Erdős problem 319

What is the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function $\delta : A \to \{-1, 1\}$ such that $$\sum_{n\in A} \frac{\delta n}{n} = 0$$ and $$\sum_{n\in A'}\frac{\delta n}{n} \neq 0$$ for all non-empty $A'\subsetneq A$.