{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T16:18:05Z","timestamp":1775837885384,"version":"3.50.1"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2011,9,5]],"date-time":"2011-09-05T00:00:00Z","timestamp":1315180800000},"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":[[2011,11]]},"abstract":"<jats:p>We consider the following random graph process: starting with<jats:italic>n<\/jats:italic>isolated vertices, add edges uniformly at random provided no such edge creates a copy of<jats:italic>C<\/jats:italic><jats:sub>4<\/jats:sub>. We show that, with probability tending to 1 as<jats:italic>n<\/jats:italic>\u2192 \u221e, the final graph produced by this process has maximum degree<jats:italic>O<\/jats:italic>((<jats:italic>n<\/jats:italic>log<jats:italic>n<\/jats:italic>)<jats:sup>1\/3<\/jats:sup>) and consequently size<jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>4\/3<\/jats:sup>(log<jats:italic>n<\/jats:italic>)<jats:sup>1\/3<\/jats:sup>), which are sharp up to constants. This confirms conjectures of Bohman and Keevash and of Osthus and Taraz, and improves upon previous bounds due to Bollob\u00e1s and Riordan and Osthus and Taraz.<\/jats:p>","DOI":"10.1017\/s0963548311000368","type":"journal-article","created":{"date-parts":[[2011,9,5]],"date-time":"2011-09-05T09:13:13Z","timestamp":1315213993000},"page":"939-955","source":"Crossref","is-referenced-by-count":10,"title":["The Final Size of the<i>C<\/i><sub>4<\/sub>-Free Process"],"prefix":"10.1017","volume":"20","author":[{"given":"MICHAEL E.","family":"PICOLLELLI","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2011,9,5]]},"reference":[{"key":"S0963548311000368_ref4","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511814068"},{"key":"S0963548311000368_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-010-0247-x"},{"key":"S0963548311000368_ref6","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240060217"},{"key":"S0963548311000368_ref9","doi-asserted-by":"publisher","DOI":"10.1002\/1098-2418(200101)18:1<61::AID-RSA5>3.0.CO;2-T"},{"key":"S0963548311000368_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2009.02.018"},{"key":"S0963548311000368_ref18","unstructured":"[18] Wolfovitz G. Triangle-free subgraphs in the triangle-free process. Random Struct. Alg., to appear. arXiv:0903.1756"},{"key":"S0963548311000368_ref7","unstructured":"[7] Gerke S. and Makai T. (2010) No dense subgraphs appear in the triangle-free graph process. Manuscript. arXiv:1002.2316"},{"key":"S0963548311000368_ref1","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1999.1910"},{"key":"S0963548311000368_ref10","unstructured":"[10] Picollelli M. (2010) The diamond-free process. Manuscript. arXiv:1010.5207"},{"key":"S0963548311000368_ref8","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240070302"},{"key":"S0963548311000368_ref13","unstructured":"[13] Spencer J. Maximal trianglefree graphs and Ramsey R(3, k). Unpublished manuscript. www.cs.nyu.edu\/spencer\/papers\/ramsey3k.pdf."},{"key":"S0963548311000368_ref17","unstructured":"[17] Wolfovitz G. (2010) The K 4-free process. Manuscript. arXiv:1008.4044"},{"key":"S0963548311000368_ref16","doi-asserted-by":"crossref","DOI":"10.37236\/93","article-title":"Lower bounds for the size of maximal H-free graphs","volume":"16","author":"Wolfovitz","year":"2009","journal-title":"Electron. J. Combin."},{"key":"S0963548311000368_ref11","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300000183"},{"key":"S0963548311000368_ref14","unstructured":"[14] Warnke L. Dense subgraphs in the H-free process. Discrete Math., to appear. arXiv:1003.0220"},{"key":"S0963548311000368_ref5","doi-asserted-by":"crossref","DOI":"10.37236\/1496","article-title":"Constrained graph processes","volume":"7","author":"Bollob\u00e1s","year":"2000","journal-title":"Electron. J. Combin."},{"key":"S0963548311000368_ref19","first-page":"73","volume-title":"Lectures on Approximation and Randomized Algorithms","author":"Wormald","year":"1999"},{"key":"S0963548311000368_ref15","unstructured":"[15] Warnke L. (2010) When does the K 4-free process stop? Manuscript. arXiv:1007.3037"},{"key":"S0963548311000368_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(90)90070-D"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548311000368","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,6,23]],"date-time":"2020-06-23T03:53:02Z","timestamp":1592884382000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548311000368\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,9,5]]},"references-count":19,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2011,11]]}},"alternative-id":["S0963548311000368"],"URL":"https:\/\/doi.org\/10.1017\/s0963548311000368","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,9,5]]}}}