{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:55Z","timestamp":1753893835592,"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>A family $\\mathcal{F}\\subseteq 2^{[n]}$ of sets is said to be $l$-trace $k$-Sperner if for any $l$-subset $L \\subset [n]$ the family $\\mathcal{F}|_L=\\{F|_L:F \\in \\mathcal{F}\\}=\\{F \\cap L: F \\in \\mathcal{F}\\}$ is $k$-Sperner, i.e. does not contain any chain of length $k+1$. The maximum size that an $l$-trace $k$-Sperner family $\\mathcal{F} \\subseteq 2^{[n]}$ can have is denoted by $f(n,k,l)$.For pairs of integers $l&lt;k$, if in a family $\\mathcal{G}$ every pair of sets satisfies $||G_1|-|G_2||&lt;k-l$, then $\\mathcal{G}$ possesses the $(n-l)$-trace $k$-Sperner property. Among such families, the largest one is $\\mathcal{F}_0=\\{F\\in 2^{[n]}: \\lfloor \\frac{n-(k-l)}{2}\\rfloor+1 \\le |F| \\le \\lfloor \\frac{n-(k-l)}{2}\\rfloor +k-l\\}$ and also $\\mathcal{F}'_0=\\{F\\in 2^{[n]}: \\lfloor \\frac{n-(k-l)}{2}\\rfloor \\le |F| \\le \\lfloor \\frac{n-(k-l)}{2}\\rfloor +k-l-1\\}$ if $n-(k-l)$ is even.In an earlier paper, we proved that this is asymptotically optimal for all pair of integers $l&lt;k$, i.e. $f(n,k,n-l)=(1+o(1))|\\mathcal{F}_0|$. In this paper we consider the case when $l=1$, $k\\ge 2$, and prove that $f(n,k,n-1)=|\\mathcal{F}_0|$ provided $n$ is large enough. We also prove that the unique $(n-1)$-trace $k$-Sperner family with size $f(n,k,n-1)$ is $\\mathcal{F}_0$ and also $\\mathcal{F}'_0$ when $n+k$ is odd.<\/jats:p>","DOI":"10.37236\/2543","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:59:01Z","timestamp":1578711541000},"source":"Crossref","is-referenced-by-count":0,"title":["Families that Remain $k$-Sperner Even After Omitting an Element of their Ground Set"],"prefix":"10.37236","volume":"20","author":[{"given":"Bal\u00e1zs","family":"Patk\u00f3s","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,2,12]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i1p32\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i1p32\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:27:05Z","timestamp":1579260425000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i1p32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,2,12]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2013,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/2543","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,2,12]]},"article-number":"P32"}}