On minimal coverings of the Boolean cube by centered antichains Original Research Article

We consider coverings of the Boolean n-cube Bn={0,1}n by families of its subsets of the special form which are called centered antichains. Each such subset consists of pairwise incomparable strings which have at least one common unit component; the collection of centered antichains also includes the one-element set containing the only string 0̃n consisting entirely of zeros. It is established that for each n⩾1 the minimum number of centered antichains, the union of which covers the n-cube, equals n⌊log2 n⌋+2(n−2⌊log2 n⌋)+2. For each n, a minimal covering is constructed in an explicit form.