On Rodl's Theorem for Cographs

A theorem of R & ouml;dl states that for every fixed F and epsilon > 0 there is delta = delta(F)(epsilon) so that every induced F-free graph contains a vertex set of size delta n whose edge density is either at most epsilon or at least 1 - epsilon. R & ouml;dl'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 epsilon, 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.