Richard Stanley




Thursday, January 24, 2019 - 4:00pm to 5:00pm



RH 306

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 $S_n$.