{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T03:28:00Z","timestamp":1771471680820,"version":"3.50.1"},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:00:00Z","timestamp":1760054400000},"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":[[2026,1]]},"abstract":"Abstract<\/jats:title>\n \n Here we consider the hypergraph Tur\u00e1n problem in uniformly dense hypergraphs as was suggested by Erd\u0151s and S\u00f3s. Given a\n \n \n \n $3$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -graph\n \n \n \n $F$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , the uniform Tur\u00e1n density\n \n \n \n $\\pi _{\\boldsymbol{\\therefore }}(F)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of\n \n \n \n $F$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is defined as the supremum over all\n \n \n \n $d\\in [0,1]$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for which there is an\n \n \n \n $F$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -free uniformly\n \n \n \n $d$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -dense\n \n \n \n $3$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -graph, where uniformly\n \n \n \n $d$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -dense means that every linearly sized subhypergraph has density at least\n \n \n \n $d$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Recently, Glebov, Kr\u00e1l\u2019, and Volec and, independently, Reiher, R\u00f6dl, and Schacht proved that\n \n \n \n $\\pi _{\\boldsymbol{\\therefore }}(K_4^{(3)-})=\\frac {1}{4}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , solving a conjecture by Erd\u0151s and S\u00f3s. Despite substantial attention, the uniform Tur\u00e1n density is still only known for very few hypergraphs. In particular, the problem due to Erd\u0151s and S\u00f3s to determine\n \n \n \n $\\pi _{\\boldsymbol{\\therefore }}(K_4^{(3)})$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n remains wide open.\n <\/jats:p>\n \n In this work, we determine the uniform Tur\u00e1n density of the\n \n \n \n $3$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -graph on five vertices that is obtained from\n \n \n \n $K_4^{(3)-}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n by adding an additional vertex whose link forms a matching on the vertices of\n \n \n \n $K_4^{(3)-}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Further, we point to two natural intermediate problems on the way to determining\n \n \n \n $\\pi _{\\boldsymbol{\\therefore }}(K_4^{(3)})$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and solve the first of these.\n <\/jats:p>","DOI":"10.1017\/s0963548325100199","type":"journal-article","created":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T07:18:11Z","timestamp":1760080691000},"page":"59-70","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["Beyond the broken tetrahedron"],"prefix":"10.1017","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-8312-6591","authenticated-orcid":false,"given":"August Y.","family":"Chen","sequence":"first","affiliation":[{"name":"Cornell University"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1621-4030","authenticated-orcid":false,"given":"Bjarne","family":"Sch\u00fclke","sequence":"additional","affiliation":[{"name":"Institute for Basic Science"}]}],"member":"56","published-online":{"date-parts":[[2025,10,10]]},"reference":[{"key":"S0963548325100199_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/j.procs.2021.11.050"},{"key":"S0963548325100199_ref14","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139004114.004"},{"key":"S0963548325100199_ref20","doi-asserted-by":"publisher","DOI":"10.1112\/jlms.12095"},{"key":"S0963548325100199_ref19","doi-asserted-by":"publisher","DOI":"10.4171\/jems\/784"},{"key":"S0963548325100199_ref21","first-page":"436","article-title":"Eine Extremalaufgabe aus der Graphentheorie","volume":"48","author":"Tur\u00e1n","year":"1941","journal-title":"Mat. Fiz. Lapok"},{"key":"S0963548325100199_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-023-2554-0"},{"key":"S0963548325100199_ref13","first-page":"228","article-title":"On a problem of Tur\u00e1n in the theory of graphs","volume":"15","author":"Katona","year":"1964","journal-title":"Mat. Lapok"},{"key":"S0963548325100199_ref4","doi-asserted-by":"publisher","DOI":"10.1090\/tran\/8873"},{"key":"S0963548325100199_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-019-3981-8"},{"key":"S0963548325100199_ref10","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548305006784"},{"key":"S0963548325100199_ref7","first-page":"51","article-title":"A limit theorem in graph theory","volume":"1","author":"Erd\u0151s","year":"1966","journal-title":"Studia Sci. Math. Hungar."},{"key":"S0963548325100199_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579235"},{"key":"S0963548325100199_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-005-0034-2"},{"key":"S0963548325100199_ref9","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1946-08715-7"},{"key":"S0963548325100199_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-015-1267-4"},{"key":"S0963548325100199_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/s40305-025-00619-7"},{"key":"S0963548325100199_ref17","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548323000196"},{"key":"S0963548325100199_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-72905-8_2"},{"key":"S0963548325100199_ref1","first-page":"21","volume-title":"Hypergraph Tur\u00e1n Problems in, ell2-Norm, Surveys in Combina- Torics 2022","author":"Balogh","year":"2022"},{"key":"S0963548325100199_ref5","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1999.1938"},{"key":"S0963548325100199_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2020.103117"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548325100199","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T08:49:23Z","timestamp":1768207763000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548325100199\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,10]]},"references-count":21,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["S0963548325100199"],"URL":"https:\/\/doi.org\/10.1017\/s0963548325100199","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,10]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. 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