{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,23]],"date-time":"2025-12-23T10:44:07Z","timestamp":1766486647722},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"06","license":[{"start":{"date-parts":[[2019,6,27]],"date-time":"2019-06-27T00:00:00Z","timestamp":1561593600000},"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 said to be <jats:italic>intersecting<\/jats:italic> if any two sets in the family have non-empty intersection. In 1973, Erd\u0151s raised the problem of determining the maximum possible size of a union of <jats:italic>r<\/jats:italic> different intersecting families of <jats:italic>k<\/jats:italic>-element subsets of an <jats:italic>n<\/jats:italic>-element set, for each triple of integers (<jats:italic>n<\/jats:italic>, <jats:italic>k<\/jats:italic>, <jats:italic>r<\/jats:italic>). We make progress on this problem, proving that for any fixed integer <jats:italic>r<\/jats:italic> \u2a7e 2 and for any <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:href=\"S096354831900004X_inline1\" xlink:type=\"simple\" \/><jats:tex-math>$$k \\le ({1 \\over 2} - o(1))n$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, if <jats:italic>X<\/jats:italic> is an <jats:italic>n<\/jats:italic>-element set, and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:href=\"S096354831900004X_inline2\" xlink:type=\"simple\" \/><jats:tex-math>$${\\cal F} = {\\cal F}_1 \\cup {\\cal F}_2 \\cup \\cdots \\cup {\\cal F}_r $$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where each <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:href=\"S096354831900004X_inline3\" xlink:type=\"simple\" \/><jats:tex-math>$$ {\\cal F}_i $$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is an intersecting family of <jats:italic>k<\/jats:italic>-element subsets of <jats:italic>X<\/jats:italic>, then <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:href=\"S096354831900004X_inline4\" xlink:type=\"simple\" \/><jats:tex-math>$$|{\\cal F}| \\le \\left( {\\matrix{n \\cr k \\cr } } \\right) - \\left( {\\matrix{{n - r} \\cr k \\cr } } \\right)$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, with equality only if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:href=\"S096354831900004X_inline5\" xlink:type=\"simple\" \/><jats:tex-math>$${\\cal F} = \\{ S \\subset X:|S| = k,\\;S \\cap R \\ne \\emptyset \\} $$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for some <jats:italic>R<\/jats:italic> \u2282 <jats:italic>X<\/jats:italic> with |<jats:italic>R<\/jats:italic>| = <jats:italic>r<\/jats:italic>. This is best possible up to the size of the <jats:italic>o<\/jats:italic>(1) term, and improves a 1987 result of Frankl and F\u00fcredi, who obtained the same conclusion under the stronger hypothesis <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:href=\"S096354831900004X_inline6\" xlink:type=\"simple\" \/><jats:tex-math>$$k &amp;#x003C; (3 - \\sqrt 5 )n\/2$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, in the case <jats:italic>r<\/jats:italic> = 2. Our proof utilizes an isoperimetric, influence-based method recently developed by Keller and the authors.<\/jats:p>","DOI":"10.1017\/s096354831900004x","type":"journal-article","created":{"date-parts":[[2019,6,27]],"date-time":"2019-06-27T10:59:34Z","timestamp":1561633174000},"page":"826-839","source":"Crossref","is-referenced-by-count":4,"title":["On the union of intersecting families"],"prefix":"10.1017","volume":"28","author":[{"given":"David","family":"Ellis","sequence":"first","affiliation":[]},{"given":"Noam","family":"Lifshitz","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2019,6,27]]},"reference":[{"key":"S096354831900004X_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01651330"},{"key":"S096354831900004X_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(87)90005-7"},{"key":"S096354831900004X_ref7","first-page":"403","volume-title":"Infinite and Finite Sets","author":"Erd\u02ddos","year":"1976"},{"key":"S096354831900004X_ref6","first-page":"93","volume":"8","author":"Erd\u02ddos","year":"1965","journal-title":"Ann. Univ. Sci. Budapest"},{"key":"S096354831900004X_ref18","doi-asserted-by":"publisher","DOI":"10.1007\/BF00537230"},{"key":"S096354831900004X_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2018.12.001"},{"key":"S096354831900004X_ref17","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9800(66)80012-1"},{"key":"S096354831900004X_ref16","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548307008474"},{"key":"S096354831900004X_ref3","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/27.1.25"},{"key":"S096354831900004X_ref15","doi-asserted-by":"publisher","DOI":"10.1017\/S096354831100068X"},{"key":"S096354831900004X_ref2","volume-title":"Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability","author":"Bollob\u00e1s","year":"1986"},{"key":"S096354831900004X_ref14","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100034241"},{"key":"S096354831900004X_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(77)90017-6"},{"key":"S096354831900004X_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(86)90084-5"},{"key":"S096354831900004X_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2013.01.008"},{"key":"S096354831900004X_ref8","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/12.1.313"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096354831900004X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,7]],"date-time":"2019-10-07T08:58:55Z","timestamp":1570438735000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096354831900004X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,27]]},"references-count":16,"journal-issue":{"issue":"06","published-print":{"date-parts":[[2019,11]]}},"alternative-id":["S096354831900004X"],"URL":"https:\/\/doi.org\/10.1017\/s096354831900004x","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,6,27]]}}}