多不饱和偏序集和图与Greene-Kleitman定理
Polyunsaturated Posets and Graphs and the Greene-Kleitman Theorem
摘要：
有限偏序集的链划分给k个反链的并的大小带来了一个自然的上界。一个链划分是k-饱和的，如果达到这个界限。Greene和Kleitman证明了，对于每个k，每个有限偏序集都有一个同时k-和k+1-饱和的链划分。West证明了Greene-Kleitman定理在强意义上是最好可能的，对每个c \ge 4，展示了一个偏序集，它具有基数为c的最长链，并且对任何不同的、非连续的k，l < c，没有k-和l-饱和链划分。我们称这种偏序集为多不饱和偏序集。给出了具有给定高度、宽度和基数的多不饱和偏序集存在的充要条件。我们证明这些结果在更一般的情况下，图满足类似的格林-克莱特曼定理。最后，我们讨论了反链分拆的类似结果。
英文摘要：
A partition of a finite poset into chains places a natural upper bound on the size of a union of k antichains. A chain partition is k-saturated if this bound is achieved. Greene and Kleitman proved that, for each k, every finite poset has a simultaneously k- and k+1-saturated chain partition. West showed that the Greene-Kleitman Theorem is best-possible in a strong sense by exhibiting, for each c \ge 4, a poset with longest chain of cardinality c and no k- and l-saturated chain partition for any distinct, nonconsecutive k,l < c. We call such posets polyunsaturated. We give necessary and sufficient conditions for the existence of polyunsaturated posets with prescribed height, width, and cardinality. We prove these results in the more general context of graphs satisfying an analogue of the Greene-Kleitman Theorem. Lastly, we discuss analogous results for antichain partitions.
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