发表机构
University of Vienna（维也纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对拓扑群的分划正则性质，通过有限着色方法扩展了Hindman定理，证明了满足额外拓扑约束的有限和单色集的存在性，相关结果还应用于欧氏空间中无处稠密集的开邻域补集的着色问题。
AI 中文摘要
我们研究拓扑群的分划正则性质，证明了Hindman定理的若干扩展，其中有限和的单色集需满足额外的拓扑约束。特别地，我们的结果表明，对每个无处稠密的集合$C \subseteq \mathbb{R}^n$，存在开集$P \supseteq C$，使得对$\mathbb{Q}^n \setminus P$的任意有限着色，存在$\mathbb{Q}^n \setminus P$中的序列族$\mathcal{A}$满足以下性质：(i) 对每个$A \in \mathcal{A}$，$A$的有限和集合$\operatorname{FS}(A)$是$\mathbb{R}^n$的闭离散子集；(ii) $\bigcup_{A \in \mathcal{A}} \operatorname{FS}(A)$是单色集；(iii) $\bigcup_{A \in \mathcal{A}} \operatorname{FS}(A)$在$\mathbb{R}^n$的一个无界开子集内稠密。
英文摘要
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 $\mathbb R^n$.
