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 Kj×t contains a monochromatic copy of Gi in color i for some i, 1≤i≤n, where Kj×t 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.

The bipartite Ramsey number B(n1,n2,…,nt) is the least positive integer b, such that any coloring of the edges of Kb,b with t colors will result in a monochromatic copy of Kni,ni in the i-th color, for some i, 1≤i≤t. The values B(2,5)=17, B(2,2,2,2)=19 and B(2,2,2)=11 have been computed in several previously published papers. In this paper, we obtain the exact values of the bipartite Ramsey number B(2,2,3). In particular, we prove the conjecture on B(2,2,3) which was proposed in 2015—in fact, we prove that B(2,2,3)=17.

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Assume that [Formula: see text] is a complete, and multipartite graph consisting of [Formula: see text] partite sets and [Formula: see text] vertices in each partite set. For given graphs [Formula: see text], the multipartite Ramsey number (M-R-number) [Formula: see text], is the smallest integer [Formula: see text], such that for any [Formula: see text]-edge-coloring [Formula: see text] of the edges of [Formula: see text], [Formula: see text] contains a monochromatic copy of [Formula: see text] for at least one [Formula: see text]. The size of M-R-number [Formula: see text] for [Formula: see text], the size of M-R-number [Formula: see text] for [Formula: see text] and [Formula: see text], and the size of M-R-number [Formula: see text] for [Formula: see text] and [Formula: see text] have been computed in several papers up to now. In this paper, we determine some lower bounds for the M-R-number [Formula: see text] for each [Formula: see text], and some values of M-R-number [Formula: see text] for some [Formula: see text], and each [Formula: see text].