The Multipartite Ramsey numbers $m_j(n{K}_2,C_7)$

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 $G_1$ and $G_2$, the multipartite Ramsey number (M-R-number) $m_j(G_1,G_2)$ is the smallest integer $t$ such that any subgraph $G$ of the $K_{j\times t}$, either $G$ contains a copy of $G_1$ or its complement relative to $K_{j\times t}$ contains a copy of $G_2$. C. J. Jayawardene, E. T. Baskoro et al. $(2016)$ gave the size of M-R-numbe $m_j(n{K}_2,C_7)$ for $j\ge 2$ and $n\le 6$. Y. Rowshan et al. $(2021)$ gave the size of M-R-number $m_j(n{K}_2,C_7)$ for $j=2,3,4$ and $n\ge 2$. In this article we compute the size of M-R-number $m_j(n{K}_2,C_7)$, for each $j\ge 5$ and $n\ge 2$.

A fundamental flaw has been discovered in the proof of the main theorem (Theorem 1.2), and the theorem is not valid in general

References in corpus (2)

• The bipartite Ramsey numbers $BR(C_8,C_{2n})$
• A proof of a conjecture on Ramsey numbers $B(2,2,3)$

Cited by in corpus (1)

• The Multipartite Ramsey numbers $m_j(C_3,C_m,n_1{K}_2,n_2{K}_2,\dots ,n_i{K}_2)$