{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T21:13:39Z","timestamp":1769116419716,"version":"3.49.0"},"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 celebrated Regularity Lemma of Szemer\u00e9di asserts that every sufficiently large graph $G$ can be partitioned in such a way that most pairs of the partition sets span $\\epsilon$-regular subgraphs.  In applications, however, the graph $G$ has to be dense and the partition sets are typically very small. If only one $\\epsilon$-regular pair is needed, a much bigger one can be found, even if the original graph is sparse. In this paper we show that every graph  with density $d$ contains a large, relatively dense $\\epsilon$-regular pair. We mainly focus on a related concept of an $(\\epsilon,\\sigma)$-dense pair, for which our bound is, up to a constant, best possible.<\/jats:p>","DOI":"10.37236\/1618","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:27:45Z","timestamp":1578709665000},"source":"Crossref","is-referenced-by-count":11,"title":["Holes in Graphs"],"prefix":"10.37236","volume":"9","author":[{"given":"Yuejian","family":"Peng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vojtech","family":"R\u00f6dl","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrzej","family":"Ruci\u0144ski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2001,10,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v9i1r1\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v9i1r1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:16:40Z","timestamp":1579324600000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v9i1r1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,10,14]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2002,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1618","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001,10,14]]},"article-number":"R1"}}