Induced subgraph density. III. The pentagon and the bull

A theorem of R\"odl says that for every graph $H$, and every $\varepsilon>0$, there exists $\delta>0$ such that if $G$ is a graph that has no induced subgraph isomorphic to $H$, then there exists $X\subseteq V(G)$ with $|X|\ge \delta|G|$ such that one of $G[X],\overline{G}[X]$ has at most $\varepsilon\binom{|X|}{2}$ edges. But for fixed $H$, how does $\delta$ depends on $\varepsilon$? If the dependence is polynomial, then $H$ satisfies the Erd\H{o}s-Hajnal conjecture; and Fox and Sudakov conjectured that the dependence is polynomial for {\em every} graph $H$. This conjecture is substantially stronger than the Erd\H{o}s-Hajnal conjecture itself, and until recently it was not known to be true for any non-trivial graphs $H$. The preceding paper of this series showed that it is true for $P_4$, and all graphs obtainable from $P_4$ by vertex-substitution. Here we will show that the Fox-Sudakov conjecture is true for all the graphs $H$ that are currently known to satisfy the Erd\H{o}s-Hajnal conjecture. In other words, we will show that it is true for the bull, and the 5-cycle, and induced subgraphs of them, and all graphs that can be obtained from these by vertex-substitution. There is a strengthening of R\"odl's theorem due to Nikiforov, that replaces the hypothesis that $G$ has no induced subgraph isomorphic to $H$, with the weaker hypothesis that the density of induced copies of $H$ in $G$ is small. We will prove the corresponding ``polynomial'' strengthening of Nikiforov's theorem for the same class of graphs $H$.