Strong failures of higher analogs of Hindman’s Theorem

Joint work with David J. Fernández Bretón.

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 commutative cancellative semigroup $G$, 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 commutative cancellative semigroup $G$ 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 $\kappa$.

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Citation information:

D. J. Fernandez Breton and A. Rinot, Strong failures of higher analogs of Hindman’s Theorem, Trans. Amer. Math. Soc., 369(12): 8939-8966, 2017.