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

More constructions for Sperner partition systems