Sperner Families and Partitions of A Partially Ordered Set

This paper is a summary (without proofs) of the main results in a series of papers by the author and D.J. Kleitman [14] and the author [11, 12, 13] concerning subsets of a finite partially ordered set called Sperner k-families. If P is a finite partially ordered set, a subset A ⊆ P is a k-family if A contains no chains of length k+1 (or, equivalently, if A can be expressed as the union of k 1-families in P). Maximum-sized k-families are called Sperner k-families of P.