Strong failures of higher analogs of Hindman’s theorem

Abstract:

We show that various analogs of Hindman’s theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets:

Theorem 1. There exists a colouring $c:\mathbb R\rightarrow \mathbb Q$, such that for every $X\subseteq \mathbb R$ with $|X|=|\mathbb R|$, and every colour $\gamma \in \mathbb Q$, there are two distinct elements $x_0,x_1$ of $X$ for which $c(x_0+x_1)=\gamma$. This forms a simultaneous generalization of a theorem of Hindman, Leader and Strauss and a theorem of Galvin and Shelah.

Theorem 2. For every abelian group $G$, there exists a colouring $c:G\rightarrow \mathbb Q$ such that for every uncountable $X\subseteq G$ and every colour $\gamma$, for some large enough integer $n$, there are pairwise distinct elements $x_0,\ldots ,x_n$ of $X$ such that $c(x_0+\cdots +x_n)=\gamma$. In addition, it is consistent that the preceding statement remains valid even after enlarging the set of colours from $\mathbb Q$ to $\mathbb R$.

Theorem 3. Let $\circledast _\kappa$ assert that for every abelian group $G$ of cardinality $\kappa$, there exists a colouring $c:G\rightarrow G$ such that for every positive integer $n$, every $X_0,\ldots ,X_n \in [G]^\kappa$, and every $\gamma \in G$, there are $x_0\in X_0,\ldots , x_n\in X_n$ such that $c(x_0+\cdots +x_n)=\gamma$. Then $\circledast _\kappa$ holds for unboundedly many uncountable cardinals $\kappa$, and it is consistent that $\circledast _\kappa$ holds for all regular uncountable cardinals $\kappa$.