Given a topological Ramsey space $(\mathcal R,\leq, r)$, we extend the notionof semiselective coideal to sets $\mathcal H\subseteq\mathcal R$ and studyconditions for $\mathcal H$ that will enable us to make the structure$(\mathcal R,\mathcal H,\leq, r)$ a Ramsey space (not necessarily topological)and also study forcing notions related to $\mathcal H$ which will satisfyabstract versions of interesting properties of the corresponding forcingnotions in the realm of Ellentuck's space. This extends results of Farah, andresults of Mijares, to the most general context of topological Ramsey spaces.As applications, we prove that for every topological Ramsey space $\mathcal R$,under suitable large cardinal hypotheses every semiselective ultrafilter$\mathcal U\subseteq\mathcal R$ is generic over $L(\mathbb R)$; and that givena semiselective coideal $\mathcal H\subseteq\mathcal R$, every definable subsetof $\mathcal R$ is $\mathcal H$--Ramsey. This generalizes the correspondingresults for the case when $\mathcal R$ is equal to Ellentuck's space.
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机译：给定拓扑Ramsey空间$（\ mathcal R，\ leq，r）$，我们扩展了半选择共理想的概念以设置$ \ mathcal H \ subseteq \ mathcal R $以及$ \ mathcal H $的学习条件，这将使我们能够结构$（\ mathcal R，\ mathcal H，\ leq，r）$一个Ramsey空间（不一定是拓扑），并且还研究与$ \ mathcal H $相关的强迫概念，这将满足对应的强制概念有趣性质的抽象版本Ellentuck空间的境界。这将Farah的结果以及Mijares的结果扩展到拓扑Ramsey空间的最一般上下文。作为应用程序，我们证明了在适当的大基数假设下，对于每个拓扑Ramsey空间$ \ mathcal R $，每个半选择性超滤器$ \ mathcal U \ subseteq \ mathcal R $在$ L（\ mathbb R）$上具有通用性；并给定一个半选择的理想的$ \ mathcal H \ subseteq \ mathcal R $，$ \ mathcal R $的每个可定义子集都是$ \ mathcal H $-Ramsey。当$ \ mathcal R $等于Ellentuck的空间时，这将推广相应的结果。
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