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

## TL;DR

This paper proves that planar posets with a specific diagram property, where every minimal element is accessible from below, have a maximum dimension of 6, extending previous bounds based on height and minimal elements.

## Contribution

It establishes a new upper bound of 6 on the dimension of planar posets with a certain accessibility property in their diagram.

## Key findings

- Planar posets with the accessibility property have dimension at most 6.
- Previous bounds depended on height and number of minimal elements.
- The result generalizes known bounds for special classes of planar posets.

## 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.