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

Assume that K j × n is a complete, and multipartite graph consisting of j partite sets and n vertices in each partite set. For given graphs G 1 , G 2 , ... , G n , the multipartite Ramsey number (M-R-number) m j (G 1 , G 2 , ... , G n) , is the smallest integer t , such that for any n -edge-coloring (G 1 , G 2 , ... , G n) of the edges of K j × t , G i contains a monochromatic copy of G i for at least one i. The size of M-R-number m j (C 3 , n K 2 ,) for j , n ≥ 2 , the size of M-R-number m j (n K 2 , m K 2) for j ≥ 2 and n , m ≥ 1 , and the size of M-R-number m j (C 3 , C 3 , n K 2) for j ≥ 2 and n ≥ 1 have been computed in several papers up to now. In this paper, we determine some lower bounds for the M-R-number m j (C 3 , C 3 , n K 2 , m K 2) for each j , n , m ≥ 2 , and some values of M-R-number m j (C 3 , C 3 , n K 2 , m K 2) for some j ≥ 2 , and each n , m ≥ 1. [ABSTRACT FROM AUTHOR]