论文标题
Sperner分区系统的更多构造
More constructions for Sperner partition systems
论文作者
论文摘要
$（n，k）$ - Sperner分区系统是一组$ n $ set的分区，因此每个分区都有$ k $ nonepty零件，并且任何分区中没有任何零件是不同分区中的零件的子集。 $（n，k）$ - 斑点分区系统中的最大分区数表示$ \ mathrm {sp}（n，k）$。在本文中，我们基于将地面划分的斑点分区系统介绍了新的建筑。在许多情况下，$ \ frac {n} {k} $在$ n $变大的情况下，我们使用它来渐近地确定$ \ mathrm {sp}（n，k）$。此外，我们表明该构造可为许多小参数集$（n，k）$产生一个最大尺寸的斑点分区系统。通过扩展一个现有的构造，我们还建立了$ \ mathrm {sp}（n，k）$的渐近学，当$ n \ equiv k \ pm 1 \ pmod {2k} $几乎所有$ k $的奇数。
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 $k$.
