{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:30Z","timestamp":1753893810863,"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>Let $r(c)$ be the least positive integer $n$ such that every two coloring of the integers $1,\\ldots,n$ contains a monochromatic solution to $x_1 + x_2 + c = 3x_3$.  Verifying a conjecture of Martinelli and Schaal, we prove that $$ r(c) = \\left\\lceil{2\\lceil{2+c\\over3}\\rceil + c\\over3}\\right\\rceil, $$ for all $c \\ge 13$, and $$ r(c) = \\left\\lceil{3\\lceil{3-c\\over2}\\rceil - c\\over2}\\right\\rceil, $$ for all $c \\le -4$.<\/jats:p>","DOI":"10.37236\/194","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:27:23Z","timestamp":1578716843000},"source":"Crossref","is-referenced-by-count":0,"title":["Generalized Schur Numbers for $x_1 + x_2 + c = 3x_3$"],"prefix":"10.37236","volume":"16","author":[{"given":"Andr\u00e9 E.","family":"K\u00e9zdy","sequence":"first","affiliation":[]},{"given":"Hunter S.","family":"Snevily","sequence":"additional","affiliation":[]},{"given":"Susan C.","family":"White","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,8,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r105\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r105\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T02:41:21Z","timestamp":1579315281000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r105"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,8,14]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/194","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,8,14]]},"article-number":"R105"}}