On Rödl’s Theorem for Cographs

A theorem of Rödl states that for every fixed F and ε > 0 there is δ = δ_F (ε) so that every induced F-free graph contains a vertex set of size δn whose edge density is either at most ε or at least 1 − ε. Rödl’s proof relied on the regularity lemma, hence it supplied only a tower-type bound for δ. Fox and Sudakov conjectured that δ can be made polynomial in ε, and a recent result of Fox, Nguyen, Scott and Seymour shows that this conjecture holds when F = P₄. In fact, they show that the same conclusion holds even if G contains few copies of P₄. In this note we give a short proof of a more general statement.