Has any work been done on Singmaster's conjecture since Singmaster's work?


The conjecture says there is a finite upper bound on how many times a number other than 1 can occur as a binomial coefficient.


Wikipedia's article on it, written mostly by me, says that


* It is known that infinitely many numbers appear exactly 3 times.
* It is unknown whether any number appears an odd number of times where the odd number is bigger than 3.
* It is known that infinitely many numbers appear 2 times, 4 times, and 6 times.
* One number is known to appear 8 times. No one knows whether there are any others nor whether any number appears more than 8 times.
* Singmaster reported that Paul Erdős told him the conjecture is probably true but would probably be very hard to prove.



Singmaster's conjecture
There is an upper bound of $O\left(\frac{(\log n)(\log \log \log n)}{(\log \log n)^3}\right)$ due to Daniel Kane: see "[Improved bounds on the number of ways of expressing *t* as a binomial coefficient](http://www.emis.de/journals/INTEGERS/papers/h53/h53.pdf)," Integers 7 (2007), #A53 for details.


