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 .

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