Local dimension of a Boolean lattice

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.