Sums of triples in Abelian groups

TL;DR

This paper explores additive properties in Abelian groups, extending combinatorial partition results to new three-dimensional objects, and demonstrates the existence of specific colorings under certain set-theoretic assumptions.

Contribution

It generalizes Todorcevic's partitions to handle additional three-dimensional objects and establishes the existence of particular colorings in large Abelian groups assuming the failure of the continuum hypothesis.

Findings

Extension of combinatorial partitions to new 3D objects

Existence of colorings in large Abelian groups under set-theoretic assumptions

Implications for additive Ramsey theory

Abstract

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 $c(x+y+z)=k$.