OPEN

Let $\epsilon>0$ and $N$ be sufficiently large depending on $\epsilon$. Is there $A\subseteq\{1,\ldots,N\}$ such that no $a\in A$ divides the sum of any distinct elements of $A\backslash\{a\}$ and $\lvert A\rvert\gg N^{1/2-\epsilon}$?

It is easy to see that we must have $\lvert A\rvert \ll N^{1/2}$. Csaba has constructed such an $A$ with $\lvert A\rvert \gg N^{1/5}$.