Problem Statement
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 $h(n)$.
Categories:
Number Theory
Progress
Erdős and Szemerédi proved that\[n^{1/2} \ll h(n) \ll n^{1-c}\]for some constant $c>0$.Source: erdosproblems.com/539 | Last verified: January 15, 2026