A Ramsey type result for oriented trees

Given positive integers hh and kk, denote by r(h,k)r(h,k) the smallest integer nn such that in any kk-coloring of the edges of a tournament on more than nn vertices there is a monochromatic copy of every oriented tree on hh vertices. We prove that r(h,k)=(h−1)kr(h,k)=(h−1)k for all kk sufficiently large (k=Θ(hlogh)k=Θ(hlogh) suffices). The bound (h−1)k(h−1)k is tight. The related parameter r∗(h,k)r∗(h,k) where some color contains all oriented trees is asymptotically determined. Values of r(h,2)r(h,2) for some small hh are also established.