The size multipartite Ramsey numbers mj(C3,C3,nK2,mK2)

Abstract

Assume that [Formula: see text] is a complete, and multipartite graph consisting of j partite sets and n 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].