{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:10Z","timestamp":1753893850250,"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>We show that for any two linear homogeneous equations ${\\cal E}_0,{\\cal E}_1$, each with at least three variables and coefficients not all the same sign, any 2-coloring of ${\\Bbb Z}^+$ admits monochromatic solutions of color 0 to ${\\cal E}_0$ or monochromatic solutions of color 1 to ${\\cal E}_1$. We define the 2-color off-diagonal Rado number $RR({\\cal E}_0,{\\cal E}_1)$ to be the smallest $N$ such that $[1,N]$ must admit such solutions. We determine a lower bound for $RR({\\cal E}_0,{\\cal E}_1)$ in certain cases when each ${\\cal E}_i$ is of the form $a_1x_1+\\dots+a_nx_n=z$ as well as find the exact value of $RR({\\cal E}_0,{\\cal E}_1)$ when each is of the form $x_1+a_2x_2+\\dots+a_nx_n=z$. We then present a Maple package that determines upper bounds for off-diagonal Rado numbers of a few particular types, and use it to quickly prove two previous results for diagonal Rado numbers.<\/jats:p>","DOI":"10.37236\/971","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:49:10Z","timestamp":1578718150000},"source":"Crossref","is-referenced-by-count":1,"title":["Two Color Off-diagonal Rado-type Numbers"],"prefix":"10.37236","volume":"14","author":[{"given":"Kellen","family":"Myers","sequence":"first","affiliation":[]},{"given":"Aaron","family":"Robertson","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2007,8,4]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1r53\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1r53\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:01:58Z","timestamp":1579320118000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v14i1r53"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,8,4]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2007,1,3]]}},"URL":"https:\/\/doi.org\/10.37236\/971","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2007,8,4]]},"article-number":"R53"}}