{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T03:39:54Z","timestamp":1777520394133,"version":"3.51.4"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"5-6","license":[{"start":{"date-parts":[[2003,12,3]],"date-time":"2003-12-03T00:00:00Z","timestamp":1070409600000},"content-version":"unspecified","delay-in-days":32,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2003,11]]},"abstract":"<jats:p>The van der Waerden theorem in Ramsey theory states that, for every <jats:italic>k<\/jats:italic> and <jats:italic>t<\/jats:italic> and sufficiently large <jats:italic>N<\/jats:italic>, every <jats:italic>k<\/jats:italic>-colouring of <jats:italic>[N]<\/jats:italic> contains a monochromatic arithmetic progression of length <jats:italic>t<\/jats:italic>. Motivated by this result, Radoi\u010di\u0107 conjectured that every equinumerous 3-colouring of [3<jats:italic>n<\/jats:italic>] contains a 3-term rainbow arithmetic progression, <jats:italic>i.e.<\/jats:italic>, an arithmetic progression whose terms are coloured with distinct colours. In this paper, we prove that every 3-colouring of the set of natural numbers for which each colour class has density more than 1\/6, contains a 3-term rainbow arithmetic progression. We also prove similar results for colourings of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S096354830300587X_inline1.png\"\/>. Finally, we give a general perspective on other <jats:italic>anti-Ramsey-type<\/jats:italic> problems that can be considered.<\/jats:p>","DOI":"10.1017\/s096354830300587x","type":"journal-article","created":{"date-parts":[[2003,12,3]],"date-time":"2003-12-03T14:23:14Z","timestamp":1070461394000},"page":"599-620","source":"Crossref","is-referenced-by-count":22,"title":["Rainbow Arithmetic Progressions and Anti-Ramsey Results"],"prefix":"10.1017","volume":"12","author":[{"given":"V","family":"Jungic","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J","family":"Licht","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M","family":"Mahdian","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J","family":"Nesetril","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R","family":"Radoicic","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2003,12,3]]},"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096354830300587X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,15]],"date-time":"2020-05-15T09:22:54Z","timestamp":1589534574000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096354830300587X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,11]]},"references-count":0,"journal-issue":{"issue":"5-6","published-print":{"date-parts":[[2003,11]]}},"alternative-id":["S096354830300587X"],"URL":"https:\/\/doi.org\/10.1017\/s096354830300587x","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,11]]}}}