# The Brown-Erd\H{o}s-S\'os Conjecture in finite abelian groups

## TL;DR

This paper proves the Brown-Erdős-Sós conjecture for 3-uniform hypergraphs derived from finite abelian groups, confirming that such hypergraphs with certain properties have o(n^2) edges.

## Contribution

The paper establishes the conjecture specifically for triple systems from finite abelian groups, a significant case in extremal combinatorics.

## Key findings

- Proves the conjecture for hypergraphs from finite abelian groups
- Shows that such hypergraphs with no large dense substructures have o(n^2) edges
- Advances understanding of extremal properties in algebraic combinatorics

## Abstract

The Brown-Erd\H{o}s-S\'{o}s conjecture, one of the central conjectures in extremal combinatorics, states that for any integer $m\geq 6,$ if a 3-uniform hypergraph on $n$ vertices contains no $m$ vertices spanning at least $m-3$ edges, then the number of edges is $o(n^2).$ We prove the conjecture for triple systems coming from finite abelian groups.