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

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

2008-11-18

arxiv.org/abs/0811.3022v2

Mathematics

Combinatorics

6 pages; shortened version in which we appeal to an exact result of Erdos in place of a weaker, asymptototic version (Lemma 2)

Scientific paper

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.

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