Denote by % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSn0BKvguHDwzZbqefeKCPfgBGuLBPn % 2BKvginnfarmWu51MyVXgatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy % 0Hgip5wzaebbnrfifHhDYfgasaacH8WjY-vipgYlH8Gipec8Eeeu0x % Xdbba9frFj0-OqFfea0dXdd9vqai-hGuQ8kuc9pgc9q8qqaq-dir-f % 0-yqaiVgFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabmabaa % GcbaGaeK4eXt1aaeWaaeaacaGIUbGaaOilaiaakccacaGITbaacaGL % OaGaayzkaaaaaa!3F3E! $$ {\user1{\mathcal{T}}}{\left( {{\user2{n}}{\user2{, m}}} \right)} $$ the class of all triangle-free graphs on n vertices and m edges. Our main result is the following sharp threshold, which answers the question for which densities a typical triangle-free graph is bipartite. Fix ε > 0 and let % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSn0BKvguHDwzZbqefeKCPfgBGuLBPn % 2BKvginnfarmqr1ngBPrgitLxBI9gBamXvP5wqSXMqHnxAJn0BKvgu % HDwzZbqegm0B1jxALjhiov2DaeHbuLwBLnhiov2DGi1BTfMBaebbnr % fifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9 % pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFv % e9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacaWG0bWa % aSbaaSqaaiaaiodaaeqaaOGaeyypa0JaamiDamaaBaaaleaacaaIZa % aabeaakmaabmaabaGaamOBaaGaayjkaiaawMcaamaalaaabaWaaOaa % aeaacaaIZaaaleqaaaGcbaGaaGinaaaacaWGUbWaaWbaaSqabeaaca % aIZaGaai4laiaaikdaaaGcdaGcaaqaaiGacYgacaGGVbGaai4zaiaa % yIW7caaMi8UaamOBaaWcbeaaaaa!5713! $$ t_{3} = t_{3} {\left( n \right)}\frac{{{\sqrt 3 }}} {4}n^{{3/2}} {\sqrt {\log {\kern 1pt} {\kern 1pt} n} } $$ . If n/2 ≤ m ≤ (1 − ε) t 3, then almost all graphs in % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSn0BKvguHDwzZbqefeKCPfgBGuLBPn % 2BKvginnfarmWu51MyVXgatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy % 0Hgip5wzaebbnrfifHhDYfgasaacH8WjY-vipgYlH8Gipec8Eeeu0x % Xdbba9frFj0-OqFfea0dXdd9vqai-hGuQ8kuc9pgc9q8qqaq-dir-f % 0-yqaiVgFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabmabaa % GcbaGaeK4eXt1aaeWaaeaacaGIUbGaaOilaiaakccacaGITbaacaGL % OaGaayzkaaaaaa!3F3E! $$ {\user1{\mathcal{T}}}{\left( {{\user2{n}}{\user2{, m}}} \right)} $$ are not bipartite, whereas if m ≥ (1 + ε)t 3, then almost all of them are bipartite. For m ≥ (1 + ε)t 3, this allows us to determine asymptotically the number of graphs in % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSn0BKvguHDwzZbqefeKCPfgBGuLBPn % 2BKvginnfarmWu51MyVXgatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy % 0Hgip5wzaebbnrfifHhDYfgasaacH8WjY-vipgYlH8Gipec8Eeeu0x % Xdbba9frFj0-OqFfea0dXdd9vqai-hGuQ8kuc9pgc9q8qqaq-dir-f % 0-yqaiVgFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabmabaa % GcbaGaeK4eXt1aaeWaaeaacaGIUbGaaOilaiaakccacaGITbaacaGL % OaGaayzkaaaaaa!3F3E! $$ {\user1{\mathcal{T}}}{\left( {{\user2{n}}{\user2{, m}}} \right)} $$ . We also obtain corresponding results for C ℓ -free graphs, for any cycle C ℓ of fixed odd length.