The oriented Gyárfás–Sumner conjecture

Let $\vv F$ be an oriented forest. For a digraph $D$, let $\omega(D)$ be the clique number of its underlying graph and let $\overrightarrow{\chi}(D)$ be its dichromatic number, the minimum number of acyclic induced subdigraphs into which its vertices can be partitioned. Let $\operatorname{Forb}_{\mathrm{ind}}(\vv F)$ be the class of digraphs with no induced subdigraph isomorphic to $\vv F$. A class of digraphs is dichromatically bounded if there is a function $f$ such that every digraph $D$ in the class satisfies $\overrightarrow{\chi}(D)\leq f(\omega(D))$.

Oriented Gyárfás–Sumner conjecture. For any oriented forest $\vv F$, $\operatorname{Forb}_{\mathrm{ind}}(\vv F)$ is dichromatically bounded.

This is the directed analogue of the Gyárfás–Sumner conjecture. The paper studies this question through heroes in induced-subdigraph-free classes; the general conjecture remains open.