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 $>1$). This inequality is sharp for $i=1$, $j=p$, and $n=2p$.

This was resolved by Bergman [Be11], who proved that for any $2\leq i<j\leq n/2$ \[\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \gg n^{1/2}\frac{2^i}{i^{3/2}},\] where the implied constant is absolute.