Profile polytopes of some classes of families

The profile vector of a family F of subsets of an n-element set is (f (0),f (1),aEuro broken vertical bar,f (n) ) where f (i) denotes the number of the i-element members of F. The extreme points of the set of profile vectors for some class of families has long been studied. In this paper we introduce the notion of k-antichainpair families and determine the extreme points of the set of profile vectors of these families, extending results of Engel and P.L. ErdAs regarding extreme points of the set of profile vectors of intersecting, co-intersecting Sperner families. Using this result we determine the extreme points of the set of profile vectors for some other classes of families, including complement-free k-Sperner families and self-complementary k-Sperner families. We determine the maximum cardinality of intersecting k-Sperner families, generalizing a classical result of Milner from k = 1.