{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T20:12:52Z","timestamp":1760472772831,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The anti-van der Waerden number, denoted by $aw([n],k)$, is the smallest $r$ such that every exact $r$-coloring of $[n]$ contains a rainbow $k$-term arithmetic progression. Butler et. al. showed that $\\lceil \\log_3 n \\rceil + 2 \\le aw([n],3) \\le \\lceil \\log_2 n \\rceil + 1$, and conjectured that there exists a constant $C$ such that $aw([n],3) \\le \\lceil \\log_3 n \\rceil + C$. In this paper, we show this conjecture is true by determining $aw([n],3)$ for all $n$. We prove that for $7\\cdot 3^{m-2}+1 \\leq n \\leq 21 \\cdot 3^{m-2}$, \u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0\\begin{equation*}\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 aw([n],3)=\\left\\{\\begin{array}{ll}\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0m+2, &amp; \\mbox{if $n=3^m$} \\\\\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0m+3, &amp; \\mbox{otherwise}.\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0\\end{array}\\right.\u00a0\u00a0 \u00a0\u00a0\u00a0 \u00a0\\end{equation*}<\/jats:p>","DOI":"10.37236\/6101","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T10:54:31Z","timestamp":1578653671000},"source":"Crossref","is-referenced-by-count":4,"title":["Anti-van der Waerden Numbers of 3-Term Arithmetic Progression"],"prefix":"10.37236","volume":"24","author":[{"given":"Zhanar","family":"Berikkyzy","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alex","family":"Shulte","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Young","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2017,6,16]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i2p39\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i2p39\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:56:23Z","timestamp":1579218983000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i2p39"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,16]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,4,13]]}},"URL":"https:\/\/doi.org\/10.37236\/6101","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,6,16]]},"article-number":"P2.39"}}