  We call a family G of subsets of [n] a k-generator of (\mathbb{P}[n]) if
every (x \subset [n]) can be expressed as a union of at most k disjoint sets in
(\mathcal{G}). Frein, Leveque and Sebo conjectured that for any (n \geq k),
such a family must be at least as large as the k-generator obtained by taking a
partition of [n] into classes of sizes as equal as possible, and taking the
union of the power-sets of the classes. We generalize a theorem of Alon and
Frankl \cite{alon} in order to show that for fixed k, any k-generator of
(\mathbb{P}[n]) must have size at least (k2^{n/k}(1-o(1))), thereby verifying
the conjecture asymptotically for multiples of k.
