Some Generalized Bipartite Ramsey Numbers Involving Short Cycles

Abstract

For bipartite graphs $$G_1, G_2,\ldots ,G_k$$G1,G2,?,Gk, the bipartite Ramsey number $$b(G_1,G_2$$b(G1,G2, $$\ldots , G_k)$$?,Gk) is the least positive integer b so that any colouring of the edges of $$K_{b,b}$$Kb,b with k colours will result in a copy of $$G_i$$Gi in the ith colour for some i. In this paper, we will consider the bipartite Ramsey number $$b(C_{2t_1},C_{2t_2},\ldots ,C_{2t_k})$$b(C2t1,C2t2,?,C2tk), where $$t_{i}$$ti is an integer and $$2 \le t_{i}\le 4,$$2≤ti≤4, for all $$1\le i\le k$$1≤i≤k. In particular, we will show that $$b(C_{2t_1},C_{2t_2},\ldots ,C_{2t_k})$$b(C2t1,C2t2,?,C2tk)$$\le $$≤$$k(t_1+t_2+\cdots +t_k-k+1)$$k(t1+t2+?+tk-k+1).