Some results on the multipartite Ramsey numbers mj(C3,Cm,n1K2,n2K2,…,niK2)

Abstract

The graph Kj×t is a graph which is complete and multipartite which includes j partite sets and t vertices in each partite set. The multipartite Ramsey number (M-R-number) mj(G1,G2,…,Gn) is the smallest integer t for the mentioned graphs G1,G2,…,Gn, in a way which for each n-edge-coloring (G1,G2,…,Gn) of the edges of Kj×t, Gi contains a monochromatic copy of Gi for at least one i. The size of M-R-number mj(nK2,C7) for j≥2, n≤6, the M-R-number mj(nK2,C7) for j=2,3,4, n≥2, the M-R-number mj(nK2,C7) for each j≥5, n≥2, the M-R-number mj(C3,C3,n1K2,n2K2,…,niK2) for j≤6, and i,ni≥1, and the size of M-R-number mj(C3,C3,nK2) for j≥2 and n≥1 have been calculated in various articles hitherto. We acquire some bounds of M-R-number mj(C3,C3,n1K2,n2K2,…,niK2) in this essay in which i,j≥2, and ni≥1, also the size of M-R-number m4(C3,C4,nK2) for each n≥1 is computed in this paper.