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

Yaser Rowshan; Mostafa Gholami

paper

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

arXiv:2101.00617

Abstract

For given graphs $G_{1}, G_{2}, \dots, G_{n}$ and any integer $j$ , the size of the multipartite Ramsey number $m_{j} (G_{1}, G_{2}, \dots, G_{n})$ 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 $m_{j} (K_{1, 2}, P_{4}, n K_{2})$ for any $j, n \geq 2$ and $m_{j} (n K_{2}, C_{7})$ , for any $j \leq 4$ and $n \geq 2$ .