Intersecting families of sets, no l containing two common elements

Let H denote the set {f1, f2, ..., fn}, 2 the collection of all subsets of H and F ⊆ 2 be a family. The maximum of |F| is studied if any k subsets have a non-empty intersection and the intersection of any l distinct subsets (1 ≤ k < l) is empty. This problem is reduced to a covering problem. If we have the conditions that any two subsets have a non-empty intersection and the intersection of any l distinct subsets contains no two different elements we show that the maximum of |F| is (l − 1)n+ o(n). AMS classification: 05D05; 05B25