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All Random Solved Random Open
OPEN
Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of integers with $\sum_{n\in A}\frac{1}{n}>K$ there exists some $S\subseteq A$ such that \[1-e^{-cK} < \sum_{n\in S}\frac{1}{n}\leq 1?\]
Erdős and Graham knew this with $e^{-cK}$ replaced by $c/K^2$.
Additional thanks to: Mehtaab Sawhney