A Note on Generalized Erd\H{o}s-Rogers Problems

For a $k$-uniform hypergraph $F$ and positive integers $s$ and $N$, the generalized Erd\H{o}s-Rogers function $f^{(k)}_{F,s}(N)$ denotes the largest integer $m$ such that every $K_s^{(k)}$-free $k$-graph on $N$ vertices contains an $F$-free induced subgraph on $m$ vertices. In particular, if $F = K^{(k)}_t$, then we write $f^{(k)}_{t,s}(N)$ for $f^{(k)}_{F,s}(N)$. Mubayi and Suk (\emph{J. London. Math. Soc. 2018}) conjectured that $f^{(4)}_{5,6}(N)=(\log \log N)^{\Theta(1)}$. Motivated by this conjecture, we prove that $f^{(4)}_{5^{-},6}(N)=(\log\log N)^{\Theta(1)}$, where $5^{-}$ denotes the $4$-graph obtained from $K_5^{(4)}$ by deleting one edge. Our proof combines a probabilistic construction of a $2$-coloring of pairs with a stepping-up construction and an analysis of multi-layer local extremum structures. Furthermore, we derive an upper bound for a more general Erd\H{o}s-Rogers function, which implies the lower bound $r_4(6,n)\ge 2^{2^{cn^{1/2}}}$. By applying a variant of the Erd\H{o}s-Hajnal stepping-up lemma due to Mubayi and Suk, we also slightly improve the lower bound for $r_k(k+2,n)$.