Generating all subsets of a finite set with disjoint unions

AbstractIf X is an n-element set, we call a family G⊂PX a k-generator for X if every x⊂X can be expressed as a union of at most k disjoint sets in G. Frein, Lévêque and Sebő conjectured that for n>2k, the smallest k-generators for X are obtained by taking a partition of X into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We prove this conjecture for all sufficiently large n when k=2, and for n a sufficiently large multiple of k when k⩾3