Intersecting Sperner families and their convex hulls (Q796532)

For a family \({\mathcal F}\) of subsets of an n-element set the profile \((f_ 0\), \(f_ 1,...,f_ n)\) is the vector of the numbers \(f_ k\) of k- element subsets in \({\mathcal F}\). A family \({\mathcal F}\) is a Sperner-family if for no \(A,B\in {\mathcal F} A<B\) holds. For the convex hull in \(R^{n+1}\) of the set of profiles of Sperner families the extreme points are determined by using the LYM-inequality. An analogous theorem is obtained for intersecting Sperner-families where additionally \(A\cap B\neq \emptyset\) for all A,\(B\in {\mathcal F}\). The convex hull of the profiles of the latter families can also be described by presenting the bordering hyperplanes. From this the theorem on the extreme points is deduced by making use of the duality theorem of linear programming. The results are applied to some extremal problems for directed hypergraphs.