Another Generalization of Lindström's 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 Ai’s, where the Ai’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 Ai’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.