Sperner partition systems

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 $k$-partition system} on a set $X$ is a set of partitions of $X$ into $k$ 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 $|X| = k \ell$, then the largest Sperner $k$-partition system has size $\frac{1}{k}\binom{|X|}{\ell}$. In this paper we find bounds on the size of the largest Sperner $k$-partition system where $k$ does not divide the size of $X$, specifically, we give an exact bound when $k=2$ and upper and lower bounds when $|X| = 2k+1$, $|X|=2k+2$ and $|X| = 3k-1$.