Oriented trees in digraphs

Let f(k) be the smallest integer such that every f(k)-chromatic digraph contains every oriented tree of order k. Burr proved f(k) ≤ (k − 1)2 in general, and conjectured f(k) = 2k − 2. Burr also proved that every (8k − 9)-chromatic digraph contains every antidirected tree. We improve both of Burr’s bounds. We show that f(k) ≤ k2/2 − k/2 + 1 and that every antidirected tree of order k is contained in every (5k − 7)-chromatic digraph. We make a conjecture which explains why antidirected trees are easier to handle. It states that if |E(D)| > (k − 2)|V (D)|, then the digraph D contains every antidirected tree of order k. This is a common strengthening of both Burr’s conjecture for antidirected trees and the celebrated Erdős-Sós Conjecture. We note that the analogue of our conjecture for general trees is false, no matter what function f(k) is used in place of k − 2. We prove our conjecture for antidirected trees of diameter 3, and present some other evidence for it. Along the way, we show that every acyclic k-chromatic digraph contains every oriented tree of order k and suggest a number of approaches for making further progress on Burr’s conjecture.