Dense $H$-Free Graphs are Almost $(\chi(H)-1)$-Partite

Abstract

By using the Szemerédi Regularity Lemma, Alon and Sudakov recently extended the classical Andrásfai-Erdős-Sós theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any $(r+1)$-partite graph $H$ whose smallest part has $t$ vertices, there exists a constant $C$ such that for any given $\varepsilon>0$ and sufficiently large $n$ the following is true. Whenever $G$ is an $n$-vertex graph with minimum degree $$\delta(G)\geq\left(1-{3\over 3r-1}+\varepsilon\right)n,$$ either $G$ contains $H$, or we can delete $f(n,H)\leq Cn^{2-{1\over t}}$ edges from $G$ to obtain an $r$-partite graph. Further, we are able to determine the correct order of magnitude of $f(n,H)$ in terms of the Zarankiewicz extremal function.