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Claim. Theorem 1 of [Er55d], printed p. 124, with $f(k)$ the least integer such that the product of $f(k)$ consecutive integers, each greater than $k$, always contains a prime greater than $k$: "There is a constant $c_1>1$ so that $f(k)\le c_1\frac{k}{\log k}$." The constant is not specified. The statement is the library result page erdos_1955_consecutive_integers / theorem_1. The paper does not state a binomial form. For Problem 683 it applies to the numerator block $n-k+1,\ldots,n$ of $\binom nk$, whose $k$ members all exceed $k$ when $k\le n/2$: a prime greater than $k$ dividing a member of the block is not canceled by $k!$, so it divides $\binom nk$. Applied with the threshold $n-k$ in place of $k$, the theorem gives a prime factor greater than $n-k$, hence $P(\binom nk)\ge n-k+1$, whenever $k\ge f(n-k)$, which holds when $n-k$ is at most a constant multiple of $k\log k$; applied with a threshold $m$ of order $k\log k$ satisfying $f(m)\le k$, it gives $P(\binom nk)>m$ otherwise. Together, $P(\binom nk)\gg\min(n-k+1,k\log k)$ for $k\le n/2$, the form the formal-conjectures file records. The site prints the bound without the minimum, and that form fails at $n=2k$, where $P(\binom{2k}k)\le2k$.

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

Depends on. No page of this wiki; the result rests on the cited paper, whose statement the library result page above records.

Acceptance. Refereed: P. Erdős, On consecutive integers, Nieuw Arch. Wisk. (3) 3 (1955), 124--128, a journal paper; the issue carries no month, so the page is dated to the year. The site's commentary credits the result to this paper, but the site labels the problem OPEN, so the remark is not acceptance of a settling claim and no reviewed evidence is listed. The formal-conjectures statement file FormalConjectures/ErdosProblems/683.lean records the bound with the minimum as the variant erdos_683.variant.erdos_log, tagged research solved with a sorry body; a statement file is not a formalization of the result.