Erdős Problem 84: asymptotics of number of cycle sets in graphs

### What is the conjecture

https://www.erdosproblems.com/84

The cycle set of a graph $G$ on $n$ vertices is a set $A\subseteq \{3,\ldots,n\}$ such that there is a cycle in $G$ of length $\ell$ if and only if $\ell \in A$. Let $f(n)$ count the number of possible such $A$. Prove that $f(n)=o(2^n)$. Prove that $f(n)/2^{n/2}\to \infty$.

### Prerequisites needed
Needs definition of "cycle set", but not much else.

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

Start with the conjecture statement at https://www.erdosproblems.com/84 and compare nearby formalized conjectures in the repository; the issue names no file, test, or entry point. Done means formalizing the cycle-set definition and proving both stated asymptotic claims.

Written by the indexing model from the issue text.