A disjoint system of type (?, ?, k, n) is a collection ?? = {??1,…, ??m} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair ??i and ??j of distinct members of ?? and for every A ? ??i there exists a B ? ??j 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,. This settles a problem of Ahlswede, Cai, and Zhang. Several related problems are considered as well.
