Intersecting Systems

An intersecting system of type (∃, ∀, k, n) is a collection double-struck F sign = {ℱ₁,...,ℱ_m} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair ℱ_i and ℱ_j of distinct members of double-struck F sign there exists an A ∈ ℱ_i that intersects every B ∈ ℱ_j. Let I_n(∃, ∀, 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 I_n(∃, ∀, 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.