More constructions for Sperner partition systems

TL;DR

This paper introduces a new construction for Sperner partition systems, enabling asymptotic determination of their maximum size in many cases and establishing optimality for various small parameters and specific congruence conditions.

Contribution

It presents a novel construction based on dividing the set into equal parts and extends existing methods to determine the maximum size of Sperner partition systems asymptotically and for specific parameter sets.

Findings

Asymptotically determined $ ext{SP}(n,k)$ for many cases with bounded $n/k$.

Constructed maximum size Sperner systems for numerous small $(n,k)$.

Extended existing constructions to cover cases where $n ot ot ot ot k$ mod $2k$ for odd $k$.

Abstract

An $(n,k)$-Sperner partition system is a set of partitions of some $n$-set such that each partition has $k$ nonempty parts and no part in any partition is a subset of a part in a different partition. The maximum number of partitions in an $(n,k)$-Sperner partition system is denoted $\mathrm{SP}(n,k)$. In this paper we introduce a new construction for Sperner partition systems based on a division of the ground set into many equal-sized parts. We use this to asymptotically determine $\mathrm{SP}(n,k)$ in many cases where $\frac{n}{k}$ is bounded as $n$ becomes large. Further, we show that this construction produces a Sperner partition system of maximum size for numerous small parameter sets $(n,k)$. By extending a separate existing construction, we also establish the asymptotics of $\mathrm{SP}(n,k)$ when $n \equiv k \pm 1 \pmod{2k}$ for almost all odd values of $k$.