External problems for finite sets and convex hulls – A survey

Let F be a family of distinct subsets of an n-element set. Define p(i)(F) (0 less than or equal to i less than or equal to n) as the number of i-element members of F. Consider the profile vectors (p(0)(F),...,p(n)(F)) for all families F belonging to a certain class A (e.g. A can be the class of all families where any two members have a non-empty intersection). Let epsilon(A) denote the set of extreme points of the convex hull of the set of these profile vectors. Results determining epsilon(A) for some classes A are surveyed. Facets and edges of these convex hulls are also described for some A. Connections to the classical extremal problems are shown.