Multipartite Ramsey number $m_j(K_m, nK_2)$

Yaser Rowshan

paper

Multipartite Ramsey number $m_{j} (K_{m}, n K_{2})$

arXiv:2201.04336

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 $G_{1}, G_{2}, \dots, G_{n}$ , the multipartite Ramsey number (M-R-number) $m_{j} (G_{1}, G_{2}, \dots, G_{n})$ is the smallest integer $t$ such that for any $n$ -edge-coloring $(G^{1}, G^{2}, \dots, G^{n})$ of the edges of $K_{j \times t}$ , $G^{i}$ contains a monochromatic copy of $G_{i}$ for at least one $i$ . The size of M-R-number $m_{j} (n K_{2}, C_{m})$ for $j, n \geq 2$ and $4 \leq m \leq 6$ , the size of M-R-number $m_{j} (n K_{2}, C_{7})$ for $j \geq 2$ and $n \geq 2$ , the size of M-R-number $m_{j} (n K_{2}, K_{3})$ , for each $j, n \geq 2$ , the size of M-R-number $m_{j} (K_{3}, K_{3}, n_{1} K_{2}, n_{2} K_{,} \dots, n_{i} K_{2})$ for $j \leq 6$ and $i, n_{i} \geq 1$ and the size of M-R-number $m_{j} (K_{3}, K_{3}, n K_{2})$ for $j \geq 2$ and $n \geq 1$ have been computed in several papers up to now. In this article we obtain the values of M-R-number $m_{j} (K_{m}, n K_{2})$ , for each $j, n \geq 2$ and each $m \geq 4$ .

arXiv admin note: text overlap with arXiv:2109.12210

Multipartite Ramsey number mj​(Km​,nK2​)

Abstract

Multipartite Ramsey number $m_{j} (K_{m}, n K_{2})$