Erdős problem 539

For $|A| = n$, how small can the cofactor set $Q(A) = \{a / \gcd(a,b) : a, b \in A\}$ be? The answer is $h(n) = n^{1/2 + o(1)}$: a new upper bound $h(n) \le n^{1/2} \exp(O(\sqrt{\log n}))$ matches the classical lower bound.