AI 中文总结
本文研究有限个交换Borel函数生成的Borel图，通过标记集等分解其自由部分，推导相关性质并改进边色数上界，还为相关定理提供新证明并得出完美匹配存在条件。
AI 中文摘要
本文研究由有限个交换Borel函数生成的Borel图，基于标记集和标记区域对这类图的自由部分进行几何分析。假设存在具有有界 syndeticity 的 r-向前独立击中集，我们将自由部分分解为具有可控几何性质的无根区域和有根区域。作为应用，我们推导了有限Borel渐近维数和超有限性，并获得了Borel边色数的上界，该上界改进了之前的已知结果。对于每个交换Borel函数均为有界到一的情况，我们验证了存在具有 syndeticity Cr（C为某常数）的r-向前独立击中集，这为Naryshkin-Shinko-Weilacher-Yu的最新定理提供了另一种证明，还用于证明：若其中一个交换Borel函数是单射，另一个是有界到一且恰好偶到一，则该图存在Borel完美匹配。
英文摘要
In this paper we study Borel graphs generated by finitely many commuting Borel functions. We give a geometric analysis of the free part of such graphs based on marker sets and marker regions. Assuming the existence of $r$-forward-independent hitting sets with bounded syndeticity, we obtain marker decompositions of the free part into rootless and rooted regions with controlled geometry. As applications, we derive finite Borel asymptotic dimension and hyperfiniteness, and obtain upper bounds for Borel edge chromatic numbers which improve previously known results. For the case in which each of the commuting Borel functions is bounded-to-one, we verify the existence of $r$-forward-independent hitting sets with syndeticity $Cr$ for some constant $C$. This gives another proof of a recent theorem of Shinko-Weilacher-Yu, and is used to show that if one of the commuting Borel functions is injective and another one is bounded-to-one and exactly even-to-one, then the graph has a Borel perfect matching.
