We construct Borel graphs which settle several questions in descriptive graph combinatorics. These include “Can the Baire measurable chromatic number of a locally finite Borel graph exceed the usual chromatic number by more than one?” and “Can marked groups with isomorphic Cayley graphs have Borel chromatic numbers for their shift graphs which differ by more than one?” We also provide a new bound for Borel chromatic numbers of graphs whose connected components all have two ends.

中文翻译：

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局部有限和处处两端图的描述色数

我们构建了 Borel 图，它解决了描述性图组合学中的几个问题。其中包括“局部有限 Borel 图的 Baire 可测量色数能否超过通常的色数一个以上？” 和“具有同构凯莱图的标记组是否可以为其移位图的 Borel 色数相差不止一个？” 我们还为连通分量都具有两端的图的 Borel 色数提供了一个新的界限。