Erdős problem 131

Let $F(N)$ be the maximal size of $A\subseteq\{1,\ldots,N\}$ such that no $a\in A$ divides the sum of any nonempty subset of $A\setminus\{a\}$. Estimate $F(N)$. The lower bound $F(N)\gg N^{1/5}$ is classical, from constructions of Erdős and Csaba, and every non-dividing set is non-averaging, which gave $F(N)\leq N^{1/4+o(1)}$. The claimed new result is the matching upper bound $F(N)\leq N^{1/5+o(1)}$, obtained by running the Pham-Zakharov density-increment argument one dimension lower through a projective normalization, hence $F(N)=N^{1/5+o(1)}$.