Problem #1063: Let $k\geq 2$ and define $n_k\geq 2k$ to be the least value...

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Progress

A problem of Erdős and Selfridge posed in [ErSe83]. Erdős and Selfridge noted (and a proof can be found in [Mo85]) that if $n\geq 2k$ then there must exist at least one $0\leq i<k$ such that $n-i$ does not divide $\binom{n}{k}$.

We have $n_2=4$, $n_3=6$, $n_4=9$, and $n_5=12$. Monier [Mo85] observed that $n_k\leq k!$ for $k\geq 3$, since $\binom{k!}{k}$ is divisible by $k!-i$ for $1\leq i<k$. Cambie observes in the comments that this can be improved to\[n_k\leq k[2,3,\ldots,k-1]\leq e^{(1+o(1))k},\]where $[\cdots]$ is the least common multiple.

This is discussed in problem B31 of Guy's collection [Gu04].