The local dimension of suborders of the Boolean lattice

TL;DR

This paper establishes asymptotic bounds on the local dimension of layers in the Boolean lattice, showing it is approximately n/log n, and improves bounds on the maximum local dimension of n-element posets.

Contribution

It provides tight asymptotic bounds on the local dimension of specific Boolean lattice layers and improves the lower bound for the maximum local dimension of n-element posets.

Findings

Asymptotic local dimension of middle layers is n/log n.

Upper and lower bounds on local dimension of lattice layers.

Maximum local dimension of n-element posets is at least (1/4 - o(1)) n/log n.

Abstract

We prove upper and lower bounds on the local dimension of any pair of layers of the Boolean lattice, and show that the local dimension of the first and middle layers of the $n$-dimensional Boolean lattice is asymptotically $\frac{n}{\log_2 n}$ as $n\to\infty$. Previously, all that was known was a lower bound of $\Omega(n/\log n)$ and an upper bound of $n$. Improving a result of Kim, Martin, Masa\v{r}\'{i}k, Shull, Smith, Uzzell, and Wang, we also prove that that the maximum local dimension of an $n$-element poset is at least $\left(\frac{1}{4}-o(1)\right)\frac{n}{\log_2 n}$.