The maximum size of intersecting and union families of sets

We consider the maximal size of families of k-element subsets of an n element set [n] = {1, 2, . . . , n} that satisfy the properties that every r subsets of the family have non-empty intersection, and no ` subsets contain [n] in their union. We show that for large enough n, the largest such family is the trivial one of all ( n−2 k−1 ) subsets that contain a given element and do not contain another given element. Moreover we show that unless such a family is such that all subsets contain a given element, or all subsets miss a given element, then it has size at most .9 ( n−2 k−1 ) . We also obtain versions of these statements for weighted non-uniform families.