{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,3]],"date-time":"2023-09-03T18:49:53Z","timestamp":1693766993923},"reference-count":29,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2019,10,9]],"date-time":"2019-10-09T00:00:00Z","timestamp":1570579200000},"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":[[2020,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Motivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle <jats:italic>C<\/jats:italic><jats:sub>5<\/jats:sub>. We show that every graph of order <jats:italic>n<\/jats:italic> and size <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000257_inline1\" \/><jats:tex-math>$ (1 - 1\/k) \\left( {\\matrix{n \\cr 2 }} \\right) $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where k \u2265 3 is an integer, contains at least\n<jats:disp-formula id=\"S0963548319000257_udisp1\"><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548319000257_eqn1\" \/><jats:tex-math>$$({1 \\over {10}} - {1 \\over {2k}} + {1 \\over {{k^2}}} - {1 \\over {{k^3}}} + {2 \\over {5{k^4}}}){n^5} + o({n^5})$$<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula>\ncopies of <jats:italic>C<\/jats:italic><jats:sub>5<\/jats:sub>. This bound is optimal, since a matching upper bound is given by the balanced complete <jats:italic>k<\/jats:italic>-partite graph. The proof is based on the flag algebras framework. We also provide a stability result. An SDP solver is not necessary to verify our proofs.<\/jats:p>","DOI":"10.1017\/s0963548319000257","type":"journal-article","created":{"date-parts":[[2019,10,9]],"date-time":"2019-10-09T02:46:32Z","timestamp":1570589192000},"page":"44-67","source":"Crossref","is-referenced-by-count":3,"title":["Minimizing the number of 5-cycles in graphs with given edge-density"],"prefix":"10.1017","volume":"29","author":[{"given":"Patrick","family":"Bennett","sequence":"first","affiliation":[]},{"given":"Andrzej","family":"Dudek","sequence":"additional","affiliation":[]},{"given":"Bernard","family":"Lidick\u00fd","sequence":"additional","affiliation":[]},{"given":"Oleg","family":"Pikhurko","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2019,10,9]]},"reference":[{"key":"S0963548319000257_ref22","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548316000110"},{"key":"S0963548319000257_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2012.12.008"},{"key":"S0963548319000257_ref18","first-page":"283","article-title":"On a problem of Tur\u00e1n","volume":"7","author":"Moon","year":"1962","journal-title":"Magyar Tud. 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