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. Erdős and Hajnal conjectured (1972) that h(k)(s) can be calculated precisely using a recursive formula and Erdős 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 Rödl 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.