The Brown-Erd\H{o}s-S\'os conjecture in dense triple systems

TL;DR

This paper proves the Brown-Erdős-Sós conjecture for dense linear 3-uniform hypergraphs with density greater than 4/5, marking the first such bound and advancing understanding of hypergraph structure.

Contribution

It establishes the conjecture for the first time for hypergraphs with density exceeding 4/5, a significant step in hypergraph extremal combinatorics.

Findings

Proves the conjecture for  > 4/5

First bound of this kind for the conjecture

Advances understanding of dense hypergraph structures

Abstract

The famous Brown-Erd\H{o}s-S\'os conjecture from 1973 states, in an equivalent form, that for any fixed $\delta>0$ and integer $k\geq 3$ every sufficiently large linear $3$-uniform hypergraph of size $\delta n^2$ contains some $k$ edges spanning at most $k+3$ vertices. We prove it to hold for $\delta>4/5$, establishing the first bound of this kind.