Multipartite Ramsey Numbers

For a graph G, a partiteness k ≥ 2 and a number of colours c, we define the multipartite Ramsey number r k (G) as the minimum value m such that, given any colouring using c colours of the edges of the complete balanced k-partite graph with m vertices in each partite set, there must exist a monochromatic copy of G. We show that the question of the existence of r k (G) is tied up with what monochromatic subgraphs are forced in a ccolouring of the complete graph Kk. We then calculate the values for some small G including r 3 (C4) = 3, r 2 4 (C4) = 2, r 3 3 (C4) = 7 and r 2 3 (C6) = 3.