On the existence of dense substructures in finite groups

Abstract

Fix $k\ge 6$. We prove that any large enough finite group $G$ contains $k$ elements which span quadratically many triples of the form $(a,b,ab)\in S\times G$, given any dense set $S\subseteq G\times G$. The quadratic bound is asymptotically optimal. In particular, this provides an elementary proof of a special case of a conjecture of Brown, Erdős and Sós. We remark that the result was recently discovered independently by Nenadov, Sudakov and Tyomkyn.