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 he conjectured f ( k ) = 2 k − 2 . Burr also proved that every ( 8 k − 7 ) -chromatic digraph contains every antidirected tree. We improve both of Burr’s bounds. We show that f ( k ) ≤ k 2 / 2 − k / 2 + 1 and that every antidirected tree of order k is contained in every ( 5 k − 9 ) -chromatic digraph. e a conjecture that 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. 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. 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.