The Multipartite Ramsey numbers $m_j(C_3,C_m,n_1{K}_2,n_2{K}_2,\dots ,n_i{K}_2)$

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 $G_1,G_2,\dots ,G_n$, the multipartite Ramsey number (M-R-number) $m_j(G_1,G_2,\dots ,G_n)$ is the smallest integer $t$ such that for any $n$-edge-coloring $(G^1,G^2,\dots ,G^n)$ 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 $m_j(n{K}_2,C_7)$ for $j\ge 2$ and $n\le 6$. Y. Rowshan et al. $(2021)$ gave the size of M-R-number $m_j(n{K}_2,C_7)$ for $j=2,3,4$ and $n\ge 2$. Y. Rowshan $(2021)$ gave the size of M-R-number $m_j(n{K}_2,C_7)$, for each $j\ge 5$ and $n\ge 2$. In this article we compute the size of M-R-number $m_j(C_3,C_3,n{K}_2)$ for each $j\ge 7$, $n\ge 1$, $m_j(C_3,C_3,n_1{K}_2,n_2{K}_{,}\dots ,n_i{K}_2)$ for each $2\le j\le 6$, $i\ge 2,n_i\ge 1$, and M-R-number $m_j(C_3,C_4,n{K}_2)$, for each $n\ge 1$, and small $j$.

arXiv admin note: text overlap with arXiv:2109.02257

References in corpus (3)

• The bipartite Ramsey numbers $BR(C_8,C_{2n})$
• A proof of a conjecture on Ramsey numbers $B(2,2,3)$
• The Multipartite Ramsey numbers $m_j(n{K}_2,C_7)$