Two-color multipartite Ramsey numbers of matchings and cycles

Abstract

Let $K_{j \times n}$ denote a complete multipartite graph with $j$ partite sets, each containing $n$ vertices. For given graphs $G_1$ and $G_2$, the multipartite Ramsey number (denoted $m_j(G_1, G_2)$) is the smallest integer $t$ such that any subgraph $G$ of $K_{j \times t}$ either contains a copy of $G_1$, or its complement relative to $K_{j \times t}$ contains a copy of $G_2$. Determining the exact value of Ramsey numbers is a challenging problem, with many cases providing only bounds rather than exact values. Due to the broad applications of these numbers, finding exact values for specific graph classes is highly valuable. In recent years, researchers have studied the multipartite Ramsey numbers $m_j(nK_2, C_m)$. The exact values of these numbers have been determined for various specific cases of $j$ and $m$. Notably, the problem is fully resolved for $m \leq 6$, and for $m = 7$, it has been solved when $j \leq 4$. In this article, we compute the M-R-number $m_j(nK_2, C_7)$ for all remaining unsolved cases, specifically for $j \geq 5$ and $n \geq 2$.