 Sperner's theorem

Every antichain in $\mathcal{P}([n])$ has at most $\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$ members. Equality is attained only by a full middle level, with either middle level possible when $n$ is odd.

Sperner's theorem is a result in combinatorics that deals with families of subsets of a finite set. Specifically, it states that if you have a set \( S \) with \( n \) elements, the largest family of subsets of \( S \) that can be chosen such that no one subset is contained within another (i.e.