Erdős Problem #683 — Largest Prime Factor of a Binomial Coefficient

Let $P(n,k)$ be the largest prime factor of $\binom nk$. Does there exist a real constant $c>0$ such that for every $n,k\in\mathbb N$ with $0<k<n$,

$$P(n,k)>\min\bigl\{n-k+1,\,k^{1+c}\bigr\}?$$