Monochromatic Sumsets in Countable Colourings of Abelian Groups

Fernández-Bretón, 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.