{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:40Z","timestamp":1753893820217,"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 two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \\cup X_2$, and $\\mathcal{F}$ is a family of subsets of $X$ such that\u00a0 no two sets $A, B \\in \\mathcal{F}$\u00a0 satisfy $A \\subset B$ (or $B \\subset A$) and $A \\cap X_i=B\\cap X_i$ for some $i$, then $|\\mathcal{F}| \\le {n \\choose \\lfloor n\/2\\rfloor}$. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfy various combinations of these properties on one or both parts $X_1$, $X_2$. Along the way, we prove the following\u00a0 new result which may be of independent interest: let $\\mathcal{F},\\mathcal{G}$ be intersecting families of subsets of an $n$-element set that are additionally cross-Sperner, meaning that if $A \\in\\mathcal{F}$ and $B \\in \\mathcal{G}$, then $A \\not\\subset B$ and $B \\not\\subset A$. Then\u00a0 $|\\mathcal{F}| +|\\mathcal{G}| \\le 2^{n-1}$ and there are exponentially many examples showing that this bound is tight.<\/jats:p>","DOI":"10.37236\/2067","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T07:35:15Z","timestamp":1578728115000},"source":"Crossref","is-referenced-by-count":0,"title":["Two-Part Set Systems"],"prefix":"10.37236","volume":"19","author":[{"given":"P\u00e9ter L.","family":"Erd\u0151s","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"D\u00e1niel","family":"Gerbner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nathan","family":"Lemons","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dhruv","family":"Mubayi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cory","family":"Palmer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bal\u00e1zs","family":"Patk\u00f3s","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2012,3,9]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p52\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p52\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:41:25Z","timestamp":1579300885000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i1p52"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,3,9]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2012,2,15]]}},"URL":"https:\/\/doi.org\/10.37236\/2067","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2012,3,9]]},"article-number":"P52"}}