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Claim. The Sylvester-Schur theorem in the binomial form Erdős proves in [Er34], printed p. 283: "If $n\ge2k$, then $\binom nk$ contains a prime divisor greater than $k$." In the notation of Problem 683 this is $P(\binom nk)>k$ for $k\le n/2$. The paper opens with the theorem in its interval form, that for $n>k$ the block $n,n+1,\ldots,n+k-1$ contains an integer with a prime divisor greater than $k$, and says that Sylvester first proved it and that Schur rediscovered and reproved it; Erdős's own proof is elementary and avoids Chebyshev's theorem. The statement is the library result page erdos_1934_theorem_sylvester_schur / theorem.

Covers. Since $\binom nk=\binom n{n-k}$, the theorem applied to $n-k$ gives $P(\binom nk)>n-k$, that is $P(\binom nk)\ge n-k+1$, whenever $n-k\le n/2$. So for $n/2\le k\le n-1$ the inequality of the problem, read as its Formulation states, holds for every $c>0$. For $k<n/2$ the theorem gives only $P(\binom nk)>k$, which is weaker than the bound asked, and it says nothing about a power of $k$; 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, A theorem of Sylvester and Schur, J. London Math. Soc. 9 (1934), no. 4, 282--288, DOI 10.1112/jlms/s1-9.4.282; the page is dated to the issue month, October 1934, as the publisher's record gives it. The site's commentary cites the theorem from 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 theorem as the variant erdos_683.variant.sylvester_schur, tagged research solved with a sorry body; a statement file is not a formalization of the result.