A Generalization of Sperner's Theorem on Compressed Ideals

Abstract

Let $[n]=\{1,2,\ldots,n\}$ and $\mathscr{B}_n=\{A: A\subseteq [n]\}$. A family $\mathscr{A}\subseteq \mathscr{B}_n$ is a Sperner family if $A\nsubseteq B$ and $B\nsubseteq A$ for distinct $A,B\in\mathscr{A}$. Sperner's theorem states that the density of the largest Sperner family in $\mathscr{B}_n$ is $\binom{n}{\left\lceil{n/2}\right\rceil}/2^n$. The objective of this note is to show that the same holds if $\mathscr{B}_n$ is replaced by compressed ideals over $[n]$.