A theorem of Rödl states that for every fixed $F$ and $\varepsilon>0$ there is $δ=δ_F(\varepsilon)$ so that every induced $F$-free graph contains a vertex set of size $δn$ whose edge density is either at most $\varepsilon$ or at least $1-\varepsilon$. 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 $\varepsilon$, 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.