Induced subgraph density. II. Sparse and dense sets in cographs

A well-known theorem of Rödl says that for every graph $H$, and every $\varepsilon >0$, there exists $\delta >0$ such that if $G$ does not contain an induced copy of $H$, then there exists $X\subseteq V\left(G\right)$ with $\left|X\right|\ge \delta \left|G\right|$ such that one of $G\left[X\right],\overline{G}\left[X\right]$ has edge-density at most $\varepsilon$. But how does $\delta$ depend on $ϵ$? Fox and Sudakov conjectured that the dependence is at most polynomial: that for all $H$ there exists $c>0$ such that for all $\varepsilon$ with $0<\varepsilon \le 1/2$, Rödl’s theorem holds with $\delta ={\varepsilon }^c$. This conjecture implies the Erdős–Hajnal conjecture, and until now it had not been verified for any non-trivial graphs $H$. Our first result shows that it is true when $H=P_4$. Indeed, in that case we can take $\delta =\varepsilon$, and insist that one of $G\left[X\right],\overline{G}\left[X\right]$ has maximum degree at most ${\varepsilon }^2\left|G\right|$).

Second, we will show that every graph $H$ that can be obtained by substitution from copies of $P_4$ satisfies the Fox–Sudakov conjecture. To prove this, we need to work with a stronger property. Let us say $H$ is viral if there exists $c>0$ such that for all $\varepsilon$ with $0<\varepsilon \le 1/2$, if $G$ contains at most ${\varepsilon }^c{\left|G\right|}^{\left|H\right|}$ copies of $H$ as induced subgraphs, then there exists $X\subseteq V\left(G\right)$ with $\left|X\right|\ge {\varepsilon }^c\left|G\right|$ such that one of $G\left[X\right],\overline{G}\left[X\right]$ has edge-density at most $\varepsilon$. We will show that $P_4$ is viral, using a “polynomial $P_4$-removal lemma” of Alon and Fox. We will also show that the class of viral graphs is closed under vertex-substitution.

Finally, we give a different strengthening of Rödl’s theorem: we show that if $G$ does not contain an induced copy of $P_4$, then its vertices can be partitioned into at most $480{\varepsilon }^{-4}$ subsets $X$ such that one of $G\left[X\right],\overline{G}\left[X\right]$ has maximum degree at most $\varepsilon \left|X\right|$.