Claims

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2011_02_04_grzesik: Theorem 3 of Grzesik (J. Combin. Theory Ser. B 2012) bounds the number of five-cycles in a triangle-free graph on n vertices by (n/5)^5, which at n = 5m is the catalog's bound m^5; accepted on the refereed publication.

2011_02_08_hatami_hladky_kral_norine_razborov: Corollary 3.3 of Hatami, Hladký, Král', Norine and Razborov (J. Combin. Theory Ser. A 2013) bounds the pentagons of a triangle-free graph on n vertices by (n/5)^5, with equality only for the balanced blow-up of C_5.

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