# On the existence of dense substructures in finite groups

## TL;DR

This paper proves that large finite groups contain dense substructures with many product triples, providing an elementary proof of a special case of a conjecture related to dense configurations in groups.

## Contribution

It establishes the existence of dense substructures with quadratic many triples in large finite groups, confirming an asymptotically optimal bound and offering an elementary proof of a conjecture.

## Key findings

- Large finite groups contain dense substructures with many triples.
- The number of triples is quadratically related to the size of the set.
- The result confirms a special case of a conjecture by Brown, Erdős, and Sós.

## 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.