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 $n$.

Sperner Partition Systems

“…This paper continues the work from that established Sperner‐type theorems for partitions. Two subsets from a set X are call incomparable if neither set is contained in the other.…”

“…A k ‐partition system is an almost‐uniform partition system if every partition in the system is almost uniform. In , it was conjectured that the largest Sperner k ‐partition system on an n ‐set is an almost‐uniform partition system. Theorem confirms this conjecture for the case where .…”

“…Sperner partition systems were introduced by Meagher, Moura, and Stevens who established bounds on the number of partitions in such a system. There have been extensions of other results in extremal set theory to partitions, for example, there are two versions of the Erdős‐Ko‐Rado theorem for partitions, see and .…”

“…With this motivation, we seek to determine bounds on the size of for certain values of n and k . In , the exact value of is found when k divides n . Theorem Let be integers, then Moreover, a Sperner partition system meets this bound only if every class of every partition in the system has exactly ℓ elements .…”

“…For general, n only an upper bound on the size of Sperner partition system was determined in . Theorem Let , and r be integers with and .…”