{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,27]],"date-time":"2023-10-27T05:40:42Z","timestamp":1698385242551},"reference-count":24,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":4120,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Random Struct Algorithms"],"published-print":{"date-parts":[[1995,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>If <jats:italic>G<\/jats:italic> and <jats:italic>H<\/jats:italic> are graphs, we write <jats:italic>G<\/jats:italic>\u2192 <jats:italic>H<\/jats:italic> (respectively, <jats:italic>G<\/jats:italic>\u2192 <jats:italic>TH<\/jats:italic>) if for any proper edge\u2010coloring \u03b3 of <jats:italic>G<\/jats:italic> there is a subgraph <jats:italic>H'<\/jats:italic> \u2282 <jats:italic>G<\/jats:italic> of <jats:italic>G<\/jats:italic> isomorphic to <jats:italic>H<\/jats:italic> (respectively, isomorphic to a subdivision of <jats:italic>H<\/jats:italic>) such that \u03b3 is injective on <jats:italic>E<\/jats:italic>(<jats:italic>H'<\/jats:italic>). Let us write <jats:italic>C<jats:sup>l<\/jats:sup><\/jats:italic> for the cycle of length <jats:italic>l<\/jats:italic>. Spencer (cf. Erd\u00f6s 10]) asked whether for any <jats:italic>g<\/jats:italic> \u2a7e 3 there is a graph <jats:italic>G<\/jats:italic> = <jats:italic>G<jats:sub>g<\/jats:sub><\/jats:italic> such that (i) <jats:italic>G<\/jats:italic> has girth <jats:italic>g<\/jats:italic>(<jats:italic>G<\/jats:italic>) at least <jats:italic>g<\/jats:italic> and (ii) <jats:italic>G<\/jats:italic>\u2192 <jats:italic>TC<\/jats:italic><jats:sup>3<\/jats:sup>. Recently, R\u00f6dl and Tuza [22] answered this question in the affirmative by proving, using nonconstructive methods, a result that implies that, for any <jats:italic>t<\/jats:italic> \u2a7e 1, there is a graph <jats:italic>G<\/jats:italic> = <jats:italic>G<jats:sub>t<\/jats:sub><\/jats:italic> of girth <jats:italic>t<\/jats:italic> + 2 such that <jats:italic>G<\/jats:italic>\u2192 <jats:italic>C<\/jats:italic><jats:sup>2<jats:italic>t<\/jats:italic>+2<\/jats:sup>. In particular, condition (ii) may be strengthened to (iii) <jats:italic>G<\/jats:italic>\u2192 <jats:italic>C<jats:sup>\ud835\udcc1<\/jats:sup><\/jats:italic> for some <jats:italic>l<\/jats:italic> = \ud835\udcc1(<jats:italic>G<\/jats:italic>). For <jats:italic>G<\/jats:italic> = <jats:italic>G<jats:sub>t<\/jats:sub><\/jats:italic> above \ud835\udcc1 = \ud835\udcc1(<jats:italic>G<\/jats:italic>) = 2<jats:italic>t<\/jats:italic> + 2 = 2<jats:italic>g<\/jats:italic>(<jats:italic>G<\/jats:italic>) \u2212 2. Here, we show that suitable Ramanujan graphs constructed by Lubotzky, Phillips, and Sarnak [18] are explicit examples of graphs <jats:italic>G<\/jats:italic> = <jats:italic>G<jats:sub>g<\/jats:sub><\/jats:italic> satisfying (i) and (iii) above. For such graphs, \ud835\udcc1 = <jats:italic>l<\/jats:italic>(<jats:italic>G<\/jats:italic>) in (iii) may be taken to be roughly equal to (3\/2)<jats:italic>g<\/jats:italic>(<jats:italic>G<\/jats:italic>), thus considerably improving the value 2<jats:italic>g<\/jats:italic>(<jats:italic>G<\/jats:italic>) \u2212 2 given in the result of R\u00f6dl and Tuza. It is not known whether there are graphs <jats:italic>G<\/jats:italic> of arbitrarily large girth for which (iii) holds with \ud835\udcc1 = \ud835\udcc1(<jats:italic>G<\/jats:italic>) = <jats:italic>g<\/jats:italic>(<jats:italic>G<\/jats:italic>).<\/jats:p>","DOI":"10.1002\/rsa.3240060405","type":"journal-article","created":{"date-parts":[[2007,5,26]],"date-time":"2007-05-26T18:51:43Z","timestamp":1180205503000},"page":"417-431","source":"Crossref","is-referenced-by-count":5,"title":["On an anti\u2010Ramsey property of Ramanujan graphs"],"prefix":"10.1002","volume":"6","author":[{"given":"P. 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