{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:03Z","timestamp":1753893783188,"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>For a given positive integer $k$ we say that a family of subsets of $[n]$ is $k$-antichain saturated if it does not contain $k$ pairwise incomparable sets, but whenever we add to it a new set, we do find $k$ such sets. The size of the smallest such family is denoted by $\\text{sat}^*(n, \\mathcal A_{k})$. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan conjectured that $\\text{sat}^*(n, \\mathcal A_{k})=(k-1)n(1+o(1))$, and proved this for $k\\leq4$. In this paper we prove this conjecture for $k=5$ and $k=6$. Moreover, we give the exact value for $\\text{sat}^*(n, \\mathcal A_5)$ and $\\text{sat}^*(n, \\mathcal A_6)$. We also give some open problems inspired by our analysis.<\/jats:p>","DOI":"10.37236\/11262","type":"journal-article","created":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T09:27:13Z","timestamp":1673602033000},"source":"Crossref","is-referenced-by-count":2,"title":["Saturation for Small Antichains"],"prefix":"10.37236","volume":"30","author":[{"given":"Irina","family":"\u00d0ankovi\u0107","sequence":"first","affiliation":[]},{"given":"Maria-Romina","family":"Ivan","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2023,1,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i1p3\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i1p3\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T09:27:13Z","timestamp":1673602033000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v30i1p3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1,13]]}},"URL":"https:\/\/doi.org\/10.37236\/11262","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2023,1,13]]},"article-number":"P1.3"}}