The Erdős--Hajnal hypergraph Ramsey problem for $r_4(6,n)$

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 $n$. Erdős and Hajnal conjectured that for every fixed $s>k\ge 4$, one has $r_k(s,n)\ge {\mathrm{twr}}_{k-1}(\mathrm{\Omega }(n))$. This conjecture was independently verified by Mubayi and Suk, and by Conlon, Fox and Sudakov, for $k\ge 4$ 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 Erdős–Hajnal conjecture for $r_k(k+2,n)$ for all fixed $k\ge 4$.