The Erd\H{o}s--Hajnal hypergraph Ramsey problem for $r_4(6,n)$

Abstract

The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. Erd\H{o}s and Hajnal conjectured that for every fixed $s>k\ge 4$, one has $r_k(s,n)\ge \operatorname{twr}_{k-1}(\Omega(n))$. This conjecture was independently verified by Mubayi and Suk, and by Conlon, Fox and Sudakov, for $k\ge4$ and $s\ge k+3$. In this paper, we prove that $r_4(6,n)\ge 2^{2^{cn}}$ for some absolute constant $c>0$, improving upon our previous bound. Consequently, we confirm the Erd\H{o}s--Hajnal conjecture for $r_k(k+2,n)$ for all fixed $k\ge4$.