Open, the site's label. The site credits two classical results, each an accepted partial claim with refereed evidence: the Sylvester-Schur theorem in the binomial form of [Er34], Erdős 1934, which settles the range $n/2\le k\le n-1$ for every $c$, and Theorem 1 of [Er55d], Erdős 1955, which gives $P(\binom nk)\gg\min(n-k+1,k\log k)$ for $k\le n/2$ and settles the instances with $n-k$ at most a constant multiple of $k\log k$. Neither gives a bound of the form $k^{1+c}$, so no claim settles or pends to settle the problem and the derived standing is open with claim none.