Sperner partition systems
P. C. Li, Karen Meagher

TL;DR
This paper investigates the maximum size of Sperner $k$-partition systems, providing exact bounds for specific cases and new bounds for others, expanding understanding of these combinatorial structures.
Contribution
It establishes new bounds on Sperner $k$-partition systems for cases where $k$ does not divide $|X|$, including exact bounds for $k=2$ and bounds for specific $|X|$ values.
Findings
Exact bound for $k=2$ case.
Upper and lower bounds for $|X|=2k+1$, $2k+2$, and $3k-1$.
Abstract
A \textsl{Sperner -partition system} on a set is a set of partitions of into classes such that the classes of the partitions form a Sperner set system (so no class from a partition is a subset of a class from another partition). These systems were defined by Meagher, Moura and Stevens in \cite{MMS} who showed that if , then the largest Sperner -partition system has size . In this paper we find bounds on the size of the largest Sperner -partition system where does not divide the size of , specifically, we give an exact bound when and upper and lower bounds when , and .
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Taxonomy
TopicsLimits and Structures in Graph Theory · graph theory and CDMA systems · Italy: Economic History and Contemporary Issues
