{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T03:36:08Z","timestamp":1777692968781,"version":"3.51.4"},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2018,6,26]],"date-time":"2018-06-26T00:00:00Z","timestamp":1529971200000},"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":[[2019,3]]},"abstract":"<jats:p>We prove that the number of multigraphs with vertex set {1, .\u00a0.\u00a0.,<jats:italic>n<\/jats:italic>} such that every four vertices span at most nine edges is<jats:italic>a<\/jats:italic><jats:sup><jats:italic>n<\/jats:italic><jats:sup>2<\/jats:sup>+<jats:italic>o<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>2<\/jats:sup>)<\/jats:sup>where<jats:italic>a<\/jats:italic>is transcendental (assuming Schanuel's conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit (somewhat natural) question in extremal graph theory whose solution is transcendental. These results may shed light on a question of Razborov, who asked whether there are conjectures or theorems in extremal combinatorics which cannot be proved by a certain class of finite methods that include Cauchy\u2013Schwarz arguments.<\/jats:p><jats:p>Our proof involves a novel application of Zykov symmetrization applied to multigraphs, a rather technical progressive induction, and a straightforward use of hypergraph containers.<\/jats:p>","DOI":"10.1017\/s0963548318000299","type":"journal-article","created":{"date-parts":[[2018,6,26]],"date-time":"2018-06-26T03:28:00Z","timestamp":1529983680000},"page":"303-324","source":"Crossref","is-referenced-by-count":6,"title":["An Extremal Graph Problem with a Transcendental Solution"],"prefix":"10.1017","volume":"28","author":[{"given":"DHRUV","family":"MUBAYI","sequence":"first","affiliation":[]},{"given":"CAROLINE","family":"TERRY","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2018,6,26]]},"reference":[{"key":"S0963548318000299_ref20","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2018.01.002"},{"key":"S0963548318000299_ref14","unstructured":"Mubayi D. and Terry C. (2015) Discrete metric spaces: Structure, enumeration, and 0\u20131 laws. J. Symbolic Logic, accepted. arXiv:1502.01212"},{"key":"S0963548318000299_ref12","volume-title":"Introduction to Transcendental Numbers","author":"Lang","year":"1966"},{"key":"S0963548318000299_ref19","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-014-0562-8"},{"key":"S0963548318000299_ref15","unstructured":"Mubayi D. and Terry C. (2016) An extremal graph problem with a transcendental solution. arXiv:1607.07742"},{"key":"S0963548318000299_ref21","unstructured":"Waldschmidt M. (2014) Schanuel's conjecture: algebraic independence of transcendental numbers. In Colloquium De Giorgi 2013 and 2014, Vol. 5 of Colloquia, Ed. Norm., Pisa, pp. 129\u2013137."},{"key":"S0963548318000299_ref5","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":"S0963548318000299_ref9","unstructured":"Gelfond A. (1934) Sur le septi\u00e8me probl\u00e8me de Hilbert. In Bulletin de l'Acad\u00e9mie des Sciences de l'URSS: Classe des sciences math\u00e9matiques et naturelles, pp. 623\u2013634."},{"key":"S0963548318000299_ref6","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1946-08715-7"},{"key":"S0963548318000299_ref11","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-2010-00687-X"},{"key":"S0963548318000299_ref8","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.10012"},{"key":"S0963548318000299_ref7","unstructured":"Falgas-Ravry V. , O'Connell K. , Str\u00f6mberg J. and Uzzell A. (2016) Multicolour containers and the entropy of decorated graph limits. arXiv:1607.08152"},{"key":"S0963548318000299_ref3","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0118(199708)25:4<267::AID-JGT4>3.0.CO;2-I"},{"key":"S0963548318000299_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2016.01.001"},{"key":"S0963548318000299_ref2","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-2014-00816-X"},{"key":"S0963548318000299_ref16","unstructured":"Mubayi D. and Terry C. (2016) Extremal theory of locally sparse multigraphs. arXiv:1608.08948"},{"key":"S0963548318000299_ref18","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1203350785"},{"key":"S0963548318000299_ref17","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-014-0031-5"},{"key":"S0963548318000299_ref4","first-page":"19","volume-title":"Colloquio Internazionale sulle Teorie Combinatorie","author":"Erd\u0151s","year":"1976"},{"key":"S0963548318000299_ref13","doi-asserted-by":"crossref","first-page":"R26","DOI":"10.37236\/750","article-title":"Extremal problems for t-partite and t-colorable hypergraphs","volume":"15","author":"Mubayi","year":"2008","journal-title":"Electron. J. Combin."},{"key":"S0963548318000299_ref1","unstructured":"Alon N. (2016) Personal communication."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548318000299","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,11,4]],"date-time":"2020-11-04T08:29:11Z","timestamp":1604478551000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548318000299\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,26]]},"references-count":21,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,3]]}},"alternative-id":["S0963548318000299"],"URL":"https:\/\/doi.org\/10.1017\/s0963548318000299","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,6,26]]}}}