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