{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T18:12:55Z","timestamp":1787508775020,"version":"build-2736575974"},"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>\n                    It is well known that an intersecting family of subsets of an\n                    <jats:italic>n<\/jats:italic>\n                    -element set can contain at most 2\n                    <jats:sup>\n                      <jats:italic>n<\/jats:italic>\n                      \u22121\n                    <\/jats:sup>\n                    sets. It is natural to wonder how \u2018close\u2019 to intersecting a family of size greater than 2\n                    <jats:sup>\n                      <jats:italic>n<\/jats:italic>\n                      \u22121\n                    <\/jats:sup>\n                    can be. Katona, Katona and Katona introduced the idea of a \u2018most probably intersecting family\u2019. Suppose that\n                    <jats:private-char>\n                      <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000387_char1\"\/>\n                    <\/jats:private-char>\n                    is a family and that 0 &lt;\n                    <jats:italic>p<\/jats:italic>\n                    &lt; 1. Let\n                    <jats:private-char>\n                      <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000387_char1\"\/>\n                    <\/jats:private-char>\n                    (\n                    <jats:italic>p<\/jats:italic>\n                    ) be the (random) family formed by selecting each set in\n                    <jats:private-char>\n                      <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000387_char1\"\/>\n                    <\/jats:private-char>\n                    independently with probability\n                    <jats:italic>p<\/jats:italic>\n                    . A family\n                    <jats:private-char>\n                      <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000387_char1\"\/>\n                    <\/jats:private-char>\n                    is\n                    <jats:italic>most probably intersecting<\/jats:italic>\n                    if it maximizes the probability that\n                    <jats:private-char>\n                      <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000387_char1\"\/>\n                    <\/jats:private-char>\n                    (\n                    <jats:italic>p<\/jats:italic>\n                    ) is intersecting over all families of size |\n                    <jats:private-char>\n                      <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000387_char1\"\/>\n                    <\/jats:private-char>\n                    |.\n                  <\/jats:p>\n                  <jats:p>\n                    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\n                    <jats:italic>p<\/jats:italic>\n                    provided that\n                    <jats:italic>n<\/jats:italic>\n                    is sufficiently large.\n                  <\/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":[{"vocabulary":"crossref","role":"author"}]},{"given":"MARK","family":"WALTERS","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"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]]}}}