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{ℵ₀,2^ℵ₀,2^2ℵ₀,...}. This complements a result of Hindman, Leader and Strauss, who showed that the result does not hold for dimension below ℵ_ω. So our result is best possible under GCH.

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