Quasi-treeings are treeable: a streamlined proof

TL;DR

This paper provides a simplified proof demonstrating that equivalence relations generated by certain locally-finite Borel graphs with quasi-tree components are treeable, using a canonical construction involving ultrafilter-like objects and Borel trees.

Contribution

It offers a streamlined, more accessible proof of the treeability of specific Borel graph equivalence relations, extending previous results with a canonical construction method.

Findings

Proves treeability of equivalence relations from quasi-tree components.

Introduces a canonical method replacing graph components with ultrafilter-like trees.

Shows that dense cuts lead to a Borel treeing union.

Abstract

We present a streamlined exposition of a construction by R. Chen, A. Poulin, R. Tao, and A. Tserunyan, which proves the treeability of equivalence relations generated by any locally-finite Borel graph such that each component is a quasi-tree. More generally, we show that if each component of a locally-finite Borel graph admits a finitely-separating Borel family of cuts, then we may 'canonically' replace each component of the graph by a tree of special ultrafilter-like objects on cuts called orientations; moreover, if the cuts are dense towards ends, then the union of these trees is a Borel treeing.