A Sperner-type theorem for set-partition systems
The electronic journal of combinatorics, Tome 12 (2005)
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 $n$.
@article{10_37236_1987,
author = {Karen Meagher and Lucia Moura and Brett Stevens},
title = {A {Sperner-type} theorem for set-partition systems},
journal = {The electronic journal of combinatorics},
year = {2005},
volume = {12},
doi = {10.37236/1987},
zbl = {1077.05097},
url = {http://geodesic.mathdoc.fr/articles/10.37236/1987/}
}
Karen Meagher; Lucia Moura; Brett Stevens. A Sperner-type theorem for set-partition systems. The electronic journal of combinatorics, Tome 12 (2005). doi: 10.37236/1987
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