It is easy to deduce from Ramsey's Theorem that, given positive integers a1, a2, …, am, and a finite colouring of the set N of positive integers, there exists an injective sequence (xi) with all sums of the form a1xr1+a2xr2+ …+amxrm (r1<r2<…<rm) lying in the same colour class. The consistency version of this result, namely that, given positive integers a1, a2, …, am and b1, b2, …, bn, and a finite colouring of N, there exist injective sequences (xi) and (yi) with all sums of the form a1xr1+a2xr2+ …+amxrm (r1<r2<…<rm) and all sums of the form b1yr1+b2yr2+ …+bnyrn (r1<r2<…<rn) 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 βN of ultrafilters on N. Our aim in this note is to present a short proof of this result which does not use properties of βN. Our proof also gives various results not obtainable by the previous method of proof.
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