{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T15:42:02Z","timestamp":1773848522112,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Combining ideas of Rankin, Elkin, Green &amp; Wolf, we give constructive lower bounds for $r_k(N)$, the largest size of a subset of $\\{1,2,\\dots,N\\}$ that does not contain a $k$-element arithmetic progression: For every $\\epsilon&gt;0$, if $N$ is sufficiently large, then $$r_3(N)  \\geq  N \\left(\\frac{6\\cdot 2^{3\/4} \\sqrt{5}}{e \\,\\pi^{3\/2}}-\\epsilon\\right) \\exp\\left({-\\sqrt{8\\log N}+\\tfrac14\\log\\log N}\\right),$$ $$r_k(N) \\geq N \\, C_k\\,\\exp\\left({-n 2^{(n-1)\/2} \\sqrt[n]{\\log N}+\\tfrac{1}{2n}\\log\\log N}\\right),$$  where $C_k&gt;0$ is an unspecified constant, $\\log=\\log_2$, $\\exp(x)=2^x$, and $n=\\lceil{\\log k}\\rceil$. These are currently the best lower bounds for all $k$, and are an improvement over previous lower bounds for all $k\\neq4$.<\/jats:p>","DOI":"10.37236\/546","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:52:31Z","timestamp":1578714751000},"source":"Crossref","is-referenced-by-count":19,"title":["Sets of Integers that do not Contain Long Arithmetic Progressions"],"prefix":"10.37236","volume":"18","author":[{"given":"Kevin","family":"O'Bryant","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,3,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p59\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p59\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:15:28Z","timestamp":1579302928000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p59"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,3,11]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/546","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,3,11]]},"article-number":"P59"}}