The local dimension of suborders of the Boolean lattice

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 $\left(\frac{1}{4}-o(1)\right)\frac{n}{\log_2 n}$.