Unsolved Problems

If a graph is the union of $n$ cliques of size $n$, no two of which share more than one vertex, then the chromatic number is $n$....

No packing of congruent spheres in three dimensions has density greater than $\frac{\pi}{\sqrt{18}} \approx 0.74048$....

Every topological manifold can be triangulated....

The only solution to $x^p - y^q = 1$ in natural numbers x, y > 0 and p, q > 1 is $3^2 - 2^3 = 1$....

For a matroid of rank $n$ with $n$ disjoint bases $B_1, \ldots, B_n$, can we always find an $n \times n$ matrix whose rows are the bases and whose col...

If $n$ complete graphs, each with $n$ vertices, have the property that every pair of complete graphs shares at most one vertex, can the entire graph b...

Can every tiling of $\mathbb{R}^n$ by unit hypercubes have two cubes that share a complete $(n-1)$-dimensional face?...

For any linear hypergraph with $n$ edges, each of size $n$, can the vertices be colored with $n$ colors such that no edge is monochromatic?...

For every graph $G$, is the list chromatic number equal to the chromatic number?...

Is the chromatic number of a graph at most its clique cover number times the maximum chromatic number of its neighborhoods?...

Is the number of sum-free subsets of $\{1, 2, \ldots, n\}$ equal to $O(2^{n/2})$?...

Does there exist a covering system of congruences using only odd distinct moduli?...