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 $n$....

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 $A \subset \mathbb{F}_2^n$...

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 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...

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