A Ramsey type result for oriented trees. (February 2017) - Yuster, Raphael
Abstract: Given positive integers h and k, denote by r ( h, k ) the smallest integer n such that in any k -coloring of the edges of a tournament on more than n vertices there is a monochromatic copy of every oriented tree on h vertices. We prove that r ( h, k ) = ( h − 1 ) k for all k sufficiently large ( k = Θ ( h log h ) suffices). The bound ( h − 1 ) k is tight. The related parameter r ∗ ( h, k ) where some color contains all oriented trees is asymptotically determined. Values of r ( h, 2 ) for some small h are also established.