Asymptotic separation index is a parameter that measures how easily a Borel graph can be approximated by its subgraphs with finite components. In contrast to the more classical notion of hyperfiniteness, asymptotic separation index is well-suited for combinatorial applications in the Borel setting. The main result of this paper is a Borel version of the Lov\'asz Local Lemma -- a powerful general-purpose tool in probabilistic combinatorics -- under a finite asymptotic separation index assumption. As a consequence, we show that locally checkable labeling problems that are solvable by efficient randomized distributed algorithms admit Borel solutions on bounded degree Borel graphs with finite asymptotic separation index. From this we derive a number of corollaries, for example a Borel version of Brooks's theorem for graphs with finite asymptotic separation index.

翻译：渐近分离指数是衡量Borel图如何被其有限分支子图逼近的参数。与更经典的超有限性概念相比，渐近分离指数特别适用于Borel环境中的组合应用。本文的主要结果是在有限渐近分离指数假设下，证明了Lovász局部引理——概率组合学中一个强大的通用工具——的Borel版本。作为推论，我们证明：在具有有限渐近分离指数的有界度Borel图上，可通过高效随机分布式算法求解的局部可检验标记问题存在Borel解。由此我们推导出若干推论，例如对于具有有限渐近分离指数的图而言，Brooks定理的Borel版本成立。