On the orthogonal product of simplices and direct products of truncated Boolean lattices Original Research Article

The initial point of this paper are two Kruskal–Katona type theorems. The first is the Colored Kruskal–Katona Theorem which can be stated as follows: Direct products of the form Bk11×Bk21×⋯×Bkn1 belong to the class of Macaulay posets, where Bkt denotes the poset consisting of the t+1 lowest levels of the Boolean lattice Bk. The second one is a recent result saying that also the products Bk1k1−1×Bk2k2−1×⋯×Bknkn−1 are Macaulay posets. The main result of this paper is that the natural common generalization to products of truncated Boolean lattices does not hold, i.e. that (Bkt)n is a Macaulay poset only if t∈{0,1,k−1,k}.