Monochromatic Sumsets in Countable Colourings of Abelian Groups

TL;DR

This paper proves that in abelian groups without elements of order 4, it is possible to color the group countably such that no monochromatic sumset of size 2 exists, extending previous results about groups with elements of order 4.

Contribution

It demonstrates that the presence of elements of order 4 is crucial for monochromatic sumset properties in countably colored abelian groups.

Findings

Groups without elements of order 4 can be colored to avoid monochromatic sumsets of size 2.

The result confirms the necessity of elements of order 4 for certain monochromatic sumset phenomena.

Extends understanding of coloring and sumset properties in abelian groups.

Abstract

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.