Unsolved Problems

Is the nim-sequence of Grundy's game eventually periodic?...

What is the optimal strategy for two agents to meet on a network without communication?...

Does the Ibragimov-Iosifescu conjecture hold for φ-mixing sequences?...

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, ...

For n > 14 points (except n=24), what is the maximum minimum distance between points on a unit sphere?...

Does every convex, closed, twice-differentiable surface in 3D Euclidean space have at least two umbilical points?...

Does the isoperimetric inequality extend to Cartan-Hadamard manifolds (complete simply-connected manifolds of nonpositive curvature)?...

Does the Euler characteristic of a compact affine manifold vanish?...

What are the relationships between curvature and Euler characteristic for higher-dimensional Riemannian manifolds?...

Is the first eigenvalue of the Laplace-Beltrami operator on an embedded minimal hypersurface of $S^{n+1}$ equal to $n$?...

Can every $n$-dimensional convex body be covered by at most $2^n$ smaller positively homothetic copies?...

What is the minimum number $g(n)$ of points in general position in the plane guaranteeing a convex $n$-gon?...

What configuration of $n$ points in the unit square maximizes the area of the smallest triangle they determine?...

Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?...

What is the maximum number of 3-point lines attainable by a configuration of $n$ points in the plane?...

How many pairs of points at unit distance can be determined by $n$ points in the Euclidean plane?...

What is the shortest path that guarantees reaching the boundary of a given shape, starting from an unknown point with unknown orientation?...

Can three unknotted space curves (not all circles) be arranged as Borromean rings?...

Do Danzer sets of bounded density or bounded separation exist?...

Can the sum of eigenvalues of the Laplacian matrix of a graph be bounded by the number of edges?...

Does there exist a graph where the dominating number equals the eternal dominating number and both are less than the clique covering number?...

Is the pebbling number of the Cartesian product of two graphs at least the product of their pebbling numbers?...

Is the cop number of a connected n-vertex graph $O(\sqrt{n})$?...

If Alice has a winning strategy for the vertex coloring game with k colors, does she have one for k+1 colors?...

Does every k-regular graph on 2n vertices admit a 1-factorization when k ≥ n (or k ≥ n-1 for even n)?...

Does every complete graph on an even number of vertices admit a perfect 1-factorization?...

For k-degenerate graphs, can any (k+2)-coloring be transformed to any other in polynomial steps via single-vertex recolorings?...

What is the maximum chromatic number of biplanar graphs?...

Is every graph class defined by excluding one fixed tree as an induced subgraph χ-bounded?...

Does every bridgeless cubic graph have a cycle-continuous mapping to the Petersen graph?...

For every graph, does the list chromatic index equal the chromatic index?...

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?...

Is the total chromatic number of every graph at most Δ + 2, where Δ is the maximum degree?...

Can the crossing number of a graph be lower-bounded by the crossing number of a complete graph with the same chromatic number?...

Does every thrackle have at most as many edges as vertices?...

Do minor-closed graph families have $\ell_1$ embeddings with bounded distortion?...

Can every planar graph be drawn with integer edge lengths?...

Does every graph with a planar cover have a projective-plane embedding?...

What is the minimum crossing number of the complete bipartite graph $K_{m,n}$?...

Is the crossing number of the complete graph $K_n$ equal to the value given by Guy's formula?...

Do planar graphs have universal point sets of subquadratic size?...

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?...

Does there exist a strongly regular graph with parameters (99,14,1,2)?...

For given maximum degree d and diameter k, what is the largest possible number of vertices in a graph?...

Does a Moore graph with girth 5 and degree 57 exist?...

Does every cubic bipartite three-connected planar graph have a Hamiltonian cycle?...

Is there a constant t such that every t-tough graph is Hamiltonian?...

Does every bridgeless graph have a collection of cycles that covers each edge exactly twice?...

Does every graph with minimum degree 3 contain cycles of lengths that are powers of 2?...

Does every graph family defined by a forbidden induced subgraph have polynomial-sized cliques or independent sets?...