Generating all subsets of a finite set with disjoint unions

TL;DR

This paper proves a conjecture about the minimal construction of families of subsets that can generate all subsets of a finite set through disjoint unions, for large enough set sizes and specific parameters.

Contribution

It confirms the conjecture by Frein, Leveque, and Sebo for large n, establishing the structure of minimal k-generators in these cases.

Findings

Confirmed the conjecture for k=2 and large n.

Extended the proof to cases where n is a large multiple of k.

Provided structural characterization of minimal k-generators.

Abstract

If X is an n-element set, we call a family G of subsets of X a k-generator for X if every subset of X can be expressed as a union of at most k disjoint sets in G. Frein, Leveque and Sebo 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 > 2.