On R\"odl's Theorem for Cographs

TL;DR

This paper provides a concise proof of a generalized version of R"odl's theorem for cographs, improving understanding of induced $F$-free graphs and their vertex subsets with specific edge densities.

Contribution

It offers a shorter proof of a broader statement related to R"odl's theorem, extending results to cographs and graphs with few $P_4$ copies.

Findings

Established a more general version of R"odl's theorem for cographs.

Provided a shorter proof technique for the theorem.

Extended the theorem's applicability to graphs with few $P_4$ copies.

Abstract

A theorem of R\"odl states that for every fixed $F$ and $\varepsilon>0$ there is $\delta=\delta_F(\varepsilon)$ so that every induced $F$-free graph contains a vertex set of size $\delta n$ whose edge density is either at most $\varepsilon$ or at least $1-\varepsilon$. R\"odl's proof relied on the regularity lemma, hence it supplied only a tower-type bound for $\delta$. Fox and Sudakov conjectured that $\delta$ can be made polynomial in $\varepsilon$, and a recent result of Fox, Nguyen, Scott and Seymour shows that this conjecture holds when $F=P_4$. In fact, they show that the same conclusion holds even if $G$ contains few copies of $P_4$. In this note we give a short proof of a more general statement.