Year: 2026
Type: preprint
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 $f_{\ell,s}(n)\geq n^{\frac{\ell}{2s}+O_\ell(s^{-2})}$, while a recent result of Bradač shows that $f_{\ell,s}(n)\leq n^{\frac{\ell-1}{s-1}+o(1)}$. In this paper, we close this gap asymptotically by proving that $f_{\ell,s}(n)= n^{\frac{\ell}{2s}+O_\ell(s^{-2})}$. More precisely, we prove that for all $2\leq \ell
Source: arXiv (Cornell University)
Authors: Bradač, Domagoj, Janzer, Oliver, Sarkar, Rik
Institution: École Polytechnique Fédérale de Lausanne
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214989240