Extensions of Hindman's theorem via finite colorings of topological groups

Abstract

We study the partition regular properties of topological groups, proving several extensions of Hindman theorem where monochromatic sets of finite sums are required to satisfy additional topological constraints. In particular, our results imply that for every nowhere dense set $C \subseteq \mathbb{R}^n$, there exists an open set $P \supseteq C$ such that for any finite coloring of $\mathbb{Q}^n \setminus P$, there is a family $\mathcal{A}$ of sequences in $\mathbb{Q}^n \setminus P$ which satisfies the following properties: (i) for each $A\in\mathcal A$, the set $\operatorname{FS}(A)$ of finite sums of $A$ is a closed discrete subset of $\mathbb R^n$; (ii) the set $\bigcup_{A\in\mathcal A}\operatorname{FS}(A)$ is monochromatic; and (iii) the set $\bigcup_{A\in\mathcal A}\operatorname{FS}(A)$ is dense in an open unbounded subset of $\mathbb R^n$.