{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T04:14:09Z","timestamp":1768968849705,"version":"3.49.0"},"reference-count":23,"publisher":"Cambridge University Press (CUP)","issue":"06","license":[{"start":{"date-parts":[[2019,6,25]],"date-time":"2019-06-25T00:00:00Z","timestamp":1561420800000},"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":[[2019,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A family of sets is <jats:italic>intersecting<\/jats:italic> if no two of its members are disjoint, and has the <jats:italic>Erd\u0151s\u2013Ko\u2013Rado property<\/jats:italic> (or <jats:italic>is EKR<\/jats:italic>) if each of its largest intersecting subfamilies has non-empty intersection.<\/jats:p><jats:p>Denote by <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:href=\"S0963548319000117_inline1\" xlink:type=\"simple\"\/><jats:tex-math>${{\\cal H}_k}(n,p)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> the random family in which each <jats:italic>k<\/jats:italic>-subset of {1, \u2026, <jats:italic>n<\/jats:italic>} is present with probability <jats:italic>p<\/jats:italic>, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks:\n<jats:disp-formula>\n<jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548319000117_eqn1\" xlink:type=\"simple\"\/><jats:tex-math>\n\\begin{equation} {\\rm{For what }}p = p(n,k){\\rm{is}}{{\\cal H}_k}(n,p){\\rm{likely to be EKR}}? \\end{equation}\n<\/jats:tex-math><\/jats:alternatives>\n<\/jats:disp-formula>\n<\/jats:p><jats:p>Here, for fixed <jats:italic>c<\/jats:italic> &amp;lt; 1\/4, and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:href=\"S0963548319000117_inline2\" xlink:type=\"simple\"\/><jats:tex-math>$k \\lt \\sqrt {cn\\log n} $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> we give a precise answer to this question, characterizing those sequences <jats:italic>p<\/jats:italic> = <jats:italic>p<\/jats:italic>(<jats:italic>n<\/jats:italic>, <jats:italic>k<\/jats:italic>) for which\n<jats:disp-formula>\n<jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548319000117_eqn2\" xlink:type=\"simple\"\/><jats:tex-math>\n\\begin{equation} {\\mathbb{P}}({{\\cal H}_k}(n,p){\\rm{is EKR}}{\\kern 1pt} ) \\to 1{\\rm{as }}n \\to \\infty . \\end{equation}\n<\/jats:tex-math><\/jats:alternatives>\n<\/jats:disp-formula>\n<\/jats:p>","DOI":"10.1017\/s0963548319000117","type":"journal-article","created":{"date-parts":[[2019,6,25]],"date-time":"2019-06-25T13:44:40Z","timestamp":1561470280000},"page":"881-916","source":"Crossref","is-referenced-by-count":4,"title":["On Erd\u0151s\u2013Ko\u2013Rado for random hypergraphs I"],"prefix":"10.1017","volume":"28","author":[{"given":"A.","family":"Hamm","sequence":"first","affiliation":[]},{"given":"J.","family":"Kahn","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2019,6,25]]},"reference":[{"key":"S0963548319000117_ref19","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s2-30.1.264"},{"key":"S0963548319000117_ref20","doi-asserted-by":"publisher","DOI":"10.2307\/2152833"},{"key":"S0963548319000117_ref17","doi-asserted-by":"publisher","DOI":"10.1007\/BF01200906"},{"key":"S0963548319000117_ref16","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548307008474"},{"key":"S0963548319000117_ref15","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032718"},{"key":"S0963548319000117_ref9","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/12.1.313"},{"key":"S0963548319000117_ref14","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/18.1.369"},{"key":"S0963548319000117_ref8","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199809)13:2&lt;99::AID-RSA1&gt;3.0.CO;2-M"},{"key":"S0963548319000117_ref13","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100034241"},{"key":"S0963548319000117_ref12","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548318000433"},{"key":"S0963548319000117_ref6","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20535"},{"key":"S0963548319000117_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01788087"},{"key":"S0963548319000117_ref5","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2016.184.2.2"},{"key":"S0963548319000117_ref10","first-page":"17","volume":"5","author":"Erd\u0151s","year":"1960","journal-title":"Publ. 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