Set-coloring of Edges and Multigraph Ramsey Numbers

Abstract

Edge-colorings of multigraphs are studied where a generalization of Ramsey numbers is given. Let \({M_n^{(r)}}\) be the multigraph of order n, in which there are r edges between any two different vertices. Suppose q ₁, q ₂, . . . , q _k and r are positive integers, and q _i ≥ 2(1 ≤ i ≤ k), k > r. Let the multigraph Ramsey number \({f^{(r)} (q_1 ,q_2 , \ldots ,q_k )}\) be the minimum positive integer n such that in any k-edge coloring of \({M_n^{(r)}}\) (every edge is colored with one among k given colors, and edges between the same pair of vertices are colored with different colors), there must be \({i \in \{1,2,\ldots,k\}}\) such that \({M_n^{(r)}}\) has such a complete subgraph of order q _i , of which all the edges are in color i. By Ramsey’s theorem it is easy to show \({f^{(r)} (q_1 ,q_2 , \ldots ,q_k )}\) exists for given q ₁ ,q ₂, . . . , q _k and r. Lower and upper bounds for some multigraph Ramsey numbers are given.