The Multipartite Ramsey numbers $m_j(nK_2,C_7)$

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 $G_1$ and $G_2$, the multipartite Ramsey number (M-R-number) $m_j(G_1, G_2)$ is the smallest integer $t$ such that any subgraph $G$ of the $K_{j\times t}$, either $G$ contains a copy of $G_1$ or its complement relative to $K_{j\times t}$ contains a copy of $G_2$. C. J. Jayawardene, E. T. Baskoro et al. $(2016)$ gave the size of M-R-numbe $m_j(nK_2, C_7)$ for $j \geq 2 $ and $n\leq 6$. Y. Rowshan et al. $(2021)$ gave the size of M-R-number $m_j(nK_2, C_7)$ for $j = 2,3, 4$ and $n\geq 2$. In this article we compute the size of M-R-number $m_j(nK_2,C_7)$, for each $j\geq 5$ and $n\geq 2$.