Erdos proved that, $$\gcd\left(\binom{2n}{n},pq\right)=1$$ For infinitely many $n\in\mathbf{N}$. Where $p,q$ are primes. Later the following idea was conjectured by Graham, That, $$\gcd\left(\binom{2n}{n},105\right)=1$$ for infinitely many $n$
What are recent advances made towards the problem?
Here is a reference.