Show that there is some B so that no integer appears more than B times among the binomial coefficients

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For integers $n\ge0$ and $0\le m\le n$, define $\binom nm=n!/(m!(n-m)!)$. For an integer $t\ge2$, let \[N(t)=\#\{(n,m)\in\mathbb Z^2:1\le m<n,\ \binom nm=t\}.\] The symmetric positions $m$ and $n-m$ are counted separately. The boundary entries $\binom n0=\binom nn=1$ occur infinitely often; Singmaster's conjecture concerns values greater than one, as specified in the original statement cited below.

Statement recorded in the supplied source.

Is there an absolute finite constant $B$ such that $N(t)\le B$ for every integer $t\ge2$? [S2]

Short English statement

Is there a uniform bound on the number of occurrences in Pascal’s triangle of any integer greater than one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3100064 (AMR-030-0064). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao, Terence Tao and Joni Teräväinen, Singmaster's conjecture in the interior of Pascal's triangle (2021), Section1, Conjecture1.1 and the following boundary-entry convention, page1. https://arxiv.org/abs/2106.03335.