{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:11Z","timestamp":1753893851893,"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>The trace of a family of sets ${\\cal A}$ on a set $X$ is  ${\\cal A}|_X=\\{A\\cap X:A\\in {\\cal A}\\}$.  If ${\\cal A}$ is a family of $k$-sets from an $n$-set such that for any $r$-subset $X$ the trace ${\\cal A}|_X$ does not contain a maximal chain, then how large can ${\\cal A}$ be?  Patk\u00f3s conjectured that, for $n$ sufficiently large, the size of ${\\cal A}$ is at most ${n-k+r-1\\choose r-1}$. Our aim in this paper is to prove this conjecture.<\/jats:p>","DOI":"10.37236\/465","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:15:58Z","timestamp":1578716158000},"source":"Crossref","is-referenced-by-count":0,"title":["Traces Without Maximal Chains"],"prefix":"10.37236","volume":"17","author":[{"given":"Ta Sheng","family":"Tan","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,3,15]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1n16\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1n16\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:57:37Z","timestamp":1579305457000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1n16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,3,15]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/465","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,3,15]]},"article-number":"N16"}}