Problem #728: Let $C>0$ and $\epsilon>0$ be sufficiently small

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Progress

A question of Erdős, Graham, Ruzsa, and Straus [EGRS75].
Erdős [Er68c] proved that if $a!b!\mid n!$ then $a+b\leq n+O(\log n)$.

This problem can be rephrased (taking $k=a+b-n$ and $N=a+b$) as asking for $\binom{N}{k}\mid \binom{N}{a}$.

This problem is ambiguous, and there are a number of trivial solutions to the problem as written. for example, the AlphaProof team has noted that there are solutions with very large $a$ and $b$ - for example, $a=n+w+1$ and $b=\frac{(n+w+1)!}{n!}-1$ for some $w\geq \max(C\log n,\epsilon n)$.

From context presumably the condition $a,b\leq n$ was intended, but here we also have trivial solutions: for example one can take $a=b=n$, or $b=n-1$ and $a$ any large divisor of $n$. No doubt the authors had in mind some condition such as $a,b\leq (1-\epsilon)n$.

Barreto and ChatGPT-5.2 have proved that, for any $0<C_1<C_2$, there are infinitely many $a,b,n$ with $b=n/2$, $a=n/2+O(\log n)$, and\[C_1\log n< a+b-n< C_2\log n\]such that\[a!b!\mid n!(a+b-n)!.\]This appears to answer the question in the spirit it was intended.

See also [729].