# The Sperner property

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In 1927 Emanuel Sperner proved that if $S_1,\dots,S_m$ are distinct

subsets of an $n$-element set such that we never have $S_i\subset S_j$,

then $m\leq \binom{n}{\lfloor n/2\rfloor}$. Moreover, equality is achieved

by taking all subsets of $S$ with $\lfloor n/2\rfloor$ elements. This

result spawned a host of generalizations, most conveniently stated in the

language of partially ordered sets. We will survey some of the highlights

of this subject, including the use of linear algebra and the cohomology of

certain complex projective varieties. An application is a proof of a

conjecture of Erd\H{o}s and Moser, namely, for all integers $n\geq 1$ and

real numbers $\alpha\geq 0$, the number of subsets with element sum

$\alpha$ of an $n$-element set of positive real numbers cannot exceed the

number of subsets of $\{1,2,\dots,n\}$ whose elements sum to $\lfloor

\frac 12\binom n2\rfloor$. We will conclude by discussing two recent

proofs of a 1984 conjecture of Anders Bj\"orner on the weak Bruhat order

of the symmetric group $S_n$.