{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:01Z","timestamp":1753893841255,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $n\\geqslant 4$ be a natural number, and let $K$ be a set $K\\subseteq [n]:=\\{1,2,\\dots,n\\}$. We study the problem of finding the smallest possible size of a maximal family $\\mathcal{A}$ of subsets of $[n]$ such that $\\mathcal{A}$ contains only sets whose size is in $K$, and $A\\not\\subseteq B$ for all $\\{A,B\\}\\subseteq\\mathcal{A}$, i.e. $\\mathcal{A}$ is an antichain. We present a general construction of such antichains for sets $K$ containing 2, but not 1. If $3\\in K$ our construction asymptotically yields the smallest possible size of such a family, up to an $o(n^2)$ error. We conjecture our construction to be asymptotically optimal also for $3\\not\\in K$, and we prove a weaker bound for the case $K=\\{2,4\\}$. Our asymptotic results are straightforward applications of the graph removal lemma to an equivalent reformulation of the problem in extremal graph theory, which is interesting in its own right.<\/jats:p>","DOI":"10.37236\/2736","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:14:21Z","timestamp":1578719661000},"source":"Crossref","is-referenced-by-count":4,"title":["Maximal Antichains of Minimum Size"],"prefix":"10.37236","volume":"20","author":[{"given":"Thomas","family":"Kalinowski","sequence":"first","affiliation":[]},{"given":"Uwe","family":"Leck","sequence":"additional","affiliation":[]},{"given":"Ian T.","family":"Roberts","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,4,9]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p3\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p3\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:23:05Z","timestamp":1579260185000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i2p3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,4,9]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2013,4,9]]}},"URL":"https:\/\/doi.org\/10.37236\/2736","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,4,9]]},"article-number":"P3"}}