Let $n\ge 2$ and $\pi (n)<k\le n$. Let $f(k,n)$ be the smallest integer $r$ such that in any $A\subseteq \{1,\dots ,n\}$ of size $|A|=k$ there exist primes $p_1,\dots ,p_r$ such that $>r$ many $a\in A$ are only divisible by primes from $\{p_1,\dots ,p_r\}$. Is it true that$$2\pi (n^{1/2})-f(\pi (n)+1,n)\to \infty$$as $n\to \infty$?In general, estimate $f(k,n)$, particularly when $\pi (n)+1<k=o(n)$.