Descriptive Chromatic Numbers of Locally Finite and Everywhere Two Ended Graphs

Borel Vizing's Theorem for 2-Ended Groups

We show that Vizing's Theorem holds in the Borel context for graphs induced by actions of 2-ended groups, and ask whether it holds more generally for everywhere two ended Borel graphs.

A descriptive construction of trees and Stallings' theorem

We give a descriptive construction of trees for multi-ended graphs, which yields yet another proof of Stallings' theorem on ends of groups. Even though our proof is, in principle, not very different from already existing proofs and it draws ideas from a paper of Krön, it is written in a way that easily adapts to the setting of countable Borel equivalence relations, leading to a free decomposition result and a sufficient condition for treeability.

One-ended spanning trees and definable combinatorics

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 mea…

局部有限 Borel 图的不友好划分

Unfriendly partitions of locally finite Borel graphs

本文否证了局部有限 Borel 图存在 Borel 不友好划分的猜想，构造无界度反例，并证明最大度至多四时存在不友好着色。