  Fern\'andez-Bret\'on, Sarmiento and Vera showed that whenever a direct sum of
sufficiently many copies of ${\mathbb Z}_4$, the cyclic group of order 4, is
countably coloured there are arbitrarily large finite sets $X$ whose sumsets
$X+X$ are monochromatic. They asked if the elements of order 4 are necessary,
in the following strong sense: if $G$ is an abelian group having no elements of
order 4, is it always the case there there is a countable colouring of $G$ for
which there is not even a monochromatic sumset $X+X$ with $X$ of size 2? Our
aim in this short note is to show that this is indeed the case.
