On the existence of dense substructures in finite groups

Abstract

Fix $k \geq 6$. 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, Erd\H{o}s and S\'{o}s. We remark that the result was recently discovered independently by Nenadov, Sudakov and Tyomkyn.