{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T18:37:17Z","timestamp":1648579037001},"reference-count":6,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2012,10,9]],"date-time":"2012-10-09T00:00:00Z","timestamp":1349740800000},"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":[[2013,1]]},"abstract":"<jats:p>It is well known that an intersecting family of subsets of an <jats:italic>n<\/jats:italic>-element set can contain at most 2<jats:sup><jats:italic>n<\/jats:italic>\u22121<\/jats:sup> sets. It is natural to wonder how \u2018close\u2019 to intersecting a family of size greater than 2<jats:sup><jats:italic>n<\/jats:italic>\u22121<\/jats:sup> can be. Katona, Katona and Katona introduced the idea of a \u2018most probably intersecting family\u2019. Suppose that <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000387_char1\" \/><\/jats:private-char> is a family and that 0 &lt; <jats:italic>p<\/jats:italic> &lt; 1. Let <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000387_char1\" \/><\/jats:private-char>(<jats:italic>p<\/jats:italic>) be the (random) family formed by selecting each set in <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000387_char1\" \/><\/jats:private-char> independently with probability <jats:italic>p<\/jats:italic>. A family <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000387_char1\" \/><\/jats:private-char> is <jats:italic>most probably intersecting<\/jats:italic> if it maximizes the probability that <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000387_char1\" \/><\/jats:private-char>(<jats:italic>p<\/jats:italic>) is intersecting over all families of size |<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000387_char1\" \/><\/jats:private-char>|.<\/jats:p><jats:p>Katona, Katona and Katona conjectured that there is a nested sequence consisting of most probably intersecting families of every possible size. We show that this conjecture is false for every value of <jats:italic>p<\/jats:italic> provided that <jats:italic>n<\/jats:italic> is sufficiently large.<\/jats:p>","DOI":"10.1017\/s0963548312000387","type":"journal-article","created":{"date-parts":[[2012,10,9]],"date-time":"2012-10-09T08:32:05Z","timestamp":1349771525000},"page":"146-160","source":"Crossref","is-referenced-by-count":2,"title":["Probably Intersecting Families are Not Nested"],"prefix":"10.1017","volume":"22","author":[{"given":"PAUL A.","family":"RUSSELL","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MARK","family":"WALTERS","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2012,10,9]]},"reference":[{"key":"S0963548312000387_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(80)90062-3"},{"key":"S0963548312000387_ref6","first-page":"423","article-title":"On a problem of Ahlswede and Katona","volume":"46","author":"Wagner","year":"2009","journal-title":"Studia Sci. Math. Hungar."},{"key":"S0963548312000387_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01902206"},{"key":"S0963548312000387_ref4","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548311000587"},{"key":"S0963548312000387_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(77)90056-5"},{"key":"S0963548312000387_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548311000472"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548312000387","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,24]],"date-time":"2019-04-24T16:05:52Z","timestamp":1556121952000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548312000387\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,9]]},"references-count":6,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2013,1]]}},"alternative-id":["S0963548312000387"],"URL":"https:\/\/doi.org\/10.1017\/s0963548312000387","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,10,9]]}}}