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arXiv · 1901.06029

Polynomial to exponential transition in Ramsey theory

Dhruv Mubayi·Alexander Razborov
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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 $k\geq 4$ and $s\geq k+2$. Here we settle the conjecture for all $s \ge k \ge 4$. We also answer a question of Bhat and R\"odl 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.

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Dhruv Mubayi, Alexander Razborov. 2019-01-17. Polynomial to exponential transition in Ramsey theory. https://doi.org/10.1112/plms.12320

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