论文标题
本地有限图的Borel组合学
Borel Combinatorics of Locally Finite Graphs
论文作者
论文摘要
我们为旨在非专家的borel组合学提供了一个轻柔的介绍，该组合学研究了拓扑空间的可定义图。这是组合学和描述性设置理论之间的边界上的一个新兴领域，与许多其他领域有着深厚的联系。 After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Borel versions of greedy algorithms and augmenting procedures, local rules, Borel transversals, etc. Also, we present the construction of Andrew Marks of acyclic Borel graphs for which the greedy bound $Δ+1$ on the Borel chromatic number is best possible. 在本文的其余部分中，我们简要讨论了各种主题，例如与本地算法的关系，霍尔的婚姻定理的可衡量版本和洛瓦斯斯本地引理，对等分可调节性的应用，等等。
We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to many other areas. After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Borel versions of greedy algorithms and augmenting procedures, local rules, Borel transversals, etc. Also, we present the construction of Andrew Marks of acyclic Borel graphs for which the greedy bound $Δ+1$ on the Borel chromatic number is best possible. In the remainder of the paper we briefly discuss various topics such as relations to LOCAL algorithms, measurable versions of Hall's marriage theorem and of Lovász Local Lemma, applications to equidecomposability, etc.
