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arXiv · 1809.05413

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

Zhidan Luo·Yuejian Peng
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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.

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Zhidan Luo, Yuejian Peng. 2018-09-13. 3-colored asymmetric bipartite Ramsey number of connected matchings and cycles. https://arxiv.org/abs/1809.05413

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