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 \(b\ge 2\).