Erdős Problem #191

Let \(C>0\) be arbitrary. Is it true that, if \(n\) is sufficiently large depending on \(C\), then in any \(2\)-colouring of \(\binom{\{2,\ldots,n\}}{2}\) there exists some \(X\subseteq \{2,\ldots,n\}\) such that \(\binom{X}{2}\) is monochromatic and \[\sum_{x\in X}\frac{1}{\log x}\geq C?\] The answer is yes, which was proved by Rödl [Ro03]. In the same article Rödl also proved a lower bound for this problem, constructing, for all \(n\), a \(2\)-colouring of \(\binom{\{2,\ldots,n\}}{2}\) such that if \(X\subseteq \{2,\ldots,n\}\) is such that \(\binom{X}{2}\) is monochromatic then \[\sum_{x\in X}\frac{1}{\log x}\ll \log\log\log n.\] In the same paper Rödl proves that the answer to the main problem is negative if we consider \(3\)-colourings. This bound is best possible, as proved by Conlon, Fox, and Sudakov [CFS13], who proved that, if \(n\) is sufficiently large, then in any \(2\)-colouring of \(\binom{\{2,\ldots,n\}}{2}\) there exists some \(X\subseteq \{2,\ldots,n\}\) such that \(\binom{X}{2}\) is monochromatic and \[\sum_{x\in X}\frac{1}{\log x}\geq 2^{-8}\log\log\log n.\] A \(2\)-colouring of the pairs of \(\{2,\ldots,n\}\) is a graph `G` on this vertex set; a monochromatic set is a clique or an independent set of `