We show that there is a rational vector space V such that, whenever V is finitely coloured, there is an infinite set X whose sumset X+X is monochromatic. Our example is the rational vector space of dimension sup(ℵ0,2ℵ0, 22ℵ0,…). This complements a result of Hindman, Leader and Strauss, who showed that the result does not hold for dimension below ℵω. So our result is best possible under GCH.