Combinatorics Extremal combinatorics Theorem

Sperner's theorem

Statement

Let $E$ be a set with $n$ elements. Let $\mathcal{S}$ be a family of subsets of $E$ in which no set is a proper subset of another. Then $|\mathcal{S}| \le \binom{n}{\lfloor n/2 \rfloor}$ . Equality holds if and only if $\mathcal{S}$ is the family of all subsets of size $\lfloor n/2 \rfloor$ , or of all subsets of size $\lceil n/2 \rceil$ .

Remarks

A family with this property is a Sperner family, or an antichain of sets. For each $k$ , the subsets of size $k$ form an antichain. The binomial coefficient $\binom{n}{k}$ is largest for $k = \lfloor n/2 \rfloor$ . Lubell gave a short proof in 1966, with the Lubell–Yamamoto–Meshalkin inequality.

History

Emanuel Sperner published the theorem in 1928.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Binomial coefficient definition
Sperner's theorem statement

Tag	`0J87V9K1GE`
URI	`m2://0J87V9K1GE`
Kind	statement, theorem
Writing	W1 stated
MSC	05D05
Status	This copy of the site has no build data. Run `m2 status 0J87V9K1GE`.
Uses	Binomial coefficient
Same as	Wikidata Q2226786 Wikipedia Sperner's theorem

Sperner's theorem

Statement

Remarks

History

Reading path

Further reading

What this page needs for W2

What links here