On Some Applications of Graph Theory Iii

1 . In the first and second parts of this sequence we dealt with applications of graph theory to distance distribution in certain sets in euclidean spaces, to potential theory, to estimations of the transfinite diameter [1] and to value distribution of "triangle functionals" (e.g. perimeter, area of triangles) [2] . The basic tool is provided in all these applications by the result formulated as Lemma 2 . This, an essentially pure logical result, proves to be a very flexible and versatile instrument in applications . Here the same method is used in an abstract setting . First we deduce certain results for the density of a given family of subsets of an abstract set S in another family of subsets of the same S. Then we apply the results obtained to distance distribution in certain (e.g. totally bounded or compact) sets in metric spaces, in particular in a normed linear function space. Applications of this method to functionals on Hilbert spaces were given by Katona [3] . 2. Let S denote an infinite set and F an infinite family of its finite subsets satisfying