On the maximum $F$-free induced subgraphs in $K_t$-free graphs

For graphs $F$ and $H$, let $f_{F,H}(n)$ be the minimum possible size of a maximum $F$-free induced subgraph in an $n$-vertex $H$-free graph. This notion generalizes the Ramsey function and the Erdős–Rogers function. Establishing a container lemma for the $F$-free subgraphs, we give a general upper bound on $f_{F,H}(n)$, assuming the existence of certain locally dense $H$-free graphs. In particular, we prove that for every graph $F$ with $\mathrm{e}\mathrm{x}(m,F)=O(m^{1+\alpha })$, where $\alpha \in [0,1\mathrm{/}2)$, we have $$f_{F,K_3}(n)=O\left(n^{\frac{1}{2-\alpha }}{(\log n)}^{\frac{3}{2-\alpha }}\right)\text{and}{f}_{F,K_4}(n)=O\left(n^{\frac{1}{3-2\alpha }}{(\log n)}^{\frac{6}{3-2\alpha }}\right)\mathrm{.}$$ For the cases where $F$ is a complete multipartite graph, letting $s={\sum }_{i=1}^r{s}_i$, we prove that $$f_{K_{s_1,\dots ,s_r},K_{r+2}}(n)=O(n^{\frac{2s-3}{4s-5}}(\log n{)}^3)\mathrm{.}$$ We also make an observation which improves the bounds of $\mathrm{e}\mathrm{x}(G(n,p),C_4)$ by a polylogarithmic factor.