Ramsey numbers and a general Erdős-Rogers function

Given a graph F, let $L\left(F\right)$ be a fixed finite family of graphs consisting of a $C_4$ and some bipartite graphs relying on an s-partite subgraph partitioning of edges of F. Define $(m,n,a,b)$-graph by an $m\times n$ bipartite graph with $n\ge m$ such that all vertices in the part of size n have degree a and all vertices in the part of size m have degree $b\ge a$. In this paper, building upon the work of Janzer and Sudakov (2023⁺) and combining with the idea of Conlon, Mattheus, Mubayi and Verstraëte (2023⁺) we obtain that for each $s\ge 2$, if there exists an $L\left(F\right)$-free $(m,n,a,b)$-graph, then there exists an F-free graph $H^{⁎}$ with at least $n{a}^{-\frac{1}{s-1}}-1$ vertices in which every vertex subset of size $m{a}^{-\frac{s}{s-1}}{\log }^3\left(an\right)$ contains a copy of $K_s$. As applications, we obtain some upper bounds of general Erdős-Rogers functions for some special graphs of F. Moreover, we obtain the multicolor Ramsey numbers $r_{k+1}(C_5;t)=\tilde{\Omega }\left(t^{\frac{3k}{7}+1}\right)$ and $r_{k+1}(C_7;t)=\tilde{\Omega }\left(t^{\frac{k}{4}+1}\right)$, which improve that by Xu and Ge (2022) [24].