A directed Andrásfai-Erdős-Sós theorem and chromatic profiles of oriented cycles

The chromatic profile of a digraph $H$, denoted by ${\delta }_{\chi }^{+}(H,k)$, is the infimum $d$ such that any $H$-free digraph $D$ on $n$ vertices with minimum out-degree ${\delta }^{+}(D)\ge dn$ must be $k$-colorable. We determine the exact chromatic profile for several fundamental classes of digraphs. Our main result is a directed analogue of the Andrásfai-Erdős-Sós theorem, stating that ${\delta }_{\chi }^{+}(T_r,r-1)=\frac{3r-7}{3r-4}$, where $T_r$ is the transitive tournament on $r$ vertices. We then determine the chromatic profile for directed odd cycles, showing that ${\delta }_{\chi }^{+}({\overrightarrow{C}}_{2\mathrm{\ell }+1},2)=1\mathrm{/}2$ for all $\mathrm{\ell }\ge 1$. Finally, we resolve the profile for the three remaining orientations of the pentagon, establishing that ${\delta }_{\chi }^{+}(C_5^{\mathrm{\prime }},2)={\delta }_{\chi }^{+}(C_5^{\mathrm{\prime }\mathrm{\prime }},2)={\delta }_{\chi }^{+}(C_5^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }},2)=1\mathrm{/}3$.