Erdős Problem #45

Let \(k\geq 2\). Is there an integer \(n_k\) such that, if \(D=\{ 1<d<n_k : d\mid n_k\}\), then for any \(k\)-colouring of \(D\) there is a monochromatic subset \(D'\subseteq D\) such that \(\sum_{d\in D'}\frac{1}{d}=1\)? 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.