Oriented trees in $O(k\sqrt{k})$-chromatic digraphs, a subquadratic bound for Burr's conjecture

In 1980, Burr conjectured that every directed graph with chromatic number $2k-2$ contains any oriented tree of order $k$ as a subdigraph. Burr showed that chromatic number $(k-1{)}^2$ suffices, which was improved in 2013 to $\frac{k^2}{2}-\frac{k}{2}+1$ by Addario-Berry et al. We give the first subquadratic bound for Burr’s conjecture, by showing that every directed graph with chromatic number $8\sqrt{\frac{2}{15}}k\sqrt{k}+O(k)$ contains any oriented tree of order $k$. Moreover, we provide improved bounds of $\sqrt{\frac{4}{3}}k\sqrt{k}+O(k)$ for arborescences, and $(b-1)(k-3)+3$ for paths on $b$ blocks, with $b\ge 2$.