The field of descriptive combinatorics investigates to what extent classical combinatorial results and techniques can be made topologically or measure-theoretically well behaved. This paper examines a class of coloring problems induced by actions of countable groups on Polish spaces, with the requirement that the desired coloring be Baire measurable. We show that the set of all such coloring problems that admit a Baire measurable solution for a particular free action $\unicode[STIX]{x1D6FC}$ is complete analytic (apart from the trivial situation when the orbit equivalence relation induced by $\unicode[STIX]{x1D6FC}$ is smooth on a comeager set); this result confirms the ‘hardness’ of finding a topologically well-behaved coloring. When $\unicode[STIX]{x1D6FC}$ is the shift action, we characterize the class of problems for which $\unicode[STIX]{x1D6FC}$ has a Baire measurable coloring in purely combinatorial terms; it turns out that closely related concepts have already been studied in graph theory with no relation to descriptive set theory. We remark that our framework permits a wholly dynamical interpretation (with colorings corresponding to equivariant maps to a given subshift), so this article can also be viewed as a contribution to generic dynamics.

中文翻译：

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关于集体行动的贝尔可测量着色

的领域描述性组合学研究在多大程度上可以使经典组合结果和技术在拓扑或测量理论上表现良好。本文研究了由可数群在波兰空间上的动作引起的一类着色问题，要求所需的着色是 Baire 可测量的。我们证明了所有此类着色问题的集合，这些问题允许特定自由动作的 Baire 可测解$\unicode[STIX]{x1D6FC}$是完全解析的（除了当轨道等价关系由$\unicode[STIX]{x1D6FC}$在comager set上是平滑的）；这一结果证实了寻找拓扑良好着色的“难度”。什么时候$\unicode[STIX]{x1D6FC}$是移位动作，我们描述了问题的类别$\unicode[STIX]{x1D6FC}$具有纯组合术语的 Baire 可测量着色；事实证明，图论中已经研究了密切相关的概念，与描述性集合论无关。我们注意到我们的框架允许完全动态解释（颜色对应于给定子移位的等变映射），因此本文也可以被视为对通用动力学的贡献。