{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T15:01:33Z","timestamp":1773068493386,"version":"3.50.1"},"reference-count":26,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2017,3,29]],"date-time":"2017-03-29T00:00:00Z","timestamp":1490745600000},"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":[[2017,5]]},"abstract":"<jats:p>Given a family of<jats:italic>r<\/jats:italic>-uniform hypergraphs<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548316000444_inline1\"\/><jats:tex-math>${\\cal F}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>(or<jats:italic>r<\/jats:italic>-graphs for brevity), the Tur\u00e1n number ex(<jats:italic>n<\/jats:italic>,<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548316000444_inline1\"\/><jats:tex-math>${\\cal F})$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>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=\"S0963548316000444_inline1\"\/><jats:tex-math>${\\cal F}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is the maximum number of edges in an<jats:italic>r<\/jats:italic>-graph on<jats:italic>n<\/jats:italic>vertices that does not contain any member 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=\"S0963548316000444_inline1\"\/><jats:tex-math>${\\cal F}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. A pair {<jats:italic>u,v<\/jats:italic>} is<jats:italic>covered<\/jats:italic>in a hypergraph<jats:italic>G<\/jats:italic>if some edge of<jats:italic>G<\/jats:italic>contains {<jats:italic>u, v<\/jats:italic>}. Given an<jats:italic>r<\/jats:italic>-graph<jats:italic>F<\/jats:italic>and a positive integer<jats:italic>p<\/jats:italic>\u2a7e<jats:italic>n<\/jats:italic>(<jats:italic>F<\/jats:italic>), where<jats:italic>n<\/jats:italic>(<jats:italic>F<\/jats:italic>) denotes the number of vertices in<jats:italic>F<\/jats:italic>, let<jats:italic>H<jats:sup>F<\/jats:sup><jats:sub>p<\/jats:sub><\/jats:italic>denote the<jats:italic>r<\/jats:italic>-graph obtained as follows. Label the vertices of<jats:italic>F<\/jats:italic>as<jats:italic>v<\/jats:italic><jats:sub>1<\/jats:sub>,.\u00a0.\u00a0.,<jats:italic>v<jats:sub>n<\/jats:sub><\/jats:italic>(<jats:italic>F<\/jats:italic>). Add new vertices<jats:italic>v<jats:sub>n(F)+1<\/jats:sub><\/jats:italic>,.\u00a0.\u00a0.,<jats:italic>v<jats:sub>p<\/jats:sub><\/jats:italic>. For each pair of vertices<jats:italic>v<jats:sub>i<\/jats:sub>, v<jats:sub>j<\/jats:sub><\/jats:italic>not covered in<jats:italic>F<\/jats:italic>, add a set<jats:italic>B<jats:sub>i,j<\/jats:sub><\/jats:italic>of<jats:italic>r<\/jats:italic>\u2212 2 new vertices and the edge {<jats:italic>v<jats:sub>i<\/jats:sub>, v<jats:sub>j<\/jats:sub><\/jats:italic>} \u222a<jats:italic>B<jats:sub>i,j<\/jats:sub><\/jats:italic>, where the<jats:italic>B<jats:sub>i,j<\/jats:sub><\/jats:italic>are pairwise disjoint over all such pairs {<jats:italic>i, j<\/jats:italic>}. We call<jats:italic>H<jats:sup>F<\/jats:sup><jats:sub>p<\/jats:sub>the expanded p-clique with an embedded F<\/jats:italic>. For a relatively large family of<jats:italic>F<\/jats:italic>, we show that for all sufficiently large<jats:italic>n<\/jats:italic>, ex(<jats:italic>n,H<jats:sup>F<\/jats:sup><jats:sub>p<\/jats:sub><\/jats:italic>) = |<jats:italic>T<jats:sub>r<\/jats:sub><\/jats:italic>(<jats:italic>n, p<\/jats:italic>\u2212 1)|, where<jats:italic>T<jats:sub>r<\/jats:sub><\/jats:italic>(<jats:italic>n, p<\/jats:italic>\u2212 1) is the balanced complete (<jats:italic>p<\/jats:italic>\u2212 1)-partite<jats:italic>r<\/jats:italic>-graph on<jats:italic>n<\/jats:italic>vertices. We also establish structural stability of near-extremal graphs. Our results generalize or strengthen several earlier results and provide a class of hypergraphs for which the Tur\u00e1n number is exactly determined (for large<jats:italic>n<\/jats:italic>).<\/jats:p>","DOI":"10.1017\/s0963548316000444","type":"journal-article","created":{"date-parts":[[2017,3,29]],"date-time":"2017-03-29T06:48:55Z","timestamp":1490770135000},"page":"367-405","source":"Crossref","is-referenced-by-count":27,"title":["Stability and Tur\u00e1n Numbers of a Class of Hypergraphs via Lagrangians"],"prefix":"10.1017","volume":"26","author":[{"given":"AXEL","family":"BRANDT","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"DAVID","family":"IRWIN","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"TAO","family":"JIANG","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2017,3,29]]},"reference":[{"key":"S0963548316000444_ref25","first-page":"279","volume-title":"Theory of Graphs: Proc. 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Tur\u00e1n numbers of extensions of some sparse hypergraphs via Lagrangians, in preparation. arXiv:1609.08983"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548316000444","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,4]],"date-time":"2020-10-04T15:27:30Z","timestamp":1601825250000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548316000444\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,29]]},"references-count":26,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,5]]}},"alternative-id":["S0963548316000444"],"URL":"https:\/\/doi.org\/10.1017\/s0963548316000444","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,3,29]]}}}