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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, and solved by Liu and Montgomery [LiMo20]. In [Er81] Erdős asks whether the $a_i$ must in fact have positive upper density. The lower density of the set can be $0$ since there are graphs of arbitrarily large chromatic number and girth.

See also [65].