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# 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 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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##### Article: Multicolor Ramsey Numbers for Complete Bipartite Versus Complete Graphs
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ABSTRACT: Let H_1, ..., H_k be graphs. The multicolor Ramsey number r(H_1,...,H_k) is the minimum integer r such that in every edge-coloring of K_r by k colors, there is a monochromatic copy of H_i in color i for some 1 <= i <= k. In this paper, we investigate the multicolor Ramsey number $r(K_{2,t},...,K_{2,t},K_m)$, determining the asymptotic behavior up to a polylogarithmic factor for almost all ranges of t and m. Several different constructions are used for the lower bounds, including the random graph and explicit graphs built from finite fields. A technique of Alon and R\"odl using the probabilistic method and spectral arguments is employed to supply tight lower bounds. A sample result is $c_1 m^2t/\log^4(mt) \leq r(K_{2,t},K_{2,t},K_m) \leq c_2 m^2t/\log^2 m$ for any t and m, where c_1 and c_2 are absolute constants.
Journal of Graph Theory 01/2012; 18(2). · 0.63 Impact Factor
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##### Article: A bipartite Ramsey problem and the Zarankiewicz numbers
Glasgow Mathematical Journal 12/1977; 19(01):13 - 26. · 0.44 Impact Factor
• ##### Article: Ramsey theory
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ABSTRACT: An abstract is not available.
01/1996;