Erdős Problem #319 — What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that $\sum_{n\in A}\frac{\delta_n}{n}=0$ and…

What is the size of the largest $A\subseteq \{1,\ldots,N\}$ 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 $A'\subsetneq A$?