Uwe Leck
Another generalization of Lindströms theorem on subcubes of a cube
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock

MSC:

06A07 Combinatorics of partially ordered sets

05D05 Extremal set theory

Abstract: We consider the poset $P(N;A_1,A_2,\dots,A_m)$ 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 $P^*$ of $P$
became 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 $P^*$ are additive.
Keywords: Macaulay posets, shadow minimization, Kruskal-Katona theorem, orthogonal product of simplices