Proof of the Brown–Erdös–Sós conjecture in groups

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.