{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:42:53Z","timestamp":1753893773376,"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>Our aim in this note is to show that, for any $\\epsilon&gt;0$, there exists a union-closed family $\\mathcal F$ with (unique) smallest set $S$ such that no element of $S$ belongs to more than a fraction $\\epsilon$ of the sets in $\\mathcal F$. More precisely, we give an example of a union-closed family with smallest set of size $k$ such that no element of this set belongs to more than a fraction $(1+o(1))\\frac{\\log_2 k}{2k}$ of the sets in $\\mathcal F$.\r\nWe also give explicit examples of union-closed families containing 'small' sets for which we have been unable to verify the Union-Closed Conjecture.<\/jats:p>","DOI":"10.37236\/11004","type":"journal-article","created":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T09:27:27Z","timestamp":1673602047000},"source":"Crossref","is-referenced-by-count":0,"title":["Small Sets in Union-Closed Families"],"prefix":"10.37236","volume":"30","author":[{"given":"David","family":"Ellis","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Imre","family":"Leader","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maria-Romina","family":"Ivan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2023,1,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i1p8\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i1p8\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T09:27:27Z","timestamp":1673602047000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v30i1p8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1,13]]}},"URL":"https:\/\/doi.org\/10.37236\/11004","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2023,1,13]]},"article-number":"P1.8"}}