Determine the true order of the number of cycle sets

Determine how close the upper bound on the number of cycle sets of n-vertex graphs, namely 2^{n - \Omega(\sqrt{n}/\log^{3/2} n)}, is to the true number of distinct cycle sets.

Background

The paper proves an upper bound of 2^{n - \Omega(\sqrt{n}/\log^{3/2} n)} for the number of distinct cycle sets of n-vertex graphs. It also recalls Faudree’s construction, which yields at least 2^{n/2} different cycle sets when n is even. The gap between these bounds leaves the actual asymptotic magnitude unresolved, and the authors explicitly state that they do not know how far their theorem is from the truth.