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Claim. Both answers to Problem 378 are yes: for every $r\ge0$ the integers $n$ for which $\binom{n}{k}$ is squarefree for at least $r$ values of $1\le k<n$ have an asymptotic density, and that density is positive. This follows from Theorem 5 of A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), no. 1, 73–107, carded at granville_1996_explicit_bounds_exponential_sums_scarcity_squarefree, which the paper introduces as the answer to a question of Erdős and Graham on p. 72 of their 1980 problem book, the problem's source. The journal record dates the issue to June 1996 without a day, so this page is dated to the first of that month. Theorem 5 states that for each $m\ge0$ the integers $n$ whose $n$th row of Pascal's triangle has exactly $2m+2$ squarefree entries, the two end entries $1$ included, have an asymptotic density $\eta_m$, and that $0<\eta_m\ll\exp(-\tau\sqrt m/\log(2m))$ for $m\ge1$ with an absolute $\tau>0$; its proof (Section 6 of the paper) fixes any $m\ge0$. The count of squarefree entries among $1\le k<n$ is even for $n>8$, since $\binom{n}{k}$ and $\binom{n}{n-k}$ are equal and the middle entry $\binom{n}{n/2}$ of an even row is squarefree only for $n\in\{2,4,8\}$ by the paper's Theorem 1, so the rows with fewer than $r$ squarefree entries in $1\le k<n$ are, up to finitely many, those counted by $\eta_m$ with $2m<r$, and the density the problem asks for is

$$1-\sum_{0\le m<r/2}\eta_m,$$

which exists; it is positive because it is at least $\eta_m$ for any $m\ge1$ with $2m\ge r$, and every such $\eta_m$ is positive. For $r=0$ the set is all integers.

Earlier postings. Anay Aggarwal pointed out in the site's discussion on 22 August 2025 that Theorem 5 resolves the problem, and Stijn Cambie gave the derivation above there on 30 August 2025; the result is Granville and Ramaré's, so their paper's date and names name this page.

Depends on. No page of this wiki.

Acceptance. The paper appeared in Mathematika, a refereed journal, and its acknowledgments thank an anonymous referee. Thomas Bloom, the site's curator, marks the problem proved and credits the paper, with Aggarwal and Cambie's observation, on the problem page (last edited 28 October 2025); the community database records the problem as proved from 31 August 2025. The paper has no Lean formalization known here.