In this talk I will discuss some conjectures on which values of n we have that $\binom{2n}{n}$ is not divisible by sets of primes $p_1,p_2,...,p_r$, including a conjecture of R. L. Graham. We do not resolve these conjectures, but have made some progress on some of these conjectures. This is joint with Hamed Mousavi and Maxie Schmidt.