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

Csaba Biró, Bartłomiej Bosek, Heather C. Smith, William T. Trotter, Ruidong Wang, Stephen J. Young

Published 2019-06-19Version 1

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.