On Some Three Color Ramsey Numbers for Paths, Cycles, Stripes and Stars

Abstract

For given graphs G_{1}, G_{2},ldots , G_{k}, k ge 2, the multicolor Ramsey numberR(G_{1}, G_{2},ldots , G_{k}) is the smallest integer n such that if we arbitrarily color the edges of the complete graph of order n with k colors, then it contains a monochromatic copy of G_{i} in color i, for some 1 le i le k. The main result of the paper is a theorem which establishes the connection between the multicolor Ramsey number and the appropriate multicolor bipartite Ramsey number together with the ordinary Ramsey number. The remaining part of the paper consists of a number of corollaries which are derived from the main result and from known results for Ramsey numbers and bipartite Ramsey numbers. We provide some new exact values or generalize known results for multicolor Ramsey numbers of paths, cycles, stripes and stars versus other graphs.