Monochromatic infinite sumsets

Abstract

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 $\sup \{{\aleph }_0,2^{{\aleph }_0},2^{2^{{\aleph }_0}},\dots \}$. This complements a result of Hindman, Leader and Strauss, who showed that the result does not hold for dimension below ${\aleph }_{\omega }$. So our result is best possible under GCH.