Directed Ramsey number for trees

Given an oriented graph H, the k-colour oriented Ramsey number of H, denoted by r→(H,k), is the least integer n, for which every k-edge-coloured tournament on n vertices contains a monochromatic copy of H. We show that r→(T,k)≤ck|T|k for any oriented tree T, which, in general, is tight up to a constant factor. We also obtain a stronger bound, when H is an arbitrarily oriented path.