Off-diagonal Ramsey numbers for slowly growing hypergraphs

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 $\mathit{\text{slowly}}\mathit{\text{growing}}$ if there is an ordering $e_1,e_2,\dots ,e_t$ of its edges such that $\mathrm{\mid }{e}_i\setminus {\bigcup }_{j=1}^{i-1}{e}_j\mathrm{\mid }\le 1$ 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 $n\ge 2$, $$r(F,n)=\mathrm{\Omega }\left(\frac{n^k}{(\log n{)}^{2k-2}}\right)\mathrm{.}$$ In particular, we deduce that the off-diagonal Ramsey number $r(F_5,n)$ is of order $n^3\mathrm{/}polylog(n)$, 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.