Improved bound on the number of cycle sets

The cycle set of a graph $G$ is the set consisting of all sizes of cycles in $G$. Answering a conjecture of Erd\H{o}s and Faudree, Verstra\"{e}te showed that there are at most $2^{n - n^{1/10}}$ different cycle sets of graphs with $n$ vertices. We improve this bound to $2^{n - n^{1/2 - o(1)}}$. Our proof follows the general strategy of Verstra\"{e}te of reducing the problem to counting cycle sets of Hamiltonian graphs with many chords or a large maximum degree. The key new ingredients are near-optimal container lemmata for cycle sets of such graphs.