Disjoint systems (Extended abstract)

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 ε C _i there exists a B ε 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,

$$lim_{n \to \infty } D_n \left( {\forall ,\exists ,k} \right)\left( {\begin{array}{*{20}c}n \\k \\\end{array} } \right)^{ - 1} = \frac{1}{2}.$$

This settles a problem of Ahlswede, Cai and Zhang. Several related problems are considered as well.

Research supported in part by a United States — Israel BSF Grant