Erdős-Graham Binomial Divisor Problem

The Erdős-Graham binomial divisor problem asks whether there is a positive constant such that every binomial coefficient with has a divisor satisfying (Erdős and Graham 1976). Since , it is enough to consider . The question is problem 387 in the Erdős problems collection (Bloom 2026).

Bui et al. (2026) refuted the conjecture in general while proving a strong positive result when is sufficiently large. For every sufficiently small and all sufficiently large , if

then has a divisor in the interval . In the other direction, for every sufficiently large fixed and sufficiently small , there are infinitely many ordered pairs with

for which has no divisor in the interval . The latter result rules out a universal positive constant .

The main ideas in the proof of an intermediate covering theorem were developed by the authors in interactive sessions with ChatGPT 5.5 Pro (Bui et al. 2026).