Planar Posets that are Accessible from Below Have Dimension at Most 6

2019/06/19 by Biró, Csaba, Bosek, Bartłomiej, Smith, Heather C. +3
#05C35 #06A07 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1906.08145

Abstract

Planar posets can have arbitrarily large dimension. However, a planar poset of height h has dimension at most 192h+96, while a planar poset with t minimal elements has dimension at most 2t+1. In particular, a planar poset with a unique minimal element has dimension at most 3. In this paper, we extend this result by showing that a planar poset has dimension at most 6 if it has a plane diagram in which every minimal element is accessible from below.

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