The ExactlyN problem in the number-on-forehead (NOF) communication setting asks k players, each of whom can see every input but their own, if the k input numbers add up to N. Introduced by Chandra, Furst and Lipton in 1983, ExactlyN is important for its role in understanding the strength of ... more >>>

We improve the best known upper bounds on the density of corner-free sets over quasirandom groups from inverse poly-logarithmic to quasi-polynomial. We make similarly substantial improvements to the best known lower bounds on the communication complexity of a large class of permutation functions in the 3-player Number-on-Forehead model. Underpinning both ... more >>>

Let G be a finite abelian group and A be a subset of G \times G which is corner--free, meaning that there are no x, y \in G and d \in G \setminus \{0\} such that (x, y), (x+d, y), (x, y+d) \in A. We prove that
$|A| \le |G|^2 ... more >>>