This follows from the colouring result of Croot [Cr03]. Croot's result allows for $n_k \leq e^{C^k}$ for some constant $C>1$ (simply taking $n_k$ to be the lowest common multiple of some interval $[1,C^k]$). Sawhney has observed that there is also a doubly exponential lower bound, and hence this bound is essentially sharp.

Indeed, we must trivially have $\sum_{d|n_k}1/d \geq k$, or else there is a greedy colouring as a counterexample. Since $\prod_{p}(1+1/p^2)$ is finite we must have $\prod_{p|n_k}(1+1/p)\gg k$. To achieve the minimal $\prod_{p|n_k}p$ we take the product of primes up to $T$ where $\prod_{p\leq T}(1+1/p)\gg k$; by Mertens theorems this implies $T\geq C^{k}$ for some constant $C>1$, and hence $n_k\geq \prod_{p\mid n_k}p\geq \exp(cC^k)$ for some $c>0$.