{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T23:34:23Z","timestamp":1773444863199,"version":"3.50.1"},"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>For a given number of $k$-sets, how should we choose them so as to minimize the union-closed family that they generate? Our main aim in this paper is to show that, if $\\mathcal{A}$ is a family of $k$-sets of size $\\binom{t}{k}$, and $t$ is sufficiently large, then the union-closed family generated by $\\mathcal{A}$ has size at least that generated by the family of all $k$-sets from a $t$-set. This proves (for this size of family) a conjecture of Roberts. We also make some related conjectures, and give some other results, including a new proof of the result of Leck, Roberts and Simpson that exactly determines this minimum (for all sizes of the family) when $k=2$, as well as resolving the conjecture of Roberts when the size of the family is very close to $\\binom{t}{k}$.<\/jats:p>","DOI":"10.37236\/13702","type":"journal-article","created":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T22:22:06Z","timestamp":1773440526000},"source":"Crossref","is-referenced-by-count":0,"title":["Minimizing the Number of Unions"],"prefix":"10.37236","volume":"33","author":[{"given":"\u017darko","family":"Ran\u0111elovi\u0107","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2026,3,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p47\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p47\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T22:22:14Z","timestamp":1773440534000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i1p47"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/13702","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,13]]},"article-number":"P1.47"}}