Monochromatic Sumsets in Countable Colourings of Abelian Groups

Fern\'andez-Bret\'on, Sarmiento and Vera showed that whenever a direct sum of sufficiently many copies of Z₄, 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.