Polynomial To Exponential Transition In Ramsey Theory

Given s > k > 3, let h(k)(s) be the minimum t such that there exist arbitrarily large k-uniform hypergraphs H whose independence number is at most polylogarithmic in the number of vertices and in which every s vertices span at most t edges. Erdos and Hajnal conjectured (1972) that h(k)(s) can be calculated precisely using a recursive formula and Erdos offered $500 for a proof of this. For k=3, this has been settled for many values of s including powers of three but it was not known for any k > 4 and s > k+2.Here we settle the conjecture for all s > k > 4. We also answer a question of Bhat and Rodl by constructing, for each k > 4, a quasirandom sequence of k-uniform hypergraphs with positive density and upper density at most k!/(kk-k). This result is sharp.