{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T21:13:54Z","timestamp":1774646034258,"version":"3.50.1"},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2015,12,7]],"date-time":"2015-12-07T00:00:00Z","timestamp":1449446400000},"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":[[2016,7]]},"abstract":"<jats:p>Answering a question raised by Dudek and Pra\u0142at, we show that if <jats:italic>pn<\/jats:italic> \u2192 \u221e, w.h.p., whenever <jats:italic>G<\/jats:italic> = <jats:italic>G<\/jats:italic>(<jats:italic>n, p<\/jats:italic>) is 2-edge-coloured there is a monochromatic path of length (2\/3 + <jats:italic>o<\/jats:italic>(1))<jats:italic>n<\/jats:italic>. This result is optimal in the sense that 2\/3 cannot be replaced by a larger constant.<\/jats:p><jats:p>As part of the proof we obtain the following result. Given a graph <jats:italic>G<\/jats:italic> on <jats:italic>n<\/jats:italic> vertices with at least <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548315000279_inline1\"\/><jats:tex-math>$(1-\\varepsilon)\\binom{n}{2}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> edges, whenever <jats:italic>G<\/jats:italic> is 2-edge-coloured, there is a monochromatic path of length at least <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548315000279_inline2\"\/><jats:tex-math>$(2\/3 - 110\\sqrt{\\varepsilon})n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. This is an extension of the classical result by Gerencs\u00e9r and Gy\u00e1rf\u00e1s which says that whenever <jats:italic>K<jats:sub>n<\/jats:sub><\/jats:italic> is 2-coloured there is a monochromatic path of length at least 2<jats:italic>n<\/jats:italic>\/3.<\/jats:p>","DOI":"10.1017\/s0963548315000279","type":"journal-article","created":{"date-parts":[[2015,12,8]],"date-time":"2015-12-08T12:16:59Z","timestamp":1449577019000},"page":"612-622","source":"Crossref","is-referenced-by-count":33,"title":["Path Ramsey Number for Random Graphs"],"prefix":"10.1017","volume":"25","author":[{"given":"SHOHAM","family":"LETZTER","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2015,12,7]]},"reference":[{"key":"S0963548315000279_ref8","first-page":"331","volume-title":"The Theory and Applications of Graphs: Kalamazoo, MI, 1980","author":"Erd\u0151s","year":"1981"},{"key":"S0963548315000279_ref1","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190070115"},{"key":"S0963548315000279_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2014.01.003"},{"key":"S0963548315000279_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511814068"},{"key":"S0963548315000279_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548312000090"},{"key":"S0963548315000279_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-72905-8_4"},{"key":"S0963548315000279_ref9","first-page":"167","article-title":"On Ramsey-type problems.","volume":"10","author":"Gerencs\u00e9r","year":"1967","journal-title":"Ann. 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Orsay, Orsay, 1976","author":"Szemer\u00e9di","year":"1978"},{"key":"S0963548315000279_ref4","doi-asserted-by":"publisher","DOI":"10.1090\/cbms\/062"},{"key":"S0963548315000279_ref10","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548311000599"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548315000279","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,18]],"date-time":"2019-04-18T22:14:59Z","timestamp":1555625699000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548315000279\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,12,7]]},"references-count":13,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2016,7]]}},"alternative-id":["S0963548315000279"],"URL":"https:\/\/doi.org\/10.1017\/s0963548315000279","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,12,7]]}}}