A bound on measurable chromatic numbers of locally finite Borel graphs
A bound on measurable chromatic numbers of locally finite Borel graphs
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局部有限Borel图可测色数的界
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
B. D. Miller
中科院分区：
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作者：
Clinton T. Conley;B. D. Miller
集合\(X\)上的一个图是\(X\times X\)的一个反自反、对称的子集\(G\)。如果每个点仅有有限多个\(G\)-邻点，那么这样的图是局部有限的。这样一个图的（\(\kappa\)-）着色是一个函数\(c:X\rightarrow\kappa\)，具有性质：对于任意\((x,y)\in G\)，\(c(x)\neq c(y)\)。这样一个图的色数，或\(\chi(G)\)，是存在这样一个\(\kappa\)-着色的最小基数\(\kappa\)。注意任何局部有限图都可以用可数多种颜色着色。在本文中，我们考虑这些概念的可测类似物，由于它与描述集合论的二分法以及群作用的动力学性质相关，在过去几年中这是一个越来越受关注的主题。拓扑空间的一个子集如果在由底层拓扑生成的\(\sigma\)-代数中，那么它是博雷尔集，如果拓扑空间之间的一个函数的开集的原像都是博雷尔集，那么这个函数是博雷尔函数。一个波兰空间是一个可分的拓扑空间，它允许一个相容的完备度量。虽然这几乎不是标准术语，但我们使用术语“波兰基数”来指代配备了波兰拓扑的基数。因此波兰基数恰好是集合\(\{0,1,\ldots,\aleph_0,2^{\aleph_0}\}\)中的那些，其中两个无穷基数支持各种拓扑。
A graph on a set X is an irreflexive, symmetric set G ⊆ X ×X. Such a graph is locally finite if every point has only finitely many G-neighbors. A (κ-)coloring of such a graph is a function c : X → κ with the property that ∀(x, y) ∈ G c(x) = c(y). The chromatic number of such a graph, or χ(G), is the least cardinal κ for which there is such a κ-coloring. Note that any locally finite graph may be colored with countably many colors. In this paper, we consider measurable analogs of these notions, a subject of increasing interest over the last few years due to its connections with descriptive set-theoretic dichotomies and dynamical properties of group actions. A subset of a topological space is Borel if it is in the σ-algebra generated by the underlying topology, and a function between topological spaces is Borel if pre-images of open sets are Borel. A Polish space is a separable topological space which admits a compatible complete metric. While it is hardly standard terminology, we use the term Polish cardinal to refer to a cardinal equipped with a Polish topology. Thus the Polish cardinals are exactly those in the set {0, 1, . . . ,א0, 2א0}, with the two infinite cardinals supporting various topologies.