SOLVED
If $G$ is a graph with infinite chromatic number and $a_1<a_2<\cdots $ are lengths of the odd cycles of $G$ then $\sum \frac{1}{a_i}=\infty$.
Conjectured by Erdős and Hajnal
[ErHa66], and solved by Liu and Montgomery
[LiMo20]. In
[Er81] Erdős asks whether the $a_i$ must in fact have positive upper density, and in
[Er95d] he speculates the upper density (or even upper logarithmic density) must be $\geq 1/2$.
The lower density of the set can be $0$ since there are graphs of arbitrarily large chromatic number and girth.
See also [65].
