New bounds on the maximum size of Sperner partition systems

Abstract

An $(n,k)$-Sperner partition system is a collection of partitions of some $n$-set, each into $k$ nonempty classes, such that no class of any partition is a subset of a class of any other. 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 and use it to asymptotically determine $\mathrm{SP}(n,k)$ in many cases as $\frac{n}{k}$ becomes large. We also give a slightly improved upper bound for $\mathrm{SP}(n,k)$ and exhibit an infinite family of parameter sets $(n,k)$ for which this bound is tight.