  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 $\aleph_2$, there
exists a coloring $c:G\rightarrow\mathbb Z$ such that for every uncountable
$X\subseteq G$ and every integer $k$, there are three distinct elements $x,y,z$
of $X$ such that $c(x+y+z)=k$.
