Problem 387

Erdős Problem #387: declared status 'open'. Formalized: yes. Is there an absolute constant $c>0$ such that, for all $1\leq k< n$, the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn,n]$? Current best: Faulkner [Fa66] proved that, if $p$ is the least prime $>2k$ and $n\geq p$, then $\binom{n}{k}$ has a prime divisor $\geq p$ (except $\binom{9}{2}$ and $\binom{10}{3}$). Prize: no. Tags: binomial coefficients, number theory.