3-colored asymmetric bipartite Ramsey number of connected matchings and cycles

TL;DR

This paper determines the exact 3-colored asymmetric bipartite Ramsey number for connected matchings and cycles, using Szemerédi's Regularity Lemma to relate matchings to cycles.

Contribution

It provides the complete exact value of the 3-colored asymmetric bipartite Ramsey number for connected matchings and applies a regularity lemma technique to asymptotically determine the number for cycles.

Findings

Exact value of r(k,l,m) for connected matchings

Asymptotic determination of bipartite Ramsey number for cycles

Application of Szemerédi's Regularity Lemma in bipartite Ramsey theory

Abstract

Let $k,l,m$ be integers and $r(k,l,m)$ be the minimum integer $N$ such that for any red-blue-green coloring of $K_{N,N}$, there is a red matching of size at least $k$ in a component, or a blue matching of at least size $l$ in a component, or a green matching of size at least $m$ in a component. In this paper, we determine the exact value of $r(k,l,m)$ completely. Applying a technique originated by {\L}uczak that applies Szemer\'edi's Regularity Lemma to reduce the problem of showing the existence of a monochromatic cycle to show the existence of a monochromatic matching in a component, we obtain the 3-colored asymmetric bipartite Ramsey number of cycles asymptotically.