Bipartite Ramsey theorems for multiple copies of K2,2

Abstract

For bipartite graphs G₁ , G₂ , ... ,G_k , the bipartite Ramsey number b(G₁ , G₂ , ... , G_k) is the least positive integer b so that any colouring of the edges of K_b,b with k colours will result in a copy of G_i in the ith colour for some i. When G_i = G for all i, we write b_k(G) = 6(G₁ , G₂ , ... , G_k), and we write b(G) = b₂(G). For all integers n ≥ 2, we show that 6(nK2,2) = 4n - 1; that is, any 2-colouring of the edges of K_4n-1,4n-1 contains a monochromatic nK_2,2.