Extremal hypergraph problems and convex hulls
Let X be a finite set of n elements and F be a family of its subsets. Then \(F_ k\) denotes the subfamily of the k-element subsets in F and \(| F_ k| =f_ k\). The vector \((f_ 0,f_ 1,...,f_ n)\) in the \((n+1)\)-dimensional Euclidean space \(R^{n+1}\) is called the profile of F. If c is a finite set in \(R^{n+1}\), the convex hull of c is the set of all convex linear combinations of the elements of c. \(p\in c\) is an extreme point of c iff p is not a convex linear combination of elements of c different from p. The determination of the convex hull of a set is equivalent to finding its extreme points. k-Sperner-family has no \(k+1\) different members satisfying \(A_ 1\subset...\subset A_{k+1}\). A family F is intersecting if A,B\(\in F\) implies \(A\cap B\neq \emptyset.\) The present paper tries to determine the extreme points of the set of profiles of the simplest known classes of families. The effort is successful for 3 classes: (1) intersecting families, (2) k-Sperner- families, (3) \(G_ 1,...,G_ t\) are not necessarily disjoint families, where \(A\in G_ i\), \(B\in G_ j\), \(i\neq j\), \(A\neq B\) imply \(A\not\subset B\). The results contain many old theorems of extremal set theory as particular cases (Sperner, Erdős-Ko-Rado, Daykin-Frankl- Green-Hilton).
scientific article; zbMATH DE number 3926951