Problem 312

Erdős Problem #312: declared status 'open'. Formalized: yes. 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$$1-e^{-cK} < \sum_{n\in S}\frac{1}{n}\leq 1?$$ Prize: no. Tags: number theory, unit fractions.