SETS PARTITION REGULAR FOR n EQUATIONS NEED NOT SOLVE n + 1

An n x m rational matrix A is said to be partition regular if for every finite colouring oi \ there is a monochromatic vector x s N with Ax = 0. A set D c f̂J is said to be partition regular for A (or for the system of equations Ax = 0) if for every finite colouring of D there is a monochromatic x e D' with Ax = 6. In this paper we show that for every n there is a set that is partition regular for every partition regular system of n equations but not for every system of n + 1 equations. We give several related results and we also prove a 'uniform" extension of this result: for each n we give a set D which is uniformly partition regular for n equations in the sense that given any finite colouring of D some one class solves all partition regular systems of n equations, but D is not partition regular for (and in fact contains no solution to) a particular partition regular system of n t 1 equations, namely that system describing a length n •+• 2 arithmetic progression with its increment. We give applications to the algebraic structure of /3\, the Stone-Cech compactification of the discrete set ^J.