{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T16:50:28Z","timestamp":1781196628828,"version":"3.54.1"},"reference-count":5,"publisher":"Cambridge University Press (CUP)","issue":"1-2","license":[{"start":{"date-parts":[[2012,2,2]],"date-time":"2012-02-02T00:00:00Z","timestamp":1328140800000},"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":[[2012,3]]},"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=\"S0963548311000587_char1\"\/><\/jats:private-char> be a family of subsets of an <jats:italic>n<\/jats:italic>-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the inequality |<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548311000587_char1\"\/><\/jats:private-char>| \u2264 2<jats:sup><jats:italic>n<\/jats:italic>\u22121<\/jats:sup>. Suppose that |<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548311000587_char1\"\/><\/jats:private-char>|=2<jats:sup><jats:italic>n<\/jats:italic>\u22121<\/jats:sup> + <jats:italic>i<\/jats:italic>. Choose the members of <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548311000587_char1\"\/><\/jats:private-char> independently with probability <jats:italic>p<\/jats:italic> (delete them with probability 1 \u2212 <jats:italic>p<\/jats:italic>). The new family is intersecting with a certain probability. We try to maximize this probability by choosing <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548311000587_char1\"\/><\/jats:private-char> appropriately. The exact maximum is determined in this paper for some small <jats:italic>i<\/jats:italic>. The analogous problem is considered for families consisting of <jats:italic>k<\/jats:italic>-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548311000587_inline1\"><jats:alt-text>$\\lfloor \\frac{n}{ 2}\\rfloor$<\/jats:alt-text><\/jats:inline-graphic>-element subsets. We determine the most probably inclusion-free family too, when the number of members is <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548311000587_inline2\"><jats:alt-text>$\\binom{n}{ \\lfloor \\frac{n}{ 2}\\rfloor} +1$<\/jats:alt-text><\/jats:inline-graphic>.<\/jats:p>","DOI":"10.1017\/s0963548311000587","type":"journal-article","created":{"date-parts":[[2012,3,19]],"date-time":"2012-03-19T15:20:59Z","timestamp":1332170459000},"page":"219-227","source":"Crossref","is-referenced-by-count":8,"title":["Most Probably Intersecting Families of Subsets"],"prefix":"10.1017","volume":"21","author":[{"given":"GYULA O. H.","family":"KATONA","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"GYULA Y.","family":"KATONA","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"ZSOLT","family":"KATONA","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2012,2,2]]},"reference":[{"key":"S0963548311000587_ref2","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/18.1.369"},{"key":"S0963548311000587_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01171114"},{"key":"S0963548311000587_ref1","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/12.1.313"},{"key":"S0963548311000587_ref4","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548311000472"},{"key":"S0963548311000587_ref3","first-page":"215","volume-title":"Theory of Graphs: Proc. Coll. Tihany 1966","author":"Kleitman","year":"1968"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548311000587","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,26]],"date-time":"2019-04-26T01:24:58Z","timestamp":1556241898000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548311000587\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,2,2]]},"references-count":5,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2012,3]]}},"alternative-id":["S0963548311000587"],"URL":"https:\/\/doi.org\/10.1017\/s0963548311000587","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,2,2]]}}}