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arXiv · 1811.03731

New bounds on the maximum size of Sperner partition systems

Yanxun Chang·Charles J. Colbourn·Adam Gowty·Daniel Horsley·Junling Zhou
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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.

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Yanxun Chang, Charles J. Colbourn, Adam Gowty, Daniel Horsley, Junling Zhou. 2020-04-15. New bounds on the maximum size of Sperner partition systems. https://doi.org/10.1016/j.ejc.2020.103165

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