Note on generating all subsets of a finite set with disjoint unions (Q1028810)

Summary: We call a family \({\mathcal G} \subset {\mathbb P}[n]\) 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 \({\mathcal G}\). \textit{Y. Frein}, \textit{Lévêque} and \textit{A. Sebő} [Comb. Probab. Comput. 17, No. 5, 641--660 (2008; Zbl 1191.05010)] 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 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 \(k\).