The tournament Erdős–Hajnal conjecture

A tournament is a directed graph in which exactly one of $uv$ and $vu$ is an edge for every pair of distinct vertices. A tournament $T$ is $S$-free if it has no subtournament isomorphic to $S$. Let $\alpha(T)$ be the maximum number of vertices in a transitive subtournament of $T$.

Tournament Erdős–Hajnal conjecture. For every tournament $S$, there exists a constant $\delta(S)>0$ such that every $S$-free tournament $T$ satisfies

$$\alpha(T)\geq |V(T)|^{\delta(H)}.$$

The source states this as equivalent to the graph Erdős–Hajnal conjecture. Transitive subtournaments provide a single analogue of cliques and stable sets, but the conjecture remains open in general; the exponent in the source is written as $\delta(H)$ despite the quantified tournament being $S$.