{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T23:36:32Z","timestamp":1780443392112,"version":"3.54.1"},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2010,7,13]],"date-time":"2010-07-13T00:00:00Z","timestamp":1278979200000},"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":[[2011,3]]},"abstract":"<jats:p>We say that \u03b1 \u2208 [0, 1) is a <jats:italic>jump<\/jats:italic> for an integer <jats:italic>r<\/jats:italic> \u2265 2 if there exists <jats:italic>c<\/jats:italic>(\u03b1) &gt; 0 such that for all \u03f5 &gt; 0 and all <jats:italic>t<\/jats:italic> \u2265 1, any <jats:italic>r<\/jats:italic>-graph with <jats:italic>n<\/jats:italic> \u2265 <jats:italic>n<\/jats:italic><jats:sub>0<\/jats:sub>(\u03b1, \u03f5, <jats:italic>t<\/jats:italic>) vertices and density at least \u03b1 + \u03f5 contains a subgraph on <jats:italic>t<\/jats:italic> vertices of density at least \u03b1 + <jats:italic>c<\/jats:italic>.<\/jats:p><jats:p>The Erd\u0151s\u2013Stone\u2013Simonovits theorem [4, 5] implies that for <jats:italic>r<\/jats:italic> = 2, every \u03b1 \u2208 [0, 1) is a jump. Erd\u0151s [3] showed that for all <jats:italic>r<\/jats:italic> \u2265 3, every \u03b1 \u2208 [0, <jats:italic>r<\/jats:italic>!\/<jats:italic>r<jats:sup>r<\/jats:sup><\/jats:italic>) is a jump. Moreover he made his famous \u2018jumping constant conjecture\u2019, that for all <jats:italic>r<\/jats:italic> \u2265 3, every \u03b1 \u2208 [0, 1) is a jump. Frankl and R\u00f6dl [7] disproved this conjecture by giving a sequence of values of non-jumps for all <jats:italic>r<\/jats:italic> \u2265 3.<\/jats:p><jats:p>We use Razborov's flag algebra method [9] to show that jumps exist for <jats:italic>r<\/jats:italic> = 3 in the interval [2\/9, 1). These are the first examples of jumps for any <jats:italic>r<\/jats:italic> \u2265 3 in the interval [<jats:italic>r<\/jats:italic>!\/<jats:italic>r<jats:sup>r<\/jats:sup><\/jats:italic>, 1). To be precise, we show that for <jats:italic>r<\/jats:italic> = 3 every \u03b1 \u2208 [0.2299, 0.2316) is a jump.<\/jats:p><jats:p>We also give an improved upper bound for the Tur\u00e1n density of <jats:italic>K<\/jats:italic><jats:sub>4<\/jats:sub><jats:sup>\u2212<\/jats:sup> = {123, 124, 134}: \u03c0(<jats:italic>K<\/jats:italic><jats:sub>4<\/jats:sub><jats:sup>\u2212<\/jats:sup>) \u2264 0.2871. This in turn implies that for <jats:italic>r<\/jats:italic> = 3 every \u03b1 \u2208 [0.2871, 8\/27) is a jump.<\/jats:p>","DOI":"10.1017\/s0963548310000222","type":"journal-article","created":{"date-parts":[[2010,7,13]],"date-time":"2010-07-13T05:50:37Z","timestamp":1279000237000},"page":"161-171","source":"Crossref","is-referenced-by-count":65,"title":["Hypergraphs Do Jump"],"prefix":"10.1017","volume":"20","author":[{"given":"RAHIL","family":"BABER","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"JOHN","family":"TALBOT","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2010,7,13]]},"reference":[{"key":"S0963548310000222_ref4","first-page":"51","article-title":"A limit theorem in graph theory","volume":"1","author":"Erd\u0151s","year":"1966","journal-title":"Studia Sci. Math. Hung. Acad."},{"key":"S0963548310000222_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579215"},{"key":"S0963548310000222_ref8","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1203350785"},{"key":"S0963548310000222_ref5","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1946-08715-7"},{"key":"S0963548310000222_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(71)90002-1"},{"key":"S0963548310000222_ref9","doi-asserted-by":"crossref","unstructured":"[9] Razborov A. A. (2010) On 3-hypergraphs with forbidden 4-vertex configurations. http:\/\/people.cs.uchicago.edu\/~razborov\/files\/turan.pdf","DOI":"10.1137\/090747476"},{"key":"S0963548310000222_ref2","first-page":"5","article-title":"Extension of a theorem of Moon and Moser on complete subgraphs","volume":"16","author":"de Caen","year":"1983","journal-title":"Ars Combinatoria"},{"key":"S0963548310000222_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2006.05.004"},{"key":"S0963548310000222_ref1","doi-asserted-by":"publisher","DOI":"10.1080\/10556789908805765"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548310000222","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T15:17:51Z","timestamp":1556378271000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548310000222\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,7,13]]},"references-count":9,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2011,3]]}},"alternative-id":["S0963548310000222"],"URL":"https:\/\/doi.org\/10.1017\/s0963548310000222","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,7,13]]}}}