Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 10 / 1 / i

If an $m$-term arithmetic progression contains no red pair, all of its unordered pairs are blue, since there are only two colours. This gives the first alternative, including the vacuous singleton case. Otherwise the position-pair finite colouring in the root solution is defined everywhere, and the construction below forces the second alternative.