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)≤ck∣T∣k for any oriented
tree T. This is a generalisation of a similar result for directed paths by
Chv\'atal and by Gy\'arf\'as 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)≤ck∣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 Gy\'arf\'as and Lehel who determined
the 2-colour directed Ramsey number of directed paths.Comment: 27 pages, 14 figure