{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:26Z","timestamp":1753893866311,"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>We say that the system of equations $Ax = b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax = b.$ Rado proved that the system $Ax = b$ is partition regular if and only if it has a constant solution.\r\nByszewski and Krawczyk asked if this remains true when the integers are replaced by a general (commutative) ring $R$. Our aim in this note is to answer this question in the affirmative. The main ingredient is a new 'direct' proof of Rado\u2019s result.<\/jats:p>","DOI":"10.37236\/7972","type":"journal-article","created":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T00:52:02Z","timestamp":1593132722000},"source":"Crossref","is-referenced-by-count":0,"title":["Inhomogeneous Partition Regularity"],"prefix":"10.37236","volume":"27","author":[{"given":"Imre","family":"Leader","sequence":"first","affiliation":[]},{"given":"Paul A.","family":"Russell","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2020,6,26]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p57\/8114","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p57\/8114","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T00:52:17Z","timestamp":1593132737000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i2p57"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,26]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,4,3]]}},"URL":"https:\/\/doi.org\/10.37236\/7972","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,6,26]]},"article-number":"P2.57"}}