On the Hypercube Subset Partitioning Varieties

In this paper, the problem of a quantitative description of partitions (QDP) of arbitrary m-subsets of the n-dimensional unit cube is considered for a given m, 0 ≤ m ≤ 2n. A necessary condition for the existence of a given QDP-subset is achieved in terms of minimal and maximal layers that are known by earlier publications. It is shown that QDP are in a correspondence to the upper homogeneous area elements of the n-cube and to the monotone Boolean functions. The NP-hardness of the QDP problem is proved. QDP singular points on different layers of the cube are described.