The size, multipartite ramsey numbers for nk2 versus path-path and cycle

Abstract

For given graphs G1, G2,…, Gn and any integer j, the size of the multipartite Ramsey number mj(G1, G2,…, Gn) is the smallest positive integer t such that any n-coloring of the edges of Kjxt contains a monochromatic copy of Gi in color i for some i, 1 ≤ i ≤ n, where Kjxt denotes the complete multipartite graph having j classes with t vertices per each class. In this paper, we computed the size of the multipartite Ramsey numbers mj(K1,2, P4, nK2) for any j, n ≥ 2 and mj(nK2, C7), for any j ≤ 4 and n ≤ 2.