{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T06:29:45Z","timestamp":1773815385562,"version":"3.50.1"},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2017,10,16]],"date-time":"2017-10-16T00:00:00Z","timestamp":1508112000000},"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":[[2018,1]]},"abstract":"<jats:p>A family of subsets of {1,.\u00a0.\u00a0.,<jats:italic>n<\/jats:italic>} is called<jats:italic>intersecting<\/jats:italic>if any two of its sets intersect. A classical result in extremal combinatorics due to Erd\u0151s, Ko and Rado determines the maximum size of an intersecting family of<jats:italic>k<\/jats:italic>-subsets of {1,.\u00a0.\u00a0.,<jats:italic>n<\/jats:italic>}. In this paper we study the following problem: How many intersecting families of<jats:italic>k<\/jats:italic>-subsets of {1,.\u00a0.\u00a0.,<jats:italic>n<\/jats:italic>} are there? Improving a result of Balogh, Das, Delcourt, Liu and Sharifzadeh, we determine this quantity asymptotically for<jats:italic>n<\/jats:italic>\u2265 2<jats:italic>k<\/jats:italic>+2+2<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548317000529_inline1\"\/><jats:tex-math>$\\sqrt{k\\log k}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and<jats:italic>k<\/jats:italic>\u2192 \u221e. Moreover, under the same assumptions we also determine asymptotically the number of<jats:italic>non-trivial<\/jats:italic>intersecting families, that is, intersecting families for which the intersection of all sets is empty. We obtain analogous results for pairs of cross-intersecting families.<\/jats:p>","DOI":"10.1017\/s0963548317000529","type":"journal-article","created":{"date-parts":[[2017,10,16]],"date-time":"2017-10-16T02:24:59Z","timestamp":1508120699000},"page":"60-68","source":"Crossref","is-referenced-by-count":9,"title":["Counting Intersecting and Pairs of Cross-Intersecting Families"],"prefix":"10.1017","volume":"27","author":[{"given":"PETER","family":"FRANKL","sequence":"first","affiliation":[]},{"given":"ANDREY","family":"KUPAVSKII","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2017,10,16]]},"reference":[{"key":"S0963548317000529_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2015.01.003"},{"key":"S0963548317000529_ref12","unstructured":"Kupavskii A. and Zakharov D. Regular bipartite graphs and intersecting families. Accepted at J. Combin. Theory Ser. 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