Directed Ramsey number for trees

In this paper, we study Ramsey-type problems for directed graphs. We first consider the k-colour oriented Ramsey number of H, denoted by r→(H,k), which is the least n for which every k-edge-coloured tournament on n vertices contains a monochromatic copy of H. We prove that r→(T,k)≤c _k |T| ^k for any oriented tree T. This is a generalisation of a similar result for directed paths by Chvátal and by Gyárfás and Lehel, and answers a question of Yuster. In general, it is tight up to a constant factor. We also consider the k-colour directed Ramsey number r↔(H,k) of H, which is defined as above, but, instead of colouring tournaments, we colour the complete directed graph of order n. Here we show that r↔(T,k)≤c _k |T| ^k−1 for any oriented tree T, which is again tight up to a constant factor, and it generalises a result by Williamson and by Gyárfás and Lehel who determined the 2-colour directed Ramsey number of directed paths.