Consistency for partition regular equations

Abstract: It is easy to deduce from Ramsey''s theorem that given positive integers and a finite colouring of the set of positive integers, there exists an injective sequence with all sums of the form () lying in the same colour class. The consistency version of this result, namely that given positive integers and , and a finite colouring of , there exist injective sequences and with all sums of the form and all sums of the form () 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 of ultrafilters on . Our aim in this note is to present a short proof of this result which does not use properties of . Our proof also gives various results not obtainable by the previous method of proof. [Copyright &y& Elsevier]