{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,2]],"date-time":"2025-06-02T13:28:24Z","timestamp":1748870904002},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2023,10,2]],"date-time":"2023-10-02T00:00:00Z","timestamp":1696204800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2024,1]]},"abstract":"Abstract<\/jats:title>Let \n$\\mathcal{F}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> be an intersecting family. A \n$(k-1)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-set \n$E$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is called a unique shadow if it is contained in exactly one member of \n$\\mathcal{F}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Let \n${\\mathcal{A}}=\\{A\\in \\binom{[n]}{k}\\colon |A\\cap \\{1,2,3\\}|\\geq 2\\}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In the present paper, we show that for \n$n\\geq 28k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, \n$\\mathcal{A}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is the unique family attaining the maximum size among all intersecting families without unique shadow. Several other results of a similar flavour are established as well.<\/jats:p>","DOI":"10.1017\/s0963548323000305","type":"journal-article","created":{"date-parts":[[2023,10,2]],"date-time":"2023-10-02T09:17:43Z","timestamp":1696238263000},"page":"91-109","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["Intersecting families without unique shadow"],"prefix":"10.1017","volume":"33","author":[{"given":"Peter","family":"Frankl","sequence":"first","affiliation":[]},{"given":"Jian","family":"Wang","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,10,2]]},"reference":[{"key":"S0963548323000305_ref4","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-35.1.85"},{"key":"S0963548323000305_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2021.112757"},{"key":"S0963548323000305_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01904851"},{"key":"S0963548323000305_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/BF01897141"},{"key":"S0963548323000305_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2014.09.005"},{"key":"S0963548323000305_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(76)90004-6"},{"key":"S0963548323000305_ref20","doi-asserted-by":"publisher","DOI":"10.37236\/9604"},{"key":"S0963548323000305_ref3","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/12.1.313"},{"key":"S0963548323000305_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(78)90003-1"},{"key":"S0963548323000305_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2022.103665"},{"key":"S0963548323000305_ref14","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/18.1.369"},{"key":"S0963548323000305_ref17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2016.06.011"},{"key":"S0963548323000305_ref13","doi-asserted-by":"publisher","DOI":"10.1201\/9780429440809"},{"key":"S0963548323000305_ref19","doi-asserted-by":"publisher","DOI":"10.1137\/140977138"},{"key":"S0963548323000305_ref16","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(74)90018-1"},{"key":"S0963548323000305_ref7","first-page":"81","article-title":"The shifting technique in extremal set theory","volume":"123","author":"Frankl","year":"1987","journal-title":"Surv. 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