{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:04Z","timestamp":1753893844211,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A linear equation is $r$-regular, if, for every $r$-coloring of the positive integers, there exist positive integers of the same color which satisfy the equation. In 2005, Fox and Radoi\u0107i\u010d conjectured that the equation $x_1 + 2x_2 + \\cdots + 2^{n-2}x_{n-1} - 2^{n-1}x_n = 0$, for any $n \\geq 2$, has a degree of regularity of $n-1$, which would verify a conjecture of Rado from 1933. Rado's conjecture has since been verified with a different family of equations. In this paper, we show that Fox and Radoi\u0107i\u010d's family of equations indeed have a degree of regularity of $n-1$. We also prove a few extensions of this result.<\/jats:p>","DOI":"10.37236\/4321","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:56:13Z","timestamp":1578704173000},"source":"Crossref","is-referenced-by-count":3,"title":["Resolving a Conjecture on Degree of Regularity of Linear Homogeneous Equations"],"prefix":"10.37236","volume":"21","author":[{"given":"Noah","family":"Golowich","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2014,8,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i3p28\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i3p28\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:49:18Z","timestamp":1579258158000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v21i3p28"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,8,21]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2014,7,3]]}},"URL":"https:\/\/doi.org\/10.37236\/4321","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2014,8,21]]},"article-number":"P3.28"}}