Boolean dimension and tree-width

TL;DR

This paper proves that posets with cover graphs of bounded tree-width have bounded boolean dimension, contrasting with the existence of posets with higher dimension and the same tree-width, and explores implications for planar posets.

Contribution

It establishes a bound on boolean dimension for posets with cover graphs of bounded tree-width, advancing understanding of poset complexity measures.

Findings

Posets with bounded tree-width cover graphs have bounded boolean dimension.

There exist posets with tree-width three and arbitrarily large dimension.

The result may help resolve whether planar posets have bounded boolean dimension.

Abstract

The dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if $P$ has dimension $d$, then to know whether $x \leq y$ in $P$ it is enough to check whether $x\leq y$ in each of the $d$ linear extensions of a witnessing realizer. Focusing on the encoding aspect Ne\v{s}et\v{r}il and Pudl\'{a}k defined a more expressive version of dimension. A poset $P$ has boolean dimension at most $d$ if it is possible to decide whether $x \leq y$ in $P$ by looking at the relative position of $x$ and $y$ in only $d$ permutations of the elements of $P$. We prove that posets with cover graphs of bounded tree-width have bounded boolean dimension. This stays in contrast with the fact that there are posets with cover graphs of tree-width three and arbitrarily large dimension. This result might be a step towards a resolution of the long-standing open problem: Do planar posets have bounded boolean dimension?