Question

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.

Answer 1

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," Integers 7 (2007), #A53 for details.