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 \b[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.

Solution

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

 Solution (source code)