A Sperner-Type Theorem for Set-Partition Systems

Abstract

A Sperner partition system is a system of set partitions such that any two set partitions $P$ and $Q$ in the system have the property that for all classes $A$ of $P$ and all classes $B$ of $Q$, $A \not\subseteq B$ and $B \not\subseteq A$. A $k$-partition is a set partition with $k$ classes and a $k$-partition is said to be uniform if every class has the same cardinality $c=n/k$. In this paper, we prove a higher order generalization of Sperner's Theorem. In particular, we show that if $k$ divides $n$ the largest Sperner $k$-partition system on an $n$-set has cardinality ${n-1 \choose n/k-1}$ and is a uniform partition system. We give a bound on the cardinality of a Sperner $k$-partition system of an $n$-set for any $k$ and $n$.