Claims

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1986_12_01_erdos_frankl_rodl: Theorem 1.6 of Erdős, Frankl and Rödl (Graphs Combin. 1986) counts the G-free graphs on n vertices as 2^{(1+o(1)) ex(n;G)} whenever G has chromatic number at least 3; accepted on the refereed publication.

2013_09_11_morris_saxton: Proposition 1.4 of Morris and Saxton (Adv. Math. 2016) gives a constant c > 0 and infinitely many n with at least 2^{(1+c) ex(n;C_6)} C_6-free graphs on n vertices, so the question fails for G = C_6; accepted.

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