Intersecting Systems

An intersecting system of type (∃,∀, k, n) is a collection F = {F1, . . . ,Fm} pf pairwise disjoint families of k–subsets of an n–element set satisfying the following condition. For every ordered pair Fi and Fj of distinct members of F there exists an A ∈ Fi that intersects every B ∈ Fj . Let In(∃,∀, k) denote the maximum possible cardinality of an intersecting system of type (∃,∀, 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.