Monochromatic Sumsets in Countable Colourings of Abelian Groups
Imre Leader, Kada Williams

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 , the cyclic group of order 4, is countably coloured there are arbitrarily large finite sets whose sumsets are monochromatic. They asked if the elements of order 4 are necessary, in the following strong sense: if is an abelian group having no elements of order 4, is it always the case there there is a countable colouring of for which there is not even a monochromatic sumset with of size 2? Our aim in this short note is to show that this is indeed the case.
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Taxonomy
Topicsadvanced mathematical theories
