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arXiv · 1909.11370

The Boolean Rainbow Ramsey Number of Antichains, Boolean Posets, and Chains

Hong-Bin Chen·Yen-Jen Cheng·Wei-Tian Li·Chia-An Liu
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Abstract

Motivated by the paper of Axenovich and Walzer [2], we study the Ramsey-type problems on the Boolean lattices. Given posets $P$ and $Q$, we look for the smallest Boolean lattice $\mathcal{B}_N$ such that any coloring on elements of $\mathcal{B}_N$ must contain a monochromatic $P$ or a rainbow $Q$. This number $N$ is called the Boolean rainbow Ramsey number of $P$ and $Q$ in the paper. Particularly, we determine the exact values of the Boolean rainbow Ramsey number for $P$ and $Q$ being the antichains, the Boolean posets, or the chains. From these results, we also give some general upper and lower bounds of the Boolean rainbow Ramsey number for general $P$ and $Q$ in terms of the poset parameters.

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Hong-Bin Chen, Yen-Jen Cheng, Wei-Tian Li, Chia-An Liu. 2019-09-25. The Boolean Rainbow Ramsey Number of Antichains, Boolean Posets, and Chains. https://arxiv.org/abs/1909.11370

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