Ramsey numbers in complete balanced multipartite graphs. Part I: Set numbers

The notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi-coloured and the sought after monochromatic subgraphs are complete, balanced, multipartite graphs, instead of complete graphs as in the classical de4nition. We previously con4ned our attention to diagonal multipartite Ramsey numbers. In this paper the de4nition of a multipartite Ramsey number is broadened still further, by incorporating o6-diagonal numbers, 4xing the number of vertices per partite set in the larger graph and then seeking the minimum number of such partite sets that would ensure the occurrence of certain speci4ed monochromatic multipartite subgraphs. c © 2004 Elsevier B.V. All rights reserved. MSC: 05C55; 05D10