On Dedekind's problem, a sparse version of Sperner's theorem, and antichains of a given size in the Boolean lattice

Dedekind’s problem, dating back to 1897, asks for the total number $\psi (n)$ of antichains contained in the Boolean lattice $B_n$ on $n$ elements. We study Dedekind’s problem using a recently developed method based on the cluster expansion from statistical physics and as a result, obtain several new results on the number and typical structure of antichains in $B_n$. We obtain detailed estimates for both $\psi (n)$ and the number of antichains of size $\beta \left(\genfrac{}{}{0px}{}{n}{\lfloor n\mathrm{/}2\rfloor }\right)$ for any fixed $\beta >0$. We also establish a sparse version of Sperner’s theorem: we determine the sharp threshold and scaling window for the property that almost every antichain of size $m$ is contained in a middle layer of $B_n$.