Multipartite Ramsey Numbers

For a graph \(G\), a partiteness \(k \geq 2\) and a number of colours \(c\), we define the multipartite Ramsey number \(r^c_k(G)\) 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 \(r^c_k(G)\) is tied up with what monochromatic subgraphs are forced in a \(c\)-colouring of the complete graph \(K_k\). We then calculate the values for some small \(G\) including \(r^2_3(C_4) = 3, r^2_4(C_4) = 2, r^3_3(C_4) = 7\) and \(r^2_3(C_6) = 3\).