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 $c:\mathbb{R}\to \mathbb{Q}$, 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 $c:G\to \mathbb{Q}$ such that for every uncountable $X\subseteq G$, and every colour $\gamma$, for some large enough integer $n$, there are pairwise distinct elements $x_0,\dots ,x_n$ 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 ${⊛}_{\kappa }$ assert that for every Abelian group $G$ of cardinality $\kappa$, there exists a colouring $c:G\to G$ such that for every positive integer $n$, every $X_0,\dots ,X_n\in [G{]}^{\kappa }$, and every $\gamma \in G$, there are $x_0\in X_0,\dots ,x_n\in X_n$ such that $c(x_0+\cdots +x_n)=\gamma$. Then ${⊛}_{\kappa }$ holds for unboundedly many uncountable cardinals $\kappa$, and it is consistent that ${⊛}_{\kappa }$ holds for all regular uncountable cardinals $\kappa$.

Final accepted version. For several of the earlier results that were stated only for regular cardinals, there is now a treatment of the singular cardinal case. Also, a new partition relation for the Real Line was obtained, see Theorem C3

Cited by in corpus (3)

• Knaster and friends I: Closed colorings and precalibers
• Transformations of the transfinite plane
• Hindman-like theorems with uncountably many colours and finite monochromatic sets