Abstract :
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}.
