There is a natural way to associate with a poset P a hypergraph , called the hypergraph of critical pairs, so that the dimension of P is exactly equal to the chromatic number of . The edges of have variable sizes, but it is of interest to consider the graph G formed by the edges of that have size 2. The chromatic number of G is less than or equal to the dimension of P and the difference between the two values can be arbitrarily large. Nevertheless, there are important instances where the two parameters are the same, and we study one of these in this paper. Our focus is on a family {Snk:n≥3,k≥0}$\{{S_{n}^{k}}:n\ge 3, k\ge 0\}$ of height two posets called crowns. We show that the chromatic number of the graph Gnk${G_{n}^{k}}$ of critical pairs of the crown Snk${S_{n}^{k}}$ is the same as the dimension of Snk${S_{n}^{k}}$, which is known to be ⌈2(n + k)/(k + 2)⌉. In fact, this theorem follows as an immediate corollary to the stronger result: The independence number of Gnk${G_{n}^{k}}$ is (k + 1)(k + 2)/2. We obtain this theorem as part of a comprehensive analysis of independent sets in Gnk${G_{n}^{k}}$ including the determination of the second largest size among the maximal independent sets, both the reversible and non-reversible types.

中文翻译：

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冠的临界对图

有一种自然的方式将偏序 P 与超图相关联，称为临界对超图，因此 P 的维数恰好等于 的色数。的边具有可变的大小，但考虑由大小为 2 的边形成的图 G 是有趣的。 G 的色数小于或等于 P 的维数，并且两个值之间的差异可以任意大。然而，存在两个参数相同的重要实例，我们在本文中研究其中之一。我们的重点是一个家庭 {Snk:n≥3,k≥0}$\{{S_{n}^{k}}:n\ge 3, k\ge 0\}$ 的高度两个称为冠的偏序集。我们表明，冠 Snk${S_{n}^{k}}$ 的关键对的图 Gnk${G_{n}^{k}}$ 的色数与 Snk$ 的维数相同{S_{n}^{k}}$, 已知为 ⌈2(n + k)/(k + 2)⌉。事实上，这个定理是更强结果的直接推论：Gnk${G_{n}^{k}}$ 的独立数是 (k + 1)(k + 2)/2。我们获得这个定理作为对 Gnk${G_{n}^{k}}$ 中独立集的综合分析的一部分，包括确定最大独立集（可逆和不可逆类型）中的第二大尺寸。