{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:56Z","timestamp":1753893836122,"version":"3.41.2"},"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>The classical van der Waerden Theorem says that for every every finite set $S$ of natural numbers and every $k$-coloring of the natural numbers, there is a monochromatic set of the form $aS+b$ for some $a&gt;0$ and $b\\geq 0$. I.e., monochromatism is obtained by a dilation followed by a translation. We investigate the effect of reversing the order of dilation and translation. $S$ has the variant van der Waerden property for $k$ colors if for every $k$-coloring there is a  monochromatic set of the form $a(S+b)$ for some $a&gt;0$ and $b\\geq 0$. On the positive side it is shown that every two-element set has the variant van der Waerden property for every $k$. Also, for every finite $S$ and $k$ there is an $n$  such that $nS$ has the variant van der Waerden property for $k$ colors.  This extends the classical van der Waerden Theorem.  On the negative side it is shown that if $S$ has at least three elements, the variant van der Waerden property fails for a sufficiently large $k$. The counterexamples to the variant van der Waerden property are constructed by  specifying colorings as Thue-Morse sequences.<\/jats:p>","DOI":"10.37236\/1454","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:00:42Z","timestamp":1578708042000},"source":"Crossref","is-referenced-by-count":0,"title":["A van der Waerden Variant"],"prefix":"10.37236","volume":"6","author":[{"given":"Kevin J.","family":"Compton","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[1999,4,2]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v6i1r22\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v6i1r22\/appendix","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v6i1r22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:33:08Z","timestamp":1579325588000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v6i1r22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,4,2]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[1999,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1454","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[1999,4,2]]},"article-number":"R22"}}