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An n × m rational matrix A is said to be partition regular if for every finite colouring of N there is a monochromatic vector $$\overline{x}\in {N}^{m}$$ with $$A\overline{x}=\overline{0}$$. A set $$D\subseteq N$$ is said to be partition regular for A (or for the system of equations $$A\overline{x}=\overline{0}$$) if for every finite colouring of D there is a monochromatic $$\overline{x}\in {D}^{m}$$ with $$A\overline{x}=\overline{0}$$ 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 + 1 equations, namely that system describing a length n + 2 arithmetic progression with its increment. We give applications to the algebraic structure of βN, the Stone-Čech compactification of the discrete set N.