Off-diagonal Ramsey numbers for slowly growing hypergraphs

Abstract

For a $k$-uniform hypergraph $F$ and a positive integer $n$, the Ramsey number $r(F,n)$ denotes the minimum $N$ such that every $N$-vertex $F$-free $k$-uniform hypergraph contains an independent set of $n$ vertices. A hypergraph is $\text{slowly growing}$ if there is an ordering $e_1,e_2,\dots ,e_t$ of its edges such that $|e_i\setminus {\bigcup }_{j=1}^{i-1}{e}_j|\le 1$ for each $i\in \{2,\dots ,t\}$. We prove that if $k\ge 3$ is fixed and $F$ is any non $k$-partite slowly growing $k$-uniform hypergraph, then for $n\ge 2$, \[ r(F,n) = Ω\Bigl(\frac{n^k}{(\log n)^{2k - 2}}\Bigr).\] In particular, we deduce that the off-diagonal Ramsey number $r(F_5,n)$ is of order $n^{3}/\mbox{polylog}(n)$, where $F_5$ is the triple system $\{123,124,345\}$. This is the only 3-uniform Berge triangle for which the polynomial power of its off-diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs, martingales, and hypergraph containers.

11 pages, 2 figures