Polynomial to exponential transition in Ramsey theory

Abstract

Given $s\ge k\ge 3$, 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. Erd\H os and Hajnal conjectured (1972) that $h^{(k)}(s)$ can be calculated precisely using a recursive formula and Erd\H os 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 $k\ge 4$ and $s\ge k+2$. Here we settle the conjecture for all $s\ge k\ge 4$. We also answer a question of Bhat and Rödl by constructing, for each $k\ge 4$, a quasirandom sequence of $k$-uniform hypergraphs with positive density and upper density at most $k!/(k^k-k)$. This result is sharp.

27 pages

References in corpus (2)

• A bound on the inducibility of cycles
• Hypergraphs with vanishing Turán density in uniformly dense hypergraphs

Cited by in corpus (3)

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• Inducibility of rainbow graphs