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{0,20,220,...}. This complements a result of Hindman, Leader and Strauss, who showed that the result does not hold for dimension below ω. So our result is best possible under GCH.
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机译：我们表明，有一个有理的矢量空间V，使得每当V是有限的彩色时，有一个无限的组x，其SUMSET X + X是单色的。我们的示例是维度Sup {0,20,220，...}的合理矢量空间。这补充了Hindman，Leader和Strauss的结果，他们表明结果不持有低于ω的尺寸。所以我们的结果是最好的GCH。
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