Note on Generating All Subsets of a Finite Set with Disjoint Unions

We call a family G ⊂ P[n] a k-generator of P[n] if every x ⊂ [n] can be expressed as a union of at most k disjoint sets in G. Frein, Lévêque and Sebő [1] conjectured that for any n ≥ 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 [2] in order to show that for fixed k, any k-generator of P[n] must have size at least k2n/k(1 − o(1)), thereby verifying the conjecture asymptotically for multiples of k.