Binomial coefficients with divisors avoiding an interval

We investigate a fifty-year-old conjecture of Erd\H{o}s and Graham concerning whether the binomial coefficient ${n \choose k}$ with $1 \leq k \leq \frac{n}{2}$ must always have a divisor $\leq n$ that is ``close'' to $n$: that is, bigger than a constant times $n$. We show this is the case when $k$ is sufficiently large as a function of $n$. However, we show (under the Generalized Riemann Hypothesis) it is possible to find binomial coefficients ${n \choose k}$, where $k$ is small compared to $n$, such that ${n \choose k}$ does not have divisors $\leq n$ close to $n$. This settles the conjecture of Erd\H{o}s and Graham, under GRH. This latter, more substantial argument involves a restricted covering problem with residue classes, sieve methods, and various exponential sum estimates.