More on monochromatic-rainbow Ramsey type theorems

Abstract

An edge colored graph is called a rainbow if no two of its edges have the same color. Let H and G be two families of graphs. Denote by RM(H, G) the smallest integer R, if it exists, having the property that every coloring of the edges of K R by an arbitrary number of colors implies that either there is a monochromatic subgraph of K R that is isomorphic to a graph in H or there is a rainbow subgraph of K R that is isomorphic to a graph in G. T n is the set of all trees on n vertices. T n (k) denotes all trees on n vertices with diam(T n (k)) ≤ k. In this paper, we investigate RM(T n , 4K 2 ), RM(T n , K 1,4 ) and RM(T n (4), K 3 ).