Mykhaylo Tyomkyn

The Brown-Erdős-Sós conjecture in groups

The conjecture of Brown, Erdős and Sos 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, but for all k ≥ 4 the conjecture remains open.

Solymosi suggested to study the Brown-Erdős-Sós conjecture when H consists of triples (a, b, ab) in some finite group Γ. In this case he proved that the conjecture holds also for k = 4. We prove it in a strong form for all k, and establish a connection to isoperimetric problems on lattices in the process.

Joint work with R. Nenadov and B. Sudakov