A theorem of Sylvester and Schur (see [Er34]) states that $P(\binom{n}{k})>k$ if $k\leq n/2$. Erdős [Er55d] proved that there exists some $c>0$ such that \[P(\binom{n}{k})>\min(n-k+1, ck\log k).\]

Erdős [Er79d] writes it 'seems certain' that this holds for every $c>0$, with only a finite number of exceptions (depending on $c$). Standard heuristics on prime gaps suggest that the largest prime divisor of $\binom{n}{k}$ is in fact \[> \min(n-k+1, e^{c\sqrt{k}})\] for some constant $c>0$.