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 ℵ 2 , there exists a coloring such that for every uncountable and every integer k , there are three distinct elements of X such that .