{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:55Z","timestamp":1753893835118,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We call a family ${\\cal G} \\subset {\\Bbb P}[n]$ a $k$-generator of ${\\Bbb P}[n]$ if every $x \\subset [n]$ can be expressed as a union of at most $k$ disjoint sets in ${\\cal G}$. Frein, L\u00e9v\u00eaque and Seb\u0151 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 ${\\Bbb P}[n]$ must have size at least $k2^{n\/k}(1-o(1))$, thereby verifying the conjecture asymptotically for multiples of $k$.<\/jats:p>","DOI":"10.37236\/254","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:31:26Z","timestamp":1578717086000},"source":"Crossref","is-referenced-by-count":1,"title":["Note on Generating All Subsets of a Finite Set with Disjoint Unions"],"prefix":"10.37236","volume":"16","author":[{"given":"David","family":"Ellis","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,5,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1n16\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1n16\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T02:55:08Z","timestamp":1579316108000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1n16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,5,20]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/254","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,5,20]]},"article-number":"N16"}}