{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T13:27:33Z","timestamp":1769520453473,"version":"3.49.0"},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2020,12,3]],"date-time":"2020-12-03T00:00:00Z","timestamp":1606953600000},"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":[[2021,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>G<\/jats:italic> be a graph of minimum degree at least <jats:italic>k<\/jats:italic> and let <jats:italic>G<\/jats:italic><jats:sub><jats:italic>p<\/jats:italic><\/jats:sub> be the random subgraph of <jats:italic>G<\/jats:italic> obtained by keeping each edge independently with probability <jats:italic>p<\/jats:italic>. We are interested in the size of the largest complete minor that <jats:italic>G<\/jats:italic><jats:sub><jats:italic>p<\/jats:italic><\/jats:sub> contains when <jats:italic>p<\/jats:italic> = (1 + <jats:italic>\u03b5<\/jats:italic>)\/<jats:italic>k<\/jats:italic> with <jats:italic>\u03b5<\/jats:italic> &gt; 0. We show that with high probability <jats:italic>G<\/jats:italic><jats:sub><jats:italic>p<\/jats:italic><\/jats:sub> contains a complete minor of order <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548320000607_inline1.png\"\/><jats:tex-math>\n$\\tilde{\\Omega}(\\sqrt{k})$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where the ~ hides a polylogarithmic factor. Furthermore, in the case where the order of <jats:italic>G<\/jats:italic> is also bounded above by a constant multiple of <jats:italic>k<\/jats:italic>, we show that this polylogarithmic term can be removed, giving a tight bound.<\/jats:p>","DOI":"10.1017\/s0963548320000607","type":"journal-article","created":{"date-parts":[[2020,12,3]],"date-time":"2020-12-03T08:05:23Z","timestamp":1606982723000},"page":"619-630","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":4,"title":["Large complete minors in random subgraphs"],"prefix":"10.1017","volume":"30","author":[{"given":"Joshua","family":"Erde","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mihyun","family":"Kang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Krivelevich","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2020,12,3]]},"reference":[{"key":"S0963548320000607_ref9","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032718"},{"key":"S0963548320000607_ref3","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20215"},{"key":"S0963548320000607_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579141"},{"key":"S0963548320000607_ref8","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1963.10500830"},{"key":"S0963548320000607_ref16","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20114"},{"key":"S0963548320000607_ref10","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2010.22"},{"key":"S0963548320000607_ref14","first-page":"115","volume-title":"Series","author":"Krivelevich","year":"2019"},{"key":"S0963548320000607_ref6","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177706098"},{"key":"S0963548320000607_ref21","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.2000.2013"},{"key":"S0963548320000607_ref20","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100061521"},{"key":"S0963548320000607_ref19","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20571"},{"key":"S0963548320000607_ref11","first-page":"37","article-title":"The minimum Hadwiger number for graphs with a given mean degree of vertices","volume":"38","author":"Kostochka","year":"1982","journal-title":"Metody Diskretnogo Analiza"},{"key":"S0963548320000607_ref13","doi-asserted-by":"publisher","DOI":"10.1137\/17M1128721"},{"key":"S0963548320000607_ref2","doi-asserted-by":"publisher","DOI":"10.37236\/6761"},{"key":"S0963548320000607_ref17","doi-asserted-by":"publisher","DOI":"10.37236\/3198"},{"key":"S0963548320000607_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548313000308"},{"key":"S0963548320000607_ref1","volume-title":"Mathematics","author":"Bollob\u00e1s","year":"2001"},{"key":"S0963548320000607_ref4","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781316339831"},{"key":"S0963548320000607_ref15","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20508"},{"key":"S0963548320000607_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548316000237"},{"key":"S0963548320000607_ref18","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20470"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548320000607","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,10]],"date-time":"2021-06-10T14:35:37Z","timestamp":1623335737000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548320000607\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,3]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,7]]}},"alternative-id":["S0963548320000607"],"URL":"https:\/\/doi.org\/10.1017\/s0963548320000607","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,3]]},"assertion":[{"value":"\u00a9 The Author(s), 2020. 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