Let $n\geq 2$ and $\pi(n)<k\leq n$. Let $f(k,n)$ be the smallest integer $r$ such that in any $A\subseteq \{1,\ldots,n\}$ of size $\lvert A\rvert=k$ there exist primes $p_1,\ldots,p_r$ such that $>r$ many $a\in A$ are only divisible by primes from $\{p_1,\ldots,p_r\}$.

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 $\pi(n)+1<k=o(n)$.