Polynomial to exponential transition in Ramsey theory

Joint School of Mathematics and ACO Colloquium
Thursday, February 14, 2019 - 11:00am
1 hour (actually 50 minutes)
Skiles 006
University of Illinois at Chicago
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 $500 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 Bhatt and Rodl about the maximum upper density of quasirandom hypergraphs. This is joint work with Alexander Razborov.