Posets with cover graph of pathwidth two have bounded dimension

TL;DR

This paper proves that posets with cover graphs of pathwidth two have a bounded dimension, specifically at most 17, and establishes a lower bound on cover graph treewidth for certain subposets.

Contribution

It confirms the existence of a universal bound on the dimension of posets with cover graphs of pathwidth two, answering a previously open question.

Findings

Posets with cover graph of pathwidth two have dimension at most 17.

Posets containing S_5 as a subposet have cover graphs with treewidth at least 3.

The result provides a concrete bound for a class of posets with restricted cover graph structure.

Abstract

Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant $d$ such that if $P$ is a poset with cover graph of $P$ of pathwidth at most $2$, then $\dim(P)\leq d$. We answer this question in the affirmative by showing that $d=17$ is sufficient. We also show that if $P$ is a poset containing the standard example $S_5$ as a subposet, then the cover graph of $P$ has treewidth at least $3$.