ON TWO SPERNER-TYPE CONDITIONS

Publisher Summary 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 F k ( P ) denote the set of all k -families of P, and S k ( 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 ∈ S k ( P ) for special orders, and properties of a lattice-ordering defined on k -families. It is assumed E t denote the set {0 , 1 , …, t− 1}. Two conditions on the set E n t 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.