{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T03:19:02Z","timestamp":1772507942401,"version":"3.50.1"},"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 show that symmetric Venn diagrams for $n$ sets exist for every prime $n$, settling an open question. Until this time, $n=11$ was the largest prime for which the existence of such diagrams had been proven, a result of Peter Hamburger.  We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice  ${\\cal B}_n$, and prove that such decompositions exist for all prime $n$. A consequence of the approach is a constructive proof that the quotient poset of ${\\cal B}_n$, under the relation \"equivalence under rotation\", has a symmetric chain decomposition whenever $n$ is prime. We also show how symmetric chain decompositions can be used to construct, for all $n$, monotone Venn diagrams with the minimum number of vertices, giving a simpler existence proof.<\/jats:p>","DOI":"10.37236\/1755","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:24:49Z","timestamp":1578702289000},"source":"Crossref","is-referenced-by-count":15,"title":["Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice"],"prefix":"10.37236","volume":"11","author":[{"given":"Jerrold","family":"Griggs","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Charles E.","family":"Killian","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carla D.","family":"Savage","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2004,1,2]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r2\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r2\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T00:05:17Z","timestamp":1579305917000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i1r2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1,2]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2004,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1755","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,1,2]]},"article-number":"R2"}}