{"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":1753893817792,"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>Let $\\Omega$ be a finite set and let $\\mathcal{S} \\subseteq \\mathcal{P}(\\Omega)$ be a set system on $\\Omega$. For $x\\in \\Omega$, we denote by $d_{\\mathcal{S}}(x)$ the number of members of $\\mathcal{S}$ containing $x$. A long-standing conjecture of Frankl states that if $\\mathcal{S}$ is union-closed then there is some $x\\in \\Omega$ with $d_{\\mathcal{S}}(x)\\geq \\frac{1}{2}|\\mathcal{S}|$. We consider a related question. Define the weight of a family $\\mathcal{S}$ to be $w(\\mathcal{S}) := \\sum_{A \\in \\mathcal{S}} |A|$. Suppose $\\mathcal{S}$ is union-closed. How small can $w(\\mathcal{S})$ be? Reimer showed $$w(\\mathcal{S}) \\geq \\frac{1}{2} |\\mathcal{S}| \\log_2 |\\mathcal{S}|,$$ and that this inequality is tight.  In this paper we show how Reimer's bound may be improved if we have some additional information about the domain $\\Omega$ of $\\mathcal{S}$: if $\\mathcal{S}$ separates the points of its domain, then $$w(\\mathcal{S})\\geq \\binom{|\\Omega|}{2}.$$ This is stronger than Reimer's Theorem when $\\vert \\Omega \\vert &gt; \\sqrt{|\\mathcal{S}|\\log_2 |\\mathcal{S}|}$. In addition we construct a family of examples showing the combined bound on $w(\\mathcal{S})$ is tight except in the region $|\\Omega|=\\Theta (\\sqrt{|\\mathcal{S}|\\log_2 |\\mathcal{S}|})$, where it may be off by a multiplicative factor of $2$. Our proof also gives a lower bound on the average degree: if $\\mathcal{S}$ is a point-separating union-closed family on $\\Omega$, then $$ \\frac{1}{|\\Omega|} \\sum_{x \\in \\Omega} d_{\\mathcal{S}}(x) \\geq \\frac{1}{2} \\sqrt{|\\mathcal{S}| \\log_2 |\\mathcal{S}|}+ O(1),$$ and this is best possible except for a multiplicative factor of $2$.<\/jats:p>","DOI":"10.37236\/582","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:48:15Z","timestamp":1578714495000},"source":"Crossref","is-referenced-by-count":3,"title":["Minimal Weight in Union-Closed Families"],"prefix":"10.37236","volume":"18","author":[{"given":"Victor","family":"Falgas-Ravry","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,4,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p95\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p95\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:12:55Z","timestamp":1579302775000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p95"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,4,21]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/582","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,4,21]]},"article-number":"P95"}}