OPEN
This is open, and cannot be resolved with a finite computation.
Let $h(n)$ be such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set\[\left\{ \frac{a}{(a,b)}: a,b\in A\right\}\]has size at least $h(n)$. Estimate $h(n)$.
Erdős and Szemerédi proved that\[n^{1/2} \ll h(n) \ll n^{1-c}\]for some constant $c>0$.
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #539, https://www.erdosproblems.com/539, accessed 2026-01-16
