{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,15]],"date-time":"2023-09-15T05:36:37Z","timestamp":1694756197950},"reference-count":23,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2012,5,14]],"date-time":"2012-05-14T00:00:00Z","timestamp":1336953600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2012,9]]},"abstract":"For c<\/jats:italic> \u2208 (0,1) let <\/jats:private-char>n<\/jats:italic><\/jats:sub>(c<\/jats:italic>) denote the set of n<\/jats:italic>-vertex perfect graphs with density c<\/jats:italic> and let <\/jats:private-char>n<\/jats:italic><\/jats:sub>(c<\/jats:italic>) denote the set of n<\/jats:italic>-vertex graphs without induced C<\/jats:italic>5<\/jats:sub> and with density c<\/jats:italic>.<\/jats:p>We show that\n<\/jats:disp-formula>\nwith <\/jats:private-char> otherwise, where H<\/jats:italic> is the binary entropy function.<\/jats:p>Further, we use this result to deduce that almost all graphs in <\/jats:private-char>n<\/jats:italic><\/jats:sub>(c<\/jats:italic>) have homogeneous sets of linear size. 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