A step towards the Erd\H{o}s-Rogers problem

For $2\le k\le t<s$, the Erd\H{o}s-Rogers function $f^{(k)}_{t,s}(N)$ denotes the largest $m$ such that every $K^{(k)}_s$-free $k$-graph on $N$ vertices contains a $K^{(k)}_t$-free induced subgraph on $m$ vertices. Mubayi and Suk (J. London Math. Soc. 2018) conjectured that $f^{(k)}_{k+1,k+2}(N)=(\log_{(k-2)}N)^{\Theta(1)}$ for $k\ge 4$, where $\log_{(i)}$ denotes the $i$-fold iterated logarithm. This is equivalent to the statement that $f^{(k)}_{k+1,s}(N)=(\log_{(k-2)}N)^{\Theta(1)}$ for every $s\ge k+2$. In this paper, we introduce multi-color patterns into a random construction of a $2$-graph to build a $4$-graph, and for the first time, combine them with multi-layer extremum structures to prove that $f^{(4)}_{5,s}(N)=(\log \log N)^{\Theta(1)}$ for every $s\ge 11$. More generally, using a variant of the Erd\H{o}s-Hajnal stepping-up lemma, we also establish that $f^{(k)}_{k+1,s}(N)=(\log_{(k-2)}N)^{\Theta(1)}$ for every $s\ge k+7$.