Multipartite Ramsey number $m_j(K_m, nK_2)$

TL;DR

This paper determines the multipartite Ramsey numbers for complete graphs and multiple copies of edges, extending known results to new parameter ranges and providing exact values for these complex combinatorial configurations.

Contribution

It computes the multipartite Ramsey number $m_j(K_m, nK_2)$ for all relevant parameters, filling gaps in existing literature.

Findings

Exact values of $m_j(K_m, nK_2)$ for all $j,n ext{≥}2$ and $m ext{≥}4$

Extension of known multipartite Ramsey number results

Provides a comprehensive set of values 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 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 $m\geq 4$.