The $m$-bipartite Ramsey number $BR_m(K_{2,2},K_{5,5})$

TL;DR

This paper investigates the $m$-bipartite Ramsey number for the graphs $K_{2,2}$ and $K_{5,5}$, determining its size for certain values of $m$, extending previous results in bipartite Ramsey theory.

Contribution

The paper computes the size of $BR_m(K_{2,2}, K_{5,5})$ for specific $m \\geq 2$, advancing understanding of bipartite Ramsey numbers involving these graphs.

Findings

Determined $BR_m(K_{2,2}, K_{5,5})$ for some $m \\geq 2$

Extended known results on bipartite Ramsey numbers involving $K_{2,2}$ and $K_{5,5}$

Provided new bounds and exact values for specific $m$

Abstract

The bipartite Ramsey number $BR(H_1,H_2,\ldots,H_k)$, is the smallest positive integer $b$, such that each $k$-decomposition of $E(K_{b,b})$ contains $H_i$ in the $i$-th class for some $i, 1\leq i\leq k$. As another view of bipartite Ramsey numbers, for given two bipartite graphs $H_1$ and $H_2$ and a positive integer $m$, the $m$-bipartite Ramsey number $BR_m(H_1, H_2)$, is defined as the least integer $n$, such that any subgraph of $K_{m,n}$ say $H$, results in $H_1\subseteq H$ or $H_2\subseteq \overline{H}$. The size of $BR_m(K_{2,2}, K_{3,3})$, $BR_m(K_{2,2}, K_{4,4})$ for each $m$, and the size of $BR_m(K_{3,3}, K_{3,3})$ for some $m$, have been determined in several papers up to now. Also, it is shown that $BR(K_{2,2}, K_{5,5})=17$. In this article, we compute the size of $BR_m(K_{2,2}, K_{5,5})$ for some $m\geq 2$.