{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:21:51Z","timestamp":1758824511911},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2020,2,3]],"date-time":"2020-02-03T00:00:00Z","timestamp":1580688000000},"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":[[2020,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some <jats:italic>k<\/jats:italic>, or the entire multiplication table of a certain large abelian group, as a subgrid. As a consequence, we show that triples systems coming from a finite group contain configurations with <jats:italic>t<\/jats:italic> triples spanning <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548319000427_inline1.png\" \/><jats:tex-math>\n$ O(\\sqrt t )$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> vertices, which is the best possible up to the implied constant. We confirm that for all <jats:italic>t<\/jats:italic> we can find a collection of <jats:italic>t<\/jats:italic> triples spanning at most <jats:italic>t<\/jats:italic> + 3 vertices, resolving the Brown\u2013Erd\u0151s\u2013S\u00f3s conjecture in this context. The proof applies well-known arithmetic results including the multidimensional versions of Szemer\u00e9di\u2019s theorem and the density Hales\u2013Jewett theorem.<\/jats:p><jats:p>This result was discovered simultaneously and independently by Nenadov, Sudakov and Tyomkyn [5], and a weaker result avoiding the arithmetic machinery was obtained independently by Wong [11].<\/jats:p>","DOI":"10.1017\/s0963548319000427","type":"journal-article","created":{"date-parts":[[2020,2,3]],"date-time":"2020-02-03T09:24:16Z","timestamp":1580721856000},"page":"633-640","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["A note on the Brown\u2013Erd\u0151s\u2013S\u00f3s conjecture in groups"],"prefix":"10.1017","volume":"29","author":[{"given":"Jason","family":"Long","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,2,3]]},"reference":[{"key":"S0963548319000427_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2019.10.007"},{"key":"S0963548319000427_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/BF02790016"},{"key":"S0963548319000427_ref2","doi-asserted-by":"publisher","DOI":"10.1093\/imrn\/rnt041"},{"key":"S0963548319000427_ref1","volume-title":"New Directions in the Theory of Graphs","author":"Brown","year":"1973"},{"key":"S0963548319000427_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004119000203"},{"key":"S0963548319000427_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-005-0006-6"},{"key":"S0963548319000427_ref7","first-page":"939","article-title":"Triple systems with no six points carrying three triangles","volume":"18","author":"Ruzsa","year":"1978","journal-title":"Coll. Math. Soc. J. Bolyai"},{"key":"S0963548319000427_ref9","doi-asserted-by":"crossref","first-page":"680","DOI":"10.1017\/S0963548314000856","article-title":"The (7, 4) -conjecture in finite groups","volume":"24","author":"Solymosi","year":"2015","journal-title":"Combin. Probab. Comput"},{"key":"S0963548319000427_ref11","unstructured":"[11] Wong, C. (2019) On the existence of dense substructures in finite groups. Preprint.10.1016\/j.disc.2020.112025"},{"key":"S0963548319000427_ref6","first-page":"372","volume-title":"The Mathematics of Paul Erd\u0151s I","author":"Pyber","year":"1997"},{"key":"S0963548319000427_ref4","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/0012-365X(89)90089-7","article-title":"A density version of the Hales\u2013Jewett Theorem for k = 3","volume":"75","author":"Furstenberg","year":"1989","journal-title":"Discrete Math."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548319000427","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,13]],"date-time":"2020-10-13T13:36:26Z","timestamp":1602596186000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548319000427\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,3]]},"references-count":11,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2020,7]]}},"alternative-id":["S0963548319000427"],"URL":"https:\/\/doi.org\/10.1017\/s0963548319000427","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,3]]},"assertion":[{"value":"\u00a9 Cambridge University Press 2020","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}