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arXiv · 2108.08033

Ramsey Properties for $V$-shaped Posets in the Boolean Lattices

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

Given posets $\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k$, let the {\em Boolean Ramsey number} $R(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k)$ be the minimum number $n$ such that no matter how we color the elements in the Boolean lattice $\mathbf{B}_n$ with $k$ colors, there always exists a poset $\mathbf{P}_i$ contained in $\mathbf{B}_n$ whose elements are all colored with $i$. This function was first introduced by Axenovich and Walzer~\cite{AW}. Recently, many results on determining $R(\mathbf{B}_m,\mathbf{B}_n)$ have been published. In this paper, we will study the function $R(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k)$ for each $\mathbf{P}_i$'s being the $V$-shaped poset. That is, a poset obtained by identifying the minimal elements of two chains. Another major result presented in the paper is to determine the minimal posets $\mathbf{Q}$ contained in $\mathbf{B}_n$, when $R(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k)=n$ is determined, having the Ramsey property described in the previous paragraph. In addition, we define the {\em Boolean rainbow Ramsey number} $RR(\mathbf{P},\mathbf{Q})$ the minimum number $n$ such that when arbitrarily coloring the elements in $\mathbf{B}_n$, there always exists either a monochromatic $\mathbf{P}$ or a rainbow $\mathbf{Q}$ contained in $\mathbf{B}_n$. The upper bound for $RR(\mathbf{P},\mathbf{A}_k)$ was given by Chang, Li, Gerbner, Methuku, Nagy, Patkos, and Vizer for general poset $\mathbf{P}$ and $k$-element antichain $\mathbf{A}_k$. We study the function for $\mathbf{P}$ being the $V$-shaped posets in this paper as well.

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Hong-Bin Chen, Wei-Han Chen, Yen-Jen Cheng, Wei-Tian Li, Chia-An Liu. 2021-08-18. Ramsey Properties for $V$-shaped Posets in the Boolean Lattices. https://arxiv.org/abs/2108.08033

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