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 $b\ge 2$.

翻译：1980年，Burr 猜想每个色数为 $2k-2$ 的有向图包含任意阶为 $k$ 的有向树作为子有向图。Burr 证明了色数 $(k-1)^2$ 足以保证该性质，后于2013年由 Addario-Berry 等人改进为 $\frac{k^2}{2} - \frac{k}{2} + 1$。我们给出了 Burr 猜想的首个次二次界，证明每个色数为 $8\sqrt{\frac{2}{15}} k \sqrt{k} + O(k)$ 的有向图包含任意阶为 $k$ 的有向树。此外，针对有向根树，我们提供了 $\sqrt{\frac{4}{3}} k \sqrt{k}+O(k)$ 的改进界；对于由 $b$ 个块构成的路径（$b\ge 2$），界为 $(b-1)(k-3)+3$。