子集平面最大反链的大小

Order Pub Date : 2024-06-27 DOI:10.1007/s11083-024-09675-9

Jerrold R. Griggs, Thomas Kalinowski, Uwe Leck, Ian T. Roberts, Michael Schmitz

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摘要

本文是两篇论文中的第二篇，研究在布尔网格 \(B_n\)（\([n]:=\{1,2,\dots ,n\})的幂集，按包含排序）中，对于哪些正整数 m 存在大小为 m 的最大反链。在第一部分中，已经描述了最大反链的大小。在这里，我们提供了另一种构造，它的好处是表明了几乎所有最大反链的大小都可以通过只包含 l 个集合和某个 l 的 \((l+1)\)集合的反链来获得。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Sizes of Flat Maximal Antichains of Subsets

This is the second of two papers investigating for which positive integers m there exists a maximal antichain of size m in the Boolean lattice \(B_n\) (the power set of \([n]:=\{1,2,\dots ,n\}\), ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only l-sets and \((l+1)\)-sets for some l.

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