{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:19Z","timestamp":1753893859037,"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 set partition of $[n]$ is a collection of pairwise\u00a0disjoint nonempty subsets (called blocks) of $[n]$ whose\u00a0union is $[n]$. Let $\\mathcal{B}(n)$ denote the family of all\u00a0set partitions of $[n]$. A family\u00a0$\\mathcal{A} \\subseteq \\mathcal{B}(n)$ is said to be $m$-intersecting if any two of its members have at least $m$ blocks\u00a0in\u00a0common. For any set partition $P \\in \\mathcal{B}(n)$, let $\\tau(P) = \\{x: \\{x\\} \\in P\\}$ denote the union of its singletons. Also, let $\\mu(P) = [n] -\\tau(P)$ denote the set of elements that do not appear as a singleton in $P$. Let  \\begin{align*} {\\mathcal F}_{2t} &amp; =\\left\\{P \\in \\mathcal{B}(n)\\ : \\ \\vert \\mu (P)\\vert\\leq t\\right\\};\\\\{\\mathcal F}_{2t+1}(i_0) &amp; =\\left\\{P \\in \\mathcal{B}(n)\\ : \\ \\vert\\mu (P)\\cap ([n]\\setminus \\{i_0\\})\\vert\\leq t\\right\\}.\\end{align*}  In this paper, we show that for $r\\geq 3$, there exists a $n_0=n_0(r)$ depending on $r$ such that for all $n\\geq n_0$, if $\\mathcal{A} \\subseteq\\mathcal{B}(n)$ is $(n-r)$-intersecting, then \\[ |\\mathcal{A}| \\leq \\begin{cases} \\vert {\\mathcal F}_{2t} \\vert, &amp; \\text{if $r=2t$};\\\\ \\vert {\\mathcal F}_{2t+1}(1) \\vert, &amp; \\text{if $r=2t+1$}.\\end{cases}\\]Moreover, equality holds if and only if \\[ \\mathcal{A}= \\begin{cases} {\\mathcal F}_{2t}, &amp; \\text{if $r=2t$};\\\\ {\\mathcal F}_{2t+1}(i_0), &amp; \\text{if $r=2t+1$},\\end{cases}\\]for some $i_0\\in [n]$.<\/jats:p>","DOI":"10.37236\/4987","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:20:10Z","timestamp":1578702010000},"source":"Crossref","is-referenced-by-count":0,"title":["A Deza\u2013Frankl Type Theorem for Set Partitions"],"prefix":"10.37236","volume":"22","author":[{"given":"Cheng Yeaw","family":"Ku","sequence":"first","affiliation":[]},{"given":"Kok Bin","family":"Wong","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,3,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p84\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p84\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:23:05Z","timestamp":1579256585000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i1p84"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3,30]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/4987","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,3,30]]},"article-number":"P1.84"}}