Note on generating all subsets of a finite set with disjoint unions
| dc.creator | Ellis, David | |
| dc.date | 2008-11-18 | |
| dc.date | 2008-11-21 | |
| dc.date.accessioned | 2026-07-07T10:19:41Z | |
| dc.date.available | 2026-07-07T10:19:41Z | |
| dc.description | 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. | |
| dc.description | 6 pages; shortened version in which we appeal to an exact result of Erdos in place of a weaker, asymptototic version (Lemma 2) which we proved in version 1 of the paper | |
| dc.identifier | https://arxiv.org/abs/0811.3022 | |
| dc.identifier | http://arxiv.org/abs/0811.3022 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/174597 | |
| dc.subject | Combinatorics | |
| dc.subject | 05D05 | |
| dc.title | Note on generating all subsets of a finite set with disjoint unions | |
| dc.type | text |