A new bound for the Brown–Erdős–Sós problem

David Conlon, Lior Gishboliner, Yevgeny Levanzov, Asaf Shapira

Research output: Contribution to journalArticlepeer-review

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+⌊log2⁡e⌋,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).

Original languageEnglish
Pages (from-to)1-35
Number of pages35
JournalJournal of Combinatorial Theory. Series B
Volume158
DOIs
StatePublished - Jan 2023

Keywords

  • Brown-Erdős-Sós conjecture
  • Hypergraph removal lemma

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