Generalized Erdős--Rogers problems for $r$-uniform hypergraphs

Let $F$ and $G$ be $r$-uniform hypergraphs, and let $f_{F,G}(n)$ be the largest integer $m$ such that every $n$-vertex $G$-free $r$-graph contains an induced $F$-free subgraph on $m$ vertices. We prove that, if $r\ge 3$, $F$ is nonempty, $G$ is $2$-tightly connected, and there is no homomorphism from $G$ to $F$, then $$f_{F,G}(n)\le C(\log n{)}^{{\beta }_F},{\beta }_F=\underset{\begin{array}{c}\mathrm{\varnothing }\mathrm{\ne }P\subseteq {\mathrm{\partial }}_{2}F\end{array}}{\max }\frac{e(P)}{v(P)-1}\mathrm{.}$$ For $r=3$, this confirms a conjecture of He and Nie for tightly connected $3$-graphs, sharpening their earlier bound by replacing the exponent ${\max }_{\begin{array}{c}\mathrm{\varnothing }\mathrm{\ne }P\subseteq {\mathrm{\partial }}_{2}F\end{array}}\frac{e(P)+1}{v(P)-1}$ with ${\beta }_F$. When $F=K_r^r$, our result recovers the Ramsey lower bound $r(G,K_n^r)\ge 2^{\mathrm{\Omega }(n^{2\mathrm{/}r})}$ whenever $G$ is $2$-tightly connected and non-$r$-partite.