连续组合数学中的概率构造和通向分布式算法的桥梁
Probabilistic constructions in continuous combinatorics and a bridge to distributed algorithms
摘要：
概率方法是一种证明组合存在结果的技术，通过显示随机选择的对象具有正概率的期望属性。一个特别强大的概率工具是Lovász局部引理（简称LLL），它是由埃尔德什和Lovász在20世纪70年代中期提出的。在这里，我们开发了一个版本的LLL，可以用来证明连续着色的存在。然后，我们给出了几个应用程序在Borel和拓扑动力学。* Seward和Tucker-Drob证明了可数群$Γ$的每一个自由Borel作用$Γ\curvearrowright X$允许一个等变Borel映射$π\colon X \to Y$到一个自由子移位$Y \subset 2^Γ$。我们给出了这个结果的一个新的简单证明。* 证明了对于可数群$Γ$，$\mathrm{Free}(2^Γ)$在Elek意义下弱包含于$Γ$在零维波兰空间上的每个自由连续作用中.这一事实类似于阿贝尔特和韦斯定理的概率测度保持行动，并有一些后果，在连续组合。特别地，我们推导出一个着色问题承认一个连续的解决方案在$\mathrm{Free}(2^Γ)$当且仅当它可以解决的有限子图的凯莱图的$Γ$的有效的确定性分布式算法（这一事实也证明了独立和使用不同的方法由Seward）。这在连续组合学和分布式计算中独立研究的问题之间建立了正式的对应关系。
英文摘要：
The probabilistic method is a technique for proving combinatorial existence results by means of showing that a randomly chosen object has the desired properties with positive probability. A particularly powerful probabilistic tool is the Lovász Local Lemma (the LLL for short), which was introduced by Erdős and Lovász in the mid-1970s. Here we develop a version of the LLL that can be used to prove the existence of continuous colorings. We then give several applications in Borel and topological dynamics. * Seward and Tucker-Drob showed that every free Borel action $Γ\curvearrowright X$ of a countable group $Γ$ admits an equivariant Borel map $π\colon X \to Y$ to a free subshift $Y \subset 2^Γ$. We give a new simple proof of this result. * We show that for a countable group $Γ$, $\mathrm{Free}(2^Γ)$ is weakly contained, in the sense of Elek, in every free continuous action of $Γ$ on a zero-dimensional Polish space. This fact is analogous to the theorem of Abért and Weiss for probability measure-preserving actions and has a number of consequences in continuous combinatorics. In particular, we deduce that a coloring problem admits a continuous solution on $\mathrm{Free}(2^Γ)$ if and only if it can be solved on finite subgraphs of the Cayley graph of $Γ$ by an efficient deterministic distributed algorithm (this fact was also proved independently and using different methods by Seward). This establishes a formal correspondence between questions that have been studied independently in continuous combinatorics and in distributed computing.
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