**MSC:**- 06A07 Combinatorics of partially ordered sets
- 06A12 Semilattices, See also {20M10}

The first is the Colored Kruskal-Katona Theorem which can be stated as

follows: Direct products of the form $B_{k_1}^1\times B_{k_2}^1\times

\cdots\times B_{k_n}^1$ belong to the class of Macaulay posets, where

$B_k^t$ denotes the poset consisting of the $t+1$ lowest levels of the

Boolean lattice $B_k$. The second one is a recent result saying that

also the products $B_{k_1}^{k_1-1}\times B_{k_2}^{k_2-1}\times

\cdots\times B_{k_n}^{k_n-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 $(B_k^t)^n$ is a

Macaulay poset only if $t\in\{0,1,k-1,k\}$.