Let $f(n)$ be maximal such that, for every $m\ge 1$, there exists some $S\subseteq \{1,\dots ,n\}$ with $|S|=f(n)$ such that $m\ne {\sum }_{a\in A}a$ for all $A\subseteq S$. Is it true that$$f(n)=\left(\frac{1}{2}+o(1)\right)\frac{n}{\log n}?$$