Erdős Problem #698

Is there some \(h(n)\to \infty\) such that for all \(2\leq i<j\leq n/2\) \[\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \geq h(n)?\] 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. The linked formal proof (van Doorn and Aristotle, see `erdos_698.variants.bergman`) gives the explicit bound \(\gcd > \frac{2^i \sqrt n}{4 i \sqrt{i - 1}}\), so \(h(n) = \lfloor \sqrt n / 4 \rfloor\) works since \(i \sqrt{i - 1} \le 2^i\) for \(i \ge 2\).