局部有限 Borel 图的不友好划分
Unfriendly partitions of locally finite Borel graphs
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中文总结 AI 辅助
本文否证了局部有限 Borel 图存在 Borel 不友好划分的猜想，构造无界度反例，并证明最大度至多四时存在不友好着色。
中文摘要 AI 辅助
我们否证了由 Conley 记录、Conley--Marks--Unger 和 Conley--Tamuz 提及的 Thomas 问题，即是否每个局部有限 Borel 图都承认一个 Borel 不友好划分。我们的反例具有无界度，并且在零维 Polish 空间上是闭的；其连通关系是超有限的，且其连通分量是二部图且一端无限的。每个不友好着色都是正常的。结合一个奇偶性障碍，这种刚性排除了在余稠密集上不友好的 Baire 可测着色，以及对于有限平均度的拟不变概率测度几乎处处不友好的可测着色。在正面方向上，一个最大度至多为四的 Borel 图，只要每个连通分量包含一个环或一个度至多为二的顶点，就承认一个 Borel 不友好着色。这将最大度为三的 Borel 问题归结为三次森林，并且结合 Conley--Marks--Unger 的一个定理，为所有最大度至多为四的 Borel 图给出了 Baire 可测的不友好着色。
英文摘要
We answer in the negative the question of Thomas, recorded by Conley, Conley--Marks--Unger, and Conley--Tamuz, of whether every locally finite Borel graph admits a Borel unfriendly partition. Our counterexample has unbounded degree and is closed on a zero-dimensional Polish space; its connectedness relation is hyperfinite, and its components are bipartite and one-ended. Every unfriendly colouring is proper. Together with a parity obstruction, this rigidity rules out Baire measurable colourings that are unfriendly on a comeager set, and measurable colourings that are unfriendly almost everywhere for a quasi-invariant probability of finite average degree. In the positive direction, a Borel graph of maximum degree at most four admits a Borel unfriendly colouring whenever each component contains a cycle or a vertex of degree at most two. This reduces the Borel problem in maximum degree three to cubic forests and, with a theorem of Conley--Marks--Unger, gives Baire measurable unfriendly colourings for all Borel graphs of maximum degree at most four.
