Generating All Sets With Bounded Unions

We consider the problem of minimizing the size of a family of sets G such that every subset of {1,...,n} can be written as a disjoint union of at most k members of G, where k and n are given numbers. This problem originates in a real-world application aiming at the diversity of industrial production. At the same time, the question of finding the minimum of \G\ so that every subset of {1,...,n} is the union of two sets in G was asked by Erd (o) over tildes and studied recently by Furedi and Katona without requiring the disjointness of the sets. A simple construction providing a feasible solution is conjectured to be optimal for this problem for all values of it and k and regardless of the disjointness requirement; we prove this conjecture in special cases including all (n,k) for which n <= 3k holds, and some individual values of n and k.