Multipartite Ramsey number $m_j(K_m, nK_2)$

Abstract

Assume that $K_{j\times n}$ be a complete, multipartite graph consisting of $j$ partite sets and $n$ vertices in each partite set. For given graphs $G_1, G_2,\ldots, G_n$, the multipartite Ramsey number (M-R-number) $m_j(G_1, G_2, \ldots,G_n)$ is the smallest integer $t$ such that for any $n$-edge-coloring $(G^1,G^2,\ldots, G^n)$ of the edges of $K_{j\times t}$, $G^i$ contains a monochromatic copy of $G_i$ for at least one $i$. The size of M-R-number $m_j(nK_2, C_m)$ for $j, n\geq 2$ and $4\leq m\leq 6$, the size of M-R-number $m_j(nK_2, C_7)$ for $j \geq 2$ and $n\geq 2$, the size of M-R-number $m_j(nK_2,K_3)$, for each $j,n\geq 2$, the size of M-R-number $m_j(K_3,K_3, n_1K_2,n_2K_,\ldots,n_iK_2)$ for $j \leq 6$ and $i,n_i\geq 1$ and the size of M-R-number $m_j(K_3,K_3, nK_2)$ for $j \geq 2$ and $n\geq 1$ have been computed in several papers up to now. In this article we obtain the values of M-R-number $m_j(K_m, nK_2)$, for each $j,n\geq 2$ and each $m\geq 4$.