Bipartite rainbow Ramsey numbers

Let G and H be graphs. A graph with colored edges is said to be monochromatic if all its edges have the same color and rainbow if no two of its edges have the same color. Given two bipartite graphs G1 and G2, the bipartite rainbow ramsey number BRR(G1; G2) is the smallest integer N such that any coloring of the edges of KN;N with any number of colors contains a monochromatic copy of G1 or a rainbow copy of G2. It is shown that BRR(G1; G2) exists if and only if G1 is a star or G2 is a star forest. Exact values and bounds for BRR(G1; G2) for various pairs of graphs G1 and G2 for which the bipartite ramsey number is de2ned are established. c © 2003 Elsevier B.V. All rights reserved.