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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    ABSTRACT: A coloring of a complete bipartite graph is shuffle-preserved if it is the case that assigning a color $c$ to edges $(u, v)$ and $(u', v')$ enforces the same color assignment for edges $(u, v')$ and $(u',v)$. (In words, the induced subgraph with respect to color $c$ is complete.) In this paper, we investigate a variant of the Ramsey problem for the class of complete bipartite multigraphs. (By a multigraph we mean a graph in which multiple edges, but no loops, are allowed.) Unlike the conventional m-coloring scheme in Ramsey theory which imposes a constraint (i.e., $m$) on the total number of colors allowed in a graph, we introduce a relaxed version called m-local coloring which only requires that, for every vertex $v$, the number of colors associated with $v$'s incident edges is bounded by $m$. Note that the number of colors found in a graph under $m$-local coloring may exceed m. We prove that given any $n \times n$ complete bipartite multigraph $G$, every shuffle-preserved $m$-local coloring displays a monochromatic copy of $K_{p,p}$ provided that $2(p-1)(m-1) < n$. Moreover, the above bound is tight when (i) $m=2$, or (ii) $n=2^k$ and $m=3\cdot 2^{k-2}$ for every integer $k\geq 2$. As for the lower bound of $p$, we show that the existence of a monochromatic $K_{p,p}$ is not guaranteed if $p> \lceil \frac{n}{m} \rceil$. Finally, we give a generalization for $k$-partite graphs and a method applicable to general graphs. Many conclusions found in $m$-local coloring can be inferred to similar results of $m$-coloring.
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    ABSTRACT: In this paper we establish a connection between polarized partition relations and the functionAq (n, d) of coding theory, which implies some known results and new examples of polarized relations. A connection between Aq(n, d) and the Zarankiewicz numbers is also discussed.
    Eur. J. Comb. 01/2001; 22:351-356.

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