Consistency for partition regular equations

It is easy to deduce from Ramseyʹs theorem that given positive integers image and a finite colouring of the set image of positive integers, there exists an injective sequence image with all sums of the form image (image) lying in the same colour class. The consistency version of this result, namely that given positive integers image and image, and a finite colouring of image, there exist injective sequences image and image with all sums of the form image and all sums of the form image (image) in the same colour class, was open for some time, being recently proved by Hindman, Leader and Strauss. The proof is long and relies heavily on the structure of the semigroup image of ultrafilters on image. Our aim in this note is to present a short proof of this result which does not use properties of image. Our proof also gives various results not obtainable by the previous method of proof.