{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:53Z","timestamp":1753893833047,"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 set $F$ on a another set $X$ is $F|_X=F \\cap X$ and the trace of a family ${\\cal F}$ of sets on $X$ is ${\\cal F}_X=\\{F|_X: F \\in {\\cal F}\\}$. In this note we prove that if a $k$-uniform family ${\\cal F} \\subset {[n] \\choose k}$ has the property that for any $k$-subset $X$ the trace ${\\cal F}|_X$ does not contain a maximal chain (a family $C_0 \\subset C_1 \\subset ... \\subset C_k$ with $|C_i|=i$), then $|{\\cal F}| \\leq {n-1 \\choose k-1}$. This bound is sharp as shown by $\\{F \\in {[n] \\choose k}, 1 \\in F\\}$. Our proof gives also the stability of the extremal family.<\/jats:p>","DOI":"10.37236\/246","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:33:13Z","timestamp":1578717193000},"source":"Crossref","is-referenced-by-count":2,"title":["Traces of Uniform Families of Sets"],"prefix":"10.37236","volume":"16","author":[{"given":"Bal\u00e1zs","family":"Patk\u00f3s","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,3,4]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1n8\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1n8\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T03:05:44Z","timestamp":1579316744000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1n8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,3,4]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/246","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,3,4]]},"article-number":"N8"}}