The $m$-bipartite Ramsey number $BR_m(H_1,H_2)$

TL;DR

This paper determines the exact values of the $m$-bipartite Ramsey number for the pair $(K_{2,2}, K_{4,4})$ across all $m \, \geq 2$, extending known results for other bipartite graphs.

Contribution

The paper provides the first exact determination of $BR_m(K_{2,2}, K_{4,4})$ for all $m \, \geq 2$, filling a gap in bipartite Ramsey number research.

Findings

Exact values of $BR_m(K_{2,2}, K_{4,4})$ for all $m \, \geq 2$

Extension of known bipartite Ramsey numbers to new graph pairs

Complements previous results on $BR_m(K_{2,2}, K_{2,2})$, $BR_m(K_{2,2}, K_{3,3})$, and $BR_m(K_{3,3}, K_{3,3})$

Abstract

In a $(G^1,G^2)$ coloring of a graph $G$, every edge of $G$ is in $G^1$ or $G^2$. For two bipartite graphs $H_1$ and $H_2$, the bipartite Ramsey number $BR(H_1, H_2)$ is the least integer $b\geq 1$, such that for every $(G^1, G^2)$ coloring of the complete bipartite graph $K_{b,b}$, results in either $H_1\subseteq G^1$ or $H_2\subseteq G^2$. As another view, for bipartite graphs $H_1$ and $H_2$ and a positive integer $m$, the $m$-bipartite Ramsey number $BR_m(H_1, H_2)$ of $H_1$ and $H_2$ is the least integer $n$, such that every subgraph $G$ of $K_{m,n}$ results in $H_1\subseteq G$ or $H_2\subseteq \overline{G}$. The size of $m$-bipartite Ramsey number $BR_m(K_{2,2}, K_{2,2})$, the size of $m$-bipartite Ramsey number $BR_m(K_{2,2}, K_{3,3})$ and the size of $m$-bipartite Ramsey number $BR_m(K_{3,3}, K_{3,3})$ have been computed in several articles up to now. In this paper we determine the exact value of $BR_m(K_{2,2}, K_{4,4})$ for each $m\geq 2$.