An intersecting system of type (∃, ∀, k, n) is a collection []={[Fscr]1, ...,[Fscr]m} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair [Fscr]i and [Fscr]j of distinct members of [] there exists an A∈[Fscr]i that intersects every B∈[Fscr]j. Let In(∃, ∀, k) denote the maximum possible cardinality of an intersecting system of type (∃, ∀, k, n). Ahlswede, Cai and Zhang conjectured that for every k≥1, there exists an n0(k) so that In(∃, ∀, k)=(n−1/k−1) for all n>n0(k). Here we show that this is true for k≤3, but false for all k≥8. We also prove some related results.