OPEN

Estimate $n_k$, the smallest integer such that $\prod_{1\leq i\leq k}(n_k-i)$ has no prime factor in $(k,2k)$.

Erdős and Graham write 'we can prove $n_k>k^{1+c}$ but no doubt much more is true'.

In [Er79d] Erdős writes that probably $n_k<e^{o(k)}$ but $n_k>k^d$ for all constant $d$.