ON TWO SPERNER-TYPE CONDITIONS

Abstract

This chapter describes the two Sperner-type conditions. It is assumed that if (P, ≤) is a finite partially ordered set, a subset A⊆ P is a k-family if A contains no chain of length k + 1, maximum-sized k-families are called Sperner k-families. It is supposed that Fk(P) denote the set of all k-families of P, and Sk(P) denote the set of all Sperner k-families of P. The results can be classified in several ways that include bounds on the size of k-families for special orders, characterization of the elements M ∈ Sk(P) for special orders, and properties of a lattice-ordering defined on k-families. It is assumed Et denote the set {0, 1 , …, t−1}. Two conditions on the set Ent each of which contain (for t = 2) the Sperner property as a special case. It is found that for one of these conditions, an analog of the 1-families is considered.