Constrained Ramsey numbers for rainbow matching

Abstract

The Ramsey number Rk(G) of a graph G is the minimum number N, such that any edge coloring of KN with k colors contains a monochromatic copy of G. The constrained Ramsey number f(G, T) of the graphs G and T is the minimum number N, such that any edge coloring of KN with any number of colors contains a monochromatic copy of G or a rainbow copy of T. We show that these two quantities are closely related when T is a matching. Namely, for almost all graphs G, f(G, tK2) = Rt − 1(G) for t≥2. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:91-95, 2011 © 2011 Wiley Periodicals, Inc.