The measurable Hall theorem fails for treeings
Abstract
We construct, for every , a -regular acyclic measurably bipartite graphing that admits no measurable perfect matching, resolving a problem of Kechris and Marks. A dense variant of our construction yields a coupling of two standard Borel probability measure spaces whose support contains no deterministic coupling, though the conditional probabilities of the coupling measure are atomless. This refutes a conjecture of Gurel-Gurevich and Peled.
Keywords
Cite
@article{arxiv.2106.02013,
title = {The measurable Hall theorem fails for treeings},
author = {Gábor Kun},
journal= {arXiv preprint arXiv:2106.02013},
year = {2024}
}
Comments
We refute in this version the conjecture of Gurel-Gurevich and Peled on deterministic couplings besides the Kechris-Marks problem on measurable matchings. We also solve further open questions including separation of local (so-called Locally Checkable Labeling) problems. The method in the previous versions has been extended using Lov\'asz' terminology on flows in measurable graphs