Let $\alpha >0$ and $N\ge 1$. Is it true that for any $A\subseteq \{1,\dots ,N\}$ with $|A|\ge \alpha N$ there exists some $S\subseteq A$ such that$$\frac{a}{b}=\sum _{n\in S}\frac{1}{n}$$with $a\le b=O_{\alpha }(1)$?