Motivated by a problem in additive Ramsey theory, we extend Todorcevic'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 w2, there exists a coloring c of G by the integers, such that for every uncountable subset X of G, and every integer k, there are three distinct elements x,y,z of X such that c(x+y+z)=k.

This is joint work with Ido Feldman.