Generating all subsets of a finite set with disjoint unions

Abstract

If X is an n-element set, we call a family $\mathcal{G}\subset \mathcal{P}X$ a k-generator for X if every $x\subset X$ can be expressed as a union of at most k disjoint sets in $\mathcal{G}$. 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 $k⩾3$.