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 $r\ge3$, $F$ is nonempty, $G$ is $2$-tightly connected, and there is no homomorphism from $G$ to $F$, then $$f_{F,G}(n)\le C(\log n)^{\beta_F}, \qquad \beta_F= \max_{\substack{\emptyset\ne P\subseteq\partial_2F}} \frac{e(P)}{v(P)-1}.$$ For $r=3$, this confirms a conjecture of He and Nie for tightly connected $3$-graphs, sharpening their earlier bound by replacing the exponent $\max_{\substack{\emptyset\ne P\subseteq\partial_2F}} \frac{e(P)+1}{v(P)-1}$ with $\beta_F$. When $F=K_r^r$, our result recovers the Ramsey lower bound $r(G,K_n^r)\ge 2^{\Omega(n^{2/r})}$ whenever $G$ is $2$-tightly connected and non-$r$-partite.