Sharp Bounds for Sets with Distinct Subset Products

Let $A\subseteq [N]$ be such that for any pair of distinct subsets $B,C\subset A$, the products ${\prod }_{b\in B}b$ and ${\prod }_{c\in C}c$ are distinct. We prove that $\mathrm{\mid }A\mathrm{\mid }\le \pi (N)+\pi (N^{1\mathrm{/}2})+o(\pi (N^{1\mathrm{/}2}))$, where $\pi$ is the prime counting function, answering a question of Erdős.