Let me begin by quickly recapping the history of this problem. In 1975, Erdos, Graham, Ruzsa, and Strauss studied the prime factorization of binomial coefficients such as \(\binom{2n}{n}\), and posed many related questions. One such question (which was an offhand spinoff of a different question) asked whether one could find infinitely many 𝑎,𝑏,𝑛 with 𝑎,𝑏≥ε𝑛 obeying the divisibility condition 𝑎!𝑏!|𝑛!(𝑎+𝑏−𝑛)! and 𝑎+𝑏>𝑛+𝐶log𝑛. However, the question was vaguely worded in a number of respects; for instance it was initially unclear whether 𝐶 was intended to be small or large. A few months ago, a team associated with the AI tool AlphaProof observed that the problem admitted several trivial solutions, if 𝑎 or 𝑏 were allowed to be large compared with 𝑛. This technically solved the problem, but was deemed not in the spirit of the question, and an additional constraint 𝑎,𝑏≤(1−ε)𝑛 was imposed to rule out these solutions. Further AI-assisted literature search did not turn up significant work on this problem. (2/5)