For two bipartite graphs G and H, define BRR(G, H) to be the least positive integer N such that any coloring of the edges of K-N,(N) with any number of colors will result in a monochromatic copy of G or a rainbow copy of H. In this paper, we determine the exact values for BRR(K-1,K-n, B(s,t)), BRR(B(t,t), K-1,K-n)and bounds for BRR(B(s, t), K-1,K-n)BRR(B(s, t), mK(2)) and BRR(B(t, t), mK(2)).