Another Generalization of Lindstrِmʹs Theorem on Subcubes of a Cube

We consider the poset P(N;A1,A2,…,Am) consisting of all subsets of a finite set N which do not contain any of the Aiʹs, where the Aiʹs 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 Aiʹs form a partition of N, the dual P* 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* are additive.