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 ] that satisfy the properties that every r subsets of the family have non-empty intersection, and no ℓ 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 ) . o obtain versions of these statements for weighted non-uniform families.