Polytopes determined by complementfree Sperner families

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