Unsolved Problems

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere....

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 $3^2 - 2^3 = 1$....

What is the largest product-free set in the alternating group $A_n$?...

Find reasonable bounds for the maximal density of a set $A \subset \{1, \ldots, N\}$ not containing a 3-term progression with common difference a squa...

Define the 2-colour van der Waerden numbers $W(k, r)$ to be the least quantities such that if $\{1, \dots, W(k, r)\}$ is coloured red and blue then th...

What is $C$, the infimum of all exponents $c$ for which the following is true, uniformly for $0 < \alpha < 1$? Suppose that $A \subset \mathbb{F}_2^n$...

Are a positive proportion of positive integers a sum of two palindromes?...

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?...

For a monotone graph property, is the threshold for a random graph to have this property at most a constant factor away from the expectation threshold...

For a univalent function $f(z) = z + \sum_{n=2}^\infty a_n z^n$ on the unit disk, is $|a_n| \leq n$ for all $n$?...

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 $O(2^{n/2})$?...

What is the minimum number of pieces needed to perform a Banach-Tarski decomposition of the ball?...

How do values of Kazhdan-Lusztig polynomials at $1$ relate to multiplicities of irreducible representations in Verma modules?...

For a finite group $G$ and prime $p$, is the number of irreducible characters of degree not divisible by $p$ equal to the corresponding number for the...

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