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

TL;DR

This paper determines the multipartite Ramsey numbers for specific graphs, expanding known results to cases where the number of partite sets is five or more, for all relevant values of n.

Contribution

It extends previous work by computing the multipartite Ramsey number m_j(nK_2,C_7) for all j ≥ 5 and n ≥ 2, filling a gap in the existing literature.

Findings

Calculated m_j(nK_2,C_7) for j ≥ 5 and n ≥ 2.

Extended known results to larger j values.

Provides exact values for new parameter ranges.

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 $n\geq 2$.