We introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for the construction comes from the adaptation of this method to the LOCAL model of distributed computing. Our approach unifies the previous results in the area, as well as produces new ones. In particular, we show that for $\Delta>2$ it is impossible to give a simple characterization of acyclic $\Delta$-regular Borel graphs with Borel chromatic number at most $\Delta$: such graphs form a $\mathbf{\Sigma}^1_2$-complete set. This implies a strong failure of Brooks'-like theorems in the Borel context.

翻译：我们引入了一种新型的封闭度环球波纹图例,并在描述性组合法中研究其组合特性,采用典型的标记确定法。构建的动机来自将这种方法改用LOCAL分布式计算模型。我们的方法统一了该地区先前的结果,并产生了新的结果。特别是,我们显示,对于$\Delta>2$,不可能简单地描述以最多$\Delta$为单位的波罗门形数字(波罗门形数字最多为$\Delta$:这种图示形成一个$\mathbf ~Sigma_1_2$-complete 集。这意味着布鲁克斯在博尔背景下类似理论的强烈失败。