The size, multipartite Ramsey numbers for nK2 versus path-path and cycle

TL;DR

This paper determines the exact sizes of certain multipartite Ramsey numbers involving paths, cycles, and matchings, expanding understanding of edge colorings in complete multipartite graphs.

Contribution

It computes the sizes of multipartite Ramsey numbers for specific graph configurations, including $m_j(K_{1,2}, P_4, nK_2)$ and $m_j(nK_2,C_7)$, for various parameters.

Findings

Calculated $m_j(K_{1,2}, P_4, nK_2)$ for all $j,n \\geq 2$.

Determined $m_j(nK_2,C_7)$ for $j \\leq 4$ and $n \\geq 2$.

Abstract

For given graphs $G_1, G_2,\ldots, G_n$ and any integer $j$, the size of the multipartite Ramsey number $m_j(G_1, G_2,\ldots, G_n)$ is the smallest positive integer $t$ such that any $n$-coloring of the edges of $K_{j\times t}$ contains a monochromatic copy of $G_i$ in color $i$ for some $i$, $1 \leq i \leq n$, where $K_{j\times t}$ denotes the complete multipartite graph having $j$ classes with $t$ vertices per each class. In this paper we compute the size of the multipartite Ramsey number $m_j(K_{1,2}, P_4, nK_2)$ for any $j,n\geq 2$ and $m_j(nK_2,C_7)$, for any $j\leq4$ and $n\geq 2$.