Optimizing diversity 1

We consider the problem of minimizing the size of a set system 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 is originating in a real-world application aiming at the diversity of industrial production, and at the same time the k=2 case is a question of Erdős, studied recently by Füredi and Katona. We conjecture that a simple construction providing a feasible solution is optimal for this problem; we prove this conjecture in special cases, complementary to those solved by Füredi and Katona, in particular for the case n≤3k. These special cases occur to be interesting from the viewpoint of the application as well.