SOLVED

If $1<k<n-1$ then $\binom{n}{k}$ is divisible by a prime $p<n/2$ (except $\binom{7}{3}=5\cdot 7$).

A conjecture of Erdős and Selfridge. Proved by Ecklund [Ec69], who made the stronger conjecture that whenever $n>k^2$ the binomial coefficient $\binom{n}{k}$ is divisible by a prime $p<n/k$. They have proved the weaker inequality $p\ll n/k^c$ for some constant $c>0$.