A New Bound for the Brown--Erd\H{o}s--S\'os Problem

TL;DR

This paper improves the asymptotic bounds for a key problem in extremal hypergraph theory, narrowing the gap towards the conjectured minimal offset for the number of edges that force a subgraph with certain properties.

Contribution

It provides the first asymptotic improvement over the previous bounds for the minimal offset d(e), showing it is smaller than previously known, approaching the conjectured value.

Findings

Established that f(n, e + O(log e / log log e), e) = o(n^2)

Improved the bound for the minimal offset d(e) in the Brown--Erdős--Sós problem

Advances understanding of extremal properties of 3-uniform hypergraphs

Abstract

Let $f(n,v,e)$ denote the maximum number of edges in a $3$-uniform hypergraph not containing $e$ edges spanned by at most $v$ vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of edges $e \geq 3$, what is the smallest integer $d=d(e)$ so that $f(n,e+d,e) = o(n^2)$? This question has its origins in work of Brown, Erd\H{o}s and S\'os from the early 70's and the standard conjecture is that $d(e)=3$ for every $e \geq 3$. The state of the art result regarding this problem was obtained in 2004 by S\'{a}rk\"{o}zy and Selkow, who showed that $f(n,e + 2 + \lfloor \log_2 e \rfloor,e) = o(n^2)$. The only improvement over this result was a recent breakthrough of Solymosi and Solymosi, who improved the bound for $d(10)$ from 5 to 4. We obtain the first asymptotic improvement over the S\'{a}rk\"{o}zy--Selkow bound, showing that $$ f(n, e + O(\log e/ \log\log e), e) = o(n^2). $$