{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T15:54:02Z","timestamp":1774626842149,"version":"3.50.1"},"reference-count":5,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2017,3,28]],"date-time":"2017-03-28T00:00:00Z","timestamp":1490659200000},"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":[[2017,7]]},"abstract":"<jats:p>In this paper we study a question related to the classical Erd\u0151s\u2013Ko\u2013Rado theorem, which states that any family of <jats:italic>k<\/jats:italic>-element subsets of the set [<jats:italic>n<\/jats:italic>] = {1,.\u00a0.\u00a0.,<jats:italic>n<\/jats:italic>} in which any two sets intersect has cardinality at most <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548317000062_inline1\"\/><jats:tex-math>$\\binom{n-1}{k-1}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p><jats:p>We say that two non-empty families <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548317000062_inline2\"\/><jats:tex-math>${\\mathcal A}, {\\mathcal B}\\subset \\binom{[n]}{k}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> are <jats:italic>s-cross-intersecting<\/jats:italic> if, for any <jats:italic>A<\/jats:italic> \u2208 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548317000062_inline3\"\/><jats:tex-math>${\\mathcal A}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, <jats:italic>B<\/jats:italic> \u2208 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548317000062_inline4\"\/><jats:tex-math>${\\mathcal B}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, we have |<jats:italic>A<\/jats:italic> \u2229 <jats:italic>B<\/jats:italic>| \u2265 <jats:italic>s<\/jats:italic>. In this paper we determine the maximum of |<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548317000062_inline3\"\/><jats:tex-math>${\\mathcal A}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>|+|<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548317000062_inline4\"\/><jats:tex-math>${\\mathcal B}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>| for all <jats:italic>n<\/jats:italic>. This generalizes a result of Hilton and Milner, who determined the maximum of |<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548317000062_inline3\"\/><jats:tex-math>${\\mathcal A}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>|+|<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548317000062_inline4\"\/><jats:tex-math>${\\mathcal B}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>| for non-empty 1-cross-intersecting families.<\/jats:p>","DOI":"10.1017\/s0963548317000062","type":"journal-article","created":{"date-parts":[[2017,3,28]],"date-time":"2017-03-28T09:53:15Z","timestamp":1490694795000},"page":"517-524","source":"Crossref","is-referenced-by-count":9,"title":["Uniform <i>s<\/i>-Cross-Intersecting Families"],"prefix":"10.1017","volume":"26","author":[{"given":"PETER","family":"FRANKL","sequence":"first","affiliation":[]},{"given":"ANDREY","family":"KUPAVSKII","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2017,3,28]]},"reference":[{"key":"S0963548317000062_ref2","unstructured":"Frankl P. and Kupavskii A. A size-sensitive inequality for cross-intersecting families, to appear in European Journal of Combinatorics, arXiv:1603.00936"},{"key":"S0963548317000062_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511977152"},{"key":"S0963548317000062_ref4","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/18.1.369"},{"key":"S0963548317000062_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(92)90054-X"},{"key":"S0963548317000062_ref1","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\/S0963548317000062","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,17]],"date-time":"2019-04-17T20:02:29Z","timestamp":1555531349000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548317000062\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,28]]},"references-count":5,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2017,7]]}},"alternative-id":["S0963548317000062"],"URL":"https:\/\/doi.org\/10.1017\/s0963548317000062","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,3,28]]}}}