Polytopes determined by hypergraph classes (Q1073043)

This very outstanding paper studies a new direction of extremal hypergraph problems: the theory of convex hulls of hypergraph families. Most extremal hypergraph problems can be formulated in the following way. There is given a non-negative weight function depending on the cardinality of the subsets. What is the maximum of the sum of this weight function over the subsets of hypergraphs of given type. With the help of the basic theorem of convex sets these can be formulated like this in general: Let \({\mathcal F}\) be hypergraph, and p(\({\mathcal F})=(f_ 0,...,f_ n)\) be the profile of \({\mathcal F}\) (n is the cardinality of groundset, \(f_ k=\) the number of k-element set in \({\mathcal F})\). The profiles of hypergraphs of given type form an \(n+1\) dimensional point set. What is the maximum of the sum of the weight function over the extreme points of the set. This approach yields very general theorems. This method applies strong instruments originally, but the present authors applying some well-known theorems of extremal hypergraph theory give new short proofs for some older theorems.