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 ϵ 𝒜i there exists a B ϵ 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,. **image** This settles a problem of Ahlswede, Cai, and Zhang. Several related problems are considered as well. © 1995 Wiley Periodicals, Inc.