Size multipartite Ramsey numbers for stripes versus small cycles

Abstract

For simple graphs $G_1$ and $G_2$, the size Ramsey multipartite number $m_j(G_1, G_2)$ is defined as the smallest natural number $s$ such that any arbitrary two coloring of the graph $K_{j \times s}$ using the colors red and blue, contains a red $G_1$ or a blue $G_2$ as subgraphs. In this paper, we obtain the exact values of the size Ramsey numbers $m_j(nK_2, C_m)$ for $j \ge 2$ and $m \in \{3,4,5,6\}$.