  Our aim in this paper is to show that, for any $k$, there is a finite
colouring of the set of rationals whose denominators contain only the first $k$
primes such that no infinite set has all of its finite sums and products
monochromatic. We actually prove a `uniform' form of this: there is a finite
colouring of the rationals with the property that no infinite set whose
denominators contain only finitely many primes has all of its finite sums and
products monochromatic. We also give various other results, including a new
short proof of the old result that there is a finite colouring of the naturals
such that no infinite set has all of its pairwise sums and products
monochromatic.
