Canonical Ramsey Theory on Polish Spaces (Encyclopedia of Mathematics and its Applications 202) By Vladimir Kanovei, Marcin Sabok and Jindrich Zapletal
Canonical Ramsey Theory on Polish Spaces (Encyclopedia of Mathematics and its Applications 202) By Vladimir Kanovei, Marcin Sabok and Jindrich Zapletal
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波兰空间的规范拉姆齐理论（数学及其应用百科全书 202） 作者：Vladimir Kanovei、Marcin Sabok 和 Jindrich Zapletal
DOI：
10.1112/blms/bdv016
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发表时间：
2015
影响因子：
0.9
通讯作者：
Brooke-Taylor A
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文献类型：
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作者：
Brooke-Taylor A
等价关系规范化的研究可以追溯到 Erdos 和 Rado [1] 的一个漂亮结果：对于从 N 开始递增的 n 元组上的任何等价关系∼，存在 N 的无限子集 A 和 {1,..., n} 的子集 σ，使得对于 An 中的所有递增元组 a=(a1,..., an) 和 b=(b1,..., bn)，a∼ b 当且仅当对于所有 i 来说 ai= bi σ。这种缩小域以使相关关系表现良好的能力可以被视为标准拉姆齐定理的扩展；事实上，我们可以立即从这个结果推导出标准拉姆齐定理，取 n= 2，对于 N 上的图 G，考虑等价关系 (m, n)∼(o, p) 当且仅当每一对或两者都不是 G 的边（在这种情况下，σ 必须是 ∅，因为只有两个等价类）。在其他上下文中很自然地会提出类似的问题：给定集合 X 上的等价关系，是否可以移动到一个合适的子集 A⊆ X，在该子集上等价关系采用规范形式？本书针对X是波兰空间，即可以赋予完备度量的可分离拓扑空间（典型例子是R）的情况来研究这个问题。 Borel 和波兰空间上的解析等价关系的研究已成为现代描述集合论的主要焦点，部分原因是其广泛的适用性：任何可以由实数合理编码的对象都适合这个框架。例如，给定类型的可数结构（无论是群、环还是其他什么）与底层集合 N 的同构关系就是这种等价关系的一个中心示例。我们说，X 上的等价关系 E 可以 Borel 还原为 Y 上的等价关系 F，写作 E≤ B F，如果存在 Borel 函数 f: X→ Y，使得 xEy 当且仅当 f (x) Ff (y) 对于所有 x，y ∈ X。使用描述性集合论的技术来确定某些等价关系是否 Borel 可还原为其他关系，对各种分类程序产生了重要影响，例如 Thomas 和Velickovic 用于有限生成群 [5]，以及 Farah、Toms 和 Törnquist 用于 C*-代数的最新结果 [2]。有关该领域的更多信息，请参阅 Hjorth [3]。下一个问题是什么构成了要限制的“合适”子集 A⊆ X。集合 A 在某种意义上应该“不小”；这是使用我认为是“小”集系列的 σ 理想来形式化的。从描述性集合论的角度来看，集合 A 也应该是可以合理定义的； Borel 在这里被认为是合适的概念。因此，我们得出了本书的中心定义，作者以真正的拉姆齐理论风格引入了箭头符号。定义 1.15。设 E 和 F 为等价关系类，并令 I 为波兰空间 X 上的 σ-理想。我们用 E→ I F 表示这样的陈述：对于每个 I-正（即不在 I 中）Borel 集合 B⊆ X 以及 B 上的每个 E∈ E，存在一个 I-正 Borel 集合 A⊆ B，这样
The study of canonization of equivalence relations goes back to a beautiful result of Erdos and Rado [1]: for any equivalence relation∼ on the increasing n-tuples from N, there is an infinite subset A of N and a subset σ of {1,..., n} such that for all increasing tuples a=(a1,..., an) and b=(b1,..., bn) in An, a∼ b if and only if ai= bi for all i in σ. This ability to reduce the domain so that the relation in question becomes well-behaved may be seen as an extension of the standard Ramsey Theorem; in fact, one may immediately deduce the standard Ramsey Theorem from this result, taking n= 2 and for a graph G on N considering the equivalence relation (m, n)∼(o, p) if and only if each or neither pair is an edge of G (in this case, σ must be∅, since there are only two equivalence classes). It is natural to ask the analogous question in other contexts: given an equivalence relation on a set X, can one move to a suitable subset A⊆ X on which the equivalence relation takes a canonical form? The present book studies this question for the case when X is a Polish space, that is, a separable topological space that can be endowed with a complete metric (the archetypal example being R). The study of Borel and analytic equivalence relations on Polish spaces has become a major focus of modern descriptive set theory, in part because of its broad applicability: any kind of object that may be reasonably encoded by reals fits into this framework. For example, the isomorphism relation on countable structures of a given type (be they groups, rings or what have you) with underlying set N is a central example of such an equivalence relation. We say that an equivalence relation E on X is Borel reducible to an equivalence relation F on Y, written E≤ B F, if there is a Borel function f: X→ Y such that xEy if and only if f (x) Ff (y) for all x, y∈ X. Using the techniques of descriptive set theory to determine whether certain equivalence relations are Borel reducible to others has had important ramifications for various classification programmes, such as the results of Thomas and Velickovic for finitely generated groups [5], and more recent results of Farah, Toms and Törnquist for C∗-algebras [2]. For more on this area, see, for example, Hjorth [3]. The next question is what constitutes a ‘suitable’subset A⊆ X to restrict to. The set A should be in some sense ‘not small’; this is formalized using a σ-ideal I considered to be the family of ‘small’sets. The set A should also be reasonably definable from a descriptive settheoretic point of view; Borel is taken to be the appropriate notion here. We thus arrive at the central definition of the book, for which the authors in true Ramsey-theoretic style introduce an arrow notation.Definition 1.15. Let E and F be classes of equivalence relations, and let I be a σ-ideal on a Polish space X. We denote by E→ I F the statement that for every I-positive (that is, not in I) Borel set B⊆ X and every E∈ E on B there is an I-positive Borel set A⊆ B such