关于 Ramsey 度、紧致性与可逼近性
On Ramsey degrees, compactness and approximability
AI总结：
本文提出了一个统一框架以推理“直接”与对偶 Ramsey 命题，引入可逼近性概念构建通用紧致性论证，并将无限 Ramsey 定理的 $\star$-版本推广至大类关系结构。
摘要：
结构 Ramsey 理论中紧致性原理的一个推论是，小 Ramsey 度不能超过相应的大 Ramsey 度，从而证明了形容词选择的合理性。然而，由于缺乏适用于该设定的紧致性论证，目前尚不清楚在对偶 Ramsey 度的领域中会发生什么。在本文中，我们提出了一个框架，在该框架内可以以统一的方式陈述和推理“直接” Ramsey 命题与对偶 Ramsey 命题。我们引入了可逼近性的概念，该概念产生了一个足够强大的通用紧致性论证，能够证明关于“直接” Ramsey 现象和对偶 Ramsey 现象的命题。作为新策略的应用，我们将 Voigt 的无限 Ramsey 定理的 $\star$-版本推广到一大类关系结构，并推导出关于枚举 Fraïssé 极限的“松散着色”的 Ramsey 命题，以此作为本文的结论。
英文摘要：
One of the consequences of the Compactness Principle in structural Ramsey theory is that the small Ramsey degrees cannot exceed the corresponding big Ramsey degrees, thereby justifying the choice of adjectives. However, it is unclear what happens in the realm of dual Ramsey degrees due to the lack of the compactness argument that applies to that setting. In this paper we present a framework within which both "direct" and dual Ramsey statements can be stated and reasoned about in a uniform fashion. We introduce the notion of approximability which yields a general compactness argument powerful enough to prove statements about both "direct" and dual Ramsey phenomena. We conclude the paper with an application of the new strategies by generalizing Voigt's $\star$-version of the Infinite Ramsey Theorem to a large class of relational structures and deriving a Ramsey statement for "loose colorings" of enumerated Fra\"ıssé limits.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
