On the Brown-Erdos-Sos conjecture

The conjecture of Brown, Erdos 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(n^2) edges. The case k = 3 of this conjecture is the celebrated (6, 3)-theorem of Ruzsa and Szemeredi, but for k ≥ 4 the conjecture is wide open. I will discuss two natural relaxations of the Brown-Erdos-Sos conjecture:

• Solymosi proposed to study triple system coming from finite groups. Jointly with R. Nenadov and B. Sudakov we resolved this in a strong form.
• Conlon and, independently, Nenadov suggested a natural Ramsey relaxation. Jointly with A. Shapira we established it in higher uniformities.