Strong failures of higher analogs of Hindman's theorem

TL;DR

This paper demonstrates that higher analogs of Hindman's theorem fail in uncountable settings, providing explicit counterexamples and showing the limits of extending Hindman's combinatorial properties to larger cardinalities.

Contribution

The authors construct strong counterexamples to uncountable analogs of Hindman's theorem, revealing fundamental limitations in extending these combinatorial principles.

Findings

Existence of colorings with no uncountable monochromatic sumsets

Failure of uncountable Hindman-type theorems in Abelian groups

Consistency results for the validity of certain colorings at large cardinals

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 $\kappa$.