{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,30]],"date-time":"2026-06-30T23:25:14Z","timestamp":1782861914668,"version":"3.54.5"},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2009,9,1]],"date-time":"2009-09-01T00:00:00Z","timestamp":1251763200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2009,9]]},"abstract":"<jats:p>Let <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830999037X_char1\"\/><\/jats:private-char> \u2282 2<jats:sup>[<jats:italic>n<\/jats:italic>]<\/jats:sup> be a family of subsets of {1, 2,.\u00a0.\u00a0., <jats:italic>n<\/jats:italic>}. For any poset <jats:italic>H<\/jats:italic>, we say <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830999037X_char1\"\/><\/jats:private-char> is <jats:italic>H<\/jats:italic>-free if <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830999037X_char1\"\/><\/jats:private-char> does not contain any subposet isomorphic to <jats:italic>H<\/jats:italic>. Katona and others have investigated the behaviour of La(<jats:italic>n<\/jats:italic>, <jats:italic>H<\/jats:italic>), which denotes the maximum size of <jats:italic>H<\/jats:italic>-free families <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830999037X_char1\"\/><\/jats:private-char> \u2282 2<jats:sup>[<jats:italic>n<\/jats:italic>]<\/jats:sup>. Here we use a new approach, which is to apply methods from extremal graph theory and probability theory to identify new classes of posets <jats:italic>H<\/jats:italic>, for which La(<jats:italic>n<\/jats:italic>, <jats:italic>H<\/jats:italic>) can be determined asymptotically as <jats:italic>n<\/jats:italic> \u2192 \u221e for various posets <jats:italic>H<\/jats:italic>, including two-end-forks, up-down trees, and cycles <jats:italic>C<\/jats:italic><jats:sub>4<jats:italic>k<\/jats:italic><\/jats:sub> on two levels.<\/jats:p>","DOI":"10.1017\/s096354830999037x","type":"journal-article","created":{"date-parts":[[2009,8,20]],"date-time":"2009-08-20T06:38:15Z","timestamp":1250750295000},"page":"731-748","source":"Crossref","is-referenced-by-count":31,"title":["On Families of Subsets With a Forbidden Subposet"],"prefix":"10.1017","volume":"18","author":[{"given":"JERROLD R.","family":"GRIGGS","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"LINYUAN","family":"LU","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2009,9,1]]},"reference":[{"key":"S096354830999037X_ref7","first-page":"51","article-title":"A limit theorem in graph theory.","volume":"1","author":"Erd\u0151s","year":"1966","journal-title":"Studia Sci. Math. Hungar."},{"key":"S096354830999037X_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-77200-2_6"},{"key":"S096354830999037X_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s11083-007-9067-z"},{"key":"S096354830999037X_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/s11083-008-9090-8"},{"key":"S096354830999037X_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2005.01.002"},{"key":"S096354830999037X_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/BF01171114"},{"key":"S096354830999037X_ref6","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1945-08454-7"},{"key":"S096354830999037X_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2007.05.011"},{"key":"S096354830999037X_ref8","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1946-08715-7"},{"key":"S096354830999037X_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01651330"},{"key":"S096354830999037X_ref13","doi-asserted-by":"publisher","DOI":"10.1023\/A:1006051802267"},{"key":"S096354830999037X_ref3","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176994428"},{"key":"S096354830999037X_ref1","unstructured":"[1] Bukh B. (2008) Set families with a forbidden poset. Preprint: arXiv:0803.3840."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096354830999037X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,4]],"date-time":"2019-04-04T17:57:45Z","timestamp":1554400665000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096354830999037X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,9]]},"references-count":13,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2009,9]]}},"alternative-id":["S096354830999037X"],"URL":"https:\/\/doi.org\/10.1017\/s096354830999037x","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,9]]}}}