Category
Problem Set
Status
Hamiltonian Decomposition of Hypergraphs
Do complete k-uniform hypergraphs admit Hamiltonian decompositions into tight cycles?...
Characterization of Word-Representable Planar Graphs
Characterize which planar graphs are word-representable....
Word-Representable Graphs: Forbidden Subgraph Characterization
Characterize word-representable graphs in terms of forbidden induced subgraphs....
Word-Representable Near-Triangulations
Characterize word-representable near-triangulations containing K₄....
Line Graphs of Non-Word-Representable Graphs
Is the line graph of a non-word-representable graph always non-word-representable?...
Translating Graph Problems to Word Problems
Which hard graph problems can be efficiently solved by translating graphs to their word representations?...
Implicit Graph Conjecture
Do slowly-growing hereditary graph families admit implicit representations?...
Ryser's Conjecture
For r-partite r-uniform hypergraphs, is the vertex cover number at most (r-1) times the matching number?...
Second Neighborhood Problem
Does every oriented graph have a vertex with at least as many vertices at distance 2 as at distance 1?...
Woodall's Conjecture
Is the minimum dicut size equal to the maximum number of disjoint dijoins in a directed graph?...
Casas-Alvero Conjecture
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 $(x-a)^d$?...
Crouzeix's Conjecture
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)?...
Determinantal Conjecture
Characterize the determinant of the sum of two normal matrices....
Eilenberg-Ganea Conjecture
Does every group with cohomological dimension 2 have a 2-dimensional Eilenberg-MacLane space K(G,1)?...
Finite Lattice Representation Problem
Is every finite lattice isomorphic to the congruence lattice of some finite algebra?...
Hadamard Matrix Conjecture
Does a Hadamard matrix of order 4k exist for every positive integer k?...
Köthe Conjecture
If a ring has no nil two-sided ideal besides {0}, does it also have no nil one-sided ideal besides {0}?...
Rota's Basis Conjecture
Given n bases of an n-dimensional matroid, can we find n disjoint rainbow bases?...
Generalized Star Height Problem
Can all regular languages be expressed with generalized regular expressions having bounded star height?...
Henson Graphs Finite Model Property
Do Henson graphs have the finite model property?...
Infinite Minimal Field Algebraic Closure
Is every infinite minimal field of characteristic zero algebraically closed?...
Brennan Conjecture
For conformal maps f into the unit disk, when is $\int |f'(z)|^p dA < \infty$ for $p > 0$?...
Fuglede's Conjecture
Is a measurable set spectral if and only if it tiles $\mathbb{R}^d$ by translation?...
Lehmer's Conjecture
Is there a constant c > 1 such that all non-cyclotomic polynomials have Mahler measure ≥ c?...
Mean Value Problem
For any polynomial f of degree d≥2 and complex z, does there exist a critical point c with $|f(z)-f(c)| \leq |f'(z)||z-c|$?...
Pompeiu Problem
Characterize domains where nonzero functions have vanishing integrals over every congruent copy....
Sendov's Conjecture
If all roots of a polynomial lie in the unit disk, is each root within distance 1 from some critical point?...
Bloch's Constant
What is the exact value of Bloch's constant (the largest radius for which every holomorphic function contains a univalent disk)?...
Berge Conjecture
Are Berge knots the only knots in S³ admitting lens space surgeries?...
Unknotting Problem
Can unknots be recognized in polynomial time?...
Whitehead Conjecture
Is every connected subcomplex of a 2-dimensional aspherical CW complex also aspherical?...
Zeeman Conjecture
Is $K \times [0,1]$ collapsible for every finite contractible 2-dimensional CW complex K?...
Lonely Runner Conjecture
If k runners with distinct speeds run on a unit circle, will each runner be "lonely" (≥1/k away from others) at some time?...
Sunflower Conjecture
Can the minimum size for sunflowers be bounded by an exponential (not super-exponential) function of k?...
Union-Closed Sets Conjecture
For any finite union-closed family of sets, does some element appear in at least half the sets?...
Ramsey Number R(5,5)
What is the exact value of the Ramsey number R(5,5)?...
Singmaster's Conjecture
Is there a finite upper bound on multiplicities of entries >1 in Pascal's triangle?...
Quasiperfect Numbers
Do quasiperfect numbers exist?...
Odd Weird Numbers
Do odd weird numbers exist?...
Infinitude of Amicable Pairs
Are there infinitely many pairs of amicable numbers?...
Gilbreath's Conjecture
Does iterating unsigned differences on prime sequence always yield 1 as first element?...
Lander-Parkin-Selfridge Conjecture
If Σᵢ aᵢᵏ = Σⱼ bⱼᵏ with m terms on left, n on right, is m+n ≥ k?...