double-ended queue in $O(1)$ time, including multi-tape Turing machines, but is to me the \"easiest\" function for which I do not know linear-size circuits. By contrast sorting $0 < 1 < 2$, called the \"Dutch National Flag Problem\", has $O(n)$-size circuits by counting. It suffices to compute $f(x)$ when $|x|$ is a power of $2$ and exactly half the entries are $2$. For this and more see my Computational Complexity blog item, PDF file here.\n\nDiscussion links:\n- Computational Complexity blog item: http://weblog.fortnow.com/2007/07/concrete-open-problem.html\n- here: http://www.cse.buffalo.edu/%7Eregan/InfoFlow.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Linear-size circuits for stable $0,1 < 2$ sorting?\" in Theoretical Computer Science; Complexity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3401, "problem_number": "OPG-2150", "title": "Discrete Logarithm Problem", "statement": "If $p$ is prime and $g,h \\in {\\mathbb Z}_p^*$, we write $\\log_g(h) = n$ if $n \\in {\\mathbb Z}$ satisfies $g^n = h$. The problem of finding such an integer $n$ for a given $g,h \\in {\\mathbb Z}^*_p$ (with $g \\neq 1$ ) is the Discrete Log Problem.\n\nConjecture There does not exist a polynomial time algorithm to solve the Discrete Log Problem.", "background": "Source: Open Problem Garden. Original node ID: 2150. URL: http://www.openproblemgarden.org/op/discrete_logarithm_problem.\n\nSource subject path: Theoretical Computer Science > Complexity.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/discrete_logarithm_problem\n- Subject(s): Theoretical Computer Science; Complexity\n- Keywords: discrete log; NP\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 27th, 2008 by cplxphil\n\nProblem-page discussion:\nThe Discrete Logarithm Problem is a critical problem in number theory, and is similar in many ways to the integer factorization problem. If it were possible to compute discrete logs efficiently, it would be possible to break numerous thought-to-be unbreakable cryptographic schemes. However, although most mathematicians and computer scientists believe that the DLP is unsolvable, this conjecture is difficult to establish, because such a proof would imply that P!= NP...which is the most difficult open problem in theoretical computer science.\n\nAvi Wigderson has shown that there is no \"natural proof\" (in the sense of [RR]) that the DLP requires circuits of greater than half-exponential size. The key idea is that a natural proof that the DLP is hard would yield a method for breaking discrete-log-based cryptosystems, and this is a contradiction. Of course, it could still be that DLP is provably hard, but by a proof that is not \"natural.\"\n\nBibliography:\n[RR] Alexander A. Razborov and Steven Rudich, Natural proofs, Journal of Computer and System Sciences 55 (1997), 24–35.\n\nComments:\n- December 29th, 2009 | Anonymous | Discrete Logrithm is polynomial: The discrete logarithm problem is little more than an integer analog to a rotor-code cipher problem. The later is a problem in finding concurrent zeros for two periodic functions where the periodicity of one tracks the value of x and the other the value of y. The value of x being x, x^2, x^3,... x^n. The value of y being y, y+p, y+2p,..., y+np. These forms of problems are amenable to solution using a multi dimensional difference (recurrence) expression (as is the Elliptic Curve Cryptographic problem which is a direct application of DE over algebraic fields). The Discrete Logarithm problem is solvable by a deterministic polynomial time algorithm in O(n^3). Google a paper titled \"Computing a Discrete Logarithm in O(n^3)\", which can be found at Cornell's arXiv website. Example code for the algorithm is also provided by the author of that paper.\n- February 25th, 2013 | Anonymous | N!= n: The above paper solves the discrete logarithm in time O(N^3) not O(n^3), two very different things. N being the size of the modulus, n being log_2 of N (the binary length). There are many algorithms that will solve the discrete log problem much faster than this method, brute force search runs at a worst case of O(N), or in other words O(2^n). The provided algorithm in the above paper runs in (2^(3*n)).\n- December 29th, 2009 | Anonymous | A polynomial algorithm: See the paper at http://arxiv1.library.cornell.edu/abs/0912.2269.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Discrete Logarithm Problem\" in Theoretical Computer Science; Complexity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3402, "problem_number": "OPG-36892", "title": "P vs. PSPACE", "statement": "Problem Is there a problem that can be computed by a Turing machine in polynomial space and unbounded time but not in polynomial time? More formally, does P = PSPACE?", "background": "Source: Open Problem Garden. Original node ID: 36892. URL: http://www.openproblemgarden.org/op/p_vs_pspace.\n\nSource subject path: Theoretical Computer Science > Complexity.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/p_vs_pspace\n- Author(s): Folklore\n- Subject(s): Theoretical Computer Science; Complexity\n- Keywords: P; PSPACE; separation; unconditional\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 4th, 2009 by cwenner\n\nProblem-page discussion:\nIf $P \\neq NP$, then $P \\neq NP, NP^{NP}, \\dotsc, PH, P^{\\#P}, PSPACE = NPSPACE$, and a whole bunch more separations can be shown. In the light of this, if one believes that $P \\neq NP$, then it is naive to try to directly separate P from NP. In the words of the great George Polya, \"If there is a problem you can’t solve, then there is an easier problem you can solve: find it.\" The P versus PSPACE question is one of the easiest such questions and would constitute an astonishing discovery in its own right. In particular, it would be the first separation result of this kind.\n\nProblem Approaches\n\nHow do we approach this problem? I don't know, readers please contribute. My personal take would be circuit complexity, i.e. functions that are known not to be computable with circuits of polynomial size. Another would be descriptive complexity: show that there is a property that can be expressed in second-order logic with a transitive closure operation which cannot be recognized in polynomial time.[N87] Fenner offered an interesting characterization of the P versus PSPACE question in terms of reductions.[FKR89] P versus PSPACE was also as an intermediate step towards the P versus NP prize by the Clay institute.[F05]\n\nBibliography:\n[F05] Harvey M. Friedman: Clay Millenium Problem: P = NP, manuscript, 2005. [pdf]\n\n[FKR89] Fenner,, S. A. and Kurtz,, S. A. and Royer,, J. A.: Every polynomial-time 1-degree collapses iff P=PSPACE, SFCS '89: Proceedings of the 30th Annual Symposium on Foundations of Computer Science, 1988, pp. 624-629, citeseer acm [pdf]\n\n[N87] Neil Immerman: Languages that Capture Complexity Classes, SIAM Journal of Computation 16:4, 1987. citeseer [pdf]\n\nRelated:\nRelated problems\nP vs. NP\n\nBibliography links:\n- [pdf]: http://www.math.ohio-state.edu/%7Efriedman/pdf/P=NP10290512pt.pdf\n- citeseer: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.3854\n- acm: http://dx.doi.org/10.1109/SFCS.1989.63545\n- [pdf]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.29.3854&rep=rep1&type=pdf\n- citeseer: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.9176\n- [pdf]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.54.9176&rep=rep1&type=pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"P vs. PSPACE\" in Theoretical Computer Science; Complexity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3403, "problem_number": "OPG-59968", "title": "One-way functions exist", "statement": "Conjecture One-way functions exist.", "background": "Source: Open Problem Garden. Original node ID: 59968. URL: http://www.openproblemgarden.org/op/one_way_functions_exist.\n\nSource subject path: Theoretical Computer Science > Complexity.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/one_way_functions_exist\n- Subject(s): Theoretical Computer Science; Complexity\n- Keywords: one way function\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 12th, 2014 by porton\n\nProblem-page discussion:\nIn fact, their existence would prove that the complexity classes P and NP are not equal.\n\nBibliography:\nOne-way functions (Wikipedia)\n\nRelated:\nRelated problems\nP vs. NP\n\nSource links:\n- One-way functions: http://en.wikipedia.org/wiki/One-way_function\n\nBibliography links:\n- One-way functions: http://en.wikipedia.org/wiki/One-way_function\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"One-way functions exist\" in Theoretical Computer Science; Complexity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3404, "problem_number": "OPG-454", "title": "Unconditional derandomization of Arthur-Merlin games", "statement": "Problem Prove unconditionally that $\\mathcal{AM}$ $\\subseteq$ $\\Sigma_2$.", "background": "Source: Open Problem Garden. Original node ID: 454. URL: http://www.openproblemgarden.org/op/unconditional_derandomization_of_mathcal_am.\n\nSource subject path: Theoretical Computer Science > Complexity > Derandomization.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/unconditional_derandomization_of_mathcal_am\n- Author(s): Shaltiel, Ronen; Umans, Christopher\n- Subject(s): Theoretical Computer Science; Complexity; Derandomization\n- Keywords: Arthur-Merlin; Hitting Sets; unconditional\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 16th, 2007 by ormeir\n\nProblem-page discussion:\nIt is trivial to show that $\\mathcal{AM} \\subseteq \\Pi_2$. It is also known that under hardness assumptions $\\mathcal{AM}= \\mathcal{NP}$ (See [MV99]). The question is, can we prove unconditionally that $\\mathcal{AM} \\subseteq \\Sigma_2$.\n\nBibliography:\n*[GSTS03] Danny Gutfreund, Ronen Shaltiel and Amnon Ta-Shma, Uniform hardness vs. randomness for Arthur-Merlin games, Proc. of CCC 2003. Can be downloaded from Ronen Shaltiel's web site.\n\n[MV99] Peter Bro Miltersen and N. Variyam Vinodchandran, \"Derandomizing Arthur-Merlin games using hitting sets\", STOC 1999, pages 71-80. Can be downloaded from N. Variyam Vinodchandran's web site.\n\n[SU01] Ronen Shaltiel and Christopher Umans, Simple extractor for all min-entropies and new pseudo-random generator, Proc. of FOCS 2001, pages 648-657. Can be downloaded from Ronen Shaltiel's web site.\n\n[SU07] Ronen Shaltiel and Christopher Umans, Low-end uniform hardness vs. randomness tradeoffs for AM, STOC 2007. Can be downloaded from Ronen Shaltiel's web site.\n\nSource links:\n- $\\mathcal{AM}$: http://en.wikipedia.org/wiki/Arthur-Merlin_protocol\n- $\\Sigma_2$: http://en.wikipedia.org/wiki/Polynomial_hierarchy\n\nBibliography links:\n- web site: http://cs.haifa.ac.il/%7Eronen/online_papers/online_papers.html\n- web site: http://www.cse.unl.edu/%7Evinod/papers/index.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Unconditional derandomization of Arthur-Merlin games\" in Theoretical Computer Science; Complexity; Derandomization, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3405, "problem_number": "OPG-51618", "title": "P vs. BPP", "statement": "Conjecture Can all problems that can be computed by a probabilistic Turing machine (with error probability < 1/3) in polynomial time be solved by a deterministic Turing machine in polynomial time? That is, does P = BPP?", "background": "Source: Open Problem Garden. Original node ID: 51618. URL: http://www.openproblemgarden.org/op/p_vs_bpp.\n\nSource subject path: Theoretical Computer Science > Complexity > Derandomization.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/p_vs_bpp\n- Author(s): Folklore\n- Subject(s): Theoretical Computer Science; Complexity; Derandomization\n- Keywords: BPP; circuit complexity; pseudorandom generators\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 14th, 2013 by Charles R Great...\n\nProblem-page discussion:\nBPP has long been considered tractable. Many problems in BPP have been derandomized, showing that they are in fact in P. Is this true for all problems in BPP? All that is known at the moment is $P\\subseteq BPP\\subseteq NEXP.$\n\nThis problem has been shown to have deep connections to circuit complexity (see for example Impagliazzo & Wigderson). It is folklore that the existence of appropriate pseudorandom generators suffices to give P = BPP; Goldreich shows that their existence also follows from P = BPP.\n\nBibliography:\nAndrea E.\u0002 F.\u0002 Clement, Jos\u0003e D.\u0002 P.\u0002 Rolim,\u0002 and Luca Trevisan, Recent Advances Towards Proving P \u0002= BPP (1998).\n\nOded Goldreich, In a World of BPP=P, Studies in complexity and cryptography, Lecture Notes in Comput. Sci., 6650, Springer, Heidelberg, 2011, pp. 191-–232. See also the presentation.\n\nRussell Impagliazzo and Avi Wigderson, P=BPP unless E has sub-exponential circuits: derandomizing the XOR Lemma, following STOC '97.\n\nRyan Williams, Towards NEXP versus BPP?, Computer Science---Theory and Applications, Lecture Notes in Computer Science Volume 7913 (2013), pp. 174--182.\n\nRelated:\nRelated problems\nP vs. NP\n\nBibliography links:\n- Recent Advances Towards Proving P \u0002= BPP: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.5568\n- In a World of BPP=P: http://www.wisdom.weizmann.ac.il/%7Eoded/PDF/bpp.pdf\n- the presentation: http://www.wisdom.weizmann.ac.il/%7Eoded/T/bpp.ppt\n- P=BPP unless E has sub-exponential circuits: derandomizing the XOR Lemma: http://www.math.ias.edu/%7Eavi/PUBLICATIONS/MYPAPERS/IW97/proc.pdf\n- Towards NEXP versus BPP?: http://www.stanford.edu/%7Errwill/nexp-v-bpp.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"P vs. BPP\" in Theoretical Computer Science; Complexity; Derandomization, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3406, "problem_number": "OPG-36884", "title": "Refuting random 3SAT-instances on $O(n)$ clauses (weak form)", "statement": "Conjecture For every rational $\\epsilon > 0$ and every rational $\\Delta$, there is no polynomial-time algorithm for the following problem.\n\nGiven is a 3SAT (3CNF) formula $I$ on $n$ variables, for some $n$, and $m = \\floor{\\Delta n}$ clauses drawn uniformly at random from the set of formulas on $n$ variables. Return with probability at least 0.5 (over the instances) that $I$ is typical without returning typical for any instance with at least $(1 - \\epsilon)m$ simultaneously satisfiable clauses.", "background": "Source: Open Problem Garden. Original node ID: 36884. URL: http://www.openproblemgarden.org/op/refuting_random_3sat_instances_on_o_n_clauses_weak_form.\n\nSource subject path: Theoretical Computer Science > Complexity > Hardness of Approximation.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/refuting_random_3sat_instances_on_o_n_clauses_weak_form\n- Author(s): Feige, Uriel\n- Subject(s): Theoretical Computer Science; Complexity; Hardness of Approximation\n- Keywords: NP; randomness in TCS; satisfiability\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 27th, 2009 by cwenner\n\nProblem-page discussion:\nThis conjecture was presented in Average Case Complexity and Approximation Complexity by Uriel Feige as a new approach for showing inapproximabiltiy results $^\\text{[F02]}$. The conjecture is strong in that it immediately implies the optimal $8/7 - \\epsilon$-hardness of approximation of 3SAT, something shown NP-hard in 1997 with heavy applications of PCP-techniques $^\\text{[H97]}$.\n\nThe strong and weak form\n\nThe weak form of Feige's conjecture is implied by the strong form of the conjecture ( $\\epsilon = 0$ ) and therefore subjectively more likely to be true. In the weak form, the choice of ambiguities of the uniform distribution (such as choosing clauses with or without replacement) may affect the parameters of the conjecture but not the truth.\n\nSupport for and against the conjecture\n\nIf the number of clauses is large enough ( $m \\in \\Omega(n^{1.5})$ ), then the problem defined above can be solved in polynomial time $^\\text{[FO06]}$. It is believed that there is a phase transition $\\Delta_c$ in the probability of satisfying a random 3SAT instance such that for every $\\Delta$ sufficiently smaller than $\\Delta_c$, only an inverse exponential number of 3SAT instances with $m = \\Delta n$ clauses are not satisfiable; and for every $\\Delta$ sufficiently larger than $\\Delta_c$, only an inverse exponential number of 3SAT instances with $m = \\Delta n$ clauses are satisfiable. Around $\\Delta_c$, it is also believed that deciding satisfiability of instances with about $\\Delta_c n$ clauses is difficult. Feige's conjecture plausibly implies the conjecture about the hardness around $\\Delta_c$, even if $\\Delta_c$ depends on $n$. The converse does not necessarily hold, that is, hardness of deciding 3SAT at $\\Delta_c$ implying hardness of the above problem for every $\\Delta$ $^\\text{[F99]}$. (expand this section)\n\nValue of the conjecture\n\nBy assuming that this conjecture holds, a number of inapproximability results have been derived for problems that have so far resisted attacks by other conjectures and techniques such as the unique games conjecture and probabilistically checkable proofs. If approximating a problem within $f(n)$ implies that Feige's conjecture is false, then the problem is said to be R3SAT-hard to approximate within $f(n)$. It has been shown that it is R3SAT-hard to approximate Maximum Balanced Bipartite Clique for some $\\delta > 0$ within $n^{-\\delta}$, Minimum Bisection below $4/3$, Dense k-Subgraph within some constant greater than 1, and the 2-Catalog Problem below some constant greater than 1. Showing either of these results without assuming the conjecture (e.g. NP-hardness) or improving the results assuming the conjecture are also open problems (of supposed medium importance for good enough improvements).\n\nA result proving the problem defined in the conjecture true under more plausible conjectures, e.g. P $\\neq$ NP, might show the way for a host of similar results (e.g. further reductions and similar extensions for other classes), add another technique to our repertoire, and greatly expand the area studying the relation between average-case hardness and approximability hardness.\n\nUsing so-called quasi-random PCPs, Subhash Khot has shown that neither of the three above problems admit a PTAS under the assumption that $NP \\nsubseteq BPTIME(2^{n^{o(1)}})$ $^\\text{[K04]}$.\n\nRelated Problems\n\n- The phase transition of random 3SAT instances (high importance) $^\\text{[F99]}$.\n\n- Alakhnovich's conjectures about hardness of linear systems and code words (medium importance) $^\\text{[A03]}$.\n\nBibliography:\n[A03] Michael Alakhnovich. More on average case vs approximation complexity. FOCS 2003. http://www.math.ias.edu/~misha/papers/average.ps\n\n*[F02] Uriel Feige. Relations between Average Case Complexity and Approximation Complexity. STOC 2002. http://citeseer.ist.psu.edu/old/feige02relations.html\n\n[F99] Ehud Friedgut. Necessary and sufficient conditions for sharp thresholds of graph properties and the $k$-SAT problem. Journal of the Amercian Mathematical Society, 1999.\n\n[FO06] Uriel Feige, Eran Ofek. Easily refutable subformulas of large random 3CNF formulas. Theory of Comuting, Volume 3 (2007). Pages 25 through 43.\n\n[H97] Johan Håstad. Some optimal inapproximability results. STOC, Proceedings of the 29th Annual ACM Symposium on Theory of Computing, 1997. http://www.nada.kth.se/~johanh/optimalinap.ps\n\n[K04] Subhash Khot. Ruling Out PTAS for Graph Min-Bisection, Densest Subgraph and Bipartite Clique. FOCS, Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2004. http://www.cc.gatech.edu/~khot/papers/mdc-bc.ps\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 26.\n\nAttempt notes:\nTarget:\nMake progress on \"Refuting random 3SAT-instances on $O(n)$ clauses (weak form)\" in Theoretical Computer Science; Complexity; Hardness of Approximation, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3407, "problem_number": "OPG-751", "title": "S(S(f)) = S(f) for reloids", "statement": "Question $S(S(f)) = S(f)$ for every endo-reloid $f$?", "background": "Source: Open Problem Garden. Original node ID: 751. URL: http://www.openproblemgarden.org/op/s_s_f_s_f_for_reloids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/s_s_f_s_f_for_reloids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: reloid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 10th, 2008 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology, especially Connectedness of funcoids and reloids for definitions of used concepts.\n\nBibliography:\n*Victor Porton. Algebraic General Topology\n\nRelated:\nRelated problems\nS(S(f)) = S(f) for funcoids\n\nSource links:\n- reloid: http://www.wikinfo.org/index.php/Reloid\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n- Connectedness of funcoids and reloids: http://www.mathematics21.org/binaries/connectedness.pdf\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComments:\n- May 8th, 2008 | porton | Mistake fixed: There were a big mistake in http://www.mathematics21.org/binaries/connectedness.pdf where are defined some of the concepts used by this open problem. I have rewritten the article, not it should be OK.\n\n--\n\nVictor Porton - http://www.mathematics21.org\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"S(S(f)) = S(f) for reloids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3408, "problem_number": "OPG-757", "title": "Inscribed Square Problem", "statement": "Conjecture Does every Jordan curve have 4 points on it which form the vertices of a square?", "background": "Source: Open Problem Garden. Original node ID: 757. URL: http://www.openproblemgarden.org/op/inscribed_square_problem.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/inscribed_square_problem\n- Author(s): Toeplitz\n- Subject(s): Topology\n- Keywords: simple closed curve; square\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 10th, 2008 by dlh12\n\nProblem-page discussion:\nA Jordan curve is a continuous function $f$ from the closed interval $[0,1]$ to the plane $\\mathbb{R}^{2}$ with the properties that $f$ is injective on the half-open interval $[0,1)$ (i.e., $f$ is simple) and $f(0)=f(1)$ (i.e., $f$ is closed).\n\nBibliography:\n[M] Meyerson, M.D., Equilateral triangles and continuous curves, Fund. Math. 110, (1980), 1--9.\n\nSource links:\n- Jordan curve: http://en.wikipedia.org/wiki/Jordan curve\n\nComments:\n- February 25th, 2021 | Anonymous | Already solved: This was proven in https://arxiv.org/pdf/2005.09193.pdf\n- March 3rd, 2010 | Anonymous | inscription of squares in simple closed curves: There is a theorem that says:\n\nin all simple closed curves there are 4n points that are vertex of n squares (inf = > n > =1)\n\nJorge Pasin.\n- November 8th, 2010 | Anonymous | in all simple closed curves there are 4n points: Would you clarify? An obtuse triangle has only one inscribed square, so this theorem is not true for n>=2. Do you have a reference to this theorem? Strashimir Popvassilev\n- November 25th, 2009 | Anonymous | Is the conjecture known to: Is the conjecture known to be true for C^1-smooth curves?\n- June 7th, 2010 | Anonymous | Yes: Yes. Walter Stromquist, Inscribed squares and square-like quadrilaterals in closed curves, Mathematika 36: 187-197 (1989).\n- June 2nd, 2010 | Anonymous | no.: If it were true for C^1 curves, then since a Jordan curve is compact, it may be weierstrass approximated by a series of C^1 curves (indeed by curves whose component functions are polynomials) such that the series converges uniformly to the given jordan curve. Then by assumption, each curve in the sequence contains 4 points forming a square, and the sequence of squares can be regarded as (eventually) a sequence in the (sequentially) compact space of the 4-fold product of any closed epsilon enlargement of the area bounded by the original jordan curve. It follows that the sequence of squares contains a convergent subsequence, which can be shown to be a square lying on the original jordan curve.\n\nThus, proving the C^1 case proves the general case.\n- June 7th, 2010 | Anonymous | This is flawed: The approximation argument is flawed: the squares on approximating curves may have sides decreasing to 0, in which case the limiting \"square\" degenerates to a point. In fact, Stromquist's theorem covers a much wider class of curves than C^1, but not all continuous curves.\n- April 28th, 2008 | Anonymous | Quantifier: Phrasing should be changed from \"Does any...\" to \"Does every...\"\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Inscribed Square Problem\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3409, "problem_number": "OPG-1783", "title": "Rank vs. Genus", "statement": "Question Is there a hyperbolic 3-manifold whose fundamental group rank is strictly less than its Heegaard genus? How much can the two differ by?", "background": "Source: Open Problem Garden. Original node ID: 1783. URL: http://www.openproblemgarden.org/op/rank_vs_genus.\n\nSource subject path: Topology.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/rank_vs_genus\n- Author(s): Johnson, Jesse\n- Subject(s): Topology\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 7th, 2008 by Jesse Johnson\n\nProblem-page discussion:\nThe rank of a 3-manifold is the minimal number of generators needed for its fundamental group. The Heegaard genus is the smallest genus of all Heegaard splittings for that 3-manifold. A Heegaard splitting determines a generating set for the 3-manifold, so the ranks is always less than or equal to the genus.\n\nThere is a family of Seifert fibered spaces for which the rank is one less than the genus, but for most Seifert fibered spaces, the rank and genus are equal. The Seifert fibered exampels have been used to construct graph manifolds for which the rank and genus differ by more than one [1]. However, there are no hyperbolic 3-manifolds for which rank and genus are known to differ.\n\nBibliography:\nSchultens, Jennifer, Weidman, Richard, On the geometric and the algebraic rank of graph manifolds. Pacific J. Math. 231 (2007), no. 2, 481--510.\n\nComments:\n- July 13th, 2008 | Anonymous | Connection to dynamics: Abert and Nikolov have found a connection between the `Rank vs Heegard Genus' problem and the `Fixed Price' problem in dynamics. Specifically, if every countable group has `fixed price' then the ratio of the Heegard genus and the rank of a hyperbolic 3-manifold can be arbitrarily large. For details, see http://arxiv.org/abs/math/0701361.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Rank vs. Genus\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3410, "problem_number": "OPG-37123", "title": "Smooth 4-dimensional Schoenflies problem", "statement": "Problem Let $M$ be a $3$-dimensional smooth submanifold of $S^4$, $M$ diffeomorphic to $S^3$. By the Jordan-Brouwer separation theorem, $M$ separates $S^4$ into the union of two compact connected $4$-manifolds which share $M$ as a common boundary. The Schoenflies problem asks, are these $4$-manifolds diffeomorphic to $D^4$? ie: is $M$ unknotted?", "background": "Source: Open Problem Garden. Original node ID: 37123. URL: http://www.openproblemgarden.org/op/smooth_4_dimensional_schoenflies_problem.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/smooth_4_dimensional_schoenflies_problem\n- Author(s): Alexander, J\n- Subject(s): Topology\n- Keywords: 4-dimensional; Schoenflies; sphere\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 6th, 2009 by rybu\n\nProblem-page discussion:\nBy the work of Mike Freedman, $M$ separates $S^4$ into two manifolds which are homeomorphic to $D^4$. So the Schoenflies problem is only non-trivial if $D^4$ admits an exotic smooth structure, which is also an open problem. Although $D^4$ could very well have an exotic smooth structure and yet the Schoenflies problem could have a positive answer. ie: although exotic smooth $D^4$ 's might exist, perhaps none of them embed in $S^4$?\n\nMartin Scharlemann has results to the effect that the Schoenflies problem is true provided the embeddings are simple enough.\n\nThe smooth Poincare conjecture in dimension 4 is related but disjoint from this problem. For example, the Poincare conjecture could be true and $D^4$ could have an exotic smooth structure -- this would amount to saying the monoid of smooth homotopy 4-spheres has some elements with inverses.\n\nThe analogous problem in other dimensions is known to be true. Namely, all embeddings of $S^n$ in $S^{n+1}$ are unknotted (bound manifolds diffeomorphic to $D^{n+1}$ ) provided $n \\neq 3$. For $n=1$ this is due to Schoenflies. For $n=2$ it's due to Alexander (see Hatcher's 3-manifolds notes for a modern exposition). For $n \\geq 4$ the result follows from the combination of the Mazur-Brown theorem that an embedding of $S^n$ in $S^{n+1}$ bounds a manifold homeomorphic to $D^{n+1}$, plus a consequence of the H-cobordism theorem which states that $D^{n+1}$ has no exotic smooth structures which restrict to the standard smooth structure on the boundary, provided $n \\geq 4$.\n\nBibliography:\n*[A] Alexander, J, On the subdivision of space by a polyhedron. Proc. Nat. Acad. Sci. USA 10 (1924) pg 6--8.\n\n[FQ] Freedman, M. Quinn, F. Topology of 4-manifolds. Princeton University Press.\n\n[S1] Scharlemann, M. The four-dimensional Schoenflies conjecture is true for genus two imbeddings. Topology 23 (1984) 211-217.\n\n[S2] Scharlemann, M. Smooth Spheres in R4 with four critical points are standard. Inventiones Math. 79 (1985) 125-141.\n\n[S3] Scharlemann, M. Generalized Property R and the Schoenflies Conjecture, Commentarii Mathematici Helvetici, 83 (2008) 421--449.\n\n[MMEB] Marston Morse and Emilio Baiada, Homotopy and Homology Related to the Schoenflies Problem. The Annals of Mathematics, Second Series, Vol. 58, No. 1 (Jul., 1953), pp. 142-165\n\n[B] Brown, Morton. A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc., vol. 66, pp. 74–76. (1960)\n\n[MAZ] Mazur, Barry, On embeddings of spheres., Bull. Amer. Math. Soc. 65 (1959) 59--65.\n\n[H] Hatcher, A. 3-manifolds notes. [http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html]\n\nRelated:\nRelated problems\nSmooth 4-dimensional Poincare conjecture\nWhat is the homotopy type of the group of diffeomorphisms of the 4-sphere?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Smooth 4-dimensional Schoenflies problem\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3411, "problem_number": "OPG-37125", "title": "Smooth 4-dimensional Poincare conjecture", "statement": "Conjecture If a $4$-manifold has the homotopy type of the $4$-sphere $S^4$, is it diffeomorphic to $S^4$?", "background": "Source: Open Problem Garden. Original node ID: 37125. URL: http://www.openproblemgarden.org/op/smooth_4_dimensional_poincare_conjecture.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/smooth_4_dimensional_poincare_conjecture\n- Author(s): Poincare; Smale; Stallings\n- Subject(s): Topology\n- Keywords: 4-manifold; poincare; sphere\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 6th, 2009 by rybu\n\nProblem-page discussion:\nThe original Poincare conjecture was the assertion that a simply-connected compact boundaryless $3$-manifold is diffeomorphic (smooth Poincare conjecture) or homeomorphic (topological Poincare conjecture) to $S^3$. Because of Poincare duality, this is equivalent to the assertion that a $3$-manifold has the homotopy-type of $S^3$ then it is diffeomorphic/homeomorphic to $S^3$. This gave birth to the generalized Poincare conjecture -- that an $n$-manifold with the homotopy type of $S^n$ is diffeomorphic or homeomorphic to $S^n$.\n\nBy the work of Smale and Stallings, the topological Poincare conjecture was shown to be true provided $n \\geq 5$. But for $n \\geq 7$ Milnor and Kervaire showed that $S^n$ admits non-standard smooth structures so the smooth Poincare conjecture is false in general.\n\nThe generalized Poincare conjecture is an undergraduate-level point-set topology problem for $n=1$.\n\nThe $n=2$ case was proven by Poincare.\n\nThe $n=3$ case was recently proven by Perelman.\n\nThe $n=4$ case is the only outstanding case. Mike Freedman has proven that a $4$-manifold which is homotopy-equivalent to $S^4$ is homeomorphic to $S^4$, so the smooth 4-dimensional Poincare conjecture is the only outstanding problem among the generalized Poincare conjectures. Moreover, it can be considered to be reduced to the question of if $S^4$ has an exotic smooth structure.\n\nTechnically, Poincare never asserted this conjecture. He only stated it was an interesting problem. So perhaps it should be called Poincare's Egregious Problem.\n\nIt is unknown whether or not $D^4$ admits an exotic smooth structure. If not, the smooth $4$-dimensional Poincare conjecture would have an affirmative answer. Similarly, it's known that $4$-dimensional euclidean space $\\mathbb R^4$ admits a continuum of pairwise non-diffeomorphic smooth structures. But it's unknown whether or not any of these exotic smooth structures extend to $D^4$, thought of as a compactification of $\\mathbb R^4$.\n\nBibliography:\n[FQ] Freedman, M. Quinn, F. Topology of 4-manifolds. Princeton University Press.\n\n[S] Smale, S., \"On the structure of manifolds\" Amer. J. Math., 84 (1962) pp. 387–399\n\n[MT] Morgan, John W.; Gang Tian. Ricci Flow and the Poincaré Conjecture. AMS/CMI (2009)\n\nRelated:\nRelated problems\nSmooth 4-dimensional Schoenflies problem\n\nComments:\n- August 26th, 2021 | Anonymous | Poincare conj is true in some dimensions > 7.: For n equals 12, 56 and 61, Sn also has a unique smooth structure.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Smooth 4-dimensional Poincare conjecture\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3412, "problem_number": "OPG-37129", "title": "Slice-ribbon problem", "statement": "Conjecture Given a knot in $S^3$ which is slice, is it a ribbon knot?", "background": "Source: Open Problem Garden. Original node ID: 37129. URL: http://www.openproblemgarden.org/op/slice_ribbon_problem_0.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/slice_ribbon_problem_0\n- Author(s): Fox, R\n- Subject(s): Topology\n- Keywords: cobordism; knot; ribbon; slice\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\nThe definitions of slice and ribbon: slice knot ribbon knot\n\nThere is a fairly vast literature on this problem. It is closely related to the problem of determining which homology $3$-spheres bound homology $4$-balls, as both are in essence a type of $4$-dimensional cobordism problem.\n\nBibliography:\n*[F] Fox, R. H. Some problems in knot theory. 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 168--176\n\n[G] Gilmer, Patrick M. On the slice genus of knots. Invent. Math. 66 (1982), no. 2, 191--197.\n\n[H] Hass, Joel. The geometry of the slice-ribbon problem. Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 1, 101--108.\n\nAna G. Lecuona. [arXiv:0910.4601] On the Slice-Ribbon Conjecture for Montesinos knots\n\nBrendan Owens. [arXiv:0802.2109] On slicing invariants of knots.\n\nRelated:\nRelated problems\nWhich homology 3-spheres bound homology 4-balls?\n\nDiscussion links:\n- slice knot: http://en.wikipedia.org/wiki/slice knot\n- ribbon knot: http://en.wikipedia.org/wiki/ribbon knot\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Slice-ribbon problem\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3413, "problem_number": "OPG-37131", "title": "Realisation problem for the space of knots in the 3-sphere", "statement": "Problem Given a link $L$ in $S^3$, let the symmetry group of $L$ be denoted $Sym(L) = \\pi_0 Diff(S^3,L)$ ie: isotopy classes of diffeomorphisms of $S^3$ which preserve $L$, where the isotopies are also required to preserve $L$.\n\nNow let $L$ be a hyperbolic link. Assume $L$ has the further `Brunnian' property that there exists a component $L_0$ of $L$ such that $L \\setminus L_0$ is the unlink. Let $A_L$ be the subgroup of $Sym(L)$ consisting of diffeomorphisms of $S^3$ which preserve $L_0$ together with its orientation, and which preserve the orientation of $S^3$.\n\nThere is a representation $A_L \\to \\pi_0 Diff(L \\setminus L_0)$ given by restricting the diffeomorphism to the $L \\setminus L_0$. It's known that $A_L$ is always a cyclic group. And $\\pi_0 Diff(L \\setminus L_0)$ is a signed symmetric group -- the wreath product of a symmetric group with $\\mathbb Z_2$.\n\nProblem: What representations can be obtained?", "background": "Source: Open Problem Garden. Original node ID: 37131. URL: http://www.openproblemgarden.org/op/realisation_problem_for_the_space_of_knots_in_the_3_sphere.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/realisation_problem_for_the_space_of_knots_in_the_3_sphere\n- Author(s): Budney, R\n- Subject(s): Topology\n- Keywords: knot space; symmetry\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\nAn answer to this problem would give a `closed form' description of the homotopy type of the space of smooth embeddings of $S^1$ in $S^3$. This is the space of embeddings in the Whitney Topology, or $C^k$-uniform topology for any $k \\geq 1$.\n\n`Closed form' means that every component of $Emb(S^1,S^3)$ would have the description as an iterated fiber bundle over certain well-known spaces, where the fibers are inductively well-known spaces, and the monodromy would be controlled rather explicitly by this list of representations.\n\nPeripherally related are various other realization problems for $3$-manifolds. For example, Sadayoshi Kojima proved that one can realize any finite group as the group of isometries of a hyperbolic $3$-manifold.\n\nBibliography:\n*[B] Budney, R. Topology of spaces of knots in dimension 3, to appear in Proc. Lond. Math. Soc.\n\n[B2] Budney, R. A family of embedding spaces. Geometry and Topology Monographs 13 (2007).\n\n[K] Kojima, S., Isometry transformations of hyperbolic $3$-manifolds. Topology Appl. 29 (1988), no. 3, 297--307.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Realisation problem for the space of knots in the 3-sphere\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3414, "problem_number": "OPG-37145", "title": "Which homology 3-spheres bound homology 4-balls?", "statement": "Problem Is there a complete and computable set of invariants that can determine which (rational) homology $3$-spheres bound (rational) homology $4$-balls?", "background": "Source: Open Problem Garden. Original node ID: 37145. URL: http://www.openproblemgarden.org/op/which_homology_3_spheres_bound_homology_4_balls.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/which_homology_3_spheres_bound_homology_4_balls\n- Author(s): Ancient/folklore\n- Subject(s): Topology\n- Keywords: cobordism; homology ball; homology sphere\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\nDetermining which homology $3$-spheres bound homology $4$-balls is a long standing open problem in 3/4-manifold topology. Much effort has gone towards understanding the situation for the Brieskorn homology spheres. For example, the Poincare Dodecahedral space is known not to bound a homology $4$-ball since the Rochlin invariant is non-trivial -- but $M\\#(-M)$ the connect-sum of Poincare Dodecahedral space $M$ with its orientation-reverse does bound a homology 4-ball, and it has a simple construction: remove an open tubular neighbourhood of $\\{*\\} \\times I$ from $M \\times I$, this is the $4$-manifold.\n\nStandard invariants used to show homology $3$-spheres do not bound homology $4$-balls are various spin or spin^c cobordism invariants such as: the Rochlin invariant, Siebenmann's $\\overline{\\mu}$-invariant, the Oszvath-Szabo $d$-invariant, and there are many others.\n\nBibliography:\n[K] Kirby, Robion (1989), The topology of 4-manifolds, Lecture Notes in Mathematics, 1374, Springer-Verlag,\n\n[R] Rokhlin, Vladimir A, New results in the theory of four-dimensional manifolds, Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221-224.\n\n[AK] S.Akbulut, R.Kirby, \"Mazur manifolds,\" Michigan Math. J. 26 (1979), 259--284.\n\n[CH] A.Casson, J.Harer, \"Some homology lens spaces which bound rational homology balls.\" Pacific. J. Math. Vol 96, No 1, (1981) 23–36.\n\n[F] H.Fickle, \"Knots, Z-Homology 3-spheres and contractible 4-manifolds,\" pp. 467--493, Houston J. Math. Vol 10, No. 4 (1984).\n\n[FS] R.Fintushel, R.Stern, \"An exotic free involution on S^4,\" Ann. Math. (2) 113 (1981) no2, 357--365.\n\n[M] B.Mazur, \"A note on some contractible 4-manifolds\", Annals of Mathematics, (2) 73 (1961). 221–228.\n\n[S] R.Stern,\"Some Brieskorn spheres which bound contractible manifolds,\" Notices Amer. Math. Soc 25 (1978), A448.\n\n[L] Lisca, Paolo Sums of lens spaces bounding rational balls. Algebr. Geom. Topol. 7 (2007), 2141--2164.\n\nRelated:\nRelated problems\nSlice-ribbon problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Which homology 3-spheres bound homology 4-balls?\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3415, "problem_number": "OPG-37151", "title": "Fundamental group torsion for subsets of Euclidean 3-space", "statement": "Problem Does there exist a subset of $\\mathbb R^3$ such that its fundamental group has an element of finite order?", "background": "Source: Open Problem Garden. Original node ID: 37151. URL: http://www.openproblemgarden.org/op/torsion_for_subsets_of_mathbb_r_3.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/torsion_for_subsets_of_mathbb_r_3\n- Author(s): Ancient/folklore\n- Subject(s): Topology\n- Keywords: subsets of euclidean space; torsion\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\nThe corresponding problem in $\\mathbb R^2$ has a negative answer. The corresponding problem in $\\mathbb R^n$ for $n \\geq 4$ has a positive answer. If the subset of $\\mathbb R^3$ has a regular neighbourhood with a smooth boundary, the answer is negative. Similarly the homology of the subset is known to have no torsion via an Alexander duality argument. So any torsion in the fundamental group must be in the commutator subgroup.\n\nBibliography:\n[E] Eda, K. Fundamental group of subsets of the plane. Topology and its Applications Volume 84, Issues 1-3, 24 April 1998, Pages 283-306\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Fundamental group torsion for subsets of Euclidean 3-space\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3416, "problem_number": "OPG-37154", "title": "Which compact boundaryless 3-manifolds embed smoothly in the 4-sphere?", "statement": "Problem Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smoothly in the 4-sphere. This should include a constructive procedure to find an embedding if the manifold is embeddable.", "background": "Source: Open Problem Garden. Original node ID: 37154. URL: http://www.openproblemgarden.org/op/which_compact_boundaryless_3_manifolds_embed_smoothly_in_the_4_sphere.\n\nSource subject path: Topology.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/which_compact_boundaryless_3_manifolds_embed_smoothly_in_the_4_sphere\n- Author(s): Kirby\n- Subject(s): Topology\n- Keywords: 3-manifold; 4-sphere; embedding\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\nFor general 3-manifolds this problem is fairly wide-open. But for some specific families of 3-manifolds it is heavily investigated.\n\nThere are two common embedding constructions: (1) obtain your 3-manifold as 0-surgery on a link which is the disjoint union of two smooth slice links. (2) Obtain your 3-manifold as the boundary of a Mazur manifold -- where Mazur manifold is taken to be a contractible 4-manifold constructed as $S^1 \\times D^3$ union a 2-handle. In both cases the resulting 3-manifold M embeds smoothly in $S^4$. There are many other embedding constructions but no known \"uniform\" construction that works for all embeddable 3-manifolds.\n\nSince such a 3-manifold would bound two 4-manifolds on either side, the embedding problem is a type of double cobordism problem, and related to issues such as the problem of determining which homology 3-spheres bound homology 4-balls.\n\nThe smoothness in the assumption is important. Mike Freedman has proven all homology 3-spheres admit tame topological embeddings into $S^4$. These embeddings have a less combinatorial nature than smooth embeddings so it is somewhat natural to restrict to the question of smooth embeddings. For example, the Poincare Homology Sphere does not embed smoothly in $S^4$, since it has a non-trivial Rochlin invariant.\n\nBibliography:\n[B] R. Budney, Embeddings of 3-manifolds in the 4-sphere from the point of view of the $11$-tetrahedron census, arXiv preprint arXiv:0810.2346\n\n[CH] J.S. Crisp, J.A. Hillman, Embedding Seifert fibred $3$-manifolds and ${\\rm Sol\\sp 3$-manifolds in $4$-space,} Proc. London Math Soc. (3) (1998), no. {\\bf 3} 685--710.\n\n[KK] A.~Kawauchi, S.~Kojima, Algebraic classification of linking pairings on $3$-manifolds, Math. Ann. {\\bf 253} (1980), no. 1, 29--42.\n\n[FS] R.~Fintushel, R.~Stern, Rational homology cobordisms of spherical space forms, Topology, {\\bf 26} no. 3 pp. 385--393, (1987).\n\n[GL] P.M.~Gilmer, C.~Livingston, On embedding 3-manifolds in 4-space, Topology, {\\bf 22}, no. 3, pp. 241--252 (1983).\n\n*[K] Kirby, R. Problem list in low-dimensional topology. [http://math.berkeley.edu/~kirby/problems.ps.gz]\n\n[L] R.A.~Litherland, Deforming twist-spun knots, Trans. Amer. Math. Soc. {\\bf 250} (1979), 311--331.\n\nRelated:\nRelated problems\nWhich homology 3-spheres bound homology 4-balls?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Which compact boundaryless 3-manifolds embed smoothly in the 4-sphere?\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3417, "problem_number": "OPG-37159", "title": "What is the homotopy type of the group of diffeomorphisms of the 4-sphere?", "statement": "Problem $Diff(S^4)$ has the homotopy-type of a product space $Diff(S^4) \\simeq \\mathbb O_5 \\times Diff(D^4)$ where $Diff(D^4)$ is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of $Diff(D^4)$.", "background": "Source: Open Problem Garden. Original node ID: 37159. URL: http://www.openproblemgarden.org/op/what_is_the_homotopy_type_of_the_group_of_diffeomorphisms_of_the_4_sphere.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/what_is_the_homotopy_type_of_the_group_of_diffeomorphisms_of_the_4_sphere\n- Author(s): Smale, S.\n- Subject(s): Topology\n- Keywords: 4-sphere; diffeomorphisms\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\n$Diff(D^4$ ) is known to be a $5$-fold loop space. In particular there is a homotopy-equivalence known as the Cerf-Morlet Comparison theorem $Diff(D^n) \\simeq \\Omega^{n+1} (PL_n / O_n)$ where $PL_n$ is the group of PL-automorphisms of $\\mathbb R^n$ and $O_n$ is the group of linear automorphisms of $\\mathbb R^n$. Otherwise there is not much in the literature about $Diff(D^4)$. Since it is a group of diffeomorphisms it has the homotopy type of a countable CW-complex. It is unknown whether or not it is connected, or if it has any other non-trivial homotopy or homology groups.\n\n$Diff(S^n)$ is known to have the homotopy-type of $O_{n+1}$ provided $n \\leq 3$ by work of Hatcher and Smale respectively. For $n \\geq 5$ many of the groups $\\pi_0 Diff(S^n)$ were computed by Kervaire and Milnor, who further related these groups to the homotopy groups of spheres. For $n \\geq 7$ the rational homotopy groups of $Diff(D^n)$ have been computed by Farrell and Hsiang in range $0 \\leq i < \\min\\{\\frac{n-4}{3}, \\frac{n-7}{2} \\}$. They show $\\pi_i Diff(D^n) \\otimes \\mathbb Q \\simeq \\left\\{ \\begin{array}{lr} \\mathbb Q & \\text{ provided }\\ 4 | (i+1) \\\\ 0 & \\text{ otherwise } \\end{array} \\right.$.\n\nBibliography:\n[B] Budney, R. Little cubes and long knots. Topology. 46 (2007) 1--27.\n\n[FH] Farrell, F.T. Hsiang, W.C. On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds. Proc. Symp. Pure. Math. 32 (1977) 403--415.\n\n[H] Hatcher, A proof of a Smale conjecture, ${\\rm Diff}(S\\sp{3})\\simeq {\\rm O}(4)$. Ann. of Math. (2) 117 (1983), no. 3, 553--607.\n\n[KS] Kirby, R. Siebenmann, L. Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. Princeton University Press.\n\n*[S] Smale, S. Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc. 10 (1959) 621--626.\n\nRelated:\nRelated problems\nSmooth 4-dimensional Schoenflies problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 19.\n\nAttempt notes:\nTarget:\nMake progress on \"What is the homotopy type of the group of diffeomorphisms of the 4-sphere?\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3418, "problem_number": "OPG-37161", "title": "Is there an algorithm to determine if a triangulated 4-manifold is combinatorially equivalent to the 4-sphere?", "statement": "Problem Is there an algorithm which takes as input a triangulated 4-manifold, and determines whether or not this manifold is combinatorially equivalent to the 4-sphere?", "background": "Source: Open Problem Garden. Original node ID: 37161. URL: http://www.openproblemgarden.org/op/is_there_an_algorithm_to_determine_if_a_triangulated_4_manifold_is_combinatorially_equivalent_to_the_4_sphere.\n\nSource subject path: Topology.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/is_there_an_algorithm_to_determine_if_a_triangulated_4_manifold_is_combinatorially_equivalent_to_the_4_sphere\n- Author(s): Novikov\n- Subject(s): Topology\n- Keywords: 4-sphere; algorithm\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\nA 4-manifold triangulation admits a unique smoothing up to diffeomorphism, so this problem is equivalent to asking for an algorithm to determine if a 4-manifold is diffeomorphic to the 4-sphere (with standard differentiable structure). \"Combinatorial equivalence\" refers to the ability to pass from one triangulation to another via a sequence of Pachner moves.\n\nRubinstein has an algorithm to determine if a triangulated 3-manifold is combinatorially equivalent to the 3-sphere. A consequence of his algorithm is that there is an algorithm to determine if a 4-dimensional simplicial complex is a 4-manifold triangulation.\n\nIt's known that no algorithms exist to determine if a triangulated 4-manifold has a trivial fundamental group, as there is a procedure to construct a compact 4-manifold with any finitely presented fundamental group.\n\nIn dimensions 5 and higher, Novikov proved that there is no algorithm to decide whether a given triangulated $n$-manifold is combinatorially equivalent to the $n$-sphere is undecidable [N, CL]. Also, it is undecidable whether a given triangulated 4-manifold is combinatorially equivalent to a connect sum of 14 copies of $S^2 \\times S^2$.\n\nBibliography:\n[CL] Chernavsky, A. V, and Leskine, V. P., Unrecognizability of manifolds, Annals of Pure and Applied Logic 141 (2006) 325--335.\n\n[N] Novikov, P.S., On the algorithSSSR 85 (5) (19552) 709--712 (in Russian). Algorithmic unsolvability of the problem of identity, Dokl. Akad. Nauk SSSR 85 (5) (1952) 709--712 (in Russian).\n\n[T] Thompson, A. Thin position and the recognition problem for $S^3$. MRL (1994).\n\nRelated:\nRelated problems\nSmooth 4-dimensional Poincare conjecture\nSmooth 4-dimensional Schoenflies problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Is there an algorithm to determine if a triangulated 4-manifold is combinatorially equivalent to the 4-sphere?\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3419, "problem_number": "OPG-37237", "title": "Unsolvability of word problem for 2-knot complements", "statement": "Problem Does there exist a smooth/PL embedding of $S^2$ in $S^4$ such that the fundamental group of the complement has an unsolvable word problem?", "background": "Source: Open Problem Garden. Original node ID: 37237. URL: http://www.openproblemgarden.org/op/unsolvability_of_word_problem_for_2_knot_complements.\n\nSource subject path: Topology.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/unsolvability_of_word_problem_for_2_knot_complements\n- Author(s): Gordon\n- Subject(s): Topology\n- Keywords: 2-knot; Computational Complexity; knot theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 23rd, 2010 by rybu\n\nProblem-page discussion:\nIt's known that there are smooth $4$-dimensional submanifolds of $S^4$ whose fundamental groups have unsolvable word problems. The complements of classical knots ( $S^1 \\to S^3$ ) are known to have solvable word problems, as do arbitrary $3$-manifold groups.\n\nBibliography:\nA. Dranisnikov, D. Repovs, \"Embeddings up to homotopy type in Euclidean Space\" Bull. Austral. Math. Soc (1993).\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Unsolvability of word problem for 2-knot complements\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3420, "problem_number": "OPG-37245", "title": "The 4x5 chessboard complex is the complement of a link, which link?", "statement": "Problem Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observation is non-constructive, as it uses the resolution of the Poincare Conjecture. Find specific links that have the 4x5 chessboard complex as their complement.", "background": "Source: Open Problem Garden. Original node ID: 37245. URL: http://www.openproblemgarden.org/op/the_4x5_chessboard_complex_is_the_complement_of_a_link_which_link.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_4x5_chessboard_complex_is_the_complement_of_a_link_which_link\n- Author(s): David Eppstein\n- Subject(s): Topology\n- Keywords: knot theory, links, chessboard complex\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 4th, 2010 by rybu\n\nProblem-page discussion:\nSee the MathOverFlow thread: Is the 4x5 chessboard complex a link complement?.\n\nBibliography:\n* D. Eppstein MathOverFlow thread\n\nDiscussion links:\n- Is the 4x5 chessboard complex a link complement?: http://mathoverflow.net/questions/36791/is-the-4x5-chessboard-complex-a-link-complement\n\nBibliography links:\n- MathOverFlow thread: http://mathoverflow.net/questions/36791/is-the-4x5-chessboard-complex-a-link-complement\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"The 4x5 chessboard complex is the complement of a link, which link?\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3421, "problem_number": "OPG-37282", "title": "Outer reloid of restricted funcoid", "statement": "Question $( \\mathsf{RLD})_{\\mathrm{out}} (f \\cap^{\\mathsf{FCD}} ( \\mathcal{A} \\times^{\\mathsf{FCD}} \\mathcal{B})) = (( \\mathsf{RLD})_{\\mathrm{out}} f) \\cap^{\\mathsf{RLD}} ( \\mathcal{A} \\times^{\\mathsf{RLD}} \\mathcal{B})$ for every filter objects $\\mathcal{A}$ and $\\mathcal{B}$ and a funcoid $f\\in\\mathsf{FCD}(\\mathrm{Src}\\,f; \\mathrm{Dst}\\,f)$?", "background": "Source: Open Problem Garden. Original node ID: 37282. URL: http://www.openproblemgarden.org/op/outer_reloid_of_restricted_funcoid.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/outer_reloid_of_restricted_funcoid\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: direct product of filters; outer reloid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: December 3rd, 2010 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography:\n*Victor Porton. Algebraic General Topology\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Outer reloid of restricted funcoid\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3422, "problem_number": "OPG-37293", "title": "Sticky Cantor sets", "statement": "Conjecture Let $C$ be a Cantor set embedded in $\\mathbb{R}^n$. Is there a self-homeomorphism $f$ of $\\mathbb{R}^n$ for every $\\epsilon$ greater than $0$ so that $f$ moves every point by less than $\\epsilon$ and $f(C)$ does not intersect $C$? Such an embedded Cantor set for which no such $f$ exists (for some $\\epsilon$ ) is called \"sticky\". For what dimensions $n$ do sticky Cantor sets exist?", "background": "Source: Open Problem Garden. Original node ID: 37293. URL: http://www.openproblemgarden.org/op/sticky_cantor_sets.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/sticky_cantor_sets\n- Subject(s): Topology\n- Keywords: Cantor set\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 6th, 2011 by porton\n\nProblem-page discussion:\nI borrowed this conjecture from this forum thread.\n\nCertainly I understand this conjecture wrongly: $C$ is a subset of a line segment. Consider a homeomorphism which moves all points of $\\mathbb{R}^n$ orthogonally to this line segment by $\\epsilon/2$. This would be a solution of this problem. Obviously it is not what is meant.\n\nIndeed I submit the problem to OPG as is in the hope that somebody will correct my wrong understanding and adjust the formulation to not be misunderstood as by me.\n\nSource links:\n- Cantor set: http://en.wikipedia.org/wiki/Cantor set\n\nDiscussion links:\n- this forum thread: http://www.mathkb.com/Uwe/Forum.aspx/math/16972/Current-Status-of-Topology\n\nComments:\n- July 29th, 2011 | Anonymous | Misunderstanding: Your misunderstanding comes from the definition of a Cantor set. A Cantor set is a set homeomorphic to the usual middle-thirds Cantor set. In general it does not have to lie on a line segment.\n- April 10th, 2012 | Anonymous | M: \"embedded\" does not imply that it is still a subset of the line. It just says that it's one-to-one and a homeomorphism with the image. The conjecture requires to prove that there exists a Cantor which cannot be separated from itself, so showing an example where it can be separated is not relevant.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Sticky Cantor sets\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3423, "problem_number": "OPG-37295", "title": "Nonseparating planar continuum", "statement": "Conjecture Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property?\n\nA set has the fixed point property if every continuous map from it into itself has a fixed point.", "background": "Source: Open Problem Garden. Original node ID: 37295. URL: http://www.openproblemgarden.org/op/nonseparating_planar_continuum.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/nonseparating_planar_continuum\n- Subject(s): Topology\n- Keywords: fixed point\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 6th, 2011 by porton\n\nComments:\n- February 16th, 2011 | Comet | Proof; some set of disks connected by line segments: 1) Any continuous map of this set to itself must traverse these line segments, and some of them will find their fixed point within one of these line segments, as any such mapping that has a submapping that maps a line segment to itself has a fixed point within that line segment. 2) For those mappings that haven't had a fixed point within one of the line segments above, must then have a submapping that maps a part of a disk to itself. This guarantees a fixed point will be found by Brouwer's fixed point theorem. *) Some of the mapping will map separate disks to each other, and there will be no fixed point in that part of the mapping. But how are the separate disks connected? Either they are connected along a line segment, in which case the fixed point must be there (see 1) or the disks are connected by a point, in which case the fixed point must be there at that point.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Nonseparating planar continuum\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3424, "problem_number": "OPG-37297", "title": "Hilbert-Smith conjecture", "statement": "Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group.", "background": "Source: Open Problem Garden. Original node ID: 37297. URL: http://www.openproblemgarden.org/op/hilbert_smith_conjecture.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hilbert_smith_conjecture\n- Author(s): David Hilbert; Paul A. Smith\n- Subject(s): Topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 6th, 2011 by porton\n\nProblem-page discussion:\nThis conjecture at Wikipedia\n\nSource links:\n- Lie group: http://en.wikipedia.org/wiki/Lie group\n\nDiscussion links:\n- This conjecture at Wikipedia: http://en.wikipedia.org/wiki/Hilbert-Smith_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Hilbert-Smith conjecture\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3425, "problem_number": "OPG-37339", "title": "Strict inequalities for products of filters", "statement": "Conjecture $\\mathcal{A} \\times^{\\mathsf{\\ensuremath{\\operatorname{RLD}}}}_F \\mathcal{B} \\subset \\mathcal{A} \\ltimes \\mathcal{B} \\subset \\mathcal{A} \\times^{\\mathsf{\\ensuremath{\\operatorname{RLD}}}} \\mathcal{B}$ for some filter objects $\\mathcal{A}$, $\\mathcal{B}$. Particularly, is this formula true for $\\mathcal{A} = \\mathcal{B} = \\Delta \\cap \\uparrow^{\\mathbb{R}} \\left( 0; + \\infty \\right)$?\n\nA weaker conjecture:\n\nConjecture $\\mathcal{A} \\times^{\\mathsf{\\ensuremath{\\operatorname{RLD}}}}_F \\mathcal{B} \\subset \\mathcal{A} \\ltimes \\mathcal{B}$ for some filter objects $\\mathcal{A}$, $\\mathcal{B}$.", "background": "Source: Open Problem Garden. Original node ID: 37339. URL: http://www.openproblemgarden.org/op/strict_inequalities_for_products_of_filters.\n\nSource subject path: Topology.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/strict_inequalities_for_products_of_filters\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: filter products\n- Importance: Low ✭\n- Recommended for undergraduates: no\n- Posted: August 9th, 2011 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nThe first conjecture probably has no use by itself but proving it may be somehow challenging, just like Fermat Last Theorem.\n\nBibliography:\n*Victor Porton. Algebraic General Topology\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Strict inequalities for products of filters\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3426, "problem_number": "OPG-37378", "title": "Funcoidal products inside an inward reloid", "statement": "Conjecture (solved) If $a \\times^{\\mathsf{\\ensuremath{\\operatorname{RLD}}}} b \\subseteq \\left( \\mathsf{\\ensuremath{\\operatorname{RLD}}} \\right)_{\\ensuremath{\\operatorname{in}}} f$ then $a \\times^{\\mathsf{\\ensuremath{\\operatorname{FCD}}}} b \\subseteq f$ for every funcoid $f$ and atomic f.o. $a$ and $b$ on the source and destination of $f$ correspondingly.\n\nA stronger conjecture:\n\nConjecture If $\\mathcal{A} \\times^{\\mathsf{\\ensuremath{\\operatorname{RLD}}}} \\mathcal{B} \\subseteq \\left( \\mathsf{\\ensuremath{\\operatorname{RLD}}} \\right)_{\\ensuremath{\\operatorname{in}}} f$ then $\\mathcal{A} \\times^{\\mathsf{\\ensuremath{\\operatorname{FCD}}}} \\mathcal{B} \\subseteq f$ for every funcoid $f$ and $\\mathcal{A} \\in \\mathfrak{F} \\left( \\ensuremath{\\operatorname{Src}}f \\right)$, $\\mathcal{B} \\in \\mathfrak{F} \\left( \\ensuremath{\\operatorname{Dst}}f \\right)$.", "background": "Source: Open Problem Garden. Original node ID: 37378. URL: http://www.openproblemgarden.org/op/funcolidal_products_inside_an_inward_reloid.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/funcolidal_products_inside_an_inward_reloid\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: inward reloid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: January 1st, 2012 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography:\n*Victor Porton. Algebraic General Topology\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Funcoidal products inside an inward reloid\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3427, "problem_number": "OPG-37385", "title": "Upgrading a completary multifuncoid", "statement": "Let $\\mho$ be a set, $\\mathfrak{F}$ be the set of filters on $\\mho$ ordered reverse to set-theoretic inclusion, $\\mathfrak{P}$ be the set of principal filters on $\\mho$, let $n$ be an index set. Consider the filtrator $\\left( \\mathfrak{F}^n; \\mathfrak{P}^n \\right)$.\n\nConjecture If $f$ is a completary multifuncoid of the form $\\mathfrak{P}^n$, then $E^{\\ast} f$ is a completary multifuncoid of the form $\\mathfrak{F}^n$.\n\nSee below for definition of all concepts and symbols used to in this conjecture.\n\nRefer to this Web site for the theory which I now attempt to generalize.", "background": "Source: Open Problem Garden. Original node ID: 37385. URL: http://www.openproblemgarden.org/op/upgrading_a_completary_multifuncoid.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/upgrading_a_completary_multifuncoid\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 4th, 2012 by porton\n\nProblem-page discussion:\nDefinition A filtrator is a pair $\\left( \\mathfrak{A}; \\mathfrak{Z} \\right)$ of a poset $\\mathfrak{A}$ and its subset $\\mathfrak{Z}$.\n\nHaving fixed a filtrator, we define:\n\nDefinition $\\ensuremath{\\operatorname{up}}x = \\left\\{ Y \\in \\mathfrak{Z} \\hspace{0.5em} | \\hspace{0.5em} Y \\geqslant x \\right\\}$ for every $X \\in \\mathfrak{A}$.\n\nDefinition $E^{\\ast} K = \\left\\{ L \\in \\mathfrak{A} \\hspace{0.5em} | \\hspace{0.5em} \\ensuremath{\\operatorname{up}}L \\subseteq K \\right\\}$ (upgrading the set $K$ ) for every $K \\in \\mathscr{P} \\mathfrak{Z}$.\n\nDefinition Let $\\mathfrak{A}$ is a family of join-semilattice. A completary multifuncoid of the form $\\mathfrak{A}$ is an $f \\in \\mathscr{P} \\prod \\mathfrak{A}$ such that we have that:\n\n- $L_0 \\cup L_1 \\in f \\Leftrightarrow \\exists c \\in \\left\\{ 0, 1 \\right\\}^n: \\left( \\lambda i \\in n: L_{c \\left( i_{} \\right)} i \\right) \\in f$ for every $L_0, L_1 \\in \\prod \\mathfrak{A}$.\n\n- If $L \\in \\prod \\mathfrak{A}$ and $L_i = 0^{\\mathfrak{A}_i}$ for some $i$ then $\\neg f L$.\n\n$\\mathfrak{A}^n$ is a function space over a poset $\\mathfrak{A}$ that is $a\\le b\\Leftrightarrow \\forall i\\in n:a_i\\le b_i$ for $a,b\\in\\mathfrak{A}^n$.\n\nFor finite $n$ this problem is equivalent to Upgrading a multifuncoid.\n\nIt is not hard to prove this conjecture for the case $\\ensuremath{\\operatorname{card}}n \\leqslant 2$ using the techniques from this my article. But I failed to prove it for $\\ensuremath{\\operatorname{card}}n = 3$ and above.\n\nBibliography:\n* Conjecture: Upgrading a multifuncoid\n\nRelated:\nRelated problems\nUpgrading a multifuncoid\n\nSource links:\n- this Web site: http://www.mathematics21.org/algebraic-general-topology.html\n\nDiscussion links:\n- Upgrading a multifuncoid: http://www.openproblemgarden.org/?q=node/37348\n- this my article: http://www.mathematics21.org/binaries/funcoids-reloids.pdf\n\nBibliography links:\n- Conjecture: Upgrading a multifuncoid: http://portonmath.wordpress.com/2011/10/09/conjecture-upgrading-multifuncoid/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 29.\n\nAttempt notes:\nTarget:\nMake progress on \"Upgrading a completary multifuncoid\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3428, "problem_number": "OPG-37386", "title": "Atomicity of the poset of completary multifuncoids", "statement": "Conjecture The poset of completary multifuncoids of the form $(\\mathscr{P}\\mho)^n$ is for every sets $\\mho$ and $n$:\n\n- atomic;\n- atomistic.\n\nSee below for definition of all concepts and symbols used to in this conjecture.\n\nRefer to this Web site for the theory which I now attempt to generalize.", "background": "Source: Open Problem Garden. Original node ID: 37386. URL: http://www.openproblemgarden.org/op/atomicity_of_the_poset_of_multifuncoids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/atomicity_of_the_poset_of_multifuncoids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: multifuncoid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 12th, 2012 by porton\n\nProblem-page discussion:\nDefinition Let $\\mathfrak{A}$ is a family of join-semilattice. A completary multifuncoid of the form $\\mathfrak{A}$ is an $f \\in \\mathscr{P} \\prod \\mathfrak{A}$ such that we have that:\n\n- $L_0 \\cup L_1 \\in f \\Leftrightarrow \\exists c \\in \\left\\{ 0, 1 \\right\\}^n: \\left( \\lambda i \\in n: L_{c \\left( i_{} \\right)} i \\right) \\in f$ for every $L_0, L_1 \\in \\prod \\mathfrak{A}$.\n\n- If $L \\in \\prod \\mathfrak{A}$ and $L_i = 0^{\\mathfrak{A}_i}$ for some $i$ then $\\neg f L$.\n\n$\\mathfrak{A}^n$ is a function space over a poset $\\mathfrak{A}$ that is $a\\le b\\Leftrightarrow \\forall i\\in n:a_i\\le b_i$ for $a,b\\in\\mathfrak{A}^n$.\n\nBibliography:\n* Algebraic General Topology\n\nRelated:\nRelated problems\nAtomicity of the poset of multifuncoids\n\nSource links:\n- this Web site: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Atomicity of the poset of completary multifuncoids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3429, "problem_number": "OPG-37388", "title": "Atomicity of the poset of multifuncoids", "statement": "Conjecture The poset of multifuncoids of the form $(\\mathscr{P}\\mho)^n$ is for every sets $\\mho$ and $n$:\n\n- atomic;\n- atomistic.\n\nSee below for definition of all concepts and symbols used to in this conjecture.\n\nRefer to this Web site for the theory which I now attempt to generalize.", "background": "Source: Open Problem Garden. Original node ID: 37388. URL: http://www.openproblemgarden.org/op/atomicity_of_the_poset_of_multifuncoids_0.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/atomicity_of_the_poset_of_multifuncoids_0\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: multifuncoid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 12th, 2012 by porton\n\nProblem-page discussion:\nDefinition A free star on a join-semilattice $\\mathfrak{A}$ with least element 0 is a set $S$ such that $0 \\not\\in S$ and\n$$\n\\forall A, B \\in \\mathfrak{A}: \\left( A \\cup B \\in S \\Leftrightarrow A \\in S \\vee B \\in S \\right).\n$$\n\nDefinition Let $\\mathfrak{A}$ be a family of posets, $f \\in \\mathscr{P} \\prod \\mathfrak{A}$ ( $\\prod \\mathfrak{A}$ has the order of function space of posets), $i \\in \\ensuremath{\\operatorname{dom}}\\mathfrak{A}$, $L \\in \\prod \\mathfrak{A}|_{\\left( \\ensuremath{\\operatorname{dom}}\\mathfrak{A} \\right) \\setminus \\left\\{ i \\right\\}}$. Then\n$$\n\\left( \\ensuremath{\\operatorname{val}}f \\right)_i L = \\left\\{ X \\in \\mathfrak{A}_i \\hspace{0.5em} | \\hspace{0.5em} L \\cup \\left\\{ (i; X) \\right\\} \\in f \\right\\}.\n$$\n\nDefinition Let $\\mathfrak{A}$ is a family of posets. A multidimensional funcoid (or multifuncoid for short) of the form $\\mathfrak{A}$ is an $f \\in \\mathscr{P} \\prod \\mathfrak{A}$ such that we have that:\n\n- $\\left( \\tmop{val} f \\right)_i L$ is a free star for every $i \\in \\tmop{dom} \\mathfrak{A}$, $L \\in \\prod \\mathfrak{A}|_{\\left( \\tmop{dom} \\mathfrak{A} \\right) \\setminus \\left\\{ i \\right\\}}$.\n\n- $f$ is an upper set.\n\n$\\mathfrak{A}^n$ is a function space over a poset $\\mathfrak{A}$ that is $a\\le b\\Leftrightarrow \\forall i\\in n:a_i\\le b_i$ for $a,b\\in\\mathfrak{A}^n$.\n\nBibliography:\n* Algebraic General Topology\n\nRelated:\nRelated problems\nAtomicity of the poset of completary multifuncoids\n\nSource links:\n- this Web site: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 19.\n\nAttempt notes:\nTarget:\nMake progress on \"Atomicity of the poset of multifuncoids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3430, "problem_number": "OPG-37389", "title": "Graph product of multifuncoids", "statement": "Conjecture Let $F$ is a family of multifuncoids such that each $F_i$ is of the form $\\lambda j \\in N \\left( i \\right): \\mathfrak{F} \\left( U_j \\right)$ where $N \\left( i \\right)$ is an index set for every $i$ and $U_j$ is a set for every $j$. Let every $F_i = E^{\\ast} f_i$ for some multifuncoid $f_i$ of the form $\\lambda j \\in N \\left( i \\right): \\mathfrak{P} \\left( U_j \\right)$ regarding the filtrator $\\left( \\prod_{j \\in N \\left( i \\right)} \\mathfrak{F} \\left( U_j \\right); \\prod_{j \\in N \\left( i \\right)} \\mathfrak{P} \\left( U_j \\right) \\right)$. Let $H$ is a graph-composition of $F$ (regarding some partition $G$ and external set $Z$ ). Then there exist a multifuncoid $h$ of the form $\\lambda j \\in Z: \\mathfrak{P} \\left( U_j \\right)$ such that $H = E^{\\ast} h$ regarding the filtrator $\\left( \\prod_{j \\in Z} \\mathfrak{F} \\left( U_j \\right); \\prod_{j \\in Z} \\mathfrak{P} \\left( U_j \\right) \\right)$.", "background": "Source: Open Problem Garden. Original node ID: 37389. URL: http://www.openproblemgarden.org/op/graph_product_of_multifuncoids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/graph_product_of_multifuncoids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: graph-product; multifuncoid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 12th, 2012 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology, especially the theory of multifuncoids for definitions of used concepts.\n\nBibliography:\n*Victor Porton. Algebraic General Topology\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n- the theory of multifuncoids: http://www.mathematics21.org/binaries/nary.pdf\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Graph product of multifuncoids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3431, "problem_number": "OPG-37540", "title": "A conjecture about direct product of funcoids", "statement": "Conjecture Let $f_1$ and $f_2$ are monovalued, entirely defined funcoids with $\\operatorname{Src}f_1=\\operatorname{Src}f_2=A$. Then there exists a pointfree funcoid $f_1 \\times^{\\left( D \\right)} f_2$ such that (for every filter $x$ on $A$ ) $$\\left\\langle f_1 \\times^{\\left( D \\right)} f_2 \\right\\rangle x = \\bigcup \\left\\{ \\langle f_1\\rangle X \\times^{\\mathsf{FCD}} \\langle f_2\\rangle X \\hspace{1em} | \\hspace{1em} X \\in \\mathrm{atoms}^{\\mathfrak{A}} x \\right\\}.$$ (The join operation is taken on the lattice of filters with reversed order.)\n\nA positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.", "background": "Source: Open Problem Garden. Original node ID: 37540. URL: http://www.openproblemgarden.org/op/a_conjecture_about_direct_product_of_funcoids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_conjecture_about_direct_product_of_funcoids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: category theory; general topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 26th, 2012 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography:\n*Victor Porton. a blog post\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- a blog post: http://portonmath.wordpress.com/2012/07/26/conjecture-direct-product-funcoids/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"A conjecture about direct product of funcoids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3432, "problem_number": "OPG-48767", "title": "Closing Lemma for Diffeomorphism (Dynamical Systems)", "statement": "Conjecture Let $f\\in Diff^{r}(M)$ and $p\\in\\omega_{f}$. Then for any neighborhood $V_{f}\\subset Diff^{r}(M)$ there is $g\\in V_{f}$ such that $p$ is periodic point of $g$\n\nThere is an analogous conjecture for flows ( $C^{r}$ vector fields. In the case of diffeos this was proved by Charles Pugh for $r = 1$. In the case of Flows this has been solved by Sushei Hayahshy for $r = 1$. But in the two cases the problem is wide open for $r > 1$", "background": "Source: Open Problem Garden. Original node ID: 48767. URL: http://www.openproblemgarden.org/op/closing_lemma_for_diffeomorphism_dynamical_systems.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/closing_lemma_for_diffeomorphism_dynamical_systems\n- Author(s): Charles Pugh\n- Subject(s): Topology\n- Keywords: Dynamics, Pertubation\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 24th, 2013 by Jailton Viana\n\nBibliography:\nDynamics beyond uniform hyperbolicity:\\Springer [Encyclopaedia of Mathematical Sciences Volume 102, Mathematical Phisics,2005]\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Closing Lemma for Diffeomorphism (Dynamical Systems)\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3433, "problem_number": "OPG-48770", "title": "Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems)", "statement": "Conjecture Let $Diff^{r}(M)$ be the space of $C^{r}$ Diffeomorphisms on the connected, compact and boundaryles manifold M and $\\chi^{r}(M)$ the space of $C^{r}$ vector fields. There is a dense set $D\\subset Diff^{r}(M)$ ( $D\\subset \\chi^{r}(M)$ ) such that $\\forall f\\in D$ exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space $M$\n\nThis is a very Deep and Hard problem in Dynamical Systems. It present the dream of the dynamicist mathematicians.", "background": "Source: Open Problem Garden. Original node ID: 48770. URL: http://www.openproblemgarden.org/op/jacob_palis_conjecture_finitude_of_attractors_dynamical_systems.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/jacob_palis_conjecture_finitude_of_attractors_dynamical_systems\n- Subject(s): Topology\n- Keywords: Attractors, basins, Finite\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 24th, 2013 by Jailton Viana\n\nProblem-page discussion:\nDefinition: A set $\\Lambda \\subset M$ is an attractor for a Diffeomorphism (or a flow ) if it is invariant, transitive and the basin of attraction $B(\\Lambda):= \\{p\\in M / \\omega(p)\\subset \\Lambda \\}$ has positive Lebesgue Measure.\n\nBibliography:\nBonatti C, Diaz L.; Viana M.; Dynamics beyond uniform hyperbolicity, Springer[Encyclopaedia of Mathematics Sciences ], Volume 102, 2005\n\nRelated:\nRelated problems\nClosing Lemma for Diffeomorphism (Dynamical Systems)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems)\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3434, "problem_number": "OPG-56573", "title": "Decomposition of completions of reloids", "statement": "Conjecture For composable reloids $f$ and $g$ it holds\n\n- $\\operatorname{Compl} ( g \\circ f) = ( \\operatorname{Compl} g) \\circ f$ if $f$ is a co-complete reloid;\n- $\\operatorname{CoCompl} ( f \\circ g) = f \\circ \\operatorname{CoCompl} g$ if $f$ is a complete reloid;\n- $\\operatorname{CoCompl} ( ( \\operatorname{Compl} g) \\circ f) = \\operatorname{Compl} ( g \\circ ( \\operatorname{CoCompl} f)) = ( \\operatorname{Compl} g) \\circ ( \\operatorname{CoCompl} f)$;\n- $\\operatorname{Compl} ( g \\circ ( \\operatorname{Compl} f)) = \\operatorname{Compl} ( g \\circ f)$;\n- $\\operatorname{CoCompl} ( ( \\operatorname{CoCompl} g) \\circ f) = \\operatorname{CoCompl} ( g \\circ f)$.", "background": "Source: Open Problem Garden. Original node ID: 56573. URL: http://www.openproblemgarden.org/op/decomposition_of_completions_of_reloids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposition_of_completions_of_reloids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: co-completion; completion; reloid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 2nd, 2013 by porton\n\nProblem-page discussion:\nWell, in fact this is three separate problems (if we count dual formulas as one formula), but I am lazy to create three pages for them.\n\nThis conjecture is inspired by the proven fact that the above formulas hold for every composable funcoids $f$ and $g$ (instead of reloids). Properties of reloids are expected to be similar to properties of funcoids.\n\nhttp://www.packersandmoverschandigarh.co.in/\nhttp://www.packersandmoversjaipur.co.in/\nhttp://www.packersandmoversinhyderabad.co.in/\nhttp://www.packersandmoversinbangalore.co.in/\n\nBibliography:\n*Algebraic General Toplogy. Volume 1\n\nDiscussion links:\n- http://www.packersandmoverschandigarh.co.in/: http://www.packersandmoverschandigarh.co.in/\n- http://www.packersandmoversjaipur.co.in/: http://www.packersandmoversjaipur.co.in/\n- http://www.packersandmoversinhyderabad.co.in/: http://www.packersandmoversinhyderabad.co.in/\n- http://www.packersandmoversinbangalore.co.in/: http://www.packersandmoversinbangalore.co.in/\n\nBibliography links:\n- Algebraic General Toplogy. Volume 1: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposition of completions of reloids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3435, "problem_number": "OPG-57401", "title": "Every metamonovalued funcoid is monovalued", "statement": "Conjecture Every metamonovalued funcoid is monovalued.\n\nThe reverse is almost trivial: Every monovalued funcoid is metamonovalued.", "background": "Source: Open Problem Garden. Original node ID: 57401. URL: http://www.openproblemgarden.org/op/every_metamonovalued_funcoid_is_monovalued.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/every_metamonovalued_funcoid_is_monovalued\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: monovalued\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 17th, 2013 by porton\n\nBibliography:\n*Algebraic General Toplogy. Volume 1\n\nRelated:\nRelated problems\nEvery metamonovalued reloid is monovalued\n\nBibliography links:\n- Algebraic General Toplogy. Volume 1: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Every metamonovalued funcoid is monovalued\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3436, "problem_number": "OPG-57403", "title": "Every metamonovalued reloid is monovalued", "statement": "Conjecture Every metamonovalued reloid is monovalued.", "background": "Source: Open Problem Garden. Original node ID: 57403. URL: http://www.openproblemgarden.org/op/every_metamonovalued_reloid_is_monovalued.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/every_metamonovalued_reloid_is_monovalued\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 17th, 2013 by porton\n\nBibliography:\n*Algebraic General Toplogy. Volume 1\n\nRelated:\nRelated problems\nEvery metamonovalued funcoid is monovalued\nEvery monovalued reloid is metamonovalued\n\nBibliography links:\n- Algebraic General Toplogy. Volume 1: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Every metamonovalued reloid is monovalued\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3437, "problem_number": "OPG-59896", "title": "Generalized path-connectedness in proximity spaces", "statement": "Let $\\delta$ be a proximity.\n\nA set $A$ is connected regarding $\\delta$ iff $\\forall X,Y \\in \\mathscr{P} A \\setminus \\{ \\emptyset \\}: \\left( X \\cup Y = A \\Rightarrow X \\mathrel{\\delta} Y \\right)$.\n\nConjecture The following statements are equivalent for every endofuncoid $\\mu$ and a set $U$:\n\n- $U$ is connected regarding $\\mu$.\n- For every $a, b \\in U$ there exists a totally ordered set $P \\subseteq U$ such that $\\min P = a$, $\\max P = b$, and for every partion $\\{ X, Y \\}$ of $P$ into two sets $X$, $Y$ such that $\\forall x \\in X, y \\in Y: x < y$, we have $X \\mathrel{[ \\mu]^{\\ast}} Y$.", "background": "Source: Open Problem Garden. Original node ID: 59896. URL: http://www.openproblemgarden.org/op/generalized_path_connectedness_in_proximity_spaces.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/generalized_path_connectedness_in_proximity_spaces\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: connected; connectedness; proximity space\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 1st, 2014 by porton\n\nBibliography:\n*Question at math.StackExchange.com by Victor Porton\n\nSource links:\n- proximity: http://en.wikipedia.org/wiki/Proximity_space\n\nBibliography links:\n- Question at math.StackExchange.com: http://math.stackexchange.com/questions/642337/connectedness-in-proximity-spaces\n\nComments:\n- February 26th, 2014 | porton | A proposed lemma: http://math.stackexchange.com/questions/691643/a-lemma-to-solve-a-conjec...\n\n--\n\nVictor Porton - http://www.mathematics21.org\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Generalized path-connectedness in proximity spaces\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3438, "problem_number": "OPG-59900", "title": "Direct proof of a theorem about compact funcoids", "statement": "Conjecture Let $f$ is a $T_1$-separable (the same as $T_2$ for symmetric transitive) compact funcoid and $g$ is a uniform space (reflexive, symmetric, and transitive endoreloid) such that $( \\mathsf{\\tmop{FCD}}) g = f$. Then $g = \\langle f \\times f \\rangle^{\\ast} \\Delta$.\n\nThe main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.\n\nThe direct proof may be constructed by correcting all errors an omissions in this draft article.\n\nDirect proof could be better because with it we would get a little more general statement like this:\n\nConjecture Let $f$ be a $T_1$-separable compact reflexive symmetric funcoid and $g$ be a reloid such that\n\n- $( \\mathsf{\\tmop{FCD}}) g = f$;\n- $g \\circ g^{- 1} \\sqsubseteq g$.\n\nThen $g = \\langle f \\times f \\rangle^{\\ast} \\Delta$.", "background": "Source: Open Problem Garden. Original node ID: 59900. URL: http://www.openproblemgarden.org/op/direct_proof_of_a_theorem_about_compact_funcoids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/direct_proof_of_a_theorem_about_compact_funcoids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 8th, 2014 by porton\n\nBibliography:\nVictor Porton. Compact funcoids\n\nSource links:\n- this draft article: http://www.mathematics21.org/binaries/compact.pdf\n\nBibliography links:\n- Compact funcoids: http://www.mathematics21.org/binaries/compact.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Direct proof of a theorem about compact funcoids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3439, "problem_number": "OPG-59970", "title": "Another conjecture about reloids and funcoids", "statement": "Definition $\\square f = \\bigcap^{\\mathsf{RLD}} \\mathrm{up}^{\\Gamma (\\operatorname{Src} f; \\operatorname{Dst} f)} f$ for reloid $f$.\n\nConjecture $(\\mathsf{RLD})_{\\Gamma} f = \\square (\\mathsf{RLD})_{\\mathrm{in}} f$ for every funcoid $f$.\n\nNote: it is known that $(\\mathsf{RLD})_{\\Gamma} f \\ne \\square (\\mathsf{RLD})_{\\mathrm{out}} f$ (see below mentioned online article).", "background": "Source: Open Problem Garden. Original node ID: 59970. URL: http://www.openproblemgarden.org/op/another_conjecture_about_reloids_and_funcoids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/another_conjecture_about_reloids_and_funcoids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 28th, 2014 by porton\n\nProblem-page discussion:\nIt's used notation from Algebraic General Topology draft book, modified by this note about new notation for a future version of this book.\n\nBibliography:\n* blog post\n\nDiscussion links:\n- Algebraic General Topology draft book: http://www.mathematics21.org/algebraic-general-topology.html\n- this note: http://www.mathematics21.org/binaries/rewrite-plan.pdf\n\nBibliography links:\n- blog post: http://portonmath.wordpress.com/2014/11/28/some-new/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Another conjecture about reloids and funcoids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3440, "problem_number": "OPG-59973", "title": "What are hyperfuncoids isomorphic to?", "statement": "Let $\\mathfrak{A}$ be an indexed family of sets.\n\nProducts are $\\prod A$ for $A \\in \\prod \\mathfrak{A}$.\n\nHyperfuncoids are filters $\\mathfrak{F} \\Gamma$ on the lattice $\\Gamma$ of all finite unions of products.\n\nProblem Is $\\bigcap^{\\mathsf{\\tmop{FCD}}}$ a bijection from hyperfuncoids $\\mathfrak{F} \\Gamma$ to:\n\n- prestaroids on $\\mathfrak{A}$;\n- staroids on $\\mathfrak{A}$;\n- completary staroids on $\\mathfrak{A}$?\n\nIf yes, is $\\operatorname{up}^{\\Gamma}$ defining the inverse bijection? If not, characterize the image of the function $\\bigcap^{\\mathsf{\\tmop{FCD}}}$ defined on $\\mathfrak{F} \\Gamma$.\n\nConsider also the variant of this problem with the set $\\Gamma$ replaced with the set $\\Gamma^{\\ast}$ of complements of elements of the set $\\Gamma$.", "background": "Source: Open Problem Garden. Original node ID: 59973. URL: http://www.openproblemgarden.org/op/what_are_hyperfuncoids_isomorphic_to.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/what_are_hyperfuncoids_isomorphic_to\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: hyperfuncoids; multidimensional\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: December 9th, 2014 by porton\n\nProblem-page discussion:\nIt's used notation from Algebraic General Topology draft book\n\nDiscussion links:\n- Algebraic General Topology draft book: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"What are hyperfuncoids isomorphic to?\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3441, "problem_number": "OPG-60017", "title": "Infinite distributivity of meet over join for a principal funcoid", "statement": "Conjecture $f \\sqcap \\bigsqcup S = \\bigsqcup \\langle f \\sqcap \\rangle^{\\ast} S$ for principal funcoid $f$ and a set $S$ of funcoids of appropriate sources and destinations.", "background": "Source: Open Problem Garden. Original node ID: 60017. URL: http://www.openproblemgarden.org/op/infinite_distributivity_of_meet_over_join_for_a_principal_funcoid.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/infinite_distributivity_of_meet_over_join_for_a_principal_funcoid\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: distributivity; principal funcoid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 27th, 2016 by porton\n\nProblem-page discussion:\nIt's used notation from Algebraic General Topology book\n\nBibliography:\n*Victor Porton. A blog post\n\nDiscussion links:\n- Algebraic General Topology book: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- A blog post: https://portonmath.wordpress.com/2016/07/27/new-conjecture-2/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Infinite distributivity of meet over join for a principal funcoid\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3442, "problem_number": "OPG-60019", "title": "A funcoid related to directed topological spaces", "statement": "Conjecture Let $R$ be the complete funcoid corresponding to the usual topology on extended real line $[-\\infty,+\\infty] = \\mathbb{R}\\cup\\{-\\infty,+\\infty\\}$. Let $\\geq$ be the order on this set. Then $R\\sqcap^{\\mathsf{FCD}}\\mathord{\\geq}$ is a complete funcoid.\n\nProposition It is easy to prove that $\\langle R\\sqcap^{\\mathsf{FCD}}\\mathord{\\geq}\\rangle \\{x\\}$ is the infinitely small right neighborhood filter of point $x\\in[-\\infty,+\\infty]$.\n\nIf proved true, the conjecture then can be generalized to a wider class of posets.", "background": "Source: Open Problem Garden. Original node ID: 60019. URL: http://www.openproblemgarden.org/op/a_funcoid_related_to_directed_topological_spaces.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_funcoid_related_to_directed_topological_spaces\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 28th, 2016 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography:\n* Blog post\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Blog post: https://portonmath.wordpress.com/2016/07/28/funcoid-related-directed-topologies/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"A funcoid related to directed topological spaces\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3443, "problem_number": "OPG-60020", "title": "Outward reloid of composition vs composition of outward reloids", "statement": "Conjecture For every composable funcoids $f$ and $g$ $$(\\mathsf{RLD})_{\\mathrm{out}}(g\\circ f)\\sqsupseteq(\\mathsf{RLD})_{\\mathrm{out}}g\\circ(\\mathsf{RLD})_{\\mathrm{out}}f.$$", "background": "Source: Open Problem Garden. Original node ID: 60020. URL: http://www.openproblemgarden.org/op/outward_reloid_of_composition_vs_composition_of_outward_reloids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/outward_reloid_of_composition_vs_composition_of_outward_reloids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: outward reloid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 10th, 2016 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography:\n* Blog post\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Blog post: https://portonmath.wordpress.com/2016/09/10/conjecture-outward-funcoids/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Outward reloid of composition vs composition of outward reloids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3444, "problem_number": "OPG-60024", "title": "A diagram about funcoids and reloids", "statement": "Define for posets with order $\\sqsubseteq$:\n\n- $\\Phi_{\\ast} f = \\lambda b \\in \\mathfrak{B}: \\bigcup \\{ x \\in \\mathfrak{A} \\mid f x \\sqsubseteq b \\}$;\n\n- $\\Phi^{\\ast} f = \\lambda b \\in \\mathfrak{A}: \\bigcap \\{ x \\in \\mathfrak{B} \\mid f x \\sqsupseteq b \\}$.\n\nNote that the above is a generalization of monotone Galois connections (with $\\max$ and $\\min$ replaced with suprema and infima).\n\nThen we have the following diagram:\n\nWhat is at the node \"other\" in the diagram is unknown.\n\nConjecture \"Other\" is $\\lambda f\\in\\mathsf{FCD}: \\top$.\n\nQuestion What repeated applying of $\\Phi_{\\ast}$ and $\\Phi^{\\ast}$ to \"other\" leads to? Particularly, does repeated applying $\\Phi_{\\ast}$ and/or $\\Phi^{\\ast}$ to the node \"other\" lead to finite or infinite sets?", "background": "Source: Open Problem Garden. Original node ID: 60024. URL: http://www.openproblemgarden.org/op/a_diagram_about_funcoids_and_reloids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_diagram_about_funcoids_and_reloids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: Galois connections\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 26th, 2016 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nThe known part of the diagram is considered in this file.\n\nBibliography:\nBlog post\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n- this file: http://www.mathematics21.org/binaries/addons.pdf\n\nBibliography links:\n- Blog post: https://portonmath.wordpress.com/2016/11/26/new-diagram/\n\nComments:\n- November 29th, 2016 | porton | The value of node \"other\": It seems that the node \"other\" is not $\\lambda f\\in\\mathsf{FCD}: \\top$.\n\nI conjecture $\\langle \\Phi_{\\ast} (\\mathsf{RLD})_{\\operatorname{out}} \\rangle f = (\\mathsf{FCD}) f$ where $f$ is the reloid defined by the cofinite filter on $A \\times B$ and thus $\\langle (\\mathsf{FCD}) f \\rangle \\{ x \\} = \\bot$ for all singletons $\\{ x \\}$ and $\\langle (\\mathsf{FCD}) f \\rangle p = \\top$ for every nontrivial atomic filter $p$.\n\nThis is my very recent thoughts and yet needs to be checked.\n\n-- Victor Porton - http://www.mathematics21.org\n- November 26th, 2016 | porton | The diagram was with an error: My diagram was with an error. I have uploaded a corrected version of the diagram.\n\n--\n\nVictor Porton - http://www.mathematics21.org\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"A diagram about funcoids and reloids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3445, "problem_number": "OPG-60026", "title": "Which outer reloids are equal to inner ones", "statement": "Warning: This formulation is vague (not exact).\n\nQuestion Characterize the set $\\{f\\in\\mathsf{FCD} \\mid (\\mathsf{RLD})_{\\mathrm{in}} f=(\\mathsf{RLD})_{\\mathrm{out}} f\\}$. In other words, simplify this formula.\n\nThe problem seems rather difficult.", "background": "Source: Open Problem Garden. Original node ID: 60026. URL: http://www.openproblemgarden.org/op/what_outer_reloids_are_equal_to_inner_ones.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/what_outer_reloids_are_equal_to_inner_ones\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: December 1st, 2016 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography:\nBlog post\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Blog post: https://portonmath.wordpress.com/2016/12/01/open-problem/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Which outer reloids are equal to inner ones\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3446, "problem_number": "OPG-60043", "title": "Several ways to apply a (multivalued) multiargument function to a family of filters", "statement": "Problem Let $\\mathcal{X}$ be an indexed family of filters on sets. Which of the below items are always pairwise equal?\n\n1. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters $\\mathcal{X}$.\n\n2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters $\\mathcal{X}$.\n\n3. $\\bigcap_{F\\in\\operatorname{up}^{\\mathrm{FCD}}\\prod^{\\mathrm{Strd}}\\mathcal{X}}\\langle f \\rangle F$.", "background": "Source: Open Problem Garden. Original node ID: 60043. URL: http://www.openproblemgarden.org/op/several_ways_to_apply_a_multivalued_multiargument_function_to_a_family_of_filters.\n\nSource subject path: Topology.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/several_ways_to_apply_a_multivalued_multiargument_function_to_a_family_of_filters\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: funcoid; function; multifuncoid; staroid\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: December 24th, 2019 by porton\n\nBibliography:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography links:\n- Algebraic General Topology: https://mathematics21.org/algebraic-general-topology-and-math-synthesis/\n\nComments:\n- January 15th, 2020 | porton | There was an error in the problem statement: I did an error in the problem statement (corrected). We need to consider the upgraded staroid, not just staroid.\n\n-- Victor Porton - http://www.mathematics21.org\n- January 15th, 2020 | porton | The error was more severe: The first item of the problem was entirely wrong. I removed it.\n\n-- Victor Porton - http://www.mathematics21.org\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Several ways to apply a (multivalued) multiargument function to a family of filters\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } }, { "id": 3447, "problem_number": "OPG-581", "title": "Rendezvous on a line", "statement": "Problem Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum speed of 1. What is the infimum expected meeting time $R$ (first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy?", "background": "Source: Open Problem Garden. Original node ID: 581. URL: http://www.openproblemgarden.org/op/rendezvous_on_a_line.\n\nSource subject path: Unsorted.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/rendezvous_on_a_line\n- Author(s): Alpern, Steve\n- Subject(s): Unsorted\n- Keywords: game theory; optimization; rendezvous\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 21st, 2007 by mdevos\n\nProblem-page discussion:\nThis is one of a handful of rendezvous problems where two players must find one another in a certain structured domain. See [AG2] for a thorough development of this subject. This is a symmetric rendezvous problem since each player is forced to adopt the same strategy. If we drop this constraint, Alpern and Gal [AG] have shown that the inf expected meeting time is 3.25.\n\nHan, Du, Vera, and Zuluaga [HDVZ] have shown that strategies in which the players move at maximum speed and only change direction at integer times dominate among all possible strategies - thus reducing this problem to a discrete one. These same authors improve upon a series of results by tightening the upper and lower bounds, proving $4.1520 < R < 4.2574$. Further, they conjecture $R=4.25$.\n\nBibliography:\n*[A] S. Alpern, The rendezvous search problem. SIAM J. Control Optim. 33 (1995), no. 3, 673--683 MathSciNet\n\n[AG1] S. Alpern and S. Gal, Rendezvous search on the line with distinguishable players. SIAM J. Control Optim. 33 (1995), no. 4, 1270--1276. MathSciNet\n\n[AG2] S. Alpern and S. Gal, The theory of search games and rendezvous. International Series in Operations Research & Management Science, 55. Kluwer Academic Publishers, Boston, MA, 2003. MathSciNet\n\n[HDVZ] Q. Han, D. Du, J. C. Vera, and L. F. Zuluaga, Improved bounds for the symmetric rendezvous search problem on the line\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1327232\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1339065\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2005053\n- Improved bounds for the symmetric rendezvous search problem on the line: http://www.optimization-online.org/DB_FILE/2006/05/1400.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Rendezvous on a line\" in Unsorted, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.", "difficulty_level_id": 2, "status": "open", "category_id": 20, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-05-13T14:45:21.187Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-05-13T14:45:21.187Z" } } ]