{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:04Z","timestamp":1753893844674,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The union-closed sets conjecture states that if a finite family of sets $\\mathcal{A} \\neq \\{\\varnothing\\}$ is union-closed, then there is an element which belongs to at least half the sets in $\\mathcal{A}$. In 2001, D. Reimer showed that the average set size of a union-closed family, $\\mathcal{A}$, is at least $\\frac{1}{2} \\log_2 |\\mathcal{A}|$. In order to do so, he showed that all union-closed families satisfy a particular condition, which in turn implies the preceding bound. Here, answering a question raised in the context of T. Gowers' polymath project on the union-closed sets conjecture, we show that Reimer's condition alone is not enough to imply that there is an element in at least half the sets.<\/jats:p>","DOI":"10.37236\/6989","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T10:45:59Z","timestamp":1578653159000},"source":"Crossref","is-referenced-by-count":0,"title":["Note on the Union-Closed Sets Conjecture"],"prefix":"10.37236","volume":"24","author":[{"given":"Abigail","family":"Raz","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,9,8]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p53\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p53\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:46:11Z","timestamp":1579218371000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i3p53"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,9,8]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/6989","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,9,8]]},"article-number":"P3.53"}}