Planar posets have dimension at most linear in their height

TL;DR

This paper proves that the dimension of planar posets is linearly bounded by their height, improving previous exponential bounds and establishing tightness up to a constant factor.

Contribution

It establishes a linear upper bound on the dimension of planar posets in terms of height, and constructs examples showing near-optimal lower bounds.

Findings

Dimension of planar posets is at most 192h + 96.

Constructed planar posets with dimension at least (4/3)h - 2.

Abstract

We prove that every planar poset $P$ of height $h$ has dimension at most $192h + 96$. This improves on previous exponential bounds and is best possible up to a constant factor. We complement this result with a construction of planar posets of height $h$ and dimension at least $(4/3)h-2$.