The perfect-induced-subgraph formulation of the Erdős–Hajnal conjecture

All graphs are finite and simple. A graph $G$ is perfect if every induced subgraph $H$ of $G$ satisfies $\chi(H)=\omega(H)$, where $\chi(H)$ is its chromatic number and $\omega(H)$ its maximum clique size. A graph is $H$-free if it has no induced subgraph isomorphic to $H$.

Perfect-subgraph formulation. For every graph $H$, there exists a constant $\psi(H) > 0$, such that every $H$-free graph $G$ has a perfect induced subgraph with at least $|V(G)|^{\psi(H)}$ vertices.

This is an equivalent formulation of the Erdős–Hajnal conjecture: a perfect graph on $m$ vertices contains a clique or stable set of size at least $\sqrt m$. The formulation replaces two possible outcomes by one and is useful for inductive arguments; it remains open in general.