Disjoint Systems

A disjoint system of type (∀,∃, k, n) is a collection C = {A1, . . . ,Am} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair Ai and Aj of distinct members of C and for every A ∈ Ai there exists a B ∈ Aj that does not intersect A. Let Dn(∀,∃, k) denote the maximum possible cardinality of a disjoint system of type (∀,∃, k, n). It is shown that for every fixed k ≥ 2, limn→∞Dn(∀,∃, k) ( n k )−1 = 1 2 . This settles a problem of Ahlswede, Cai and Zhang. Several related problems are considered as well.