Is there some $h(n)\to \infty$ such that for all $2\leq i<j\leq n/2$ $$\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \geq h(n)?$$

A problem of Erdős and Szekeres, who observed that $$\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \geq \frac{\binom{n}{i}}{\binom{j}{i}} \geq 2^i$$ (in particular the greatest common divisor is always $>1$).