Given positive integers   h     h        and   k     k       , denote by   r(h,k)     r   (  h  ,  k  )         the smallest integer   n     n        such that in any   k     k       -coloring of the edges of a tournament on more than   n     n        vertices there is a monochromatic copy of every oriented tree on   h     h        vertices. We prove that   r(h,k)=(h−1) k      r   (  h  ,  k  )   =     (  h  −  1  )     k          for all   k     k        sufficiently large (  k=Θ(hlogh)     k  =  Θ   (  h  log  h  )         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   h     h        are also established.
['Raphael Yuster']
A Ramsey type result for oriented trees