Planarity and dimension I

TL;DR

This paper proves that for posets with planar cover graphs, the dimension is bounded polynomially by the size of the largest standard example, resolving a long-standing open problem in poset theory.

Contribution

It establishes that large dimension in such posets necessarily implies the presence of large standard examples, providing a polynomial bound on the dimension.

Findings

Posets with planar cover graphs have dimension at most polynomial in the size of their largest standard example.

Large dimension in these posets is characterized by large standard examples.

The result confirms a long-standing conjecture in poset theory.

Abstract

The dimension of a partially ordered set $P$ (poset for short) is the least positive integer $d$ such that $P$ is isomorphic to a subposet of $\mathbb{R}^d$ with the natural product order. Dimension is arguably the most widely studied measure of complexity for posets, and standard examples in posets are the canonical structure forcing dimension to be large. In many ways, dimension for posets is analogous to chromatic number for graphs with standard examples in posets playing the role of cliques in graphs. However, planar graphs have chromatic number at most four, while posets with planar diagrams may have arbitrarily large dimension. The key feature of all known constructions of such posets is that large dimension is forced by a large standard example. The question of whether every poset of large dimension and with a planar cover graph contains a large standard example has been a critical challenge in posets theory since the early 1980s, with very little progress over the years. We answer the question in the affirmative. Namely, we show that every poset $P$ with a planar cover graph has dimension $\mathcal{O}(s^8)$, where $s$ is the maximum order of a standard example in $P$.