Proof of the Brown–Erdös–Sós conjecture in groups

Rajko Nenadov, Benny Sudakov, Mykhaylo Tyomkyn

Research output: Contribution to journalArticlepeer-review


The conjecture of Brown, Erdös and Sós from 1973 states that, for any k ≥ 3, if a 3-uniform hypergraph H with n vertices does not contain a set of k + 3 vertices spanning at least k edges then it has o(n2) edges. The case k = 3 of this conjecture is the celebrated (6, 3)-theorem of Ruzsa and Szemerédi which implies Roth’s theorem on 3-term arithmetic progressions in dense sets of integers. Solymosi observed that, in order to prove the conjecture, one can assume that H consists of triples (a, b, ab) of some finite quasigroup Γ. Since this problem remains open for all k ≥ 4, he further proposed to study triple systems coming from finite groups. In this case he proved that the conjecture holds also for k = 4. Here we completely resolve the Brown–Erdös–Sós conjecture for all finite groups and values of k. Moreover, we prove that the hypergraphs coming from groups contain sets of size Θ(k) which span k edges. This is best possible and goes far beyond the conjecture.

Original languageEnglish
Pages (from-to)323-333
Number of pages11
JournalMathematical Proceedings of the Cambridge Philosophical Society
Issue number2
StatePublished - Sep 2020
Externally publishedYes


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