Polytopes Determined by Hypergraph Classes

DEFINITION 1.1. For a finite subset Xc IR n + 1 we say that D is the dominating set of X if (a) Dc X, (b) for all x E X there exist dlo d2 , ••• , dj ED and positive real numbers alo"" aj so that L. a, = I and L. ajdj dominates x, (c) D is minimal with respect to these properties. The elements of D are called dominating vertices. It is not hard to see that D consists of exactly those vertices of the convex hull of X which are not dominated by any other vertex. Most extremal hypergraph problems can be formulated in the following way. Suppose we are given a weight function w:{O, ... ,n}-'»Z+, i.e, w(i);;;'O for all i. What is the maximum ofL.;=o w(i)/; over all families $ c 2 satisfying certain properties (e.g. F n F' ¥o holds for all F, F' E fJi)? That is, we have to find the maximum of a linear function with non-negative coefficients over the set X of all possible profile vectors. Clearly, this maximum equals the maximum over all dominating vertices. Thus, for most problems, it is sufficient to determine the dominating set of possible profile vectors. This was done in [5] and [6] for some classes of hypergraphs. However, the proofs were lengthy and relied heavily on the duality theorem of linear programming. On the other hand [6] developed a unified treatment of these problems using no result of extremal set theory. Here we propose much shorter individual proofs using extremal set theory. Our theorems are generalizations of old results of extremal types. The connections and consequences can be found in [5] and [6].