The self-complementary Erdős–Hajnal conjecture

For a graph $H$, let $H^c$ denote its complement. A graph $G$ is $\{H,H^c\}$-free if it has no induced subgraph isomorphic to either $H$ or $H^c$. A clique is a set of pairwise adjacent vertices, and a stable set is a set of pairwise non-adjacent vertices.

Self-complementary Erdős–Hajnal conjecture. For every graph $H$, there exists a constant $\epsilon(H)>0$ such that every $\{H,H^c\}$-free graph $G$ has either a clique or a stable set of size at least $|V(G)|^{\epsilon(H)}$.

The source presents this as a potentially easier special-case weakening of the full Erdős–Hajnal conjecture, particularly for difficult graphs such as the five-cycle. It remains open in general.