2021/08/18 by Hongbin Chen, Chen, Hong-Bin, Wei-Han Chen +7
Mathematics · #05C55 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2108.08033
openalex publication_date 2021/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given posets P1,P2,…,Pk, let the \em Boolean Ramsey number R(P1,P2,…,Pk) be the minimum number n such that no matter how we color the elements in the Boolean lattice Bn with k colors, there always exists a poset Pi contained in Bn whose elements are all colored with i. This function was first introduced by Axenovich and Walzer~\citeAW. Recently, many results on determining R(Bm,Bn) have been published. In this paper, we will study the function R(P1,P2,…,Pk) for each Pi'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 Q contained in Bn, when R(P1,P2,…,Pk)=n is determined, having the Ramsey property described in the previous paragraph. In addition, we define the \em Boolean rainbow Ramsey number RR(P,Q) the minimum number n such that when arbitrarily coloring the elements in Bn, there always exists either a monochromatic P or a rainbow Q contained in Bn. The upper bound for RR(P,Ak) was given by Chang, Li, Gerbner, Methuku, Nagy, Patkos, and Vizer for general poset P and k-element antichain Ak. We study the function for P being the V-shaped posets in this paper as well.