A new bound for the Brown-Erdős-Sós problem

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 ≥ 3, what is the smallest integer d = d(e) such that f(n, e+d, e) = o(n²)? This question has its origins in work of Brown, Erdős and Sós from the early 70's and the standard conjecture is that d(e) = 3 for every e ≥ 3. The state of the art result regarding this problem was obtained in 2004 by Sárközy and Selkow, who showed that f(n, e+2+[log₂e], e) = o(n²). 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 Sárközy–Selkow bound, showing that f(n, e+O(log e / log log e), e) = o(n²).