Set-coloring of Edges and Multigraph Ramsey Numbers

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.