Fox–Sudakov polynomial Rödl conjecture

Let $H$ be a graph. For a graph $G$, a vertex set $S$ is $\varepsilon$-homogeneous if the edge density of the induced graph $G[S]$ is at most $\varepsilon$ or at least $1-\varepsilon$.

Fox–Sudakov's conjecture. For any graph $H$ there exists a $C_H>0$ such that for any $0<\varepsilon \le 1/2$ any $n$-vertex $H$-free graph contains an $\varepsilon$-homogeneous set with at least $\varepsilon^{C_H}n$ vertices.

This is a polynomial quantitative strengthening of Rödl's theorem and implies the Erdős–Hajnal conjecture for each fixed graph $H$. The source presents it as an open conjecture, while the paper proves equivalence with the Erdős–Hajnal and polynomial Nikiforov conjectures.