2024/07/04 by Imre Leader, Leader, Imre, Kada Williams +1
Mathematics · #05D10 #Combinatorics (math.CO) #FOS: Mathematics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2407.03938
openalex publication_date 2024/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fernández-Bretón, Sarmiento and Vera showed that whenever a direct sum of sufficiently many copies of \mathbb 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.