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

TL;DR

This paper investigates the minimal size of families of subsets that can generate all subsets of a finite set through disjoint unions, confirming a conjecture asymptotically for fixed parameters.

Contribution

It generalizes a theorem to establish a lower bound on the size of k-generators, confirming a conjecture asymptotically for multiples of k.

Findings

Established a lower bound of k·2^{n/k}(1-o(1)) for k-generators

Verified the conjecture asymptotically for multiples of k

Extended a theorem of Alon and Frankl

Abstract

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.