Local dimension of a Boolean lattice

TL;DR

This paper proves that the local dimension of the Boolean lattice of all subsets of an n-element set is less than n for all n ≥ 4, resolving a previously posed question and exploring related problems.

Contribution

It establishes a new upper bound on the local dimension of Boolean lattices, answering a specific open question in poset theory.

Findings

The local dimension of the Boolean lattice is strictly less than n for all n ≥ 4.

The paper provides bounds and properties related to the local dimension of Boolean lattices.

It explores related problems in the structure of posets and their dimensions.

Abstract

For every integer $n$ with $n \geq 4$, we prove that the local dimension of a poset consisting of all the subsets of $\{1,\dots,n\}$ equipped with the inclusion relation is strictly less than $n$, answering a question of Kim, Martin, Masa\v{r}\'ik, Shull, Smith, Uzzell, and Wang (Eur. J. Comb. 2020). We also study several related problems.