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：我们引入了一类新的有界度无环Borel图实例，并利用Marks确定性方法的推广，在描述组合学背景下研究了它们的组合性质。该构造的动机源于将此方法应用于分布式计算的LOCAL模型。我们的方法统一了该领域以往的结果，并产生了新的结论。特别地，我们证明了对于$\Delta>2$，无法给出Borel色数至多为$\Delta$的无环$\Delta$正则Borel图的简单刻画：此类图构成一个$\mathbf{\Sigma}^1_2$完全集。这一结果意味着在Borel背景下，布鲁克斯型定理的强失效性。