On the Ramsey Problem for Multicolor Bipartite Graphs

Department of Philosophy and Center for Logic and Epistemology, State University of Campinas, Campinas, SP, Brazil; Department of Mathematics, State University of Maringá, Maringá, PR, Brazil
Advances in Applied Mathematics (Impact Factor: 0.88). 01/1999; DOI: 10.1006/aama.1998.0620

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.

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