The Multipartite Ramsey numbers $m_j(C_3, C_m, n_1K_2,n_2K_2,\ldots, n_iK_2)$

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 on $i$. 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 $n\geq 2$. Y. Rowshan $(2021)$ gave the size of M-R-number $m_j(nK_2,C_7)$, for each $j\geq 5$ and $n\geq 2$. In this article we compute the size of M-R-number $m_j(C_3,C_3, nK_2)$ for each $j\geq 7$, $n\geq 1$, $m_j(C_3,C_3, n_1K_2,n_2K_,\ldots,n_iK_2)$ for each $2\leq j\leq 6$, $i\geq 2, n_i\geq 1$, and M-R-number $m_j(C_3,C_4, nK_2)$, for each $n\geq 1$, and small $j$.