[Paper Review] The measurable Hall theorem fails for treeings

Gábor Kun|arXiv (Cornell University)|Jun 3, 2021

Limits and Structures in Graph Theory19 references4 citations

TL;DR

This paper constructs, for every degree $d \geq 2$, a $d$-regular, measurably bipartite treeing (essentially acyclic graphing) that admits no measurable perfect matching, demonstrating that the measurable Hall theorem fails in the treeing setting. The construction uses an inverse limit of finite $d$-regular bipartite graphs with controlled edge orientations and measure-theoretic properties, proving that all $L^1$ circulations are almost everywhere zero, which implies the absence of measurable matchings.

ABSTRACT

We construct, for every $d \geq 3$, a $d$-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.

Motivation & Objective

• To resolve an open problem posed by Kechris and Marks on whether $d$-regular Borel treeings admit measurable perfect matchings.
• To extend known counterexamples in hyperfinite and Borel settings to the broader class of treeings, particularly for odd and even degrees.
• To demonstrate that the existence of a perfect matching does not imply the existence of a measurable one in the context of treeings, even when the graphing is regular and measurably bipartite.
• To show that all $L^1$ circulations on the constructed graphing are almost everywhere zero, which obstructs the existence of measurable matchings.
• To establish that such graphings cannot arise as Schreier graphings of free $\mathbb{Z}$-actions, linking the result to measured group theory.

Proposed method

• Constructing a sequence of finite $d$-regular bipartite graphs $\{G_n\}$ recursively using a measure-preserving projection $f_n: V(G_{n+1}) \to V(G_n)$.
• Defining an inverse limit graphing $T_0$ whose vertex set consists of sequences $\{x_n\}$ with $f_n(x_{n+1}) = x_n$, equipped with a product probability measure.
• Restricting the measure to the full measure subgraphing $T$ by removing vertices with infinitely many coordinates in the 'bad' set $V_0(G_n)$, ensuring acyclicity and regularity.
• Using a measurable orientation $\mathcal{O}$ on $T$ and a lifting of orientations from $G_n$ to $T$ to control discrepancies in edge directions.
• Applying Lemma 4 to show that any absolutely integrable circulation must vanish a.e., by approximating the orientation with finite-level orientations and bounding the measure of disagreement.
• Proving that the resulting graphing $T$ is a treeing (essentially acyclic) and $d$-regular, with no nontrivial $L^1$ circulations, implying no measurable perfect matching exists.

Experimental results

Research questions

• RQ1 Does every $d$-regular measurably bipartite treeing admit a measurable perfect matching for $d \geq 2$?
• RQ2 Can the measurable Hall theorem be extended to the class of treeings, given its failure in the hyperfinite and Borel settings?
• RQ3 What is the role of circulations in obstructing measurable matchings in regular graphings?
• RQ4 Can such counterexamples be constructed for all $d \geq 2$, including odd degrees?
• RQ5 Is it possible for a $d$-regular treeing to arise as the Schreier graphing of a free $\mathbb{Z}$-action?

Key findings

• For every $d \geq 2$, there exists a $d$-regular, measurably bipartite treeing that admits no measurable perfect matching.
• All $L^1$ circulations on the constructed graphing are zero almost everywhere, which implies the nonexistence of measurable perfect matchings.
• The graphing $T$ is a treeing, as the set of vertices with cycles in their components has measure zero, and the graph is $d$-regular almost everywhere.
• The construction ensures that the measure of disagreement between a global orientation and its finite-level approximation is arbitrarily small, enabling the use of approximation arguments.
• The graphing cannot be realized as a Schreier graphing of a free $\mathbb{Z}$-action, as such actions would induce nontrivial bounded circulations, which do not exist on $T$.
• The result resolves a long-standing open problem in descriptive set theory and measurable combinatorics, showing that the measurable Hall theorem does not extend to treeings.