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

TL;DR

This paper determines the multipartite Ramsey numbers for various small graphs and configurations, extending previous results to new parameter ranges and providing exact values for specific cases.

Contribution

It computes new multipartite Ramsey numbers involving triangles, cycles, and matchings for a wide range of parameters, filling gaps in existing literature.

Findings

Exact values of m_j(C_3,C_3, nK_2) for j ≥ 7 and n ≥ 1.

Values of m_j(C_3,C_4, nK_2) for small j and n ≥ 1.

Extended the known ranges of multipartite Ramsey numbers for specific graph configurations.

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 $j$.