Borel Matchings and Analogs of Hall's Theorem

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Borel Matchings and Analogs of Hall’s Theorem

Thesis by

Allison Yiyun Wang

In Partial Fulfillment of the Requirements for the Degree of

Bachelor of Science in Mathematics

CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California

2020

Defended May 21, 2020

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ACKNOWLEDGEMENTS

I would like to thank my mentor, Dr. Alexander Kechris, for his guidance and support throughout this project. I would also like to thank Dr. Matthias Flach, Dr.

Aristotelis Panagiotopoulos, and Dr. Omer Tamuz for their time and patience as members of my committee.

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ABSTRACT

In classical graph theory, Hall’s theorem gives a necessary and sufficient condition for a bipartite graph to have a perfect matching. The analogous statement for Borel perfect matchings is false. If we instead consider Borel perfect matchings almost everywhere or Borel perfect matchings generically, results similar to Hall’s theorem hold. We present Marks’ proof that König’s theorem, a special case of Hall’s theorem, fails in the context of Borel perfect matchings. We then discuss positive results about the existence of Borel matchings that are close to perfect in the measure theory and Baire category settings.

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INTRODUCTION

Given a standard Borel space 𝑋, a(Borel) graph on 𝑋 is some 𝑮 = (𝑋 , 𝐺) such that𝐺 ⊆ 𝑋²is a symmetric, irreflexive, (Borel) set. The elements of 𝑋 are called the vertices of 𝑮, and we think of the relation 𝐺 as defining the edges of 𝑮. If 𝑥 , 𝑦 ∈ 𝑋 satisfy(𝑥 , 𝑦) ∈𝐺, we say that𝑥and𝑦 are adjacent and that these vertices are incident to the edge{𝑥 , 𝑦}. For𝑥 ∈ 𝑋, we denote the set of vertices adjacent to 𝑥by 𝑁_𝐺(𝑥) :={𝑦 ∈ 𝑋 | (𝑥 , 𝑦) ∈𝐺}. For 𝐴 ⊆ 𝑋, we define𝑁_𝐺(𝐴) :=∪_𝑥∈𝐴𝑁_𝐺(𝑥). A set 𝐴⊆ 𝑋 is anindependent setif no two vertices in 𝐴are adjacent. Thedegree of a vertex𝑥 ∈ 𝑋is the cardinality of𝑁_𝐺(𝑥). A graph𝑮is𝑑-regularif every vertex has degree 𝑑. We say that 𝑮 hasbounded degreeif there is some𝑛 ∈Nsuch that every vertex has degree at most𝑛. We call𝑮 locally finiteif every vertex has finite degree, and we call𝑮locally countableif the degree of every vertex is countable.

A (Borel) matching of 𝑮 is a symmetric, irreflexive, (Borel) subset 𝑀 ⊆ 𝐺 such that if (𝑥 , 𝑦) ∈ 𝑀 and (𝑥 , 𝑧) ∈ 𝑀 for some𝑥 , 𝑦, 𝑧 ∈ 𝑋, then 𝑦 = 𝑧. Then 𝑀 is a subset of the edges of 𝑮 such that no vertex is incident to more than one edge in 𝑀. Let 𝑋_𝑀 denote the set of vertices that are incident to edges in 𝑀. A (Borel) matching𝑀 is a(Borel) perfect matchingif𝑋_𝑀 = 𝑋. A matching𝑀 is𝐺-invariant if𝑥 ∈ 𝑋_𝑀 implies𝑁_𝐺(𝑥) ⊆ 𝑋_𝑀.

Thechromatic numberof a graph𝑮 = (𝑋 , 𝐺), written𝜒(𝑮), is the least cardinality of a set𝑌for which there is a map𝑐: 𝑋 →𝑌 such that𝑐(𝑥) ≠𝑐(𝑦)if(𝑥 , 𝑦) ∈𝐺. A graph𝑮is calledbipartiteif𝜒(𝑮) =2. TheBorel chromatic numberof𝑮 = (𝑋 , 𝐺), denoted 𝜒_𝐵(𝑮), is the least cardinality of a standard Borel space𝑌 for which there is a Borel map𝑐: 𝑋 →𝑌 such that𝑐(𝑥) ≠𝑐(𝑦)if(𝑥 , 𝑦) ∈𝐺.

In classical graph theory, Hall’s theorem gives the following necessary and sufficient condition for a bipartite graph to have a perfect matching.

Theorem 1.0.1(Hall’s theorem). Let𝑮 be a finite bipartite graph with bipartition {𝐴, 𝐵}. Then 𝑮 has a perfect matching if and only if |𝑁_𝐺(𝐹) | ≥ |𝐹| for every 𝐹 ⊆ 𝐴and every𝐹 ⊆ 𝐵.

In the case where𝑮is𝑑-regular, Hall’s theorem specializes to König’s theorem.

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Theorem 1.0.2(König’s theorem). For all 𝑑 ≥ 2, every finite bipartite 𝑑-regular graph has a perfect matching.

We are interested in similar results about Borel matchings. However, the direct analog of König’s theorem in a Borel setting does not hold.

Theorem 1.0.3. [Mar16, Theorem 1.5] For all 𝑑 ≥ 2, there is a𝑑-regular acyclic Borel graph 𝑮 on a standard Borel space 𝑋 such that 𝜒_𝐵(𝑮) = 2 and 𝑮 has no Borel perfect matching.

We present Marks’s proof of Theorem 1.0.3 in Section 2.

Instead of Borel perfect matchings, we may consider Borel matchings that are almost perfect. In Section 3, we discuss such matchings in the context of measure theory.

Suppose 𝑋 is a standard Borel space with a Borel probability measure 𝜇, and suppose𝑮 = (𝑋 , 𝐺) is a locally countable Borel graph. We call a Borel matching 𝑀 of 𝑮 aBorel perfect matching 𝜇-almost everywhere(a.e.) if 𝑋_𝑀 is𝐺-invariant and 𝜇(𝑋_𝑀) =1. We say that 𝑮 is 𝜇-measure preservingif for every partial Borel bijection 𝑓 :𝑌 → 𝑍 between Borel subsets𝑌 , 𝑍 ⊆ 𝑋 satisfying Graph(𝑓) ⊆ 𝐺, we have𝜇(𝐴) =𝜇(𝑓(𝐴))for all Borel𝐴 ⊆𝑌. In this setting, the following theorem by Lyons and Nazarov provides a sufficient condition similar to Hall’s theorem.

Theorem 1.0.4. [LN11, Remark 2.6] Let 𝑮 = (𝑋 , 𝐺) be a Borel graph on (𝑋 , 𝜇) that is locally finite, 𝜇-measure preserving, and bipartite. Suppose there is some constant𝑐 >1such that for all Borel independent sets𝐴⊆ 𝑋,𝜇(𝑁_𝐺(𝐴)) ≥𝑐 𝜇(𝐴). Then𝑮has a Borel perfect matching 𝜇-a.e.

Finally, in Section 4, we consider Borel matchings that are almost perfect in a Baire category sense. Suppose 𝑋 is a Polish space and𝑮 = (𝑋 , 𝐺) is a Borel graph. We say that a Borel matching𝑀 of𝐺 is aBorel perfect matching genericallyif 𝑋_𝑀 is 𝐺-invariant and comeager. Marks and Unger proved the following statement, which gives a sufficient condition analogous to that in Hall’s theorem.

Theorem 1.0.5. [MU16, Theorem 1.3] Let 𝑋 be a Polish space, and let 𝑮 be a locally finite bipartite Borel graph with a bipartition {𝐵₀, 𝐵₁}. Suppose there is some𝜀 >0such that for every finite set𝐹 ⊆ 𝐵₀or𝐹 ⊆ 𝐵₁, |𝑁_𝐺(𝐹) | ≥ (1+𝜀) |𝐹|. Then𝑮admits a Borel perfect matching generically.

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C h a p t e r 2

BOREL PERFECT MATCHINGS

One important result about matchings in classical graph theory is König’s theorem.

Theorem 2.0.1(König’s theorem). For all 𝑑 ≥ 2, every finite bipartite 𝑑-regular graph has a perfect matching.

2.1 Laczkovich’s Example

In [Lac88], Laczkovich constructs the following example, which demonstrates that König’s theorem does not hold in a Borel setting for 𝑑 = 2. Fix some irrational 𝛼 ∈ (0,1). Let 𝑅denote the rectangle with vertices (0, 𝛼), (𝛼,0), (1,1−𝛼), and (1−𝛼,1), and let𝑅⁰:=𝑅∪ {(0,0),(1,1)}, as shown in the diagram below.

(0, 0)

(1, 1)

(𝛼, 0)

(1, 1−𝛼) (1−𝛼, 1)

(0,𝛼)

𝑥 𝑦₁

𝑦₂

Let 𝑋 and𝑌 be copies of [0,1], and let 𝑮 be the bipartite graph on 𝑋 t𝑌 where 𝑥 ∈ 𝑋 and 𝑦 ∈𝑌 are adjacent in 𝑮 if and only if (𝑥 , 𝑦) ∈ 𝑅⁰. For example, in the diagram above, the neighborhood of 𝑥 is {𝑦₁, 𝑦₂}. Observe that 𝑮 is a 2-regular Borel graph with 𝜒_𝐵(𝑮) = 2. Laczkovich proved that 𝑮 is a counterexample to König’s theorem in the Borel setting.

Theorem 2.1.1. [Lac88] The graph𝑮 has no Borel perfect matching.

Proof of Theorem 2.1.1. We follow Laczkovich’s proof in [Lac88].

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Let𝜆₁denote the measure on𝑅⁰such that𝜆₁(𝑝) =0 for all𝑝 ∈ 𝑅⁰and such that for any disjoint line segments 𝐴₁, 𝐴₂, . . . , 𝐴_𝑛 ⊆ 𝑅⁰,

𝜆₁(𝐴₁t𝐴₂t. . .t𝐴_𝑛) =

𝑛

Õ

𝑖=1

𝜆(𝐴_𝑖),

where𝜆gives the Euclidean length of each line segment. Define the measure𝜇on 𝑅⁰by

𝜇(𝐻) := 1 2

√ 2

𝜆₁(𝐻) for all measurable subsets𝐻 ⊆ 𝑅⁰. So 𝜇(𝑅⁰) =1.

Note that a matching in 𝑮 is equivalent to a subset 𝑀 ⊆ 𝑅⁰ that is the graph of an injective involution from 𝑋 to 𝑌. We will prove that no such 𝑀 ⊆ 𝑅⁰ is 𝜇-measurable, which will imply that𝑮does not have a Borel matching.

We will now define two functions 𝑓 : 𝑅⁰→ 𝑅⁰and𝑔 : 𝑅⁰→ 𝑅⁰. Let 𝑓(𝑥 , 𝑦) = (𝑥 , 𝑦) if𝑥 = 0 or𝑥 =1. If𝑥 ∈ (0,1), let 𝑓(𝑥 , 𝑦) := (𝑥 , 𝑧), where𝑧 is the unique point in [0,1] \ {𝑦}such that(𝑥 , 𝑧) ∈ 𝑅⁰. Similarly, let𝑔(𝑥 , 𝑦) =(𝑥 , 𝑦)if𝑦 =0 or 𝑦 =1. If 𝑦 ∈ (0,1), let𝑔(𝑥 , 𝑦) := (𝑤 , 𝑦), where𝑤is the unique point in[0,1] \ {𝑥}such that (𝑤 , 𝑦) ∈ 𝑅⁰. Note that 𝑓 = 𝑓⁻¹and𝑔=𝑔⁻¹and that 𝑓 and𝑔are measure-preserving homeomorphisms from 𝑅⁰to itself.

We claim that 𝑔◦ 𝑓 is ergodic on 𝑅. Let𝑇 = R/Z be the circle group with the Lebesgue measure. Let ℎ : 𝑅 → 𝑇 be a measure-preserving homeomorphism satisfying

ℎ(1,1−𝛼) =0;

ℎ(1−𝛼,1) = 𝛼 2; ℎ(0, 𝛼) = 1

2; ℎ(𝛼,0) = 1+𝛼

2 .

Define𝑘 := ℎ◦𝑔◦ 𝑓 ◦ℎ⁻¹. Note that 𝑘 is a measure-preserving homeomorphism from𝑇 to itself, so there is some constant𝑐 ∈𝑇 such that𝑘(𝑡) =𝑡 +𝑐for all𝑡 ∈𝑇 or𝑘(𝑡) =−𝑡+𝑐for all𝑡 ∈𝑇. Since𝑘(0)=𝛼and𝑘(¹

2) = ¹₂+𝛼, 𝑘(𝑡) =𝑡+𝛼for all 𝑡 ∈𝑇. We chose𝛼to be irrational, so 𝑘is ergodic on𝑅. Therefore,𝑔◦ 𝑓 is ergodic on𝑅.

Suppose there is a 𝜇-measurable 𝑀 ⊆ 𝑅⁰such that 𝑀 is the graph of an injective involution from 𝑋 to𝑌. Observe that 𝑀 ∩ 𝑓(𝑀), 𝑅⁰\ (𝑀 ∪ 𝑓(𝑀)), 𝑀 ∩𝑔(𝑀),

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and 𝑅⁰\ (𝑀 ∪𝑔(𝑀)) are finite sets, so𝜇(𝑀) = ¹₂. Using our earlier observation that𝑔 =𝑔⁻¹, we note that𝑀4(𝑔◦ 𝑓) (𝑀)is finite. Then𝜇(𝑀) ∈ {0,1}since𝑔◦ 𝑓 is ergodic. This is a contradiction, so there is no such 𝜇-measurable 𝑀 ⊆ 𝑅⁰, and therefore,𝑮 does not have a Borel perfect matching.

2.2 König’s Theorem in Higher Degree

Laczkovich’s example demonstrates that König’s theorem fails in the Borel setting for degree 𝑑 = 2. A theorem by Marks states that König’s theorem fails for Borel matchings for all𝑑 ≥ 2:

Theorem 2.2.1. [Mar16, Theorem 1.5] For all 𝑑 ≥ 2, there is a𝑑-regular acyclic Borel graph 𝑮 on a standard Borel space 𝑋 such that 𝜒_𝐵(𝑮) = 2 and 𝑮 has no Borel perfect matching.

We follow the proof of Theorem 2.2.1 given in [Mar16]. We first introduce some definitions. Given a standard Borel space 𝑋 and a countable group Γ, 𝑋^Γ is a standard Borel space under the product structure. We define theleft shift actionof Γon𝑋^Γby

𝛼·𝑥(𝛽) =𝑥(𝛼⁻¹𝛽)

for all 𝑥 ∈ 𝑋^Γ and 𝛼, 𝛽 ∈ Γ. We define Free(𝑋^Γ) to be the set of all𝑥 ∈ 𝑋^Γ for which𝛾 ·𝑥 ≠ 𝑥 for all𝛾 ∈ Γ\ {𝑒}. If 𝑋 and𝑌 are spaces equipped with actions ofΓ and 𝑓 : 𝑋 → 𝑌 is a function satisfying𝛾 · 𝑓(𝑥) = 𝑓(𝛾 ·𝑥) for all𝛾 ∈ Γand 𝑥 ∈ 𝑋, then we say 𝑓 isΓ-equivariant.

The proof of Theorem 2.2.1 relies on a result about equivalence relations. For an equivalence relation𝐸 on a standard Borel space 𝑋, a subset 𝐴 ⊆ 𝑋 is acomplete sectionif 𝐴intersects every𝐸-class. If𝐸 and𝐹are equivalence relations on 𝑋and there exist disjoint Borel sets 𝐴, 𝐵⊆ 𝑋 such that𝐴and𝐵are complete sections for 𝐸and𝐹, respectively, then𝐸 and𝐹haveBorel disjoint complete sections. We have the following theorem:

Theorem 2.2.2. [Mar16, Theorem 3.7] LetΓandΔ be countable groups. Define 𝐸_Γto be the equivalence relation onFree(N^Γ∗Δ)such that𝑥 𝐸_Γ𝑦if and only if there exists some𝛾 ∈Γfor which𝛾·𝑥 =𝑦. Define𝐸_Δ similarly. Then𝐸_Γand𝐸_Δ do not have Borel disjoint complete sections.

To prove Theorem 2.2.2, we use several lemmas. We need the following definitions to state these lemmas. Given an equivalence relation𝐸 on a standard Borel space

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𝑋, 𝐸 is a countable Borel equivalence relation if 𝐸 ⊆ 𝑋 × 𝑋 is a Borel set and each𝐸-class is countable. Suppose 𝐸 is a countable Borel equivalence relation on a standard Borel space 𝑋, we define the following. A set 𝐴 ⊆ 𝑋 is 𝐸-invariant if for every 𝑥 ∈ 𝐴, the orbit of 𝑥 is contained in 𝐴. We define [𝑋]^<^∞ to be the standard Borel space of finite subsets of𝑋, and we let[𝐸]^<^∞be the Borel subset of [𝑋]^<^∞ consisting of the finite subsets of𝑋 whose elements are𝐸-equivalent. The intersection graphon[𝐸]^<∞is the graph with vertex set[𝐸]^<∞where𝐴, 𝐵 ∈ [𝐸]^<∞

are adjacent exactly when𝐴∩𝐵≠0 and𝐴 ≠ 𝐵. Then we have the following lemma, which we state without proof:

Lemma 2.2.3. [KM04, Lemma 7.3] Suppose 𝐸 is a countable Borel equivalence relation on a standard Borel space 𝑋. Let 𝑮 be the intersection graph on [𝐸]^<^∞. Then𝑮has a BorelN-coloring.

We also need the following definitions. Let𝐼 ∈ {1,2, . . . ,∞}, and let {𝐸_𝑖}𝑖 < 𝐼 be a collection of equivalence relations on a standard Borel space𝑋. If, for some𝑛 ≥ 2, there is a sequence of distinct points𝑥₀, 𝑥₁, . . . , 𝑥_𝑛 ∈ 𝑋and a sequence𝑖₀, 𝑖₁, . . . , 𝑖_𝑛 ∈ Nsuch that𝑖_𝑗 ≠𝑖_𝑗+₁for 0 ≤ 𝑗 ≤ 𝑛−1,𝑖_𝑛 ≠𝑖₀, and𝑥₀𝐸_𝑖

0𝑥₁𝐸_𝑖

1𝑥₂. . . 𝑥_𝑛𝐸_𝑖

𝑛𝑥₀, then we say that the𝐸_𝑖 arenon-independent. Otherwise, we say that the𝐸_𝑖areindependent.

LetÔ

𝑖 < 𝐼𝐸_𝑖 be the smallest equivalence relation that contains every𝐸_𝑖. The𝐸_𝑖 are everywhere non-independentif for eachÔ

𝑖 < 𝐼𝐸_𝑖-class 𝐴⊆ 𝑋, the restrictions𝐸_𝑖𝐴 are not independent. We state the following lemma without proof:

Lemma 2.2.4. [Mar16, Lemma 2.3] Suppose 𝐼 ∈ {1,2, . . . ,∞}, and suppose {𝐸_𝑖}𝑖 < 𝐼 are countable Borel equivalence relations on a standard Borel space 𝑋 that are everywhere non-independent. Then there is a Borel partition{𝐴_𝑖}𝑖 < 𝐼 of 𝑋 such that for all𝑖 < 𝐼, 𝐴^𝑐

𝑖 intersects every𝐸_𝑖-class.

We use Lemma 2.2.3 and Lemma 2.2.4 to prove the following lemma:

Lemma 2.2.5. [Mar16, Lemma 2.1] Let Γ and Δ be countable groups. Suppose 𝐴 ⊆ Free(N^Γ∗Δ)is a Borel set. Then at least one of the following holds.

1. There is a continuous injective function 𝑓 : Free(N^Γ) → Free(N^Γ∗Δ) such that 𝑓 is equivariant with respect to the left shift action of Γandran(𝑓) ⊆ 𝐴. 2. There is a continuous injective function 𝑓 : Free(N^Δ) → Free(N^Γ∗Δ) such that 𝑓 is equivariant with respect to the left shift action of Δ andran(𝑓) ⊆ Free(N^Γ∗Δ) \𝐴.

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We refer to words of the form 𝛾₀𝛿₁𝛾₂. . ., where 𝛾_𝑖 ∈ Γ\ {𝑒} and 𝛿_𝑖 ∈ Δ \ {𝑒} for all 𝑖, as Γ-words. Similarly, we refer to words of the form 𝛿₀𝛾₁𝛿₂. . ., where 𝛾_𝑖 ∈Γ\ {𝑒}and𝛿_𝑖 ∈Δ\ {𝑒}, asΔ-words.

Proof of Lemma 2.2.5. First, we will define a game that produces an element 𝑦 ∈ N^Γ∗Δ, where player I defines 𝑦(𝛼) when 𝛼 is a Γ-word and player II defines 𝑦(𝛼)when𝛼is aΔ-word. Let𝛾₀, 𝛾₁, . . ., and𝛿₀, 𝛿₁, . . .be enumerations ofΓ\ {𝑒} andΔ \ {𝑒}, respectively. Let𝑌 be the subset ofN^Γ∗Δ consisting of the elements 𝑥 ∈N^Γ∗Δ such that for all𝑥⁰in the orbit of𝑥 and𝑖 ∈N,𝛾_𝑖·𝑥⁰≠ 𝑥⁰and𝛿_𝑖·𝑥⁰≠𝑥⁰. We now define a function𝑡 : Γ∗Δ\ {𝑒} →N∪ {−1}to determine which values of 𝑦are fixed on each turn of the game. Define𝑡(𝑒) :=−1. For every𝛼∈Γ∗Δ\ {𝑒}, we can write 𝛼 uniquely in the form 𝛾_𝑖

0𝛿_𝑖

1𝛾_𝑖

2. . . 𝜋_𝑖

𝑚 or 𝛿_𝑖

0𝛾_𝑖

1𝛿_𝑖

2. . . 𝜋_𝑖

𝑚, where 𝜋_𝑖

𝑚 ∈ {𝛾_𝑖

𝑚, 𝛿_𝑖

𝑚} based on the parity of 𝑚. Define 𝑡(𝛼) := max_0≤𝑗≤𝑚(𝑖_𝑗 + 𝑗).

Observe that if 𝛼 is a Δ-word or 𝛼 = 𝑒, then 𝑡(𝛾_𝑖𝛼) = max{𝑡(𝛼) +1, 𝑖}. In particular, if𝑖 ≤ 𝑛, then𝑡(𝛾_𝑖𝛼) ≤ 𝑛if and only if𝑡(𝛼) < 𝑛. Similarly, if𝑖 ≤ 𝑛and 𝛼is aΓ-word or𝛼=𝑒, then𝑡(𝛿_𝑖𝛼) ≤ 𝑛if and only if𝑡(𝛼) < 𝑛.

We will define the game 𝐺^𝐵

𝑘 for any 𝑘 ∈ N and any Borel set 𝐵 ⊆ 𝑌. The game 𝐺^𝐵

𝑘 will produce some 𝑦 ∈ N^Γ∗Δ satisfying𝑦(𝑒) = 𝑘. On each turn 𝑛 ∈N, player I defines𝑦(𝛼) for allΓ-words𝛼satisfying𝑡(𝛼) =𝑛, followed by player II defining 𝑦(𝛼) for all Δ-words 𝛼 satisfying𝑡(𝛼) = 𝑛. We now specify who wins each run of the game. If 𝑦 ∈ 𝐵, then player II wins, and if 𝑦 ∈ 𝑌 \ 𝐵, then player I wins.

Otherwise,𝑦∉𝑌, so one of the following must hold:

1. There is some𝛼 ∈ Γ∗Δand𝑖 ∈Nsuch that 𝛾_𝑖𝛼⁻¹· 𝑦 = 𝑦. In this case, we say that(𝛼,Γ)witnesses𝑦 ∉𝑌.

2. There is some𝛼 ∈ Γ∗Δ and𝑖 ∈ Nsuch that𝛿_𝑖𝛼⁻¹· 𝑦 = 𝑦. In this case, we say that(𝛼,Δ)witnesses𝑦∉𝑌.

We say that𝛼witnesses 𝑦 ∉𝑌 if either of the above statements holds. We specify that if(𝑒,Γ)witnesses𝑦 ∉𝑌, then player II wins, and if(𝑒,Δ)witnesses𝑦∉𝑌, then player I wins. If neither of these happens, then there must be a Γ-word orΔ-word witnessing𝑦 ∉𝑌. If there is someΔ-word𝛼witnessing𝑦 ∉𝑌 such that noΓ-words 𝛽with𝑡(𝛽) ≤ 𝑡(𝛼) witness𝑦∉𝑌, then player I wins. Otherwise, player II wins.

We will use games of the form𝐺^𝐵

𝑘 to construct our desired function 𝑓 for the given 𝐴 ⊆ Free(N^Γ∗Δ). Let 𝐸_Γ be the equivalence relation on𝑌 such that 𝑥 𝐸_Γ𝑦 if and

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only if𝛾·𝑥 = 𝑦for some𝛾 ∈ Γ, and let𝐸_Δ be defined similarly. Note that𝐸_Γand 𝐸_Δ are everywhere non-independent on𝑌\Free(N^Γ∗Δ). By applying Lemma 2.2.4 with𝐼 =2, there exists some Borel𝐶 ⊆𝑌\Free(N^Γ∗Δ)such that𝐶intersects every 𝐸_Δ-class on𝑌 \Free(N^Γ∗Δ) and𝐶^𝑐 intersects every𝐸_Γ-class on𝑌 \Free(N^Γ∗Δ).

Define 𝐵_𝐴 := 𝐴∪𝐶. Borel determinacy implies that for every 𝑘 ∈ N, player I or player II has a winning strategy for 𝐺

𝐵_𝐴

𝑘 . Then player I or player II must have a winning strategy for 𝐺^𝐵^𝐴

𝑘 for infinitely many 𝑘. Suppose player II has a winning strategy for𝐺^𝐵^𝐴

𝑘 for infinitely many𝑘 ∈N, and let𝑆be the set of all such𝑘. We omit the case where player I has a winning strategy for 𝐺^𝐵^𝐴

𝑘 for infinitely many 𝑘 ∈ N; a similar proof holds in that situation. For each 𝑘 ∈ 𝑆, fix a winning strategy for player II in𝐺^𝐵^𝐴

𝑘 .

There exists a continuous injective equivariant function𝑔 : Free(N^Γ) →Free(𝑆^Γ).

So if we can construct a continuous injection 𝑓 : ran(𝑔) →Free(N^Γ∗Δ) such that 𝑓 is equivariant with respect to the left shift action ofΓand ran(𝑓) ⊆ 𝐴, then 𝑓 ◦𝑔is a function satisfying the lemma.

We now define 𝑓 : ran(𝐺) → Free(N^Γ∗Δ). Consider any𝑥⁰ ∈ Free(N^Γ), and let 𝑥 := 𝑔(𝑥⁰). We will choose moves for player I in 𝐺^𝐵^𝐴

𝑥⁰(𝛾⁻1) such that the outcome of the game when player II plays according to their fixed winning strategy will be the value of𝛾 · 𝑓(𝑥). We will do so simultaneously for all 𝛾 ∈ Γsuch that 𝑓 is an equivariant function and 𝑓(𝑥) (𝛾) =𝑥⁰(𝛾)for all𝛾 ∈Γ.

We will define(𝛾· 𝑓(𝑥)) (𝛼) inductively on𝑡(𝛼). By our definition of𝐺

𝐵_𝐴

𝑥⁰(𝛾⁻¹), we have(𝛾· 𝑓(𝑥)) (𝑒) =𝑥⁰(𝛾⁻¹) for all𝛾 ∈ Γ. Suppose we have defined(𝛾· 𝑓(𝑥)) (𝛼) for all𝛾 ∈ Γand all𝛼 for which𝑡(𝛼) < 𝑛. We need to specify player I’s move on turn𝑛 in each game; equivalently, we need to define(𝛾 · 𝑓(𝑥)) (𝛽) for all Γ-words 𝛽 with𝑡(𝛽) = 𝑛. Suppose 𝛽 is aΓ-word such that𝑡(𝛽) = 𝑛and 𝛽 =𝛾_𝑖𝛼for some 𝑖 ∈Nand some𝛼with𝑡(𝛼) < 𝑛. By definition of the left shift action ofΓ, we need (𝛾· 𝑓(𝑥)) (𝛾_𝑖𝛼) = (𝛾⁻¹

𝑖 𝛾· 𝑓(𝑥)) (𝛼) for all𝛾 ∈Γ. By assumption, we have already defined the value of(𝛾⁻¹

𝑖 𝛾· 𝑓(𝑥)) (𝛼), so this determines what player I should play for (𝛾 · 𝑓(𝑥)) (𝛾_𝑖𝛼). So we can determine player I’s move on turn𝑛in each game, which determines (𝛾· 𝑓(𝑥)) (𝛽)for all𝛾 ∈ Γand allΓ-words 𝛽such that𝑡(𝛽) =𝑛. The values of(𝛾· 𝑓(𝑥)) (𝛽)forΔ-words𝛽is determined by player II’s move on turn 𝑛in each game, which is specified by the winning strategies we fixed for player II.

By induction, we can thus use the games𝐺^𝐵^𝐴

𝑥⁰(𝛾⁻¹) to define𝛾· 𝑓(𝑥) for all𝛾 ∈ Γ.

From our construction, 𝑓 is injective, continuous, and equivariant with respect to the left shift action ofΓ. We also know that each 𝑓(𝑥)is a winning outcome for player

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II in𝐺^𝐵^𝐴

𝑥⁰ . It suffices to show that ran(𝑓) ⊆ 𝐴. We will first show that ran(𝑓) ⊆𝑌. Consider any𝑥⁰ ∈Free(N^Γ), and let𝑥 =𝑔(𝑥⁰). Because 𝑓(𝑥)results from a winning strategy for player II, (𝑒,Δ) does not witness 𝑓(𝑥) ∉𝑌. Suppose (𝑒,Γ) witnesses 𝑓(𝑥) ∉𝑌. Then there exists𝑖 ∈Nsuch that for all𝛾 ∈Γ,(𝛾_𝑖· 𝑓(𝑥)) (𝛾) = 𝑓(𝑥) (𝛾). We can rewrite (𝛾_𝑖 · 𝑓(𝑥)) (𝛾) as 𝑓(𝑥) (𝛾⁻¹

𝑖 𝛾). By construction of 𝑓, we then have 𝑥⁰(𝛾⁻¹

𝑖 𝛾) = 𝑥⁰(𝛾) for all 𝛾 ∈ Γ, or equivalently, 𝛾_𝑖 ·𝑥⁰ = 𝑥⁰. But this contradicts 𝑥 ∈Free(N^Γ). We conclude that (𝑒,Γ) does not witnesses 𝑓(𝑥) ∉𝑌. Therefore,𝑒 does not witness 𝑓(𝑥) ∉𝑌.

We will show that for all 𝛼 ∈ Γ∗Δ and all 𝑥⁰ ∈ Free(N^Γ), 𝛼 does not witness 𝑓(𝑥) ∉𝑌, where𝑥 =𝑔(𝑥⁰). We will use induction on𝑡(𝛼). Suppose this statement holds for all𝛽 ∈Γ∗Δsuch that𝑡(𝛽) < 𝑛. First, we consider anyΓ-word𝛼satisfying 𝑡(𝛼) =𝑛. We can find𝛾 ∈ Γand 𝛽 ∈ Γ∗Δ such that𝛼 =𝛾 𝛽 and𝑡(𝛽) < 𝑛. Then 𝛼⁻¹· 𝑓(𝑥) = 𝛽⁻¹𝛾⁻¹ · 𝑓(𝑥), which we can rewrite as 𝛽⁻¹ · 𝑓(𝛾⁻¹· 𝑥) since 𝑓 is Γ-equivariant. By our induction hypothesis, 𝛽does not witness 𝑓(𝛾⁻¹·𝑥) ∉𝑌, so 𝛼 does not witness 𝑓(𝑥) ∉ 𝑌. Now consider any Δ-word 𝛼 satisfying𝑡(𝛼) = 𝑛. By our induction hypothesis and our argument for Γ-words, there is no Γ-word 𝛽 witnessing 𝑓(𝑥) ∉𝑌 satisfying𝑡(𝛽) ≤ 𝑛. Since 𝑓(𝑥)is consistent with player II’s winning strategy,𝛼cannot witness 𝑓(𝑥) ∉𝑌.

Therefore, we have ran(𝑓) ⊆ 𝑌. Because each 𝑓(𝑥) is the result of a winning strategy for player II in some game𝐺

𝐵_𝐴

𝑘 , we must have ran(𝑓) ⊆ 𝐵_𝐴 = 𝐴∪𝐶. Note that ran(𝑓) is Γ-invariant. By definition of𝐶, 𝐶 does not contain any non-empty Γ-invariant sets, so we must have 𝑓(𝑥) ⊆ 𝐴. We conclude that 𝑓 is a function of

form (1), as desired.

We now use these lemmas to prove Theorem 2.2.2.

Proof of Theorem 2.2.2. It suffices to show that it is impossible to find any Borel set 𝐴 ⊆ Free(N^Γ∗Δ) such that 𝐴is a complete section for𝐸_Δ and Free(N^Γ∗Δ) \𝐴 is a complete section for𝐸_Γ. Given any Borel set 𝐴 ⊆ Free(N^Γ∗Δ), we can find a function 𝑓_𝐴 as in Lemma 2.2.5. If 𝑓_𝐴 satisfies statement (1) of Lemma 2.2.5, then Free(N^Γ∗Δ) \ 𝐴cannot be a complete section for 𝐸_Γ. If 𝑓_𝐴 satisfies statement (2) of Lemma 2.2.5, then𝐴cannot be a complete section for𝐸_Δ. Therefore, we cannot have𝐴be a complete section for𝐸_Δ while Free(N^Γ∗Δ) \𝐴is a complete section for

𝐸_Γ.

Finally, we prove Theorem 2.2.1.

Proof of Theorem 2.2.1. Fix some 𝑑 ≥ 2, and let Γ := Z/𝑑Z =: Δ. Define 𝐸_Γ

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and 𝐸_Δ as in Theorem 2.2.2, and let 𝑋 be the standard Borel space consisting of equivalence classes of𝐸_Γand 𝐸_Δ. LetGbe the intersection graph on 𝑋, and note thatGis𝑑-regular, acyclic, and Borel with𝜒_𝐵(G) =2.

We claim thatGdoes not admit a Borel perfect matching. Suppose𝑀 ⊆ 𝑋× 𝑋 is a Borel perfect matching forG. Define 𝐴 ⊆ Free(N^Γ∗Δ)by

𝐴:={𝑥 ∈Free(N^Γ∗Δ) | {𝑥}= 𝑅∩𝑆for some(𝑅, 𝑆) ∈ 𝑀}.

Then 𝐴 and Free(N^Γ∗Δ) \ 𝐴 are Borel disjoint complete sections for 𝐸_Γ and 𝐸_Δ, contradicting Theorem 2.2.2. Therefore,Gdoes not admit a Borel perfect matching.

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C h a p t e r 3

BOREL MATCHINGS AND MEASURE THEORY

In graph theory, Hall’s theorem states that bipartite graphs with a certain expansion property have perfect matchings. Recall that for any set of vertices𝐹 in a graph𝑮, we write𝑁_𝑮(𝐹) to denote the set of vertices that are adjacent to𝐹. Hall’s theorem states the following:

Theorem 3.0.1(Hall’s theorem). Let𝑮 be a finite bipartite graph with bipartition {𝐴, 𝐵}. Then 𝑮 has a perfect matching if and only if |𝑁_𝐺(𝐹) | ≥ |𝐹| for every 𝐹 ⊆ 𝐴and every𝐹 ⊆ 𝐵.

Theorem 2.2.1 states that König’s theorem does not hold for Borel perfect matchings.

Since König’s theorem is a specific case of Hall’s theorem, Hall’s theorem likewise fails for Borel perfect matchings. Instead of Borel perfect matchings, we can consider Borel matchings that are perfect on “large” subsets of a graph. Lyons-Nazarov [LN11] and Marks-Unger [MU16] proved results analogous to Hall’s theorem from the perspectives of measure theory and Baire category notions, respectively.

3.1 Bipartite Graphs

To present the theorem of Lyons and Nazarov, we first recall some definitions.

Let 𝑮 = (𝑋 , 𝐺) be a locally countable Borel graph on a standard Borel space 𝑋 with some probability measure 𝜇. Recall that if 𝑀 is a Borel matching of 𝑮 such that 𝑋_𝑀 is𝐺-invariant and 𝜇(𝑋_𝑀) = 1, we call 𝑀 aBorel perfect matching 𝜇-almost everywhere (a.e.). Furthermore, recall that the graph 𝑮 is 𝜇-measure preservingif for every Borel automorphism 𝑓 : 𝑋 → 𝑋 such that Graph(𝑓) ⊆ 𝐺, 𝜇(𝐴) =𝜇(𝑓⁻¹(𝐴))for all measurable 𝐴 ⊆ 𝑋.

Lyons and Nazarov proved the following theorem, which uses a measure-theoretic concept of expansion in the setting of Borel perfect matchings𝜇-a.e.

Theorem 3.1.1. [LN11, Remark 2.6] Let𝑮 =(𝑋 , 𝐺)be a Borel graph on a standard probability space (𝑋 , 𝜇) such that 𝑮 is locally finite, 𝜇-measure preserving, and bipartite. Suppose there is some constant𝑐 > 1such that for all Borel independent sets𝐴 ⊆ 𝑋, 𝜇(𝑁_𝐺(𝐴)) ≥𝑐 𝜇(𝐴). Then𝑮has a Borel perfect matching𝜇-a.e.

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The proof of this statement relies on a result by Elek and Lippner about the lengths of augmenting paths in Borel matchings. Apath of length 𝑘 in 𝑮 = (𝑋 , 𝐺) is a sequence of vertices𝑥₀, 𝑥₁, . . . , 𝑥_𝑘 such that (𝑥_𝑖, 𝑥_𝑖+1) ∈ 𝐺 for 0 ≤ 𝑖 < 𝑘 and such that𝑥_𝑖 ≠ 𝑥_𝑗 for 𝑖 ≠ 𝑗. Given a Borel matching 𝑀 on a graph 𝑮 = (𝑥 , 𝐺), a path 𝑥₀, 𝑥₁, . . . , 𝑥₂_𝑛₊₁in𝑮is called anaugmenting pathif(𝑥₂_𝑖, 𝑥₂_𝑖₊₁) ∉𝑀 for 0 ≤𝑖 ≤ 𝑛, (𝑥₂_𝑖₊₁, 𝑥₂_𝑖₊₂) ∈ 𝑀 for 0≤ 𝑖 < 𝑛, and𝑥₀, 𝑥₂_𝑛₊₁ ∉ 𝑋_𝑀. Such a path is augmenting in the following sense: if we define 𝑀⁰to be the matching on 𝑮 that reverses which edges in𝑥₀, 𝑥₁, . . . , 𝑥₂_𝑛₊₁are contained in𝑀 and agrees with𝑀 on all other edges, then 𝑋_𝑀0 = 𝑋_𝑀 ∪ {𝑥₀, 𝑥₂_𝑛+₁}. Elek and Lippner proved the following statement about the lengths of augmenting paths in Borel matchings.

Proposition 3.1.2. [EL10, Proposition 1.1] Let 𝑋 be a standard Borel space, and let𝑮 = (𝑋 , 𝐺)be a locally finite Borel graph on 𝑋. Fix any𝑇 ≥ 1. For any Borel matching𝑀 of𝑮, there is a Borel matching𝑀⁰of𝑮such that 𝑋_𝑀 ⊆ 𝑋_𝑀0 and such that every augmenting path in𝑀⁰has length greater than2𝑇+1.

Proof of Proposition 3.1.2. We follow the proof given in [Ant13].

Let𝑌 be the set of paths of odd length at most 2𝑇 +1 in 𝑋. Let 𝑯 be the graph with vertex set𝑌 such that two paths𝑥₀, 𝑥₁, . . . , 𝑥₂_𝑛+1and𝑦₀, 𝑦₁, . . . , 𝑦₂_𝑚+1in𝑌 are adjacent in 𝑯 if and only if 𝑥_𝑖 = 𝑦_𝑗 for some 0 ≤ 𝑖 ≤ 𝑛 and 0 ≤ 𝑗 ≤ 𝑚. Since 𝑮 is locally finite, 𝑯is locally finite as well, an therefore, there is a Borel coloring 𝑐 :𝑌 →Non 𝑯.

Define 𝑀₋₁ := 𝑀. Let 𝑐₀, 𝑐₁, 𝑐₂, . . . be a sequence of elements in N such that each element ofNappears infinitely many times in this sequence. To construct the desired matching 𝑀⁰, we first construct a sequence of matchings 𝑀₀, 𝑀₁, 𝑀₂, . . . such that𝑋_𝑀

𝑛 ⊆ 𝑋_𝑀

𝑛+1for all𝑛 ∈N. We do the following for each𝑖 ∈Ninductively.

First, consider all paths in𝑐⁻¹(𝑐_𝑖). If 𝑥₀, 𝑥₁, . . . , 𝑥₂_𝑖+₁ ∈ 𝑐⁻¹(𝑐_𝑖) is an augmenting path in 𝑀_𝑖−1, switch which edges of the path lie in the matching. In other words, remove each(𝑥₂_𝑗+1, 𝑥₂_𝑗+2)from𝑀_𝑖−1for 0≤ 𝑗 < 𝑖, and add(𝑥₂_𝑗, 𝑥₂_𝑗+1)to𝑀_𝑖−1for 0≤ 𝑗 ≤ 𝑖. Since no two paths in𝑐⁻¹(𝑐_𝑖)share a vertex, the result of this procedure is still a matching; call this matching 𝑀_𝑖. Note that𝑋_𝑀

𝑖−1 ⊆ 𝑋_𝑀

𝑖 for each𝑖 ∈N. We claim that for every edge(𝑥 , 𝑦) ∈ 𝐺, there exists an𝑁such that either(𝑥 , 𝑦) ∈ 𝑋_𝑛 for all𝑛 > 𝑁 or (𝑥 , 𝑦) ∉ 𝑋_𝑛 for all𝑛 > 𝑁. Suppose not, and let (𝑥 , 𝑦) ∈ 𝐺 be a counterexample. Then (𝑥 , 𝑦) must be switched infinitely many times as part of an augmenting path. Define 𝐵 ⊆ 𝑋 to be the collection of points 𝑧 ∈ 𝑋 for which there is some 𝑘 ≤ 2𝑛 and𝑥₁, 𝑥₂, . . . , 𝑥_𝑘 ∈ 𝑋 such that𝑥 , 𝑥₁, 𝑥₂, . . . , 𝑥_𝑘, 𝑧 is a path

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in𝑮. Note that every augmenting path that contains the vertex 𝑥 must lie in𝐵, so switching the edges in any augmenting path containing𝑥 increases the number of matched vertices in𝐵by 1. Since(𝑥 , 𝑦)is switched infinitely many times,|𝑋_𝑀

𝑛∩𝐵| is unbounded as𝑛increases, implying that 𝐵is an infinite set. However, 𝐵must be finite since𝑮is locally finite. So such a counterexample cannot exist. We conclude that for every (𝑥 , 𝑦) ∈ 𝐺, there is some 𝑁 such that (𝑥 , 𝑦) ∈ 𝑋_𝑛 for all𝑛 > 𝑁 or (𝑥 , 𝑦) ∉ 𝑋_𝑛for all𝑛 > 𝑁.

Let 𝑀⁰ ⊆ 𝐺 consist of the edges (𝑥 , 𝑦) for which there exists some 𝑁 such that (𝑥 , 𝑦) ∈ 𝑀_𝑛 for all 𝑛 > 𝑁. If (𝑥 , 𝑦) and (𝑥 , 𝑧) are edges in 𝑀⁰, there is some 𝑛 for which (𝑥 , 𝑦) ∈ 𝑀_𝑛 and (𝑥 , 𝑧) ∈ 𝑀_𝑛 by definition of 𝑀⁰. Since each 𝑀_𝑛 is a matching, 𝑦 =𝑧. Therefore, 𝑀⁰is a matching as well. Since each 𝑀_𝑛is Borel, 𝑀⁰ is a Borel matching. We claim that𝑀⁰has the desired properties.

Consider any𝑥 ∈ 𝑋_𝑀. Since 𝑋_𝑀 ⊆ 𝑋_𝑀

𝑛 for all𝑛 ∈ N, we can find points {𝑦_𝑛}𝑛∈N

such that(𝑥 , 𝑦_𝑛) ∈ 𝑀_𝑛for each𝑛∈N. By local finiteness of𝑮, some𝑦_𝑚must occur infinitely many times. Our argument above implies that there is some 𝑁 such that (𝑥 , 𝑦_𝑚) ∈ 𝑀_𝑛for all𝑛 > 𝑁. Then(𝑥 , 𝑦_𝑚) ∈ 𝑀⁰, so𝑥 ∈ 𝑋_𝑀0. Therefore,𝑋_𝑀 ⊆ 𝑋_𝑀0. It remains to show that 𝑀⁰ does not contain any augmenting paths of length at most 2𝑇 +1. Suppose there is an augmenting path 𝑥₀, 𝑥₁, . . . , 𝑥₂_𝑛+1 in 𝑀⁰ with 𝑛 ≤ 𝑇. By definition of 𝑀⁰, we can find some 𝑁 such that for all 𝑛 > 𝑁, we have (𝑥₂_𝑖, 𝑥₂_𝑖₊₁) ∉ 𝑀_𝑛 for 0 ≤ 𝑖 ≤ 𝑛, and (𝑥₂_𝑖₊₁, 𝑥₂_𝑖₊₂) ∈ 𝑀_𝑛 for 0 ≤ 𝑖 < 𝑛. Let 𝑘 be the color assigned to this path by the coloring 𝑐. By definition of the sequence 𝑐₀, 𝑐₁, 𝑐₂, . . ., we can find𝑁⁰> 𝑁 such that𝑐_𝑁0 =𝑘. Then in round𝑁⁰, the edges of 𝑥₀, 𝑥₁, . . . , 𝑥₂_𝑛+₁are switched. In other words,(𝑥₂_𝑖, 𝑥₂_𝑖+₁) ∈ 𝑀_𝑁0 for 0≤ 𝑖 ≤ 𝑛, and (𝑥₂_𝑖+₁, 𝑥₂_𝑖+₂) ∈ 𝑀_𝑁0 for 0 ≤ 𝑖 < 𝑛. This is a contradiction, so no such path exists.

Therefore, 𝑀⁰does not contain any augmenting paths of length at most 2𝑇 +1.

Thus, 𝑀⁰is a Borel matching of𝑮 satisfying the desired conditions.

We now use Proposition 3.1.2 to prove Theorem 3.1.1.

Proof of Theorem 3.1.1. We follow the proof given in [Ant13].

First, we recursively define a sequence {𝑀_𝑖}_𝑖∈_N of Borel matchings in 𝑮. Let 𝑀₀ := ∅. Given the matching 𝑀_𝑖, define 𝑀_𝑖+1 to be the Borel matching obtained via the construction in the proof of Proposition 3.1.2 above, using𝑇 =𝑖+1. Then 𝑋_𝑀

𝑖 ⊆ 𝑋_𝑀

𝑖+1, and every augmenting path for𝑀_𝑖₊₁has length greater than 2(𝑖+1) +1.

We bound the measure of𝑋 \𝑋_𝑀

𝑖 using the following lemma.

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Lemma 3.1.3. [LN11] Suppose 𝑀 is a Borel matching of 𝑮 such that every aug- menting path for 𝑀 has length greater than 2𝑛+1, and let 𝐵 := 𝑋 \ 𝑋_𝑀. Then 𝜇(𝐵) ≤𝑐⁻

𝑛 2.

Proof of Lemma 3.1.3. We define a sequence of sets {𝐵_𝑖}₀_{≤𝑖≤𝑛} recursively. Let 𝐵₀:= 𝐵. Given𝐵₂_𝑘 for some𝑘 ∈N, define

𝐵₂_𝑘₊₁:=𝑁_𝐺(𝐵₂_𝑘) ={𝑥 ∈ 𝑋 | ∃𝑦 ∈𝐵₂_𝑘 (𝑥 , 𝑦) ∈𝐺}. Given𝐵₂_𝑘₊₁for some𝑘 ∈N, define

𝐵₂_𝑘₊₂:={𝑥 ∈ 𝑋 | ∃𝑦 ∈ 𝐵₂_𝑘₊₁ (𝑥 , 𝑦) ∈ 𝑀}.

First, we will show that 𝐵₂_𝑘 is an independent set for 0 ≤ 2𝑘 ≤ 𝑛. Suppose 𝑥 , 𝑦 ∈ 𝐵₂_𝑘satisfy(𝑥 , 𝑦) ∈𝐺. By construction of𝐵₂_𝑘, there is a path𝑥₀, 𝑥₁, . . . , 𝑥₂_𝑘 = 𝑥 such that 𝑥_𝑗 ∈ 𝐵_𝑗 for 0 ≤ 𝑗 ≤ 2𝑘 and such that (𝑥₂_𝑖+1, 𝑥₂_𝑖+2) ∈ 𝑀 for all 0 ≤ 𝑖 < 𝑘. Similarly, there is a path 𝑦₀, 𝑦₁, . . . 𝑦₂_𝑘 = 𝑦 such that 𝑦_𝑗 ∈ 𝐵_𝑗 for 0 ≤ 𝑗 ≤ 2𝑘 and such that (𝑦₂_𝑖₊₁, 𝑦₂_𝑖₊₂) ∈ 𝑀 for all 0 ≤ 𝑖 < 𝑘. Then observe that 𝑥₀, 𝑥₁, . . . , 𝑥₂_𝑘, 𝑦₂_𝑘, 𝑦₂_𝑘₋₁, . . . , 𝑦₁, 𝑦₀is an augmenting path for𝑀 of length 2𝑘+1, contradicting the definition of𝑀. Therefore, we conclude that𝐵₂_𝑘is an independent set.

Since each𝐵₂_𝑘is a Borel independent set, our conditions on𝑮imply that𝜇(𝐵₂_𝑘+1) ≥ 𝑐 𝜇(𝐵₂_𝑘)for each𝑘. Now observe that𝐵₂_𝑘+1 ⊆ 𝑋_𝑀: if some element of𝐵₂_𝑘+1lies in 𝑋\ 𝑋_𝑀 = 𝐵₀, then𝑮 must contain an odd cycle, contradicting the assumption that 𝑮 is bipartite. So we have 𝜇(𝐵₂_𝑘₊₁) = 𝜇(𝐵₂_𝑘₊₂) for each 𝑘 since 𝑮 is 𝜇-measure preserving. Thus, we conclude that

𝜇(𝐵)= 𝜇(𝐵₀)

≤ 𝑐⁻

𝑛 2𝜇(𝐵_𝑛)

≤ 𝑐⁻

𝑛 2.

By Lemma 3.1.3,𝜇(𝑋\𝑋_𝑀

𝑖)converges to 0 as𝑖→ ∞. Since𝑋\𝑋_𝑀

0 ⊇ 𝑋\𝑋_𝑀

1 ⊇ . . ., we have𝜇(∪𝑛∈N𝑋_𝑀

𝑛) =1.

For each𝑥 ∈ ∪𝑛∈N𝑋_𝑀

𝑛, there exists𝑁 such that𝑥 ∈ 𝑋_𝑀

𝑛 for all𝑛 > 𝑁. For𝑛 > 𝑁, let 𝑦_𝑛 ∈ 𝑋 be defined such that(𝑥 , 𝑦_𝑛) ∈ 𝑀_𝑛. Since𝑮 is locally finite, there must be some𝑦 such that 𝑦_𝑛 = 𝑦for infinitely many𝑛 > 𝑁. We claim that there is a set

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𝐴⁰ ⊆ ∪_𝑛∈_N𝑋_𝑀

𝑛of measure 1 such that for𝑥 ∈ 𝐴⁰,𝑦_𝑛 =𝑦for cofinitely many such𝑛. Let𝐶 be the set of all𝑥 ∈ ∪_𝑛∈_N𝑋_𝑀

𝑛 such that this does not hold. We wish to show that𝜇(𝐶)=0.

For each𝑛 ∈ N, define𝐶_𝑛to be the set of 𝑥 ∈ 𝑋 such that there is some𝑦 ∈ 𝑋 for which(𝑥 , 𝑦) ∈ 𝑀_𝑛and (𝑥 , 𝑦) ∉ 𝑀_𝑛₊₁. Observe that𝐶 ⊆ lim sup

𝑛→∞

𝐶_𝑛. So

𝜇(𝐶) ≤ 𝜇(lim sup

𝑛→∞

𝐶_𝑛)

≤ 𝜇(lim

𝑘→∞

Ø

𝑛≥𝑘

𝐶_𝑛)

≤ lim

𝑘→∞

Õ

𝑛≥𝑘

𝜇(𝐶_𝑛).

Suppose𝑥 ∈ 𝐶_𝑛. By the construction of 𝑀_𝑛+1 from 𝑀_𝑛 according to the proof of Proposition 3.1.2, 𝑥 must be contained in exactly one augmenting path of length at most 2𝑛 + 1 containing some element of 𝑋 \ 𝑋_𝑀

𝑛. Then because 𝑮 is 𝜇- measure preserving, 𝜇(𝐶_𝑛) ≤ (2𝑛+ 1)𝜇(𝑋 \ 𝑋_𝑀

𝑛). Applying Lemma 3.1.3, we have 𝜇(𝐶_𝑛) ≤ (2𝑛+1)𝑐⁻

𝑛

2. So lim

𝑘→∞

Í

𝑛≥𝑘𝜇(𝐶_𝑛) = 0, implying that 𝜇(𝐶) = 0.

Therefore, there is a set 𝐴⁰ ⊆ ∪_𝑛∈_N𝑋_𝑀

𝑛 of measure 1 such that for𝑥 ∈ 𝐴⁰, 𝑦_𝑛 = 𝑦 for cofinitely many such𝑛. Let𝐴 ⊆ 𝐴⁰be a𝐺-invariant set of measure 1.

We now define a matching 𝑀 on 𝐴. For 𝑥 , 𝑦 ∈ 𝐴, we define (𝑥 , 𝑦) ∈ 𝑀 if and only if there are cofinitely many𝑛such that (𝑥 , 𝑦) ∈ 𝑀_𝑛. Since each 𝑀_𝑛 is a Borel matching, 𝑀 is a Borel matching. Therefore, 𝑀 is a Borel perfect matching 𝜇-a.e.

for𝑮.

The requirement that 𝑐 > 1 in Theorem 3.1.1 is necessary. Let 𝑮 be the Cayley graph of the free part of the shift action of Zon 2^Z. Then 𝜇(𝑁_𝐺(𝐴)) ≥ 𝜇(𝐴) for all Borel independent sets 𝐴⊆ 2^Z. If𝑮 has Borel perfect matching𝜇-a.e., then we obtain a Borel set that has measure¹₂ and is invariant under applying the shift action twice, contradicting the fact that the action ofZon 2^Zis mixing. Therefore,𝑮does not have a Borel perfect matching𝜇-a.e., so the assumption𝑐 > 1 is necessary.

Furthermore, the expansion condition in Theorem 3.1.1 cannot be relaxed to one on finite sets, unlike in Hall’s theorem.

Proposition 3.1.4. [KM19, Proposition 15.1] Let (𝑋 , 𝜇) be a standard probability space. For every𝑛 ≥ 1, there is some Borel graph 𝑮 = (𝑋 , 𝐺) that is 𝜇-measure preserving and has bounded degree, satisfies 𝜒_𝐵(𝑮) =2, and has the property that

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|𝑁_𝐺(𝐹)) | ≥ 𝑛|𝐹| for every finite independent set 𝐹 ⊆ 𝑋, with no Borel perfect matching 𝜇-a.e.

Conley and Miller proved a theorem about Borel perfect matchings𝜇-a.e. when the complexity of the edge relation is restricted. Let (𝑋 , 𝜇) be a standard probability space, and let𝑮 = (𝑋 , 𝐺)be a locally countable Borel graph. If there is a𝐺-invariant Borel set 𝐴 ⊆ 𝑋 such that 𝜇(𝐴) =1 and such that, on 𝐴, the equivalence relation generated by 𝐺 can be written as the increasing union of finite Borel equivalence relations, the graph 𝑮 is 𝜇-hyperfinite. Conley and Miller proved the following result about Borel matchings in acyclic𝜇-hyperfinite graphs:

Theorem 3.1.5. [CM17, Theorem B] Let (𝑋 , 𝜇) be a standard probability space, and let 𝑮 = (𝑋 , 𝐺) be an acyclic, locally countable Borel graph. If 𝑮 is 𝜇- hyperfinite and every point in some𝐺-invariant Borel set of measure 1 has degree at least 3, then𝑮 has a Borel perfect matching𝜇-a.e.

3.2 A Shift Action

We conclude this section by presenting a theorem by Csóka and Lippner, which extends a result by Lyons and Nazarov in [LN11] by removing the assumption that the Cayley graph of the given group is bipartite.

Recall that for a standard Borel space 𝑋 and a countable groupΓ, the shift action of Γon 𝑋^Γ is given by 𝛼·𝑥(𝛽) = 𝑥(𝛼⁻¹𝛽) for all𝑥 ∈ 𝑋^Γ and𝛼, 𝛽 ∈ Γ. Suppose 𝑆 is a finite symmetric generating set of Γ, where 𝑆 does not contain the identity.

Then we define𝑮(𝑆,[0,1]^Γ)to be the graph on [0,1]^Γsuch that𝑥 , 𝑦 ∈ [0,1]^Γare adjacent exactly when there is some 𝛾 ∈ 𝑆such that 𝛾·𝑥 = 𝑦. Csóka and Lippner proved the following theorem about Borel matchings on𝑮(𝑆,[0,1]^Γ).

Theorem 3.2.1. [CL17, Theorem 1.1] LetΓbe a non-amenable group with a finite symmetric generating set𝑆not containing the identity. Let𝜇be the probability mea- sure on[0,1]^Γdefined as the product of the Lebesgue measure on each coordinate.

Then𝑮(𝑆,[0,1]^Γ) has a Borel perfect matching𝜇-a.e.

We first present several definitions.

Let 𝑮 = (𝑋 , 𝐺) be a 𝑑-regular, infinite, connected graph. A real cut for 𝑮 is a partition 𝑋 = 𝐴t 𝐴^𝑐 such that 𝐴is a finite set and |𝐴| ≥ 2. If |𝐴|is odd, we say that this partition is areal cut into odd sets. The sizeof the cut is |{(𝑥 , 𝑦) ∈ 𝐺 | 𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴^𝑐}|, the number of edges between 𝐴and 𝐴^𝑐, which is finite since𝑮 is

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𝑑-regular and 𝐴is finite. Csóka and Lippner proved the following lemma about the sizes of real cuts:

Lemma 3.2.2. [CL17, Lemma 2.5] Let 𝑮 be as above, and suppose that for every 𝑥₁, 𝑥₂ ∈ 𝑋, there is an automorphism of𝑮sending𝑥₁↦→ 𝑥₂. Then every real cut of 𝑮 has size at least𝑑, and there is a real cut with size exactly 𝑑if and only if every 𝑥 ∈ 𝑋 lies in a unique𝑑-clique.

Now let𝑋be a probability space with measure𝜇, and let𝑮 = (𝑋 , 𝐺)be a𝑑-regular, 𝜇-measure preserving, Borel graph on 𝑋. We define a measure on sets of edges as follows. For any symmetric, measurable set𝐻 ⊆ 𝐺, define

𝜇^∗(𝐻) := 1 2

∫

𝑋

𝑑_𝐻(𝑥)𝑑𝜇(𝑥),

where 𝑑_𝐻(𝑥) is the number of elements 𝑦 ∈ 𝑁_𝐺(𝑥) such that (𝑥 , 𝑦) ∈ 𝐻. For any measurable set of vertices𝑌 ⊆ 𝑋, let

𝐸(𝑌 , 𝑌^𝑐) :={(𝑥 , 𝑦) ∈𝐺 |𝑥 ∈𝑌 , 𝑦 ∈𝑌^𝑐} ∪ {(𝑦, 𝑥) ∈ 𝐺 | 𝑥 ∈𝑌^𝑐, 𝑦 ∈𝑌} be the collection of edges between 𝑌 and 𝑌^𝑐. For 𝑐₀ > 0, we say that 𝑮 is a 𝑐₀-expanderif for all measurable𝑌 ⊆ 𝑋, we have

𝜇^∗(𝐸(𝑌 , 𝑌^𝑐)) ≥ 𝑐₀𝜇(𝐻)𝜇(𝐻^𝑐).

If every real cut into odd sets for𝑮has size at least𝑑+1, and if there is some𝑐₀> 0 such that𝑮 is a𝑐₀-expander, then we say that𝑮 isadmissible. Csóka and Lippner proved the following theorem about augmenting paths in admissible graphs:

Theorem 3.2.3. [CL17, Theorem 4.2] For any 𝑐₀ > 0 and any integer 𝑑 ≥ 3, there is a constant𝑐= 𝑐(𝑐₀, 𝑑)such that the following holds: given any 𝑑-regular, 𝜇-measure preserving, Borel graph𝑮 on a probability space (𝑋 , 𝜇) such that𝑮 is admissible, and given any Borel matching 𝑀on𝑮 such that𝜇(𝑋 \𝑋_𝑀) ≥ 𝜀, there is an augmenting path for𝑀 of length at most𝑐(log¹_𝜀)³.

We can now prove Theorem 3.2.1, following [CL17].

Proof of Theorem 3.2.1. Let𝑑 :=|𝑆|, and note that𝑮 :=𝑮(𝑆,[0,1]^Γ)is𝑑-regular.

We also observe that for any 𝑥₁, 𝑥₂ ∈ [0,1]^Γ, there is an automorphism sending 𝑥₁↦→𝑥₂.

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First, suppose there is a real cut of 𝑮 with size 𝑑. Then by Lemma 3.2.2, every vertex of𝑮 lies in a unique𝑑-clique. Then define 𝑀 to be the collection of edges (𝑥 , 𝑦) ∈ 𝐺 such that𝑥 and 𝑦 are in different 𝑑-cliques. Observe that 𝑀 is a Borel perfect matching, as desired.

Now suppose every real cut of𝑮 has size at least 𝑑+1. Let 𝑀₁ := ∅, and define Borel matchings𝑀₁, 𝑀₂, 𝑀₃, . . .inductively such that𝑀_𝑛has no augmenting paths of length at most 2𝑛+1 and such that𝑀_𝑛₊₁is obtained from 𝑀_𝑛via the procedure described in the proof of Proposition 3.1.2. Let𝑈_𝑛:= 𝑋\𝑋_𝑀

𝑛. Let𝐸_𝑛 =𝑀_𝑛4𝑀_𝑛+₁ denote the set of edges in 𝐺 that are switched between 𝑀_𝑛 and 𝑀_𝑛+₁. Note that every edge in𝐸_𝑛must lie along an augmenting path of length at most 2𝑛+3. Each augmenting path has both endpoints in𝑈_𝑛, and each point in𝑈_𝑛 is contained in at most one augmenting path whose edges are switched, so𝜇^∗(𝐸_𝑛) ≤ (2𝑛+3)𝜇(𝑈_𝑛).

Suppose 𝜇(𝑈_𝑛) = 𝜀. Since Γis non-amenable, there is some 𝑐₀ > 0 such that 𝑮 is a 𝑐₀-expander [LN11, Lemma 2.3]. We are assuming that every real cut of 𝑮 has size at lest 𝑑+1, so 𝑮 is admissible. Let𝑐 = 𝑐(𝑐₀, 𝑑) be a constant satisfying Theorem 3.2.3. Then we know that𝑮 has an augmenting path for 𝑀_𝑛of length at most𝑐(log¹_𝜀)³. By definition of𝑀_𝑛, we must have

𝑐

log1 𝜀

3

>2𝑛+1, which yields

𝜀 <exp −

2𝑛+1 𝑐

¹₃! .

So by the Borel-Cantelli lemma, the set E of edges that lie in𝐸_𝑘 for infinitely many 𝑘 satisfies𝜇^∗(𝐸) =0.

Define𝑀to be the set of edges(𝑥 , 𝑦) ∈𝐺that lie in𝑀_𝑛for cofinitely many𝑛. Since each 𝑀_𝑛 is a Borel matching, 𝑀 is a Borel matching. From above, we have that 𝜇(𝑋 \ ∪_𝑛∈_N𝑋_𝑀

𝑛) =0 and𝜇^∗(𝐸) =0, so 𝑀is a Borel perfect matching𝜇-a.e.

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C h a p t e r 4

BOREL MATCHINGS AND BAIRE CATEGORY

An expansion condition on finite sets of vertices is sufficient to conclude that a Borel matching is almost perfect in a Baire category setting. Suppose 𝑮 = (𝑋 , 𝐺) is a Borel graph on a Polish space 𝑋. We recall that a Borel matching 𝑀 of𝐺 is called aBorel perfect matching genericallyif𝑋_𝑀 is𝐺-invariant and comeager. The following theorem by Marks and Unger adapts Hall’s theorem to this setting.

Theorem 4.0.1. [MU16, Theorem 1.3] Let𝑋be a Polish space, and let𝑮 =(𝑋 , 𝐺) be a locally finite bipartite Borel graph with a bipartition{𝐵₀, 𝐵₁}, which need not be Borel. Suppose there is some 𝜀 > 0 such that for every finite set 𝐹 ⊆ 𝐵₀ or 𝐹 ⊆ 𝐵₁,|𝑁_𝐺(𝐹) | ≥ (1+𝜀) |𝐹|. Then𝑮admits a Borel perfect matching generically.

To prove this result, we need several definitions. Let 𝑮 = (𝑋 , 𝐺) be a graph.

For𝑥 , 𝑦 ∈ 𝑋, let 𝑑_𝐺(𝑥 , 𝑦) be the length of the minimum-length path from 𝑥 to 𝑦. Define 𝑮₂ = (𝑋 , 𝐺₂) to be the graph on 𝑋 such that (𝑥 , 𝑦) ∈ 𝐺₂ if and only if 𝑑_𝐺(𝑥 , 𝑦) = 2. For any matching 𝑀 of 𝑮, we define the graph 𝑮 − 𝑀 to be the restriction𝑮(𝑋 \𝑋_𝑀).

Now suppose𝑮 is bipartite with some bipartition {𝐵₀, 𝐵₁}. The graph𝑮 satisfies Hall’s condition if for every finite set 𝐹 ⊆ 𝐵₀ or 𝐹 ⊆ 𝐵₁, |𝑁_𝐺(𝐹) | ≥ |𝐹|. If 𝑮 satisfies Hall’s condition, and if every𝐺₂-connected finite set𝐹 with|𝐹| ≥𝑛such that 𝐹 ⊆ 𝐵₀ or 𝐹 ⊆ 𝐵₁ satisfies |𝑁_𝐺(𝐹) | ≥ (1+𝜀) |𝐹|, we say that 𝑮 satisfies Hall𝜀,𝑛.

Marks and Unger proved the following lemma.

Lemma 4.0.2. [MU16, Lemma 3.1] Let 𝑋 be a Polish space, and let𝑮 = (𝑋 , 𝐺) be a locally finite Borel graph on 𝑋. Given any funtion 𝑓 : N → N, there is some sequence{𝐴_𝑛}_𝑛∈_N of Borel sets in𝑋 such that 𝐴:=∪_𝑛∈_N𝐴_𝑛is comeager and 𝐺-invariant, and such that𝑑_𝐺(𝑥 , 𝑦) > 𝑓(𝑛) for all distinct𝑥 , 𝑦 ∈ 𝐴_𝑛.

Proof of Lemma 4.0.2. We follow the proof in [MU16]. Let{𝑈_𝑖}_𝑖∈_N be a basis of open sets for 𝑋. For𝑖 ∈Nand𝑟 >0, let 𝐵_𝑖,𝑟 be the set consisting of exactly those 𝑥 ∈𝑈_𝑖 such that for all 𝑦 ≠ 𝑥with 𝑑_𝐺(𝑥 , 𝑦) ≤ 𝑟, 𝑦 ∉𝑈_𝑖. Given any𝑟 > 0 and any

Borel Matchings and Analogs of Hall's Theorem