On the Ramsey problem for multicolor bipartite graphs

Abstract

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. © 1999 Academic Press.