KöMaL 2022 A838

Sets $X\subset \mathbb{Z}^{+}$ and $Y\subset \mathbb{Z}^{+}$ are called comradely, if every positive integer $n$ can be written as $n=xy$ for some $x\in X$ and $y\in Y$. Let $X(n)$ and $Y(n)$ denote the number of elements of $X$ and $Y$, respectively, among the first $n$ positive integers. Let $f\colon \mathbb{Z}^{+}\to \mathbb{R}^{+}$ be an arbitrary function that goes to infinity. Prove that one can find comradely sets $X$ and $Y$ such that $\frac{X(n)}{n}$ and $\frac{Y(n)}{n}$ goes to $0$, and for all $\varepsilon>0$ exists $n \in \mathbb{Z}^+$ such that $\frac{\min\big\{X(n), Y(n)\big\}}{f(n)}<\varepsilon.$