Dimension is polynomial in height for posets with planar cover graphs

TL;DR

This paper proves that posets with planar cover graphs have polynomial dimension in terms of height, significantly improving previous exponential bounds, with planarity being a crucial factor in the analysis.

Contribution

It establishes a polynomial upper bound on the dimension of height h posets with planar cover graphs, advancing understanding of poset dimension constraints.

Findings

Dimension is polynomial in height for such posets

Planarity is essential for the polynomial bound

Exponential dimension can occur without planarity

Abstract

We show that height $h$ posets that have planar cover graphs have dimension $\mathcal{O}(h^6)$. Previously, the best upper bound was $2^{\mathcal{O}(h^3)}$. Planarity plays a key role in our arguments, since there are posets such that (1) dimension is exponential in height and (2) the cover graph excludes $K_5$ as a minor.