Let $C>0$ be arbitrary. Is it true that, if $n$ is sufficiently large depending on $C$, then in any $2$-colouring of $\left(\genfrac{}{}{0ex}{}{\{2,\dots ,n\}}{2}\right)$ there exists some $X\subseteq \{2,\dots ,n\}$ such that $\left(\genfrac{}{}{0ex}{}{X}{2}\right)$ is monochromatic and$$\sum _{x\in X}\frac{1}{\log x}\ge C?$$