Oriented trees in O(k √(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 k^2/2 - 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√(2/15) k √(k) + O(k) contains any oriented tree of order k. Moreover, we provide improved bounds of √(4/3) k √(k)+O(k) for arborescences, and (b-1)(k-3)+3 for paths on b blocks, with b≥ 2.