Article

# 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 - [show abstract] [hide abstract]

**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.06/2003; - [show abstract] [hide abstract]

**ABSTRACT:**Some bounds for G 1 --G 2 bipartite Ramsey numbers b(G 1 ; G 2 ) are given, which imply that b(K 2; 2 ; K 1;n )=n + q for the range q 2 - q +16n6q 2 , where q is a prime power. Our new construction establishes in particular that b(K 2;n ; K 2;n )=4n - 3if4n - 3 is a prime power, reinforcing a weaker form of a conjecture due to Beineke and Schwenk. Particular relationships between b(G 1 ; G 2 ) and G 1 --G 2 Ramsey numbers are also determined. c # 2000 Elsevier Science B.V. All rights reserved. MSC: 05C55 Keywords: Generalized Ramsey theory; Bipartite Ramsey number; Strongly regular graphs 1. Introduction Given G 1 and G 2 bipartite graphs, the bipartite Ramsey number b(G 1 ; G 2 )isthe least positive integer b such that if the edges of K b; b are colored with two colors, then there will always exist an isomorphic copy of G 1 in the #rst color, or a copy of G 2 in the second color. In other words, b(G 1 ; G 2 ) is the least positive integer b such that given any subgraph H...12/2000; - [show abstract] [hide abstract]

**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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