A Generalization of Sperner's Theorem on Compressed Ideals

Lili Mu,Yi Wang

doi:10.37236/5750

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 $[n]$.

Highlights

Let Bn be the poset of subsets of [n] = {1, 2, . . . , n} ordered by inclusion
For P ⊆ Bn, we say that P is a convex family if A, B ∈ P and A ⊆ C ⊆ B imply that C ∈ P
Let I be a compressed ideal in Bn and A the largest Sperner family in I

Summary

Introduction

A famous result due to Sperner [5] states that the density of the largest Sperner family in Bn is n n/2. A family I ⊆ Bn is an ideal if A ∈ I and B ⊆ A imply B ∈ I. In [3, Conjecture 1.3], Frankl conjectured that the density of the largest Sperner family in any convex subfamily of Bn is at least n n/2. For every convex family P over the set [n], there exists a Sperner family. Let C(m, Bn) be the family of the first m minimal elements of Bn with respect to. Let I be a compressed ideal in Bn and A the largest Sperner family in I

Proof of Theorem 2

Remarks