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.

中文翻译：

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Sperner分区系统最大大小的新界限

一个 $（ñ，ķ）$- Sperner隔断系统是一些分区的集合$ñ$-设置，每个成 $ķ$非空类，因此任何分区的任何类都不是任何其他类的子集。一个分区中的最大分区数$（ñ，ķ）$-Sperner分区系统表示 $\mathrm{SP}（ñ，ķ）$。在本文中，我们介绍了Sperner分区系统的新结构，并使用它渐近确定$\mathrm{SP}（ñ，ķ）$ 在许多情况下 $\frac{ñ}{ķ}$变大。我们还为$\mathrm{SP}（ñ，ķ）$ 并展示出无限个参数集 $（ñ，ķ）$ 为此，这个界限是紧密的。