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Claim. For $1\le k\le n/2$,

$$P\left(\binom nk\right)\gg\min\left(n-k+1,\frac{k\log k\log\log k}{\log\log\log k}\right).$$

This is a consequence, derived here, of the bound

$$f(m)<c_1\frac{m\log\log\log m}{\log m\log\log m}$$

of Jutila, Ramachandra and Shorey, where $f(m)$ is the least $L$ such that every block of $L$ consecutive integers, each greater than $m$, contains a multiple of a prime greater than $m$, the function of Problem 961 and of Theorem 1 of [Er55d]. The bound is display (1) on printed p. 271 of Erdős's survey Problems and results on consecutive integers, Publ. Math. Debrecen 23 (1976), 271--282 (result page), which credits it to Jutila, Ramachandra and Shorey: M. Jutila, On numbers with a large prime factor. II, J. Indian Math. Soc. (N.S.) 38 (1974), 125--130; K. Ramachandra and T. N. Shorey, On gaps between numbers with a large prime factor, Acta Arith. 24 (1973), no. 1, 99--111; and T. N. Shorey, On gaps between numbers with a large prime factor. II, Acta Arith. 25 (1974), no. 4, 365--373. None of the three states a binomial form.

The deduction is that of the 1955 claim page, with this bound in place of Erdős's $f(m)\le c_1m/\log m$. For $k\le n/2$ the $k$ members of the numerator block $n-k+1,\ldots,n$ of $\binom nk$ all exceed $n-k\ge k$, and a prime greater than $k$ dividing one of them is not canceled by $k!$, so it divides $\binom nk$. Write $g(m)$ for the right side of the bound, which increases for large $m$, and $m_k$ for the largest $m$ with $g(m)\le k$, of order $k\log k\log\log k/\log\log\log k$. If $n-k\le m_k$, then $f(n-k)<g(n-k)\le k$, so the block contains a multiple of a prime greater than $n-k$, and $P(\binom nk)\ge n-k+1$. Otherwise every member of the block exceeds $m_k$ and $f(m_k)<k$, so $P(\binom nk)>m_k$. The finitely many small $k$ are absorbed in the implied constant.

Covers. The instances with $k\le n/2$ and $n-k$ at most a constant multiple of $k\log k\log\log k/\log\log\log k$, where $P(\binom nk)\ge n-k+1$ and the inequality of the problem, read as its Formulation states, holds for every $c>0$. This widens the range of the 1955 claim by the factor $\log\log k/\log\log\log k$. Elsewhere the bound gives only $P(\binom nk)\gg k\log k\log\log k/\log\log\log k$, no bound of the form $k^{1+c}$; those instances stay open.

Depends on. The accepted claim on Problem 961, for the bound on $f$; the step to $\binom nk$ above is this page's.

Acceptance. Refereed: the bound appeared in Acta Arithmetica and in the Journal of the Indian Mathematical Society, as Erdős's 1976 survey attests. The papers are not held, so the bound's exact statements and hypotheses rest on that attestation and erdosproblems.com's, as on the Problem 961 claim page. The erdosproblems.com page for Problem 683 does not cite the papers; it calls the problem essentially equivalent to Problem 961, whose commentary credits the bound. The step from $f$ to $\binom nk$ is the elementary deduction of the 1955 claim page, made here without separate review. Bloom's catalog labeled the problem OPEN, so its commentary gives no reviewed evidence.