North-Holland POLYTOPES DETERMINED BY COMPLEMENTFREE SPERNER FAMILIES

The profile of a family of subsets of an n-element set is a vector f = (f,, , f,), where fk denotes the number of k-element sets in the family. Using a new method the extreme points of the convex hull of the profiles of all complementfree Sperner families over an n-element set are determined. LetN={l,..., n} and (7) = {Xc N: 1X( = i}. If 9 is a family of subsets of N, then let 5 = {X E 9: 1x1 = i} and A = 141, i = 0,..., n. The vector f = &.*-7.L) is called the profile of 9. If A is a class of families, let p(A) be the set of profiles of the families belonging to A, (p(A)) be its convex hull in the space [Wn+l and E(A) be the set of all extreme points (i.e. vertices) of the polytope (p(A)). This subject was first studied by P.L. Erdiis, P. Frank1 and G.O.H. Katona who determined E(A) for some special classes [4]. The determination of E(A) is motivated by the following fact: If w is any (weight-)function from (0,..., n} into [w, then (see [4] or [3, pp. 172 ff]). In this paper we will present a new method of proof that a given set of vectors is the set E(A) of all extreme points arising from the considered class A. The idea is simple: To prove that a vector f is contained in some bounded polytope P it is enough to show that f is contained in some smaller bounded polytope (e.g. a simplex) all of whose extreme points belong to P. Thereby the consideration of canonical families is useful. Let A, be the class of all Sperner families (9 is a Sperner family if X, Y E 9, X c Y imply X = Y), A2 be the class of all complementfree Sperner families (9 is compfementfree if X E 9 implies N\X 9) and, if n = 2k, let A3 be the class of all Sperner families for which fk c i(z) holds. It is easy to see that A2 E Al and