The Erdős--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 $K_s<sup>{(k)}$ or an independent set of size $n$. Erdős and Hajnal conjectured that for every fixed $s&gt;k\ge 4$, one has $r_k(s,n)\ge \operatorname{twr}_{k-1}(Ω(n))$. 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 $r_4(6,n)\ge 2<sup>{2<sup>{cn}}$ for some absolute constant $c&gt;0$, improving upon our previous bound. Consequently, we confirm the Erdős--Hajnal conjecture for $r_k(k+2,n)$ for all fixed $k\ge4$.