N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of $mathbb R$ so that no infinite sumset $X+X={x+y:x,yin X}$ is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any $c:mathbb Rto r$ with $r$ finite there is an infinite $Xsubseteq mathbb R$ so that $c$ is constant on $X+X$.
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