Singmaster's conjecture in the interior of Pascal's triangle

Singmaster's conjecture asserts that every natural number greater than one
occurs at most a bounded number of times in Pascal's triangle; that is, for any
natural number $t \geq 2$, the number of solutions to the equation
$\binom{n}{m} = t$ for natural numbers $1 \leq m < n$ is bounded. In this paper
we establish this result in the interior region $\exp(\log^{2/3+\varepsilon} n)
\leq m \leq n-\exp(\log^{2/3 + \varepsilon} n)$ for any fixed $\varepsilon >
0$. Indeed, when $t$ is sufficiently large depending on $\varepsilon$, we show
that there are at most four solutions (or at most two in either half of
Pascal's triangle) in this region. We also establish analogous results for the
equation $(n)_m = t$, where $(n)_m := n(n-1)\ldots(n-m+1)$ denotes the falling
factorial.