Sums of triples in Abelian groups

Abstract

Motivated by a problem in additive Ramsey theory, we extend Todorčević's partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then for every Abelian group G of size ℵ₂, there exists a coloring $c:G\to Z$ such that for every uncountable $X\subseteq G$ and every integer k, there are three distinct elements $x,y,z$ of X such that $c(x+y+z)=k$.