{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:07:07Z","timestamp":1758823627110},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2010,5,14]],"date-time":"2010-05-14T00:00:00Z","timestamp":1273795200000},"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":[[2011,1]]},"abstract":"<jats:p>Let<jats:italic>X<\/jats:italic>be the random variable that counts the number of triangles in the binomial random graph<jats:italic>G<\/jats:italic>(<jats:italic>n, p<\/jats:italic>). We show that for some positive constant<jats:italic>c<\/jats:italic>, the probability that<jats:italic>X<\/jats:italic>deviates from its expectation by at least \u03bbVar(<jats:italic>X<\/jats:italic>)<jats:sup>1\/2<\/jats:sup>is at most<jats:italic>e<\/jats:italic><jats:sup>\u2212<jats:italic>c<\/jats:italic>\u03bb<jats:sup>2<\/jats:sup><\/jats:sup>, provided<jats:italic>p<\/jats:italic>=<jats:italic>o<\/jats:italic>(1), \u03bb = \u03c9(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548310000106_inline1\"><jats:alt-text>$\\sqrt{\\ln n}$<\/jats:alt-text><\/jats:inline-graphic>) and \u03bb \u2264 (<jats:italic>n<\/jats:italic><jats:sup>3<\/jats:sup><jats:italic>p<\/jats:italic><jats:sup>3<\/jats:sup>+<jats:italic>n<\/jats:italic><jats:sup>4<\/jats:sup><jats:italic>p<\/jats:italic><jats:sup>5<\/jats:sup>)<jats:sup>1\/6<\/jats:sup>.<\/jats:p>","DOI":"10.1017\/s0963548310000106","type":"journal-article","created":{"date-parts":[[2010,5,14]],"date-time":"2010-05-14T10:10:22Z","timestamp":1273831822000},"page":"155-160","source":"Crossref","is-referenced-by-count":2,"title":["Sub-Gaussian Tails for the Number of Triangles in<i>G<\/i>(<i>n, p<\/i>)"],"prefix":"10.1017","volume":"20","author":[{"given":"GUY","family":"WOLFOVITZ","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2010,5,14]]},"reference":[{"key":"S0963548310000106_ref6","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.10031"},{"key":"S0963548310000106_ref10","first-page":"148","volume-title":"Surveys in Combinatorics","author":"McDiarmid","year":"1989"},{"key":"S0963548310000106_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BF00718031"},{"key":"S0963548310000106_ref14","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.10032"},{"key":"S0963548310000106_ref2","first-page":"17","article-title":"On the evolution of random graphs.","volume":"5","author":"Erd\u0151s","year":"1960","journal-title":"Magyar Tud. 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