# A note on the Brown--Erd\H{o}s--S\'os conjecture in groups

## TL;DR

This paper proves the Brown--Erd"H{o}s--S"os conjecture for triples systems derived from finite groups by showing dense subsets contain specific large configurations, using advanced combinatorial theorems.

## Contribution

It establishes the conjecture in the context of group-based triples systems, demonstrating optimal bounds and employing key arithmetic combinatorics results.

## Key findings

- Triples systems from finite groups contain configurations with (( ext{t})^{1/2}) vertices.
- For all t, t triples can be spanned by at most t+3 vertices.
- Confirms the Brown--Erd"H{o}s--S"os conjecture in this setting.

## Abstract

We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some $k$, or the entire multiplication table of a certain large abelian group, as a subgrid. As a consequence, we show that triples systems coming from a finite group contain configurations with $t$ triples spanning $\mathcal{O}(\sqrt{t})$ vertices, which is the best possible up to the implied constant. We confirm that for all $t$ we can find a collection of $t$ triples spanning at most $t+3$ vertices, resolving the Brown--Erd\H os--S\'os conjecture in this context. The proof applies well-known arithmetic results including the multidimensional versions of Szemer\'edi's theorem and the density Hales--Jewett theorem.