On the Ramsey Problem for Multicolor Bipartite Graphs

Giveni,jpositive integers, letKi,jdenote a bipartite complete graph and letRr(m,n) be the smallest integerasuch that for anyr-coloring of the edges ofKa,aone can always find a monochromatic subgraph isomorphic toKm,n. In other words, ifa?Rr(m,n) then every matrixa×awith entries in {0,1,?,r?1} always contains a submatrixm×norn×mwhose entries arei, 0?i?r?1. We shall prove thatR2(m,n)?2m(n?1)+2m?1?1, which generalizes the previous resultsR2(2,n)?4n?3 andR2(3,n)?8n?5 due to Beineke and Schwenk. Moreover, we find a class of lower bounds based on properties of orthogonal Latin squares which establishes that limr?∞Rr(2,2)r?2=1.