On the asymptotics of the Erdős-Rogers function

Abstract: The Erdős-Rogers function $f_{\ell,s}(n)$ is the largest order of a $K_\ell$-free induced subgraph guaranteed to exist in every $K_s$-free graph on $n$ vertices. While this function is well understood for $s=\ell+1$, the case where $s$ is much larger than $\ell$ has remained wide open. A long-standing lower bound of Sudakov states that $$f_{\ell,s}(n)\geq n<sup>{\frac{\ell}{2s}+O_\ell(s<sup>{-2})}$$, while a recent result of Bradač shows that $f_{\ell,s}(n)\leq n<sup>{\frac{\ell-1}{s-1}+o(1)}$. In this paper, we close this gap asymptotically by proving that $$f_{\ell,s}(n)= n<sup>{\frac{\ell}{2s}+O_\ell(s<sup>{-2})}$$. More precisely, we prove that for all $2\leq \ell&lt;s$, we have $$f_{\ell,s}(n)\leq n<sup>{\frac{\ell}{2s-\ell}+o(1)}$$. Our proof builds on Bradač's recent tight construction for off-diagonal Ramsey numbers, which can be viewed as the $\ell=2$ case of our result.