Disjoint Systems (Extended Abstract)

A disjoint system of type (∀, ∃, κ, n) is a collection C={A, ..., A} of pairwise disjoint families of κ-subsets of an n-element set satisfying the following condition. For every ordered pair A i and A j of distinct members of C and for every A e C i there exists a B e C j that does not intersect A. Let D n (∀, ∃, κ) denote the maximum possible cardinality of a disjoint system of type (∀, ∃, κ, n). It is shown that for every fixed k≥2, $$lim_{n \to \infty } D_n \left( {\forall ,\exists ,k} \right)\left( {\begin{array}{*{20}c}n \\k \\\end{array} } \right)^{ - 1} = \frac{1}{2}.$$