{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T14:44:54Z","timestamp":1774277094751,"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>Given a set $X$, a collection $\\mathcal{F}\\subseteq\\mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to $k$, then the minimum size of a saturated $k$-Sperner system $\\mathcal{F}\\subseteq\\mathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture by showing that there exists $\\varepsilon&gt;0$ such that for every $k$ and $|X| \\geq n_0(k)$ there exists a saturated $k$-Sperner system $\\mathcal{F}\\subseteq\\mathcal{P}(X)$ with cardinality at most $2^{(1-\\varepsilon)k}$.A collection $\\mathcal{F}\\subseteq \\mathcal{P}(X)$ is said to be an oversaturated $k$-Sperner system if, for every $S\\in\\mathcal{P}(X)\\setminus\\mathcal{F}$, $\\mathcal{F}\\cup\\{S\\}$ contains more chains of length $k+1$ than $\\mathcal{F}$. Gerbner et al. proved that, if $|X|\\geq k$, then the smallest such collection contains between $2^{k\/2-1}$ and $O\\left(\\frac{\\log{k}}{k}2^k\\right)$ elements. We show that if $|X|\\geq k^2+k$, then the lower bound is best possible, up to a polynomial factor.<\/jats:p>","DOI":"10.37236\/4136","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:56:31Z","timestamp":1578704191000},"source":"Crossref","is-referenced-by-count":11,"title":["On Saturated $k$-Sperner Systems"],"prefix":"10.37236","volume":"21","author":[{"given":"Natasha","family":"Morrison","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonathan A.","family":"Noel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alex","family":"Scott","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2014,8,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i3p22\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i3p22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:49:37Z","timestamp":1579258177000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v21i3p22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,8,13]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2014,7,3]]}},"URL":"https:\/\/doi.org\/10.37236\/4136","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,8,13]]},"article-number":"P3.22"}}