OPEN
Let $\epsilon,C>0$. Are there integers $a,b,n$ such that $a>\epsilon n$, $b>\epsilon n$,
\[a! b! \mid n!(a+b-n)!\]
and $a+b>n+C\log n$?
A question of Erdős, Graham, Ruzsa, and Straus
[EGRS75].
Erdős
[Er68c] proved that if $a!b!\mid n!$ then $a+b\leq n+O(\log n)$.
By Legendre's formula $a! b! \mid n!(a+b-n)!$ is true if and only if for all primes $p$
\[s_p(n)+s_p(a+b-n) \leq s_p(a)+s_p(b),\]
where $s_p(n)$ is the sum of the base $p$ digits of $n$.
See also [729].
