Abstract: After a brief introduction to classical hypergraph Ramsey numbers, I will focus on the following problem. What is the minimum t such that there exist arbitrarily large k-uniform hypergraphs whose independence number is at most polylogarithmic in the number of vertices and every s vertices span at most t edges? Erdos and Hajnal conjectured (1972) that this minimum can be calculated precisely using a recursive formula and Erdos offered a $500 prize for a proof. For k = 3, this has been settled for many values of s, but it was not known for larger k.
Here we settle the conjecture for all k at least 4. Our method also answers a question of Bhat and Rodl about the maximum upper density of quasirandom hypergraphs.
This is joint work with Alexander Razborov.