Fix $\varepsilon>0$ and a nonnull graph $H$. A well-known theorem of Rödl from the 80s says that every $G$ with no induced copy $H$ contains linear-sized $\varepsilon$-restricted set $S\subseteq V(G)$, which means $S$ induces subgraph maximum degree at most $\varepsilon |S|$ in or its complement. There are two extensions this result:
 
 quantitatively, Nikiforov (and later Fox Suda...