On homogeneous sets of positive integers (Q1873829)

This paper solves affirmatively the longstanding conjecture of Paul Erdős in Ramsey theory on two-colourations of the pairs of the first \(n\) positive integers. It shows that in any two-colouration of \([n]^2\), for sufficiently large natural number \(n,\) there exists a homogeneous set \(H \subseteq [n]\) (all pairs in \(H\) belong to the same colour class) such that the elements of \(H\) are relatively small. More precisely, \(H\) satisfies \(\min H \geq 2\) and \[ \sum_{h \in H} {1 \over \log h} \geq C {\log \log \log \log n \over \log \log \log \log \log n}. \]