TITLE: How $a+b$ can grow when $a!b! \mid n!$
QUESTION [13 upvotes]: Let $a,b,n$ be natural numbers such that $a!b! \mid n!$. I am looking for a (somehow best) upper bound of $a+b$ in terms of $n$ (for large values on $n$). For example it is clear to see that we must have $a+b \leq 2n$. But unfortunately I am looking for much smaller bound ! Any idea would be helpful.

REPLY [11 votes]: This is really a comment, but given all the interest in the question it seemed a good idea to write it as an answer.   Apparently what was discussed earlier (in the answers of Seva and GH from MO) is an old result of Erdos! The reference is Aufgabe 557, Elemente Math. (1968) 111-113, which may be found here: http://retro.seals.ch/digbib/view?rid=elemat-001:1968:23::117&id=browse&id2=browse4&id3= . I found this reference in an interesting paper of Erdos, Graham, Ruzsa and Strauss from 1975, which you can find here (see bottom of page 90): http://math.ucsd.edu/~ronspubs/75_03_prime_factors.pdf 
Erdos, Graham, Ruzsa and Strauss also comment that there is a constant $c>0$ such that for all $n$ except a set of density zero we have 
$$ 
\frac{(2n)!}{n!(n+[c\log n])!} \in {\Bbb N}.
$$ 
Thus the bound of $2n+O(\log n)$ is best possible. 
This is also in Erdos's exercise linked above. Note on the history:  Erdos proposed as an exercise to show the upper bound $a+b\le n+C\log n$ (part 1) and that this is best possible (part 2).  Part 1 was solved by P. Bundschuh  (same solution as in GH from MO's response).  The solution printed for Part 2 was that sent by the proposer Erdos.  
They also raise a "curious problem" in this paper:  Namely it is possible that $n!/(a!b!)$ is not an integer for $a+b \ge n+c\log n$ only because of small primes.  In other words for every $c$ it is asked whether there exists a $k$ such that for infinitely many (or even all sufficiently large $n$) there are suitable $a$ and $b$ with $a+b>n+c\log n$ and such that $n!/(a!b!)$ has no prime factor $>k$ in its denominator.