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

Some New Results on Monochromatic Sums and Products in the Rationals