Dimension of posets with planar cover graphs excluding two long incomparable chains

TL;DR

This paper proves that posets with planar cover graphs excluding two long incomparable chains have bounded dimension, confirming a longstanding conjecture and expanding understanding of the structure of such posets.

Contribution

It confirms Gutowski and Krawczyk's conjecture that excluding two long chains bounds the dimension of posets with planar cover graphs.

Findings

Posets with planar cover graphs excluding two long chains have bounded dimension.

The conjecture by Gutowski and Krawczyk is proven true.

Discussion on potential generalizations beyond planar cover graphs.

Abstract

It has been known for more than 40 years that there are posets with planar cover graphs and arbitrarily large dimension. Recently, Streib and Trotter proved that such posets must have large height. In fact, all known constructions of such posets have two large disjoint chains with all points in one chain incomparable with all points in the other. Gutowski and Krawczyk conjectured that this feature is necessary. More formally, they conjectured that for every $k\geq 1$, there is a constant $d$ such that if $P$ is a poset with a planar cover graph and $P$ excludes $\mathbf{k}+\mathbf{k}$, then $\dim(P)\leq d$. We settle their conjecture in the affirmative. We also discuss possibilities of generalizing the result by relaxing the condition that the cover graph is planar.