A cycle of length at least five and a forest complement have the Erdős-Hajnal property

Choose a forest $H$ that contains $F$ as an induced subgraph and also contains an induced path on $\mathrm{\ell }-3$ vertices. For instance, take the disjoint union of $F$ with a path long enough to realize that length. By [L2], the star-expansion $H^{\star }$ contains an induced copy of $C$.

If a graph $G$ is $C$-free and $\bar{F}$-free, then it is also $H^{\star }$-free and $\bar{H}$-free. Indeed, an induced copy of $H^{\star }$ would contain the induced cycle $C$ from step 1.1, and an induced copy of $\bar{H}$ would contain $\bar{F}$ because $F$ is an induced subgraph of $H$. Thus the class forbidding $\{C,\bar{F}\}$ is a hereditary subclass of the class forbidding $\{H^{\star },\bar{H}\}$.

By [L1], the larger class from step 2.1 has the Erdős-Hajnal property, so [L3] passes that property to the subclass forbidding $\{C,\bar{F}\}$.