论文标题
单跨树木和可定义的组合学
One-ended spanning trees and definable combinatorics
论文作者
论文摘要
令$（x，τ）$为波兰空间，带有borel概率度量$μ，$ g $ $ x上的本地有限的单端borel图。如果$ g $是由一个自由的Borel动作引起的（分别是多项式增长）组，那么我们显示相同的结果$μ$ -A.E。 （分别，到处都是）。我们的结果概括了Timár以及Conley，Gaboriau，Marks和Tucker-Drob的最新工作，他们在保存环境的概率度量中证明了这一点。我们应用定理以偶数图以均匀图表找到Borel方向，并在常规的两部分图中可测量的可测量和可测量的完美匹配，从而完善了以前仅闻名的定理，这些定理仅用于测量图形。特别是，我们证明了两分的$ d $常规鲍尔图允许Baire可衡量的完美匹配。
Let $(X,τ)$ be a Polish space with Borel probability measure $μ,$ and $G$ a locally finite one-ended Borel graph on $X.$ We show that $G$ admits a Borel one-ended spanning tree generically. If $G$ is induced by a free Borel action of an amenable (resp., polynomial growth) group then we show the same result $μ$-a.e. (resp., everywhere). Our results generalize recent work of Timár, as well as of Conley, Gaboriau, Marks, and Tucker-Drob, who proved this in the probability measure preserving setting. We apply our theorem to find Borel orientations in even degree graphs and measurable and Baire measurable perfect matchings in regular bipartite graphs, refining theorems that were previously only known to hold for measure preserving graphs. In particular, we prove that bipartite one-ended $d$-regular Borel graphs admit Baire measurable perfect matchings.
