Sums of triples in Abelian groups

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 (Formula presented.) such that for every uncountable (Formula presented.) and every integer k, there are three distinct elements (Formula presented.) of X such that (Formula presented.).