{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:15Z","timestamp":1753893795089,"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 central concept in Szemer\u00e9di's powerful regularity lemma is the so-called $\\epsilon$-regular pair.  A useful statement of Alon et al. essentially equates the notion of an $\\epsilon$-regular pair with degree uniformity of vertices and pairs of vertices.  The known proof of this characterization uses a clever matrix argument. This paper gives a simple proof of the characterization without appealing to the matrix argument of Alon et al. We show the $\\epsilon$-regular characterization follows from an application of Szemer\u00e9di's regularity lemma itself.<\/jats:p>","DOI":"10.37236\/1732","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T07:03:02Z","timestamp":1578639782000},"source":"Crossref","is-referenced-by-count":1,"title":["Matrix-Free Proof of a Regularity Characterization"],"prefix":"10.37236","volume":"10","author":[{"given":"A.","family":"Czygrinow","sequence":"first","affiliation":[]},{"given":"B.","family":"Nagle","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2003,10,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r39\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r39\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:06:37Z","timestamp":1579323997000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v10i1r39"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,10,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2003,1,6]]}},"URL":"https:\/\/doi.org\/10.37236\/1732","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2003,10,13]]},"article-number":"R39"}}