In 1974, Hindman proved that considering the semigroup \((\mathbb{N}, +)\), for any partition \(\mathbb{N} = S_0 \uplus S_1\), there exists an infinite \(X \subseteq \mathbb{N}\) such that the set of its finite sums, is monochromatic, that is, contained in one of the cells.

In contrast, in 2016 Komjáth showed that, for the group \((\mathbb{R}, +)\) there exists a partition \(\mathbb{R} = S_0 \uplus S_1\) such that, whenever \(X \subseteq \mathbb{R}\) is uncountable, not only is the set of finite sums not monochromatic, but already the set \(\mathrm{FS}_2(X) := \{x + y | \{x, y\} \in [X]^2\}\) is not monochromatic. These results motivate a general investigation of additive Ramsey theory in the spirit of the classical partition calculus, and which in fact for some cases are a strengthening of the classical partition calculus.

Motivated by similar problems at the level of \(\aleph_2\), we extend Todorčević’s partition of three-dimension combinatorial cube to handle additional three dimensional objects. As a corollary we prove that the failure of continuum hypothesis asserts that for every Abelian group \(G\) of size \(\aleph_2\), there exist a coloring \(c : G \to \mathbb{Z}\) such that, for every uncountable \(X \subseteq G\) and every integer \(k\), there exist three distinct elements \(x, y, z \in X\) such that \(c(x, y, z) = k\).

This is a joint work with Assaf Rinot. For further reading the article is available here: https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12200