{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:37Z","timestamp":1753893817907,"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>The union-closed sets conjecture states that if a finite family of sets $\\mathcal{F}$ is union-closed, then there must be some element contained in at least half of the sets of $\\mathcal{F}$.\u00a0 In this work we study the relationship between the union-closed sets conjecture and union-closed families that have the property of being well-graded.\u00a0 In doing so, we show how the density and other properties are affected by the extra structure contained in well-graded families, and we also give several conditions under which well-graded families satisfy the union-closed sets conjecture.<\/jats:p>","DOI":"10.37236\/8380","type":"journal-article","created":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:44:35Z","timestamp":1584578675000},"source":"Crossref","is-referenced-by-count":0,"title":["Well-Graded Families and the Union-Closed Sets Conjecture"],"prefix":"10.37236","volume":"27","author":[{"given":"Jeffrey","family":"Matayoshi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,3,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p64\/8054","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p64\/8054","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:44:36Z","timestamp":1584578676000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i1p64"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,20]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/8380","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,3,20]]},"article-number":"P1.64"}}