{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T16:08:14Z","timestamp":1774627694834,"version":"3.50.1"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2017,10,9]],"date-time":"2017-10-09T00:00:00Z","timestamp":1507507200000},"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":[[2018,3]]},"abstract":"<jats:p>This paper deals with a combinatorial problem concerning colourings of uniform hypergraphs with large girth. We prove that if<jats:italic>H<\/jats:italic>is an<jats:italic>n<\/jats:italic>-uniform non-<jats:italic>r<\/jats:italic>-colourable simple hypergraph then its maximum edge degree \u0394(<jats:italic>H<\/jats:italic>) satisfies the inequality<jats:disp-formula-group><jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548317000475_eqnU1\"\/><jats:tex-math>$$ \\Delta(H)\\geqslant c\\cdot r^{n-1}\\ffrac{n(\\ln\\ln n)^2}{\\ln n} $$<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula><\/jats:disp-formula-group>for some absolute constant<jats:italic>c<\/jats:italic>&gt; 0.<\/jats:p><jats:p>As an application of our probabilistic technique we establish a lower bound for the classical van der Waerden number<jats:italic>W<\/jats:italic>(<jats:italic>n, r<\/jats:italic>), the minimum natural<jats:italic>N<\/jats:italic>such that in an arbitrary colouring of the set of integers {1,.\u00a0.\u00a0.,<jats:italic>N<\/jats:italic>} with<jats:italic>r<\/jats:italic>colours there exists a monochromatic arithmetic progression of length<jats:italic>n<\/jats:italic>. We prove that<jats:disp-formula-group><jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548317000475_eqnU2\"\/><jats:tex-math>$$ W(n,r)\\geqslant c\\cdot r^{n-1}\\ffrac{(\\ln\\ln n)^2}{\\ln n}. $$<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula><\/jats:disp-formula-group><\/jats:p>","DOI":"10.1017\/s0963548317000475","type":"journal-article","created":{"date-parts":[[2017,10,9]],"date-time":"2017-10-09T00:43:25Z","timestamp":1507509805000},"page":"245-273","source":"Crossref","is-referenced-by-count":12,"title":["Colourings of Uniform Hypergraphs with Large Girth and Applications"],"prefix":"10.1017","volume":"27","author":[{"given":"ANDREY","family":"KUPAVSKII","sequence":"first","affiliation":[]},{"given":"DMITRY","family":"SHABANOV","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2017,10,9]]},"reference":[{"key":"S0963548317000475_ref19","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-62-02914-9"},{"key":"S0963548317000475_ref18","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2011.174.1.20"},{"key":"S0963548317000475_ref15","doi-asserted-by":"crossref","first-page":"P59","DOI":"10.37236\/546","article-title":"Sets of integers that do not contain long arithmetic progressions","volume":"18","author":"O'Bryant","year":"2011","journal-title":"Electron. 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