Use an optimal Hamiltonian container bound to obtain the target global bound

Determine whether an optimal bound of 2^{-Ω(√p)} in the Hamiltonian container lemma can be used to prove that the number of cycle sets of n-vertex graphs is at most 2^{n - Ω(√n)}, despite the obstruction posed by the family of graphs with a long cycle and only a linear number of edges.

Background

The main theorem loses logarithmic factors because the Hamiltonian container lemma provides a weighted bound of 2^{-Ω(√p / log n)} rather than the conjecturally stronger 2^{-Ω(√p)}. The authors explain that even obtaining the stronger container estimate would not immediately yield the desired 2^{n - Ω(√n)} bound, because the family G_2—graphs with a long cycle and only O(n/log n) additional edges—would still be difficult to handle. Thus the unresolved issue is not only improving the container lemma but also exploiting such an improvement in the complete counting argument.