{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:26:44Z","timestamp":1759336004355,"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>We study maximal families ${\\cal A}$ of subsets of $[n]=\\{1,2,\\dots,n\\}$ such that ${\\cal A}$ contains only pairs and triples and $A\\not\\subseteq B$ for all $\\{A,B\\}\\subseteq{\\cal A}$, i.e. ${\\cal A}$ is an antichain. For any $n$, all such families ${\\cal A}$ of minimum size are determined. This is equivalent to finding all graphs $G=(V,E)$ with $|V|=n$ and with the property that every edge is contained in some triangle and such that $|E|-|T|$ is maximum, where $T$ denotes the set of triangles in $G$. The largest possible value of $|E|-|T|$ turns out to be equal to $\\lfloor(n+1)^2\/8\\rfloor$. Furthermore, if all pairs and triples have weights $w_2$ and $w_3$, respectively, the problem of minimizing the total weight $w({\\cal A})$ of ${\\cal A}$ is considered. We show that $\\min w({\\cal A})=(2w_2+w_3)n^2\/8+o(n^2)$ for $3\/n\\leq w_3\/w_2=:\\lambda=\\lambda(n) &lt; 2$. For $\\lambda\\ge 2$ our problem is equivalent to the (6,3)-problem of Ruzsa and Szemer\u00e9di, and by a result of theirs it follows that $\\min w({\\cal A})=w_2n^2\/2+o(n^2)$.<\/jats:p>","DOI":"10.37236\/158","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:29:25Z","timestamp":1578698965000},"source":"Crossref","is-referenced-by-count":8,"title":["Maximal Flat Antichains of Minimum Weight"],"prefix":"10.37236","volume":"16","author":[{"given":"Martin","family":"Gr\u00fcttm\u00fcller","sequence":"first","affiliation":[]},{"given":"Sven","family":"Hartmann","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Kalinowski","sequence":"additional","affiliation":[]},{"given":"Uwe","family":"Leck","sequence":"additional","affiliation":[]},{"given":"Ian T.","family":"Roberts","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,5,29]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r69\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r69\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T21:53:14Z","timestamp":1579297994000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r69"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,5,29]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/158","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,5,29]]},"article-number":"R69"}}