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Another Generalization of Lindström's Theorem on Subcubes of a Cube

✍ Scribed by Uwe Leck

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
185 KB
Volume
99
Category
Article
ISSN
0097-3165

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✦ Synopsis

We consider the poset P ðN ; A 1 ; A 2 ; . . . ; A m Þ consisting of all subsets of a finite set N which do not contain any of the A i 's, where the A i 's are mutually disjoint subsets of N : The elements of P are ordered by inclusion. We show that P belongs to the class of Macaulay posets, i.e. we show a Kruskal-Katona-type theorem for P : For the case that the A i 's form a partition of N ; the dual P n of P came to be known as the orthogonal product of simplices. Since the property of being a Macaulay poset is preserved by turning to the dual, we show, in particular, that orthogonal products of simplices are Macaulay posets. Besides, we prove that the posets P and P n are additive.

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