Finitely Forcible Graphons with an Almost Arbitrary Structure by

Jakub Sosnovec Master’s Thesis Submitted to the University of Warwick in partial fulfilment of the requirements for admission to the degree of

MSc. by Research in Computer Science

Department of Computer Science December 2017

Contents Acknowledgments

ii

Declarations

iii

Abstract

iv

Chapter 1 Introduction

1

Chapter 2 Preliminaries 2.1 Graphons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Partitioned graphons and decorated constraints . . . . . . . . . . . . 2.3 Cut norm and graphon regularity . . . . . . . . . . . . . . . . . . . .

3 3 5 10

Chapter 3 Proof of the main result 3.1 Preprocessing of the graphon W⊗ . . . . 3.2 General structure of W0 . . . . . . . . . 3.3 The finitely forcible graphon W CKM . . 3.4 Lining up the vertices . . . . . . . . . . 3.4.1 The measure preserving map g . 3.4.2 Aligning the CKM graphon . . . 3.4.3 Coordinate system . . . . . . . . 3.4.4 Ordering the parts . . . . . . . . 3.5 Forcing auxiliary tiles . . . . . . . . . . 3.5.1 Checker tiles . . . . . . . . . . . 3.5.2 Dyadic square indices . . . . . . 3.5.3 Referencing dyadic squares . . . 3.6 Forcing densities . . . . . . . . . . . . . 3.7 Cleaning up . . . . . . . . . . . . . . . . 3.7.1 Finishing the tile (A ∪ · · · ∪ F)2 3.7.2 Degree balancing . . . . . . . . . 3.7.3 Degree distinguishing . . . . . .

14 14 15 19 20 20 22 23 24 25 25 26 28 28 33 33 33 35

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Acknowledgments First and foremost, I would like to thank my supervisor Dan Kr´ al’. He invited me to Warwick and thus provided me with the amazing opportunity to study and work in such a stimulating environment. Dan is always ready to give guidance and encouragement and his devotion to students is truly exceptional. My sincere gratitude goes to the Leverhulme Trust 2014 Philip Leverhulme Prize of Daniel Kr´ al’, which generously provided funding. Next I would like to thank my friends and fellow students at the University of Warwick. In particular, my gratitude goes to Jake Cooper, Ta´ısa Martins and Jordan Venters for their company and support. I would also like to express gratitude to Jon Noel for a fruitful collaboration and proof-reading of the draft. I also wish to thank my girlfriend Ad´ela for her patience and support that she provided through the long year of separation. Last but not least, I thank my parents for their continuous support and love.

ii

Declarations This thesis is submitted to the University of Warwick in support of my application for the degree of Master of Science. It has been composed by myself and has not been submitted in any previous application for any degree.

iii

Abstract Graphons are analytic objects representing (convergent sequences of) large graphs. A graphon is said to be finitely forcible if it is determined by finitely many subgraph densities, i.e., if the asymptotic structure of graphs represented by such a graphon depends only on finitely many density constraints. Such graphons appear in various scenarios, particularly in extremal combinatorics. Lov´asz and Szegedy conjectured that all finitely forcible graphons possess a simple structure. This was disproved in a strong sense by Cooper, Kr´ al’ and Martins, who showed that any graphon is a subgraphon of a finitely forcible graphon. We strengthen this result by showing for every ε > 0 that any graphon spans a 1 − ε proportion of a finitely forcible graphon.

iv

Chapter 1

Introduction The theory of graph limits is an emerging area of combinatorics, which offers analytic tools to study large graphs. The range of applications of analytic methods offered by the theory of graph limits has been constantly expanding. In particular, the graph limit theory and the closely related flag algebra method of Razborov [25] changed the landscape of extremal graph theory by providing progress on many important problems in extremal combinatorics, e.g. [1, 2, 3, 4, 14, 15, 16, 17, 18, 23, 24, 25, 26, 27]. Among other applications of the methods provided by the theory of graph limits, we would like to highlight those from computer science that are related to property and parameter testing algorithms [22]. We refer the reader to the recent monograph by Lov´asz [19] for further results on graph limits. In this thesis, we are interested in limits of sequences of dense graphs. An analytic object representing sequence of dense graphs is called a graphon. Formally, a graphon is a measurable function W from the unit square [0, 1]2 to the unit interval [0, 1] that is symmetric, i.e., W (x, y) = W (y, x) for every (x, y) ∈ [0, 1]2 . Every graphon is uniquely determined, up to weak isomorphism, by the densities of all graphs. The main object of our study are finitely forcible graphons, which are graphons that are uniquely determined by the densities of finitely many graphs. We refer the reader to Chapter 2 for further introduction to the concepts that we work with in this thesis. Results on finitely forcible graphons can be found in disguise in various setting in graph theory. For example, a classical result of Thomason [28] (see also Chung, Graham and Wilson [6]) on quasirandom graphs is equivalent to saying that the constant graphon is finitely forcible by the densities of 4-vertex subgraphs. Another source of motivation for studying finitely forcible graphons comes from extremal graph theory, see e.g. Proposition 3. Lov´asz and Szegedy [20] initiated a systematic study of properties of finitely forcible graphons and conjectured, based on examples of finitely forcible graphons known at that time, that all finitely forcible graphons must posses a simple structure. 1

Conjecture 1 (Lov´asz and Szegedy, [20, Conjecture 9]). The space of typical vertices of every finitely forcible graphon is compact. Conjecture 2 (Lov´asz and Szegedy, [20, Conjecture 10]). The space of typical vertices of every finitely forcible graphon has finite dimension. Conjectures 1 and 2 were disproved by counterexample constructions in [13] and [12], respectively. A stronger counterexample to the latter conjecture was given in [8] (if true, Conjecture 2 would imply that the minimum number of parts of a weak εregular partition of a finitely forcible graphon is bounded by a polynomial of ε−1 ; a finitely forcible graphon constructed in [8] has weak ε-regular partition with the minimum number of parts almost exponential in ε−2 for infinitely many ε > 0, which is close to the general lower bound from [7]). This line of research culminated with the following general result of Cooper, Kr´ al’ and Martins [9]. Theorem 1. For every graphon W⊗ , there exists a finitely forcible graphon W0 such that W⊗ is a subgraphon of W0 induced by a 1/13 fraction of the vertices of W0 Theorem 1 yields counterexamples to both conjectures mentioned earlier and provides a universal framework for constructing finitely forcible graphons with very complex structure. In view of Proposition 3, Theorem 1 says that problems on minimizing a linear combination of subgraph densities, which are among the problems of the simplest kind in extremal graph theory, may have unique optimal solutions with highly complex structure. Given the general nature of Theorem 1, it is surprising that the family of graphs whose densities force W0 in Theorem 1 can be chosen to be independent of W⊗ ; see [9] It is natural to ask whether the fraction 1/13 in Theorem 1 can be replaced by a larger quantity. The proof techniques from [9] allows replacing the fraction by any number smaller than 1/2. The purpose of this thesis is to show that the fraction can be replaced by any number smaller than 1. Theorem 2. For every ε > 0 and every graphon W⊗ , there exists a finitely forcible graphon W0 such that W⊗ is a subgraphon of W0 induced by a 1 − ε fraction of the vertices of W0 . The proof of Theorem 2 is based on the method of decorated constraints, which was introduced in [12, 13], and uses Theorem 1 as one of the main tools. Informally speaking, Theorem 1 is used to embed the graphon W⊗ on a small part of W0 and other auxiliary structure of W0 is then used to magnify the graphon W⊗ to the 1 − ε fraction of the vertices of W0 . We remark that, in contrast to the proof of Theorem 1, the family of graphs used to force W0 in Theorem 2 depends on ε.

2

Chapter 2

Preliminaries We now introduce the notation and terminology used in the thesis; our notation mostly follows that used in [9] in relation to graph limits. We start with some general notation. For k ∈ N, [k] denotes the set of integers {1, 2, . . . , k}. If F is a S family of sets, we use F to denote the union of all sets F ∈ F. We work with the Lebesgue measure on [0, 1]d throughout the thesis, i.e., measurability of sets and functions is meant in the Lebesgue sense. If X ⊂ Rd is a measurable set, we write |X| for its measure and for two measurable sets X, Y ⊂ Rd , we write X ⊑ Y to denote |Y \ X| = 0.

2.1

Graphons

The order of a graph G, which is denoted by |G|, is its number of vertices. The density of a graph H in G, denoted d(H, G), is the probability that a uniformly randomly chosen set of |H| vertices of G induces a graph isomorphic to H. If |H| > |G|, then we set d(H, G) to zero. A graphon is a measurable function W from the unit square [0, 1]2 to the unit interval [0, 1] that is symmetric, i.e., W (x, y) = W (y, x) for every (x, y) ∈ [0, 1]2 . A graphon W can be viewed as an infinite weighted graph on the vertex set [0, 1] with the edge (x, y) ∈ [0, 1]2 having the weight W (x, y). Following this intuition, we refer to the points of [0, 1] as vertices. To visualize the structure of a graphon, we shall use a figure that may be seen as a continuous version of the adjacency matrix. More precisely, in a figure depicting W, the domain of W is represented by the unit square [0, 1]2 with the origin in the top left corner and the values of W are represented by appropriate shades of gray, with 0 corresponding to white and 1 to black. A graphon can also be associated with a probability distribution on graphs of a fixed order. Formally, for a graphon W and an integer k ∈ N, the W -random graph of order k is a graph G obtained by the following two step procedure. First, let x1 , . . . , xk ∈ [0, 1] be k points chosen uniformly and independently at random. 3

Then, G is the graph with the vertex set [k] such that the edge ij is present with probability W (xi , xj ) for every distinct vertices i, j ∈ [k]. The density of a graph H in the graphon W is the probability that the W -random graph of order |H| is isomorphic to H; we denote this quantity by d(H, W ). Note that d(H, W ) is also the expected density of H in the W -random graph of order k for every k ≥ |H|. Consider a sequence of graphs (Gn )n∈N such that the orders |Gn | tend to infinity. We say that the sequence (Gn )n∈N is convergent if for every graph H, the sequence of the densities of H in Gn , i.e., the sequence (d(H, Gn ))n∈N , is convergent. We say that the sequence (Gn )n∈N converges to a graphon W if lim d(H, Gn ) = d(H, W )

n→∞

for every graph H. Lov´asz and Szegedy [21] showed that every convergent sequence of graphs converges to a graphon. Conversely, for every graphon, there exists a sequence of graphs converging to it. Indeed, the sequence of W -random graphs with increasing orders converges to W with probability one. For a graphon W and a vertex x ∈ [0, 1], we define the degree of x in W as Z W (x, y) dy. degW (x) = [0,1]

Note that the degree is well-defined for almost every vertex x ∈ [0, 1]. We also define the neighbourhood of a vertex x ∈ [0, 1] as the set {y ∈ [0, 1] : W (x, y) > 0} and denote it NW (x). Note that the set NW (x) is measurable for almost every x ∈ [0, 1]. When the graphon W is clear from context, we omit the subscripts. Two graphons W1 and W2 are said to be weakly isomorphic if d(H, W1 ) = d(H, W2 ) for every graph H. Weakly isomorphic graphons are limits of the same sequences of graphs. It is natural to ask how weakly isomorphic graphs differ in their structure, which was answered in [5] as follows. A function ϕ : [0, 1] → [0, 1] is measure preserving if it is measurable and |ϕ−1 (X)| = |X| for every measurable subset X ⊆ [0, 1]. It is easy to check that if ϕ : [0, 1] → [0, 1] is a measure preserving map, then the graphon W ϕ defined as W ϕ (x, y) = W (ϕ(x), ϕ(y)) is weakly isomorphic to W . It was shown in [5] that two graphons W1 and W2 are weakly isomorphic if and only if there exist measure preserving maps ϕ1 , ϕ2 : [0, 1] → [0, 1] such that W ϕ1 ≡ W ϕ2 almost everywhere. Let W1 and W2 be two graphons and X ⊆ [0, 1] a non-null measurable set. We say that W1 is a subgraphon of W2 induced by X if there exist measure preserving maps ϕ1 : X → [0, |X|) and ϕ2 : X → X such that
W1 |X|−1 · ϕ1 (x), |X|−1 · ϕ1 (y) = W2 (ϕ2 (x), ϕ2 (y)) 4

for almost every (x, y) ∈ X 2 . A graphon W is finitely forcible if there exist graphs H1 , . . . , Hm such that any graphon W ′ satisfying that d(Hi , W ′ ) = d(Hi , W ) for every i ∈ [m] is weakly isomorphic to W . In other words, finitely forcible graphons are uniquely determined by finitely many subgraph densities, up to weak isomorphism. The following proposition, stated e.g. in [9], provides a link between finitely forcible graphons and extremal graph theory. Proposition 3. A graphon W is finitely forcible if and only if there exist graphs H1 , . . . , Hm and reals α1 , . . . , αm such that for every graphon W ′, m X

αi d(Hi , W ) ≤

m X

αi d(Hi , W ′ ),

i=1

i=1

and the equality holds if and only if W and W ′ are weakly isomorphic. We close this section with two well-known measure-theoretic results which we will apply throughout the thesis known as the Monotone Reordering Theorem and the Measure Isomorphism Theorem, respectively. Proposition 4. For every measurable function h : [0, 1] → [0, 1], there exists a nondecreasing function f : [0, 1] → [0, 1] and a measure preserving map ϕ : [0, 1] → [0, 1] such that h(x) = f (ϕ(x)) for almost every x ∈ [0, 1]. Proposition 5. Let X1 , X2 ⊆ [0, 1] be two measurable sets with the same measure. Then there exist measurable subsets Y1 ⊆ X1 and Y2 ⊆ X2 with Y1 ⊑ X1 and Y2 ⊑ X2 such that there is a measure preserving bijection between Y1 and Y2 .

2.2

Partitioned graphons and decorated constraints

The most direct way of showing that a graphon W is finitely forcible is by explicitly providing the family of graphs H1 , . . . , Hm and their densities d1 , . . . , dm and analyzing all graphons W ′ such that d(Hi , W ′ ) = di . We now introduce the method of decorated constraints, which was developed in [12, 13]. This method allows us to use more advanced constraints to establish that a graphon is finitely forcible. A density expression is a formal polynomial in graphs, i.e., a real number from [0, 1] or a graph are density expressions, and if D1 and D2 are density expressions, then so are D1 + D2 and D1 · D2 . A density expression D can be evaluated with respect to a graphon W by replacing every graph H in D with d(H, W ). A constraint is an equality between two density expressions and it is satisfied by a graphon W if both density expressions evaluated with respect to W are equal. A simple example of a constraint is the equality H = d, which is satisfied by a graphon W if and only if d(H, W ) = d. 5

If C is a finite set of constraints and W is the unique graphon, up to weak isomorphism, that satisfies all of the constraints C, then W is finitely forcible. Indeed, W is the unique graphon with the density of H equal to d(H, W ) for all graphs H appearing in C. We will often be saying that the constraints C force the graphon W. A graphon W is partitioned if there exist positive reals a1 , . . . , ak summing to one and distinct reals d1 , . . . , dk ∈ [0, 1] such that the set Ai ⊆ [0, 1] of vertices of W with degree di has measure ai for all i ∈ [k]. The sets Ai are called parts and the degree of a part Ai is di . We will abuse the notation here and if W and W ′ are two partitioned graphons with parts of measures ai and degrees di , we will use the same letters to denote the corresponding parts of W and W ′ . This is technically incorrect since the part Ai can be a different subset of [0, 1] in W and W ′ but we will make sure that the graphon that we have in mind is always clear from the context. The property of being a partitioned graphon can be forced in the following sense. Lemma 6. Let a1 , . . . , ak be positive reals summing to one and d1 , . . . , dk distinct reals from [0, 1]. There exists a finite set of constraints C such that a graphon W satisfies all constraints in C if and only if W is a partitioned graphon with parts of measures a1 , . . . , ak and degrees d1 , . . . , dk . Consider a partitioned graphon W and let P be the set of its parts. The relative degree of a vertex x ∈ [0, 1] in W with respect to a set P ⊆ P of parts is defined as [ −1 Z P degW (x) = P · S W (x, y) dy. P

Similarly, the relative neighbourhood of x ∈ [0, 1] with respect to P, which is denoted S P (x), is the set N (x) ∩ by NW P. If P = {A} for some part A, then we write W A P (x) as N A (x). As before, if the graphon W is clear degP (x) as deg (x) and N W W W W from context, we omit the subscripts. For two non-empty subsets P1 , P2 ⊆ P, the S S restriction of the graphon W to P1 × P2 will be referred to as the tile P1 × P2 . If both P1 and P2 are single parts, we call the tile simple; otherwise, it is composite. We now introduce a formally stronger (but technically equivalent) version of constraints, which we call decorated constraints. These are similar to decorated constraints used in [8, 9, 12, 13] except that we will allow the vertices of graphs appearing in constraints can be assigned multiple parts, instead of a single part; we discuss the difference in more detail further in our exposition. We will always have a particular set of parts in mind when working with decorated constraints. So, fix a set P of parts. A decorated graph G is a graph with 0 ≤ m ≤ |G| distinguished vertices labelled from 1 do m, which are called roots, and with every vertex v (including the roots) assigned a non-empty subset of P, which is called the decoration of v. If the decoration of a vertex is a single element set, e.g., {A}, 6

we just write A as the decoration to simplify our notation. Two decorated graphs G1 and G2 are compatible if the subgraphs induced by their roots are isomorphic, respecting both the labels of roots and the decorations assigned to them. A decorated density expression is a formal polynomial in decorated graphs such that all graphs in the expression are mutually compatible and a decorated constraint is an equality between two decorated density expressions such that all graphs in the expression are mutually compatible. Let W be a partitioned graphon with parts P and C a decorated constraint of the form D = 0 where D is a decorated density expression. We now describe what we mean when we say that the graphon W satisfies C. Let H0 be the decorated graph induced by the roots v1 , . . . , vm of the decorated graphs in C. The graphon W satisfies the constraint C if the following holds for almost every m-tuple (x1 , . . . , xm ) ∈ [0, 1]m such that xi belongs to one of the parts that vi is decorated with, W (xi , xj ) > 0 for every edge vi vj ∈ E(H0 ) and W (xi , xj ) < 1 for every nonedge vi vj ∈ / E(H0 ): if each decorated graph H is replaced with the probability that a W -random graph of order |H| is the graph H conditioned on the event that the roots are chosen as the vertices x1 , . . . , xk and they induce the graph H0 , and that each non-root vertex is chosen from the union of the parts in its decoration, then the value of D is equal to zero. We say that W satisfies a decorated constraint of the form D = D ′ if it satisfies D − D ′ = 0. We next describe a convention of depicting decorated constraints that we use, which is analogous to that used in [9, 8]. The roots of decorated graphs will be represented by squares and the non-root vertices by circles. The decoration of every vertex will be depicted as a label inside the square or circle. The roots in all decorated graphs appearing in a constraint will be placed on the same mutual positions, i.e., the corresponding roots of different graphs in the constraint are on the same respective positions. Edges are represented as solid lines between vertices and non-edges are represented as dashed lines. The absence of any line between two root vertices indicates that the constraint should hold for both cases when the edge between the root vertices is present and when it is not present. Finally, the absence of a line between a non-root vertex and another vertex represent the sum of decorated graphs with this edge present and without this edge. Thus, if k such lines are absent in a decorated graph, the figure represent the sum of 2k decorated graphs. An example of such a figure and evaluation of decorated constraint can be found in Figure 2.1. We now give an example. We consider the graphon W depicted in the left part of Figure 2.1. The graphon W has three parts A, B1 and B2 , each of measure 1/3; the A is 2/3, the degree of B1 is 7/9 and the degree of B2 is 1. The densities between the parts are as given in the figure. We next consider a decorated graph H depicted in the right part of Figure 2.1. The graph H has two roots v1 and v2 7

that are adjacent and decorated with B1 and A, respectively, and it has two nonroot vertices v3 and v4 that are also adjacent and decorated with {B1 , B2 } and A, respectively. The vertex v3 is adjacent to both roots and v4 is adjacent only to v1 . The probability described in the previous paragraph is independent of the choice of x1 and x2 in B1 and A and is equal to 13/96. In particular, the graphon W satisfies the decorated constraint H = 13/96 depicted in Figure 2.1.

A B1 B2 A

1/2 1/2

1

B1

1/2 1/3

1

B2

A

= B1

1

1

B∗ A

13 96

1

Figure 2.1: An example of evaluating a decorated constraint. The symbol B∗ denotes the set {B1 , B2 }. In [13], decorated constraints where each vertex is decorated with a single element were considered. Let us call such decorated constraints simple, i.e., a decorated constraint is simple if all decorations appearing in it are single element sets. It was proved in [13] that simple decorated constraints are equivalent to (ordinary) constraints. Lemma 7. For every set of parts P and a simple decorated constraint C, there exists a constraint C ′ such that any partitioned graphon W with parts P satisfies C if and only if it satisfies C ′ . We now prove that every decorated constraint is equivalent to a set of simple decorated constraints. This implies that decorated constraint are in fact equivalent to (ordinary) constraints by Lemma 7. Lemma 8. For every set of parts P and a decorated constraint C, there exists a finite set of simple decorated constraints C ′ such that any partitioned graphon W with parts P satisfies C if and only if it satisfies all constraints in C ′ . Proof. Let H0 be the decorated graph induced by the roots v1 , . . . , vm of the decorated graphs in C. Since the graphs are mutually compatible, every vertex vi is decorated with a family of parts Pi ⊆ P and the decoration is the same in all graphs appearing in C. The set of simple decorated constraints C ′ will consist of constraints C P1 ,...,Pm for every m-tuple (P1 , . . . , Pm ) ∈ P1 × · · · × Pm . The simple decorated constraint C P1 ,...,Pm will treat the case (x1 , . . . , xm ) ∈ P1 × · · · × Pm from the procedure of evaluating decorated constraints. Fix some m-tuple (P1 , . . . , Pm ) ∈ P1 × · · · × Pm . Let H be a decorated graph appearing in C and u1 , . . . , um′ be the non-root vertices of H, with ui decorated with 8

P1 ,...,Pm a family Qi ⊆ P. For an m′ -tuple (Q1 , . . . , Qm′ ) ∈ Q1 × · · · × Qm′ , let HQ 1 ,...,Qm′ be the same graph as H, except every root vertex vi is decorated with Pi and every P non-root vertex ui is decorated with Qi . Finally, we denote the sum Q∈Qi |Q| by Si . The simple decorated constraint C P1 ,...,Pm has the same structure as C, except each decorated graph H is replaced by the combination ′

m Y |Qi |

X

i=1

(Q1 ,...,Qm′ )∈Q1 ×···×Qm′

Si

!

P1 ,...,Pm . HQ 1 ,...,Q ′ m

The law of total probability implies that, for a fixed m-tuple (x1 , . . . , xm ) ∈ P1 ×· · ·× Pm , the density the of the above combination is equal to the density of H. Thus, the simple decorated constraint C P1 ,...,Pm is satisfied if and only if the decorated constraint C has equal values on both sides for almost every (x1 , . . . , xm ) ∈ P1 × · · · × Pm such that W (xi , xj ) > 0 for every edge vi vj ∈ E(H0 ) and W (xi , xj ) < 1 for every non-edge vi vj ∈ / E(H0 ). Since we include a constraint C P1 ,...,Pm for every (P1 , . . . , Pm ) ∈ P1 × · · · × Pm , we are done. It has been proved that a finitely forcible graphon can be forced on a single part of a partitioned graphon without altering the structure of the rest of the graphon. We need to generalize this assertion to a union of parts. Lemma 9. Let W0 be a finitely forcible graphon, P a set of parts and P ⊆ P. There exists a finite set C of decorated constraints such that every partitioned graphon S W with parts P satisfies C if and only if the subgraphon of W induced by P is weakly isomorphic to W0 . Proof. Assume that W0 is forced by densities d(Hi , W ) = di , i ∈ [m]. The set C is then formed by the decorated constraints d(Hi′ , W ) = di , i ∈ [m], where Hi′ is the graph Hi with all vertices decorated with P. We conclude with stating a lemma, which appeared implicitly in [20, proof of Lemma 3.3], and was explicitly stated in [8, Lemma 8]. Lemma 10. Let X, Z ⊆ R be measurable non-null sets and let F : X × Z → [0, 1] be a measurable function. If there exists a constant C ∈ R such that Z F (x, z)F (x′ , z) dz = C Z

9

for almost every (x, x′ ) ∈ X 2 , then it holds that Z F (x, z)2 dz = C Z

for almost every x ∈ X.

2.3

Cut norm and graphon regularity

Let W be a graphon and U1 , U2 measurable subsets of [0, 1]. We define the density between U1 and U2 in W as Z W (x, y) dx dy. dW (U1 , U2 ) = U1 ×U2

We drop the subscript if W is clear from context. A graphon W is a step graphon if there exists a partition of the unit interval [0, 1] into non-null measurable sets U1 , . . . , Uk ⊆ [0, 1] such that W is constant on Ui × Uj for every i, j ∈ [k]. Let W1 and W2 be step graphons with partitions ′ , respectively. We say that W refines W if every for U1 , . . . , Uk and U1′ , . . . , UK 2 1 ′ every i ∈ [K], the part Ui is a subset of one of the parts U1 , . . . , Uk , and the densities between Ui and Uj in W1 and W2 are equal, i.e., dW1 (Ui , Uj ) = dW2 (Ui , Uj ) =

X

dW2 (Uu′ , Uv′ )

u,v∈[K] s.t. Uu′ ⊆Ui , Uv′ ⊆Uj

for every i, j ∈ [k]. We now recall the notion of the cut norm. Let X : [0, 1]2 → [−1, 1] be a measurable function, then the cut norm of X, denoted kXk , is defined as Z X(x, y) dx dy , sup A,B⊆[0,1]

A×B

where the supremum is taken over all pairs of measurable subsets of [0, 1]. The cut norm induces a pseudometric in the space of graphons. We say that two graphons W1 and W2 are ε-close if kW1 − W2 k ≤ ε. By [19, Section 10.5], it holds that if W1 and W2 are graphons and H is a k-vertex graph, then   k |d(H, W1 ) − d(H, W2 )| ≤ kW1 − W2 k . (2.1) 2 For a graphon W , a partition of the unit inverval [0, 1] into non-null measur-

10

able sets U1 , . . . , Uk is called ε-regular if X |Ui ∩ A||Uj ∩ B| ≤ε dW (A, B) − d (U , U ) W i j |Ui ||Uj | i,j∈[k]

for every two measurable sets A, B ⊆ [0, 1]. In other words, the step graphon with d (Ui ,Uj ) partition U1 , . . . , Uk that is equal to W|Ui ||U on Ui × Uj is ε-close to W . j We next recall the Weak Regularity Lemma of Frieze and Kannan [11], which allows us to approximate graphons by step graphons. Originally stated for finite graphs, the lemma extends to graphons as follows: for every ε > 0, there exists a constant K ∈ N such that every graphon has an ε-regular partition with at most K parts. In fact, we will need the following slightly stronger version of the statement, its proof readily follows from the proof of the standard version, e.g., the one presented in [19, Section 9.2]. Proposition 11. For every ε > 0 and k ∈ N, there exists K ∈ N such that for every graphon W and every partition U1 , . . . , Uk of [0, 1] into measurable non-null ′ of [0, 1] with K ′ ≤ K such that sets, there exists an ε-regular partition U1′ , . . . , UK ′ every part Ui′ , i ∈ [K ′ ], is a subset of one of the parts U1 , . . . , Uk . We now define an auxiliary graphon parameter, that will be useful to us due to its properties relative to the cut norm. For a graphon W , we define d(Γ4 , W ) as the integral Z W (x1 , x2 )W (x2 , x3 )W (x3 , x4 )W (x4 , x1 ) dx1 dx2 dx3 dx4 . d(Γ4 , W ) = [0,1]4

Note that d(Γ4 , W ) can be seen as the density of non-induced C4 in a W -random graph. The inequality (2.1) also holds for d(Γ4 , W ), i.e., we have |d(Γ4 , W ) − d(Γ4 , W ′ )| ≤ 6kW − W ′ k

(2.2)

for any two graphons W, W ′. It can be shown that d(Γ4 , W ) ≥ kW k4 for every graphon W , see e.g. [19, Section 8.2]. However, we will need a stronger version of this inequality for step graphons. The following lemma was proved in [9]. Lemma 12. Let W0 be a step graphon with all parts of the same size, and W a step graphon refining W0 such that all parts of W also have the same size. If kW − W0 k ≥ ε, then d(Γ4 , W ) ≥ d(Γ4 , W0 ) + ε4 /8. We use the following generalization of Lemma 12 to step graphons with parts of arbitrary sizes. Lemma 13. Let W0 and W be step graphons such that W refines W0 . If kW − W0 k ≥ ε, then d(Γ4 , W ) ≥ d(Γ4 , W0 ) + ε4 /256. 11

Proof. First, we realize that Lemma 12 immediately extends to step graphons with all parts of rational sizes. Indeed, we can refine such a partition into parts of the same size without altering the graphon itself. Also, we may assume that all parts of W and W0 are intervals, since one can invoke the Monotone Reordering Theorem with a function h : [0, 1] → [0, 1] assigning every part of W a unique value, and apply the resulting measure preserving transformation ϕ to both W and W0 . We now describe a perturbation procedure parametrized by δ > 0, a small constant chosen later. Let U1 , . . . , UK be the partition of W into intervals so that sup Ui = inf Ui+1 for every i ∈ [K − 1]; we denote sup Ui = ai . The partition ′ is obtained by perturbing the points a , . . . , a U1′ , . . . , UK 1 K−1 , along with the endpoints of the corresponding intervals, so that they become rational points and they ′ change by at most δ. The graphon W ′ is a step graphon with partition U1′ , . . . , UK so that W ′ on Ui′ × Uj′ is equal to W on Ui × Uj , for every i, j ∈ [K]. Observe that kW − W ′ k ≤ ((K − 1)δ)2 Let V1 , . . . , Vk be the partition of W0 into intervals so that sup Vi = inf Vi+1 for every i ∈ [k−1]. Since W refines W0 , each interval Vi is a union of Uj , Uj+1 , . . . , Uj+ℓ for some ℓ ∈ [K − 1] and j ∈ [K − ℓ]. The partition V1′ , . . . , Vk′ is obtained by perturbing the endpoints of V1 , . . . , Vk in accordance with the corresponding endpoints of the partition of W ′ . We define the graphon W0′ as the step graphon with partition V1′ , . . . , Vk′ that is refined by W ′ . In other words, W0′ on Vi′ × Vj′ is equal to dW ′ (Vi′ , Vj′ ) 1 = ′ ′ |Vi′ ||Vj′ | |Vi ||Vj |

X

dW ′ (Uu′ , Uv′ ).

u,v∈[K] s.t. Uu′ ⊆Vi′ ,Uv′ ⊆Vj′

Observe that for every i, j ∈ [K], we have dW (Ui , Uj ) − dW ′ (Ui′ , Uj′ ) ≤ (2δ)2 .

(2.3)

Since dW0′ (Vi′ , Vj′ ) is a sum of at most K 2 densities dW ′ (Uu′ , Uv′ ) and W refines W0 , 2.3 implies that for every i, j ∈ [k], we have ′ ′ ′ dW0 (Vi , Vj ) − dW0 (Vi , Vj ) ≤ K 2 (2δ)2 .

We obtain that the cut distance of W0 and W0′ satisfies

kW0 − W0′ k ≤ ((k − 1)δ)2 + k2 K 2 (2δ)2 . Clearly, it is possible to choose δ > 0 so that kW − W ′ k ≤ ε4 /3072 and kW0 − W0′ k ≤ ε4 /3072. It follows that kW ′ − W0′ k ≥ ε − 2ε4 /3072 ≥ ε/2. We can now apply Lemma 12 to the step graphons W ′ and W0′ with partitions appropriately

12

refined so that all parts have the same size, to obtain that d(Γ4 , W ′ ) ≥ d(Γ4 , W0′ ) + ε4 /128. By (2.2), we have |d(Γ4 , W ) − d(Γ4 − W ′ )| ≤ 6 ·

ε4 ε4 = , 3072 512

|d(Γ4 , W0 ) − d(Γ4 − W0′ )| ≤ 6 ·

ε4 ε4 = . 3072 512

It follows that d(Γ4 , W ) ≥ d(Γ4 , W0 ) + ε4 /256, as desired.

13

Chapter 3

Proof of the main result We now present the proof of Theorem 2.

3.1

Preprocessing of the graphon W⊗

In this section, we transform the given graphon W⊗ according the degree distribution, to obtain a graphon W⊗s , which is weakly isomorphic to W⊗ . The graphon W⊗s will be directly embedded into W0 as a subgraphon induced by the interval [0, 1− ε), as we describe in Section 3.2. Fix a real ε > 0 and a graphon W⊗ . We can assume that 1/ε is an integer, since otherwise we can apply the theorem to the graphon W⊗′ that contains W⊗ as a subgraphon on a set of measure (1 − ε)/(1 − ε′ ) with ε′ = 1/⌈1/ε⌉. Let M be defined as the integer 4(1/ε − 1). We partition the vertices of W⊗ into M disjoint subsets according to their degree as follows,    k−1 k Qk = x ∈ [0, 1] : degW⊗ (x) ∈ , M M for k ∈ [M ], with QM also containing the vertices of degree one. Note that Q1 , . . . , QM is thus a family of disjoint subsets of [0, 1] and their union has full measure.  Pk−1
Pk For every k ∈ [M ], let Pk denote the interval i=1 |Qi |, i=1 |Qi | and fix an arbitrary measure preserving map ψk : Pk → Qk . We then combine the functions ψ1 , . . . , ψM into a measure preserving map ψ⊗ : [0, 1) → [0, 1] defined as ψ⊗ (x) = ψk (x) for x ∈ Pk . Finally, let W⊗s be a graphon such that W⊗s (x, y) = W⊗ (ψ⊗ (x), ψ⊗ (y))

14

for every x, y ∈ [0, 1). Clearly, W⊗s is weakly isomorphic to W⊗ . One should view the graphon W⊗s as the graphon W⊗ with vertices permuted so that the sets Q1 , . . . , QM form intervals. In particular, observe that    k−1 k , (3.1) Pk = x ∈ [0, 1] : degW⊗s (x) ∈ M M for every k ∈ [M ], with PM also containing all vertices of degree one in W⊗s , where the equality is up to a null set. In fact, since we can change W⊗s on a null set, we shall assume the equalities (3.1) hold exactly.

3.2

General structure of W0

We now provide a general overview of the structure of the constructed graphon W0 . The graphon W0 is a partitioned graphon with 3M + 3⌈log 2 M ⌉ + 6 parts, denoted P, which are categorized into 7 families as follows, P = A ∪ B ∪ C ∪ D ∪ E ∪ F ∪ G. These families are given in Table 3.1. A sketch of the placement of the parts P and the structure of W0 is given in 3.1. Family of parts

Corresponding constant

A = {A1 , . . . , AMA }

MA = M

B = {B1 , . . . , BMB , B∆ , BΣ }

MB = M

C = {C1 , . . . , CMC , C⋄ }

MC = ⌈log2 M ⌉

D = {D1 , . . . , DMD , D⋄ }

MD = ⌈log2 M ⌉

E = {E1 , . . . , EME , E⋄ }

ME = ⌈log2 M ⌉ − 1

F = {F1 , . . . , FMF }

MF = M

G = {G1 , G2 } Table 3.1: The families of parts of the partitioned graphon W0 For a part X ∈ P, we will use X 0 to denote the particular subset of [0, 1] that forms the part X in W0 . Similarly, for a family of parts X ⊆ P, we let X 0 denote the family of the sets X 0 such that X ∈ X . In the graphon W0 , all the parts are placed as subintervals of [0, 1] of the

15

ε/4

1−ε

A

ε/2

ε/4

W⊗s

B C D E F

G1

G2

Figure 3.1: A sketch of the graphon W0 . Note that the composite tile A×A contains the graphon W⊗ and the composite tile B × B contains the CKM graphon. C

D

E

F

Ak or Bk

Figure 3.2: Structure of the graphon W0 on the composite tiles Ak × (C ∪ D ∪ E ∪ F) and Bk × (C ∪ D ∪ E ∪ F) for k ∈ [M ]. Note that the measure of the parts Ak or Bk may be very different from the measure of the parts C, D, E, F.

16

form [a, b) and they appear in the order as in Table 3.1. In other words, 
A01 = 0, |A1 | , 
A02 = |A1 |, |A1 | + |A2 | , .. . 
0 G2 = 1 − |G2 |, 1 .

We give the measures of the parts later in this section. The graphon W⊗s is contained in the composite tile A × A as follows. For every x, y ∈ [0, 1), we define W0 ((1 − ε) · x, (1 − ε) · y) = W⊗s (x, y). The part Ak ∈ A in W0 is then formed by vertices in the set (1 − ε)Pk , for every k ∈ [M ], i.e., A0k = (1 − ε)Pk . Note that some of the sets A0k may have measure zero–we will not treat such cases, as one can omit these parts from W0 in the end. The families other than A consist of auxiliary parts that serve to ensure that W0 is finitely forcible. In particular, the parts G1 , G2 are used to enforce the property of being a partitioned graphon with the correct partition. The part G1 , set in Subsection 3.7.2, serves to balance the degrees, i.e., we choose the values of W0 on a tile G1 × X for X ∈ A ∪ · · · ∪ F so that for every (x, y) ∈ X 0 × X 0 , we have Z Z W0 (y, z) dz. W0 (x, z) dz = [0,1]\G02

[0,1]\G02

The part G2 , set in Subsection 3.7.3, is then used to distinguish degrees of vertices in different parts, i.e., the values of W0 on a tile G2 × X for X ∈ P will be chosen so that if x, y ∈ [0, 1] belong to different parts, then deg(x) 6= deg(y). The final choice of degrees of all parts (including G1 , G2 ) is postponed until later, but it will only depend on ε and W⊗ . Instead, for every part X ∈ A∪· · ·∪F, we now set a real number from [0, 1], called the pre-degree of X and denoted pre-deg(X). In Subsection 3.7.2, it will be shown that for every x ∈ X 0 , Z W0 (x, z) dz = pre-deg(X). (3.2) [0,1]\G02

Note that (3.2) implies that for every X ∈ A ∪ · · · ∪ F and x ∈ X 0 , Z W0 (x, z) dz. deg(X) = pre-deg(X) + G02

The graphon W0 is constant on a tile G2 × X for X ∈ P, with the value chosen 17

with the sole purpose of distinguishing the degrees of different parts. In Table 3.2, we give the measures of all parts and pre-degrees of the parts from A ∪ · · · ∪ F. Part

Measure

Pre-Degree

Ak

(1 − ε)|Pk |

Bk

ε 20

ε(k+1) 4 ε 4 ε 4 ε 4 1−ε + 4ε 2k−1 1−ε + 4ε 2MC 1−ε + 4ε 2k 1−ε + 4ε 2ME +1 ε(k+1) 4

B∆ BΣ Ck , Dk C⋄ , D⋄ Ek E⋄ Fk G1 G2

1 13 · |Pk | ε 1 20 · 13 11 ε 20 · 13 ε 1 20 · 2k 1 ε 20 · 2MC ε 1 20 · 2k ε 1 20 · 2ME 1 ε 20 · MF ε 2 ε 4

·

Table 3.2: The sizes and pre-degrees of the parts P. Remark 1. We would now like to remark that the use of families of parts instead of single parts (and the use of decorated constraints) is due to technical difficulties with degree balancing. For example, the composite tiles Ak × E for k ∈ [M ] will all contain the checker graphon (see Figure 3.2; the checker graphon is formally defined S in Subsection 3.5.1), which means that the degrees of vertices E 0 will vary so much that we wouldn’t be able to balance them to a uniform degree only using the values S of W0 on G1 × E, since the measure of G1 is too small. Hence, the vertices E 0 S cannot form a single part of W0 . We resolve this by partitioning E 0 into small enough parts, so that we can balance them with suitably chosen values of G1 × E. We advise the reader to view the families C, D, E, F as single parts, which will later be partitioned according to degree in a fashion similar to the partitioning of W⊗s . Indeed, the forcing of auxiliary composite tiles will use known methods which were previously used in [8, 9, 12, 13] to force simple tiles. Once the composite tiles are established, we ensure that they are correctly partitioned into simple tiles by considering the degrees. Rather than giving a complete definition of the graphon W0 at once, we decided to present the particular details of the structure of W0 together with the decorated constraints fixing its structure in the following sections. Before proceeding, we define a convenient notation, which we will use when defining W0 on particular tiles. For a non-empty family of parts X ⊂ P such that 18

X 0 is an interval, let γX denote the linear bijective function γX : [0, 1) → mapping [ [ x 7→ x · X 0 + min X 0 . S

S

X0

If X is a single set, we may write its unique element instead of the single element set. For example, W0 (γA (x), γA (y)) = W⊗s (x, y) for every x, y ∈ [0, 1), as we defined above.

3.3

The finitely forcible graphon W CKM

In this section, we introduce a finitely forcible graphon W CKM , which is obtained from Theorem 1. The graphon W CKM will be embedded in W0 on the composite tile B × B. First, let us define the half-graphon W∆ by W∆ (x, y) = 1 if x + y ≥ 1 and W∆ (x, y) = 0 otherwise, for every x, y ∈ [0, 1]. We will be using a somewhat stronger statement than Theorem 1, which readily follows from the proof in [9]. Theorem 14. For every graphon W⊗∗ , there exists a finitely forcible graphon W CKM such that W⊗∗ is a subgraphon of W CKM . Moreover, W CKM is a partitioned graphon with two distinguished parts GCKM and P CKM that satisfy the following, (a) the part GCKM has degree 78/234, measure 1/13 and is formed by the vertices [0, 1/13) of W CKM , (b) the part P CKM has degree 79/234, measure 1/13 and is formed by the vertices [1/13, 2/13) of W CKM , (c) the graphon W⊗∗ is contained on the tile GCKM × GCKM , i.e., W CKM

x y = W⊗∗ (x, y), , 13 13

for every x, y ∈ [0, 1), (d) the tile GCKM × P CKM contains the half-graphon W∆ , i.e., W

CKM



x y 1 , + 13 13 13



=

( 1 0

if x + y ≥ 1, otherwise,

for every x, y ∈ [0, 1), (e) for every graphon W1 that is weakly isomorphic to W CKM , there exists a measure preserving map g1 : [0, 1] → [0, 1] such that W CKM (g1 (x), g1 (y)) = W1 (x, y) for almost every (x, y) ∈ [0, 1]2 . 19

The graphon W CKM in Theorem 14 is the graphon W0 constructed in [9]. The partitioning of W CKM is given in [9, Section 3.0], the part GCKM corresponds to the part G and the part P CKM corresponds to the part P . In [9], the vertices of G are not identified with the interval [0, 1/13], nor are the vertices of P identified with the interval [1/13, 2/13]; however, this can be easily arranged by applying a measure preserving map which simply permutes the parts of W CKM . The item (e) also follows from [9, Section 3.0], since if W1 is weakly isomorphic to W CKM , it also satisfies all the decorated constraints given in [9] and we can set g1 to be the function g from the cited paper. We now apply Theorem 14 with W⊗s playing the role of W⊗∗ , to obtain the finitely forcible graphon W CKM with two distinguished parts GCKM and P CKM as in the statement. The graphon W0 is defined on the composite tile B × B as W0 (γB (x), γB (y)) = W CKM (x, y) for every x, y ∈ [0, 1). Observe that by the placement of the parts B in W0 , we have
0 = γB GCKM , B10 ∪ · · · ∪ BM B

3.4


0 = γB P CKM , B∆

0 BΣ = γB ([2/13, 1)).

Lining up the vertices

We now start the proof of finite forcibility of the graphon W0 . Let W be a graphon that satisfies all of the following sets of constraints, (a) the constraints from Lemma 6 applied to the parts P introduced in Section 3.2 (the final choice of degrees of parts is postponed until Subsection 3.7.3; however, they only depend on ε and W⊗ ), (b) the decorated constraints from Lemma 9 applied to the finitely forcible graphon W CKM and the family of parts B ⊂ P, (c) the decorated constraints given in the Subsections 3.4.2-3.7.3. Our goal is to show that W is weakly isomorphic to W0 , establishing that W0 is finitely forcible.

3.4.1

The measure preserving map g

The purpose of this subsection is, informally, to construct a measure preserving map g that permutes • the vertices in the parts A ∪ C ∪ D ∪ E ∪ F of W based on their relative degree with respect to the family of parts F,

20

• the vertices in the parts B of W according to the function g1 from item (e) of Theorem 14, • the vertices in the parts G1 and G2 of W arbitrarily, and maps the permuted vertices to the corresponding families of parts of W0 . Recall that X 0 denotes the interval where the part X is placed in W0 . As we will now mostly talk about the graphon W , from now on we will identify every part X ∈ P with the set of vertices of W having the corresponding degree. Note that by definition and by Lemma 6, for every part X ∈ P, we have |X| = |X 0 | and degW (x) = degW0 (y) for every x ∈ X and y ∈ X 0 . As a first step, it is useful to “line up” the vertices of W with the vertices of W0 . First, for G, we simply fix an arbitrary measure preserving map ηGi from Gi to the interval G0i , for both i ∈ {1, 2}. Next, let us consider the vertices in families A, C, D, E and F. The Monotone Reordering Theorem implies that for every S S X ∈ {A, C ∪D ∪E, F}, there exists a measure preserving map ϕX : X → [0, | X |) S and a non-decreasing function fX : [0, | X |) → [0, 1] such that fX (ϕX (x)) = degF W (x)

S for almost every x ∈ X . We proceed with the family B. We introduce a graphon WB , which will be the subgraphon of W induced by the family of parts B. By the Measure Isomorphism S S Theorem, there exist subsets Y1 ⊑ [0, | B|) and Y2 ⊑ B and a measure preserving bijection ν : Y1 → Y2 . Let WB be the graphon defined almost everywhere on [0, 1]2 by setting  [ [ −1  −1 WB B · x, B · y = W (ν (x) , ν (y))

for every x, y ∈ Y1 . By Lemma 9, WB is weakly isomorphic to the graphon W CKM . Let g1 be the function from item (e) of Theorem 14 with W1 chosen as WB . Let S −1 νs : Y2 → [0, 1) denote the function x 7→ B · ν −1 (x). Observe that W CKM (g1 (νs (x)) , g1 (νs (y))) = WB (νs (x), νs (y)) = W (x, y)

for almost every (x, y) ∈

S

B×

S

B. Finally, we define a map ψB : Y2 →



ψB (x) = γB g1 νs (x) .

S

B 0 by

It is straightforward to verify that ψB is measure preserving. We extend ψB to a S S measure preserving map from B to B 0 by simply setting ψB (x) = 0 for x ∈ S B \ Y2 . 21

Finally, we combine the measure preserving maps described above to obtain the measure preserving map g : [0, 1) → [0, 1) defined almost everywhere as follows,  ϕA (x)      0   ϕC∪D∪E (x) + min C1 g(x) = ϕF (x) + min F10     ψB (x)     ηGi (x)

if x ∈ if x ∈ if x ∈ if x ∈

S

A,

S

(C ∪ D ∪ E),

S

B,

S

F,

if x ∈ Gi , i ∈ {1, 2}.

For a part X of W , we will use X g to refer to the set g(X). Similarly, for a family of parts X of W , X g denotes the family of the sets g(X) such that X ∈ X . Observe that by the definition of g, we have [

[ [ [ Ag = A0 , (C g ∪ D g ∪ E g ) = (C 0 ∪ D 0 ∪ E 0 ), [ [ [ [ Fg = F 0, Bg = B 0 , Gg1 = G01 , Gg2 = G02 ,

(3.3)

where all the equalities hold up to a null set. Moreover, by the Monotone Reordering S S Theorem, for almost every pair (x, y) from X × X for X ∈ {A, C ∪ D ∪ E, F}, F if g(x) ≤ g(y), then degF W (x) ≤ degW (y). Our goal will now be to show that W (x, y) = W0 (g(x), g(y)) for almost every (x, y) ∈ [0, 1)2 . Note that by the definition of W0 on the tile B × B and the construction of the map ψB , we can already conclude that W (x, y) = W0 (g(x), g(y)) S S for almost every (x, y) ∈ B × B.

3.4.2

Aligning the CKM graphon

In this subsection, we analyze the decorated constraints aligning the parts GCKM and P CKM of the graphon W CKM embedded on the composite tile B × B with the parts B∆ and BΣ . Let B∗ denote the family {B1 , B2 , . . . , BMB } = B \ {B∆ , BΣ }. B

= B∗

B

78 234

= B∆

79 234

Figure 3.3: Decorated constraints aligning the parts B∆ and BΣ . Consider the decorated constraints from Figure 3.3. The first constraint S implies that for almost every x ∈ B∗ , we have degB W (x) = 78/234. Similarly, the second constraint yields that for almost every x ∈ B∆ , it holds that degB W (x) = 79/234. By the previous subsection and items (a) and (b) of Theorem 14, we obtain

22

that [

B∗g = γB (GCKM ) =

[

B∗0 ,

g 0 B∆ = γB (P CKM ) = B∆ . S S Since we have already concluded that B g = B 0 , it follows that g 0 = γB ([2/13, 1)) = BΣ . BΣ

3.4.3

Coordinate system

In this subsection, we analyze the composite tiles F × F and X × F for X ∈ {A, B∗ , C ∪ D ∪ E}. We follow the ideas of [9, Section 3.1], however, we present the full argument here, as there are some differences. The graphon W0 on the tile F × F contains the half-graphon W∆ , defined in Section 3.3. Formally, we define W0 on F × F by W0 (γF (x), γF (y)) = W∆ (x, y) for every x, y ∈ [0, 1). The half-graphon W∆ is finitely forcible as shown in [10, 20]. Note that the degree distribution of the half-graphon satisfies that degW∆ (x) = x for every x ∈ [0, 1]. Consider the decorated constraints from Lemma 9 forcing the composite tile F × F to be weakly isomorphic to the half-graphon. Recall that for almost S S F every (x, y) ∈ F × F such that g(x) ≤ g(y), we have degF W (x) ≤ degW (y). By Lemma 9, we obtain that
−1 −1 degF W (x) = degW∆ γF (g(x)) = γF (g(x))

for almost every x ∈ S S (x, y) ∈ F × F. X

S

F. It follows that W (x, y) = W0 (g(x), g(y)) for almost every

X

X

=0 F

F

=

F

F

B∗

B∗

F

F

B∆

=0

Figure 3.4: Decorated constraints forcing the tiles X × F, X ∈ {A, B∗ , C ∪ D ∪ E}. Fix some X ∈ {A, B∗ , C ∪ D ∪ E}. We define the graphon W0 on the composite tile X × F by W0 (γX (x), γF (y)) = W∆ (x, y) for every x, y ∈ [0, 1). Consider the decorated constraints depicted in Figure 3.4. The first constraint implies that S S W (x, y) ∈ {0, 1} for almost every (x, y) ∈ X × F and that N X (x) ⊑ N X (x′ ) S S or N X (x′ ) ⊑ N X (x) for almost every (x, x′ ) ∈ F × F. The second constraint implies that −1 F degX W (x) = degW (x) = γF (g(x)) 23

S for almost every x ∈ F. S S As before, recall that for almost every (x, y) ∈ X × X such that g(x) ≤ g(y), we have degF (x) ≤ degF (y). If X ∈ {A, C ∪ D ∪ E}, by the construction of the map g, we already obtain that W (x, y) = W0 (g(x), g(y)) for almost every S S (x, y) ∈ X × F. If X = B∗ , the third constraint implies that for almost every S S (x, x′ ) ∈ B∗ × B∗ , N F (x) ⊑ N F (x′ ) if and only if N B∆ (x) ⊑ N B∆ (x′ ).

(3.4)

The composite tile B∗ × B∆ contains the half-graphon W∆ by Theorem 14 and Subsection 3.4.2, i.e.,   −1 (g(y)) (g(x)), γ W (x, y) = W0 (g(x), g(y)) = W∆ γB−1 B∆ ∗

S for almost every (x, y) ∈ B∗ ×B∆ . Combined with (3.4), we obtain that W (x, y) = S S W0 (g(x), g(y)) for almost every (x, y) ∈ B∗ × F. S We finish this subsection by observing that for almost every (x, y) ∈ X × S X with X ∈ {A, B∗ , C ∪ D ∪ E, F}, the following assertions are equivalent, (a) g(x) ≤ g(y),

(b) degF (x) ≤ degF (y), (c) N F (x) ⊒ N F (y).

3.4.4

Ordering the parts

We now introduce and analyze constraints forcing that parts of W are mapped by g to the corresponding parts of W0 , i.e., that X g = X 0 for every X ∈ P. F

=0 X

Y

Figure 3.5: The decorated constraint forcing that g(x) ≤ g(y) for almost every (x, y) ∈ X × Y . Consider the decorated constraint in Figure 3.5 with (X, Y ) such that one of the following holds, • X = Ai and Y = Ai+1 for i ∈ [MA − 1], • X = Bi and Y = Bi+1 for i ∈ [MB − 1], • X, Y ∈ C ∪ D ∪ E and sup X 0 = min Y 0 . 24

• X = Fi and Y = Fi+1 for i ∈ [MF − 1]. The constraint implies that for almost every (x, y) ∈ X×Y , we have N F (x) ⊒ N F (y). By the previous subsection, it follows that g(x) ≤ g(y) for almost every (x, y) ∈ X × Y . This forces that for every family X ∈ {A, B∗ , C ∪ D ∪ E, F} and two parts X1 , X2 ∈ X , if X1 precedes X2 in the graphon W0 , i.e., sup X10 ≤ min X20 , then X1g precedes X2g , up to a null set, i.e., g(x) ≤ g(y) for almost every (x, y) ∈ X1 × X2 . S By (3.3) in Subsection 3.4, we have already concluded that the set X g is equal S to X 0 , up to a null set. Moreover, we have Gg1 = G01 and Gg2 = G02 and, by g 0 and B g = B 0 , with all equalities up to a null set. We Subsection 3.4.2, B∆ = B∆ Σ Σ obtain that X g = X 0, up to a null set, for every part X ∈ P.

3.5 3.5.1

Forcing auxiliary tiles Checker tiles

We now consider the composite tiles E × E and E × X for X ∈ {A1 , . . . , AMA , B1 , . . . , BMB , C, D}.

(3.5)

Figure 3.6: The checker graphon WC . For k ∈ N0 , let Ik denote the interval [1 − 2−k , 1 − 2−k−1 ). The checker graphon WC is the graphon such that WC (x, y) = 1 if x, y ∈ Ik for some k ∈ N0 and WC (x, y) = 0 otherwise. The checker graphon is depicted in Figure 3.6. We set the graphon W0 on the composite tile E × E as W0 (γE (x), γE (y)) = WC (x, y) for every x, y ∈ [0, 1). For every X as in (3.5), we set the graphon on the composite tile E × X as W0 (γE (x), γX (y)) = WC (x, y) for every x, y ∈ [0, 1). The decorated constraints in Figure 3.7 force the structure of the tile E × E and the decorated constraints in Figure 3.8 force the structure of the tiles E × X for 25

F

E

=0

=0 E

E

E

E E E

E

E

E

=

F

F

E

E

F

=

E

E

1 3

Figure 3.7: The decorated constraints forcing the structure of the tile E × E. E

E

=0 E

X

=

E E

F

X

F

X

E

F

F

E

X

X

X

=0

= X

E

F

X

F

F

Figure 3.8: The decorated constraints forcing the structure of the tiles E × X for X ∈ {A1 , . . . , AMA , B1 , . . . , BMB , C, D}. X as in (3.5). The argument follows the lines of the analogous argument presented in [9, Section 3.2]; we omit the details. The structure of the tiles E × E and E × X for X ∈ A ∪ B∗ ∪ {C, D} will be used in the succeeding Subsections 3.5.2 and 3.5.3 to force the auxiliary tiles C × C, C × D, D × D, and C × X and D × X for X ∈ A ∪ B∗ .

3.5.2

Dyadic square indices

In this subsection we analyze the composite tiles C × C and C × D and D × D.

Figure 3.9: Illustration of the tiles C × C and C × D (the first picture) and D × D (the second picture).

26

For X ∈ {C, D}, we define the graphon W0 on the tile C×X as W0 (γC (x), γX (y)) = 1 for x, y ∈ [0, 1) if x and y belong to the same interval Ik , k ∈ N0 and 

   x − min Ik k y − min Ik k ·2 = ·2 , |Ik | |Ik |

and W0 (γC (x), γX (y)) = 0 otherwise. Similarly, W0 (γD (x), γD (y)) = 1 for x, y ∈ [0, 1) if x and y belong to the same interval Ik , k ∈ N0 and 

   y − min Ik 2k x − min Ik 2k ·2 ·2 = , |Ik | |Ik |

and W0 (γD (x), γD (y)) = 0 otherwise. Illustrations can be found in Figure 3.9. F

X X

X

C

!2

E

=2 C

C

D

E

=4 D

D

X

=0

=0 X

F X

C

=

C

!3

=0

X

X

∞  X

−(k+1)

2

k=0

D

=

D

∞  X

k=1

E

1 2k

1 2−(k+1) 2k 2

2 · 2k 2 · 22k

Figure 3.10: The decorated constraints forcing the tiles C × C and D × D, with X ∈ {C, D}. C

D

=0 D C

F

F

C C

=0 D

D

D

=0

C

=

C

C

D

D

C

C

F

D

=0 F

C

Figure 3.11: The decorated constraints forcing the tile C × D. The decorated constraints in Figure 3.10 force the structure of tiles C × C and D × D and the decorated constraints in Figure 3.11 force the structure of the 27

tile C × D. Again, the argument follows the lines of analogous argument presented in [9, Section 5.1]; we omit the details. The structure of the tiles C × C and C × D and D × D is used in the succeeding Subsection 3.5.3 to force the auxiliary tiles C × X and D × X for X ∈ A ∪ B∗ . Also, the structure of the tile C × D is used in the main argument in Section 3.6.

3.5.3

Referencing dyadic squares

In this subsection we analyze the composite tiles C × X and D × X for X ∈ A ∪ B∗ .

Figure 3.12: The tiles X × C and X × D for X ∈ A ∪ B∗ . For X ∈ A∪B∗ , we define the graphon W0 of the tiles C×X as W0 (γC (x), γX (y)) = 1 for x, y ∈ [0, 1) if x ∈ Ik , k ∈ N0 and 

 x − min Ik k · 2 = ⌊y · 2k ⌋ |Ik |

and W0 (γC (x), γX (y)) = 0 otherwise. Similarly, W0 (γD (x), γX (y)) = 1 for x, y ∈ [0, 1) if x ∈ Ik , k ∈ N0 and 

 x − min Ik 2k ·2 ≡ ⌊y · 2k ⌋ (mod 2k ) |Ik |

and W0 (γD (x), γX (y)) = 0 otherwise. Illustrations can be found in Figure 3.12. The decorated constraints in Figure 3.13 force the structure of tiles C × X and D × X for X ∈ A ∪ B∗ . Finally, the argument follows the lines of analogous argument used in [9, Section 5.2]; we omit the details. The structure of the auxiliary tiles C × X and D × X for X ∈ A ∪ B∗ is used in the succeeding Section 3.6 in the main part of the proof.

3.6

Forcing densities

In this section, we present the main argument of the proof. We use the auxiliary tiles established in the preceding sections to force the composite tile A × A, which contains the graphon W⊗s itself. Note that the argument is inspired by the one in [9,

28

Y

Y

F

F

X

X

X

Y

X X

E

=2 Y

X

Z

Y

=0

=0 X

Y

Y

F

=0 X

=0

Z Y

F

Y

Figure 3.13: The decorated constraints forcing the structure of the tiles C × X and D × X, where X ∈ A ∪ B∗ and (Y, Z) ∈ {(C, E), (D, C)}. Section 5.4], however, the setting is different and we are facing technical difficulties due to the increased generality. We present the argument in its entirety. Recall that W0 is defined on A×A as W0 (γA (x), γA (y)) = W⊗s (x, y) for every x, y ∈ [0, 1). Also, in Subsection 3.4.4, we concluded that Ai = (1 − ε)Pi for every i ∈ [M ]. By Subsection 3.4.1, we have
−1 −1 (g(y)) W (x, y) = W0 (g(x), g(y)) = W⊗s γB (g(x)), γB j i

(3.6)

for every i, j ∈ [M ] and almost every (x, y) ∈ Bi × Bj . Our goal now is to show that
−1 −1 W (x, y) = W⊗s γA (g(y)) (g(x)), γA j i

for every i, j ∈ [M ] and almost every (x, y) ∈ Ai × Aj . By the definition of W0 on A × A, this will imply that W (x, y) = W0 (g(x), g(y)) for almost every (x, y) ∈ S S A × A. First, we introduce some notation. For integers d ∈ N0 and s ∈ {0, 1, . . . , 2d − 1}, we use I d (s) to denote the interval [s/2d , (s + 1)/2d ). In other words, I d (s) is the s-th dyadic interval of order d. Let γPi : [0, 1) → Pi denote the bijective linear map x 7→ x · |Pi | + min Pi . We will be concerned with the partition γPi (I d (s)), i ∈ [M ], s ∈ {0, 1, . . . , 2d − 1} of the unit interval. This partition should be viewed as a refinement of the partition P1 , . . . , PM , with each interval Pi refined into dyadic intervals γPi (I d (s)), s ∈ {0, 1, . . . , 2d − 1}. Let us informally outline the argument. We shall examine the densities of s W⊗ on the squares

γPi I d (s) × γPj I d (t) for i, j ∈ [M ] and s, t ∈ {0, 1, . . . , 2d − 1}. We will use a decorated constraint to establish that for every d ∈ N0 and i, j ∈ [M ] and s, t ∈ {0, 1, . . . , 2d − 1}, this

29

density matches the density of W on the corresponding square  

 g −1 γAi I d (s) × g −1 γAj I d (t) .

Then, we employ an argument using graphon regularity and Lemma 13 to show that the subgraphon of W induced by the family of parts A is indeed weakly isomorphic to W⊗s . Ai

Aj

C

D

=

Bi

Bj

A

A

C

D

A

A

= d(Γ4 , W⊗s )

Figure 3.14: The decorated constraints forcing the structure of the tile A × A, where i, j ∈ [M ]. Fix some i, j ∈ [M ]. As outlined above, we claim that the first constraint in Figure 3.14 implies that for every d ∈ N0 and s, t ∈ {0, 1, . . . , 2d − 1},  





 dW g −1 γAi I d (s) , g−1 γAj I d (t) = (1 − ε)2 dW⊗s γPi I d (s) , γPj I d (t) . (3.7) 2 s Note that the factor (1 − ε) is simply the scaling of W⊗ as embedded in W0 . First, S −1 (g(u)) ∈ Id and choose some u ∈ D such that γD $

% −1 (g(u)) − min Id 2d γD = s · 2d + t. ·2 |Id |

S Then, choose some v ∈ C such that W (u, v) = 1. Structure of the tile D × C now implies that γC−1 (g(v)) ∈ Id and $

% γC−1 (g(v)) − min Id d · 2 = s. |Id |

S S Note there is a positive measure of such pairs (v, u) ∈ C × D. With the roots chosen as the vertices v, u, the left side of the first constraint in Figure 3.14 is then equal to Z 1 · W (x, y) dxy, |Ai ||Aj | with the integral taken over (x, y) ∈ Ai × Aj satisfying that j

k

−1 (g(x)) · 2d = s, γA i

j

k

−1 (g(y)) · 2d = γA j

30

$

% −1 (g(u)) − min Id 2d γD mod 2d = t, ·2 |Id |



which is equivalent to x ∈ g −1 γAi (I d (s)) and y ∈ g −1 γAj (I d (t)) . Therefore, the left side of the constraint is equal to 1 |Ai ||Aj |

Z





W (x, y) dx dy =

g −1 γAi (I d (s)) ×g −1 γAj (I d (t))




dW g−1 γAi (I d (s)) , g −1 γAj (I d (t)) . |Ai ||Aj |

By an analogous argument, we obtain that the right side of the constraint is equal to



dW⊗s γPi (I d (s)), γPj (I d (t)) dW g −1 γBi (I d (s)) , g−1 γBj (I d (t)) = , |Bi ||Bj | |Pi ||Pj | where the equality follows from (3.6). Finally, we recall that |Ai | = (1 − ε) · |Pi | for every i ∈ [M ], to obtain the equality (3.7). We proceed with the final argument, combining (3.7) with the second decoS rated constraint in Figure 3.14. Let π : [0, 1) → A be a measurable map such that S |X| = (1 − ε) · |π −1 (X)| for every measurable X ⊆ A. We next define a graphon WA by WA (x, y) = W (π(x), π(y)) for every x, y ∈ [0, 1). The graphon WA is thus the subgraphon of W induced by the family of parts A. Let ξ : [0, 1) → [0, 1) denote −1 (g(π(x))). Finally, we define a graphon W⊗ξ the measure preserving map x 7→ γA by W⊗ξ (x, y) = W⊗s (ξ(x), ξ(y)) for every x, y ∈ [0, 1). The graphon W⊗ξ should be viewed as the graphon W⊗s with vertices permuted according to the map π. Our goal now is to show that W⊗ξ is equal to WA almost everywhere. Suppose that kWA − W⊗ξ k = δ > 0. Let W d be the step graphon refining the partition ξ −1 (P1 ), . . . , ξ −1 (PM ) such that for every i, j ∈ [M ] and d ∈ N0 , s, t ∈ {0, 1, . . . , 2d − 1}, the value of W d on

ξ −1 γPi (I d (s)) × ξ −1 γPj (I d (t)) is equal to 

 22d · dW ξ ξ −1 γPi (I d (s)) , ξ −1 γPj (I d (t)) . ⊗ |Pi ||Pj |

The sequence (W d )d∈N0 forms a martingale on [0, 1]2 , and Doob’s Martingale Convergence Theorem implies that W d converges uniformly to W⊗ξ . Hence, there exists d ∈ N0 such that kW d − W⊗ξ k ≤ δ4 /(6 · 104 ). From now on, fix such d. Next, we apply Proposition 11 to the graphon WA and the partition of W d ,
i.e., the partition ξ −1 γPi (I d (s)) , i ∈ [M ], s ∈ {0, 1, . . . , 2d − 1}, to obtain a step 31

graphon WA′ that is δ4 /(6 · 104 )-close to WA . Crucially, we now observe that WA′ refines W d by equality (3.7):  



 dW d ξ −1 γPi (I d (s)) , ξ −1 γPj (I d (t)) = dW ξ ξ −1 γPi (I d (s)) , ξ −1 γPj (I d (t)) ⊗   d = dW⊗s γPi (I (s)), γPj (I d (t)) 
−1
 1 −1 d d = g γ (I (s)) , g · d γ (I (t)) A W A i j (1 − ε)2   

 = dWA π −1 g −1 γAi (I d (s)) , π −1 g −1 γAj (I d (t)) 

 = dWA′ ξ −1 γPi (I d (s)) , ξ −1 γPj (I d (t))

holds for every i, j ∈ [M ], s, t ∈ {0, 1, . . . , 2d − 1}. The first equality is simply the definition of W d , the second equality follows from the construction of ξ. The third equality is the crucial consequence of (3.7) and the fourth equality is implied by the definition of the map π and the graphon WA . Finally, the last equality follows from the fact that WA′ is the step graphon refining the partition of W d and approximating the graphon WA . By the triangle inequality of the cut norm, we have kW d − WA′ k ≥ δ − 2δ4 /(6 · 106 ) ≥ δ/2, which implies that d(Γ4 , WA′ ) − d(Γ4 , W d ) ≥

δ4 (δ/2)4 = 256 4096

(3.8)

by Lemma 13. On the other hand, by (2.2), we have |d(Γ4 , WA′ ) − d(Γ4 , WA )| ≤ 6kWA′ − WA k ≤ δ4 /104 , |d(Γ4 , W d ) − d(Γ4 , W⊗ξ )| ≤ 6kW d − W⊗ξ k ≤ δ4 /104 .

(3.9)

The equalities (3.8) and (3.9) yield that d(Γ4 , WA ) − d(Γ4 , W⊗ξ ) > 0. This contradicts the second constraint in Figure 3.14. Therefore, kWA − W⊗ξ k = 0, which is equivalent to that WA is equal to W⊗ξ almost everywhere. Finally, we obtain that W (π(x), π(y)) = WA (x, y) = W⊗ξ (x, y) = W⊗s (ξ(x), ξ(y)) = W0 (g(π(x)), g(π(y))) S for almost every (x, y) ∈ [0, 1)2 . Since π([0, 1)) ⊑ A, it follows that W (x, y) = S S W0 (g(x), g(y)) for almost every (x, y) ∈ A × A.

32

3.7

Cleaning up

In the remaining section, we finish the description of W0 and the proof that W0 is finitely forcible.

3.7.1

Finishing the tile (A ∪ · · · ∪ F )2

We start with the tiles A × B and (A ∪ C ∪ D ∪ E ∪ F) × {B∆ , BΣ }. Let X × Y be one of those tiles, we define the graphon W0 on X × Y as W0 (γX (x), γY (y)) = 0 for every x, y ∈ [0, 1). Y

=0 X

Figure 3.15: The decorated constraint forcing the structure of the tiles X × Y with (X , Y) ∈ {(A, B), (A ∪ C ∪ D ∪ E ∪ F, {B∆ , BΣ })}. The decorated constraint in Figure 3.15 yields that W (x, y) = W0 (g(x), g(y)) S S for almost every (x, y) ∈ X × Y. Note that we have by now completely established the structure of the composite tile (A ∪ · · · ∪ F)2 .

3.7.2

Degree balancing

It remains to guarantee that W0 is a partitioned graphon with the correct partition and show that we can force the same structure upon W. We use the vertices in the part G1 to ensure that for every X ∈ A ∪ · · · ∪ F and every x ∈ X 0 , we have Z W0 (x, y) dy = pre-deg(X), S (A0 ∪···∪F 0 )∪G01

where the values of pre-deg(X) for every part X are given in Table 3.2. First, we claim that for every part X ∈ A ∪ · · · ∪ F and every x ∈ X 0 , Z ε W0 (x, y) dy ≤ . 0 ≤ pre-deg(X) − S (3.10) 2 (A0 ∪···∪F 0 ) Since the measure of the union of B 0 ∪ · · · ∪ F 0 is equal to ε/4, it trivially holds that S for every x ∈ (A0 ∪ · · · ∪ F 0 ), we have Z ε W0 (x, y) dy ≤ . (3.11) S 0 4 (B ∪···∪F 0 ) If X = Ak for some k ∈ [M ], then, by the definition of the partition P1 , . . . , PM of 33

the vertices of W⊗s , we obtain that for every x ∈ A0k , k−1 (1 − ε) ≤ M

Z

S

A0

W0 (x, y) dy <

k (1 − ε) . M

By the definition of M and the value pre-deg(Ak ) = ε(k + 1)/4, this yields (3.10). R If X ∈ B, we have S A0 W0 (x, y) dy = 0 for every x ∈ X 0 , and, since pre-deg(X) = ε/4, we obtain (3.10) immediately by (3.11). If X = Ck for some k ∈ [MC ], then, by the structure of the tile A × C, Z 1−ε S 0 W0 (x, y) dy = 2k−1 A for every x ∈ X 0 . Since pre-deg(Ck ) = ε/4 + (1 − ε)/(2k−1 ), (3.10) again follows. If X = C⋄ , then for every x ∈ X 0 , Z ε 1−ε S 0 W0 (x, y) dy ≤ 2MC ≤ 4 , A which also implies (3.10). By analogous arguments, the families D, E, F are partitioned so that identical inequalities hold. X G1

G1



1 = |X|

ε 1− 4

Z

κX (x)2 dx

X0

Gc2

= pre-deg(X)

X

Figure 3.16: The decorated constraints forcing the structure of the simple tiles X × G1 for X ∈ A ∪ · · · ∪ F; the symbol Gc2 denotes the family A ∪ · · · ∪ F ∪ {G1 }. It is therefore possible to set W0 on the simple tiles X ×G1 for X ∈ A∪· · ·∪F as W0 (x, y) = κX (x) for every (x, y) ∈ X 0 × G01 , where 2 κX (x) = ε

pre-deg(X) −

Z

S

(A0 ∪···∪F 0 )

W0 (x, y) dy

!

.

We claim that the decorated constraints in Figure 3.16 force the structure of the tiles X × G1 for X ∈ A ∪ · · · ∪ F. The first constraint implies that almost every

34

(z, z ′ ) ∈ G1 × G1 satisfy that Z Z ′ W (x, z)W (x, z ) dx = X

X0

κX (x)2 dx.

By Lemma 10, we also obtain that for almost every z ∈ G1 , Z Z κX (x)2 dx. W (x, z)2 dx = X0

X

R By considering X (W (x, z) − W (x, z ′ ))2 dx, this yields that W (x, z) = W (x, z ′ ) for almost every (z, z ′ ) ∈ G1 × G1 and x ∈ X. Hence, there exists a function h : X → [0, 1] such that W (x, z) = h(x) for almost every (x, z) ∈ X × G1 . The second constraint then implies that Z W (x, y) dy = pre-deg(X). S (A∪···∪F )∪G1

for almost every x ∈ X. We now split the left side as follows, Z εh(x) W (x, y) dy + = pre-deg(X). S 2 (A∪···∪F ) Since we have already established that W (x, y) = W0 (g(x), g(y)) holds for almost S S every (x, y) ∈ (A ∪ · · · ∪ F) × (A ∪ · · · ∪ F), we obtain that h(x) = κX (x) for almost every x ∈ X. It follows that W (x, y) = W (g(x), g(y)) for almost every (x, y) ∈ X × G1 .

3.7.3

Degree distinguishing

Finally, we set and force the tiles G1 × G1 and X × G2 for every X ∈ P, so that degrees of different parts are distinct. We first consider the simple tile G1 × G1 . By the previous subsection, the S value of W0 (x, y) for (x, y) ∈ (A0 ∪ · · · ∪ F 0 ) × G01 do not depend on y. Hence, there exists a constant ζ ∈ [0, 1] such that for every y ∈ G01 , we have Z W0 (x, y) dx = ζ. S (A0 ∪···∪F 0 )

We set W0 on the tile G1 × G1 as W0 (x, y) = ζ ′ for every x, y ∈ G01 , where ζ ′ ∈ [0, 1] is a real number such that that ζ+

ζ′ · ε ∈ Q. 2

In other words, with this choice of values of W0 on G1 × G1 , 35

R

[0,1]\G02

W0 (x, y) dx is

the same rational number for every y ∈ G01 . We extend the definition of pre-degree to G1 by defining pre-deg(G1 ) as this very rational number. Note that the pre-degrees of all parts A ∪ · · · ∪ F are also rational. We thus have that for every part X ∈ P \ {G2 } and vertex x ∈ X 0 , it holds that Z W0 (x, y) dy ∈ Q. (3.12) pre-deg(X) = [0,1]\G02

For every part X ∈ P, choose an irrational number αX from [0, 1], so that the set {αX : X ∈ P} is linearly independent in R/Q, the vector space of the reals over the rationals. We now define W0 on the tiles X × G2 for X ∈ P as W0 (x, y) = αX for every (x, y) ∈ X 0 × G02 . Combining (3.12), the choices of αX and the fact that the sizes of all parts are also rational, it is straightforward to verify that the final degrees of all parts are distinct. In other words, for all parts X, Y ∈ P such that X 6= Y and every (x, y) ∈ X 0 × Y 0 , we have degW0 (x) = pre-deg(X) +

G1

G1

=ζ G1

αX · ε αY · ε 6= pre-deg(Y ) + = degW0 (y). 4 4 G2

′

=ζ G1

G2 2 = αX

= αX

′2

G1

X

X

X

Figure 3.17: The decorated constraints forcing the structure of the tiles G1 × G1 and X × G2 for X ∈ P. Finally, we have to show how to force the structure of the tiles G1 × G1 and X × G2 for X ∈ P. Consider the first two decorated constraints in Figure 3.17. The constraints imply that for almost every x ∈ G1 , we have Z 4 W (x, y) dy = ζ ′ ε G1 and, using Lemma 10,

4 ε

Z

W (x, y)2 dy = ζ ′2 .

G1

R By considering G1 (W (x, y) − ζ ′ )2 dx, we obtain that W (x, y) = ζ ′ for almost every (x, y) ∈ G1 × G1 . An analogous argument shows that W (x, y) = αX for every X ∈ P and almost every (x, y) ∈ X × G2 . It follows that W (x, y) = W0 (g(x), g(y)) for almost every (x, y) ∈ G1 × G1 and almost every (x, y) ∈ X × G2 , as required. This concludes the proof of Theorem 2.

36

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