{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T03:18:40Z","timestamp":1772507920119,"version":"3.50.1"},"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 $n$-cube\u00a0is the poset obtained by ordering all subsets of $\\{1,\\ldots,n\\}$ by inclusion, and it can be partitioned into $\\binom{n}{\\lfloor n\/2\\rfloor}$ chains, which is the minimum possible number. Two such decompositions of the $n$-cube are called orthogonal\u00a0if any two chains of the decompositions share at most a single element. Shearer and Kleitman conjectured in 1979 that the $n$-cube has $\\lfloor n\/2\\rfloor+1$ pairwise orthogonal decompositions into the minimum number of chains, and they constructed two such decompositions. Spink recently improved this by showing that the $n$-cube has three pairwise orthogonal chain decompositions for $n\\geq 24$. In this paper, we construct four pairwise orthogonal chain decompositions of the $n$-cube for $n\\geq 60$. We also construct five pairwise edge-disjoint\u00a0symmetric chain decompositions of the $n$-cube for $n\\geq 90$, where edge-disjointness is a slightly weaker notion than orthogonality, improving on a recent result by Gregor, J\u00e4ger, M\u00fctze, Sawada, and Wille.\r\n\u00a0<\/jats:p>","DOI":"10.37236\/8531","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T06:05:13Z","timestamp":1578636313000},"source":"Crossref","is-referenced-by-count":2,"title":["On Orthogonal Symmetric Chain Decompositions"],"prefix":"10.37236","volume":"26","author":[{"given":"Karl","family":"Da\u0308ubel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sven","family":"Ja\u0308ger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Torsten","family":"M\u00fctze","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Manfred","family":"Scheucher","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2019,9,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i3p64\/7925","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i3p64\/7925","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:08:17Z","timestamp":1579234097000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i3p64"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,27]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,7,4]]}},"URL":"https:\/\/doi.org\/10.37236\/8531","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,9,27]]},"article-number":"P3.64"}}