$χ$-bounded families of oriented graphs

Abstract: A famous conjecture of Gy\'arf\'as and Sumner states for any tree $T$ and integer $k$, if the chromatic number of a graph is large enough, either the graph contains a clique of size $k$ or it contains $T$ as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star $S$ and integer $k$, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size $k$ or it contains $S$ as an induced subgraph. As an evidence, we prove that for any oriented star $S$, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order $3$ or $S$ as an induced subdigraph. We then study for which sets ${\cal P}$ of orientations of $P_4$ (the path on four vertices) similar statements hold. We establish some positive and negative results.