Erdős Problem #387 — Is there an absolute constant $c>0$ such that, for all $1\leq k< n$, the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn,n]$?

Erdős and Graham conjectured that some divisor of every relevant binomial coefficient lies between a fixed positive fraction of $n$ and $n$. Erdős also proposed the stronger version requiring this for every fixed $c<1$ once $n$ is sufficiently large.

May 2026 paper

The paper proves that, for sufficiently large $n$, a divisor lies in $\left(n-n/(\log n)^{1/4},n\right]$ when $\exp((\log n)^{2/3+\epsilon})\leq k\leq n/2$. It also gives infinitely many small-$k$ counterexamples; the resulting failure of any fixed-multiple assertion is conditional on GRH, while computations refute stronger bounds such as $3n/4$. These are claimed advances, not an unconditional resolution of Problem $\#387$.

Current status (as of September 2026): The large-$k$ regime is claimed proved and conditional small-$k$ counterexamples are reported, but the original fixed-positive-multiple assertion remains open unconditionally.