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Elusive extremal graphs

We study the uniqueness of optimal solutions to extremal graph theory problems. Lovasz conjectured that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the resulting set is satisfied by an asymptotically unique graph. This statement is often referred to as saying that `every extremal graph theory problem has a finitely forcible optimum'. We present a counterexample to the conjecture. Our techniques also extend to a more general setting involving other types of constraints.

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04/26/2021

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1 Introduction

Many problems in extremal graph theory do not have asymptotically unique solutions. As an example, consider the problem of minimizing the sum of the induced subgraph densities of and its complement. It can be shown that this sum is minimized by any -vertex graph where all vertices have degrees equal to . For example, the complete bipartite graph , the union of two

-vertex complete graphs, or (with high probability) an Erdős-Rényi random graph

all minimize the sum. However, the structure of an optimal solution can be made unique by adding additional density constraints. In our example, setting the triangle density to be zero forces the structure to be that of the complete bipartite graph with parts of equal sizes. Alternatively, fixing the density of cycles of length four forces the structure to be that of a quasirandom graph. The most frequently quoted conjecture concerning dense graph limits is a conjecture of Lovász [30, 31, 32, 34], stated as Conjecture 1 below, which asserts that this is a general phenomenon for a large class of problems in extremal graph theory. We disprove this conjecture.

We treat Conjecture 1 in the language of the theory of graph limits, to which we provide a brief introduction. This theory has offered analytic tools to represent and analyze large graphs, and has led to new tools and views on various problems in mathematics and computer science. We refer the reader to a monograph by Lovász [32] for a detailed introduction to the theory. The theory is also closely related to the flag algebra method of Razborov [45], which changed the landscape of extremal combinatorics [48] by providing solutions and substantial progress on many long-standing open problems, see, e.g. [2, 4, 3, 5, 6, 23, 24, 25, 28, 29, 37, 38, 45, 46, 47].

The density of a -vertex graph in , denoted by , is the probability that a uniformly randomly chosen -tuple of vertices of induce a subgraph isomorphic to ; if has less than vertices, we set . A sequence of graphs is convergent if the sequence is convergent for every graph . In this paper, we only consider convergent sequences of graphs where the number of vertices tends to infinity.

A convergent sequence of graphs is finitely forcible if there exist graphs with the following property: if is another convergent sequence of graphs such that

for every , then

for every graph . For example, one of the classical results on quasirandom graphs [50, 51, 13] is equivalent to saying that a sequence of Erdős-Rényi random graphs is finitely forcible (by densities of -vertex subgraphs) with probability one. Lovász and Sós [33] generalized this result to graph limits corresponding to stochastic block models (which are represented by step graphons). Additional examples of finitely forcible sequences can be found, e.g., in [19, 34].

One of the most commonly cited problems concerning graph limits is the following conjecture of Lovász, which is often referred to as saying that “every extremal problem has a finitely forcible optimum”, see [32, p. 308]. The conjecture appears in various forms, also sometimes as a question, in the literature; we include only some of the many references with its statement below.

Conjecture 1 (Lovász [30, Conjecture 3],[31, Conjecture 9.12],[32, Conjecture 16.45], and [34, Conjecture 7]).

Let be graphs and reals. If there exists a convergent sequence of graphs with the limit density of equal to , , then there exists a finitely forcible such sequence.

Our main result (Theorem 18) implies that the conjecture is false. The proof of Theorem 18 uses analytic objects (graphons) provided by the theory of graph limits, however, unlike in earlier results concerning finitely forcible graph limits, we have parametrized a whole family of objects that represent large graphs and applied analytic tools to understand the behavior of these objects. To control the objects in the family, we have adopted the method of decorated constraints. This method builds on the flag algebra method of Razborov, which we have mentioned earlier, and was used to enforce an asymptotically unique structure of graphs. In the setting of the proof of Theorem 18, the method is used to enforce a unique structure inside a large graph while keeping some of its other parts variable.

We now explain the general strategy of the proof of Theorem 18

in more detail. By fixing finitely many subgraph densities, we will enforce the asymptotic structure of graphs except for certain specific parts. The structure of the variable parts will depend on a vector

in a controlled way, in particular, one of the variable parts of the graphs will encode the values as edge densities. We will also have a countable set of polynomial inequalities that the values will satisfy for any graph in the family that we consider; this will be enforced by the variable parts and the subgraph densities. This set of polynomial inequalities is constructed iteratively, and each time we add new constraints, the graphs in the family change in a controlled way. We eventually construct a large subset of , which is defined by a countable set of polynomial inequalities, such that any subgraph density in the graph corresponding to is determined by finitely many coordinates of . This will yield a counterexample to Conjecture 1.

While we present our arguments to avoid forcing the unique structure of a graph by subgraph densities, our approach also applies to additional graph parameters, as we discuss in Section 8. For example, one may hope that even if it is not possible to force an asymptotically unique structure by fixing a finite number of subgraph densities, it may be possible to force a unique typical structure, i.e., obtain an asymptotically unique structure maximizing the entropy. However, this hope is dismissed by Theorem 28.

The paper is organized as follows. In Section 2, we start by introducing the notation that we use. We next give a broad outline and set up some basic concepts for the proof of Theorem 18 in Section 3, including a summary of the construction of a family of graphons with finitely forcibly structure, which is parameterized by . We also state the properties of the construction that we need for the proof. We next detail the proof, assuming such a construction exists. We first present an iterative way of restricting values of well-behaved functions in countably many variables in Section 4. In Section 5, we show how this implies Theorem 18. In Section 6, we detail the construction of the family of graphons and prove that it can be forced by finitely many constraints. Finally, in Section 7, we analyze the dependence of the graphons in the family on .

2 Preliminaries

In this section, we introduce notation used throughout the paper. We start with some general notation. The set of all positive integers is denoted by , the set of all non-negative integers by , and the set of integers between and (inclusive) by . All measures considered in this paper are Borel measures on , . If a set is measurable, then we write for its measure, and if and are two measurable sets, then we write if .

We will be working with vectors with both finitely many and countably infinitely many coordinates, and sometimes with double coordinates. If and is a set, then the set is the set of vectors with coordinates from indexed by . In particular, denotes the set of all vectors with coordinates indexed by positive integers and each coordinate from the interval . We will always understand to be . If , and , then we define

If , and , then we set

If , we will omit the second index in the subscript. Finally, we use for the closure of .

We will often work with the following set of double indexed vectors:

The elements of are vectors, each having coordinates indexed by and the -th coordinate being a vector from . For example, if , then and the coordinates of are , and . We define the following bijection between the elements of and : if , we define so that . Using this bijection , we will understand the corresponding elements of and to be the same, and we use the arrow overscript to distinguish between the corresponding elements of and , i.e., whether the vector is single or double indexed.

We say that the function is totally analytic if the following two properties hold:

  • for any finite multiset of positive integers, the partial derivative

    is a continuous function according to the product topology on , and

  • for any fixed , the function

    can be extended to an analytic function of the variables on an open set containing .

We say that two totally analytic functions are -close if the functions and all their first partial derivatives differ by at most on . We say that a function is a polynomial in , , if it is a polynomial in finitely many of the variables , . Clearly, every polynomial in , , is totally analytic.

2.1 Graph limits

We now introduce some notions from the theory of graph limits, which we have not covered in Section 1. To simplify our notation, if is a graph, we will write for its order, i.e., its number of vertices.

Each convergent sequence of graphs can be associated with an analytic limit object, which is called a graphon. Formally, a graphon is a symmetric measurable function from to the unit interval , where symmetric refers to the property that for all . We remark that it is more usual to define graphons as symmetric measurable functions from the closed unit square ; however, both definitions represent the same notion and it is more convenient for us to work with half-open intervals. Graphons are often visualized by a unit square filled in with different shades of gray representing the values of (with being represented by white and by black). We will often refer to the points of as vertices, and we think of a graphon intuitively as a continuous version of an adjacency matrix. Because of this, the origin is always drawn in the top left corner in visualizations of graphons.

We next present a connection between convergent sequences of graphs and graphons. Given a graphon , a -random graph of order is a graph obtained from by sampling points from independently and uniformly at random, associating each point with one of the vertices, and joining two vertices by an edge with probability where and are the points in associated to them. The density of a graph in a graphon , denoted by , is the probability that a -random graph of order is isomorphic to . Observe that the expected density of in a -random graph of order is equal to . It also holds that the density of in a -random graph is concentrated around its expected density by standard martingale arguments. We will also use the labeled version of the density of a graph in a graphon , which is the probability that a -random graph is the graph and the -th vertex of the -random graph is the -th vertex of the graph . This probability will be denoted by and it holds that

Note that is equal to multiplied by .

A graphon is a limit of a convergent sequence of graphs if

for every graph . A standard martingale argument yields that if is a graphon, then a sequence of -random graphs with increasing orders converges to with probability one. In the other direction, Lovász and Szegedy [35] showed that every convergent sequence of graphs has a limit graphon. However, the limit is in general not unique. We say that two graphons and are weakly isomorphic if for every graph , i.e., the graphons and are limits of the same convergent sequences of graphs. Borgs, Chayes and Lovász [8] proved that two graphons and are weakly isomorphic if and only if there exists a third graphon and measure preserving maps such that for almost every and .

Because of Conjecture 1, we are interested in graphons that are uniquely determined, up to weak isomorphism, by the densities of a finite set of graphs. Formally, a graphon is finitely forcible if there exist graphs such that if a graphon satisfies for , then and are weakly isomorphic. The following characterization of finitely forcible graphons is one of the motivations coming from extremal graph theory for Conjecture 1.

Proposition 1.

A graphon is finitely forcible if and only if there exist graphs and reals such that

for every graphon , with equality if only if and are weakly isomorphic.

While the initial results on the structure of finitely forcible graphons suggested that all finitely forcible graphons could possess a simple structure [34, Conjectures 9 and 10], this turned out not to be the case [14, 20, 21] in general. In particular, every graphon can be a subgraphon of a finitely forcible graphon [16]. Here, we say that is a subgraphon of if there exists a non-null subset and a measure preserving map such that for all .

Theorem 2 (Cooper et al. [16]).

There exist graphs with the following property. For every graphon , there exist a graphon and reals such that any graphon with for all is weakly isomorphic to , and is a subgraphon of formed by a fraction of its vertices. In particular, the graphon is finitely forcible.

Figure 1: The graphon from Theorem 2.

The graphon from Theorem 2 is visualized in Figure 1. The following proposition gives some additional properties of the graphons obtained in Theorem 2, which we will need later. A dyadic square is a square of the form , where and . Recall that the standard binary representation of a real number is the binary representation such that set of digits equal to zero is not finite.

Proposition 3.

For every , there exists with the following property. Let and be any two graphons such that the densities of their dyadic squares of sizes at least agree up to the first bits after the decimal point in the standard binary representation, and let and be the corresponding graphons from Theorem 2 that contain and , respectively. The -distance between the graphons and viewed as functions in is at most .

We finish by presenting graph limit analogues of several standard graph theory notions. Let be a graphon. The degree of a vertex is defined as

When it is clear from the context which graphon we are referring to, we will omit the subscript, i.e., we will just write instead of . The neighborhood of is the set of all such that . This set is denoted by , where we again drop the subscript if it is clear from the context. Note that and the inequality can be strict. If is a non-null subset of , then the relative degree of a vertex with respect to is

i.e., the degree of with respect to normalized by the measure of . Similarly, is the relative neighborhood of with respect to . As in the previous cases, we drop the subscripts if is clear from the context.

2.2 Decorated constraints

The arguments related to forcing the structure of graphons introduced in Section 6 are based on the method of decorated constraints from [20, 21]. This method uses the language of the flag algebra method of Razborov developed in [45], which has had many significant applications in extremal combinatorics [48].

We now give a formal definition of decorated constraints. A density expression is a polynomial in graphs with real coefficients. Formally, it can be defined recursively as follows. A real number or a graph are density expressions, and if and are density expressions, then and are also density expressions. An ordinary density constraint is an equality between two density expressions. The value of a density expression for a graphon is obtained by substituting for each graph . A graphon satisfies an ordinary density constraint if the density expressions on the two sides have the same value for . Note that we obtain an equivalent notion if we use instead of when computing the value of a density of expression (adjusting the coefficients appropriately).

We next introduce a formally stronger type of density constraint: a rooted density constraint. A rooted graph is a graph with distinguished vertices, which are labeled with the elements of and are referred to as the roots. Let be the subgraph of induced by the roots. We then define to be the probability that the -random graph is , conditioned on being associated with the -th root. It follows that

where is the set of the non-root vertices of . Two rooted graphs are compatible if the subgraphs induced by their roots are isomorphic through an isomorphism preserving their labels. A rooted density expression is a density expression containing compatible rooted graphs. We define to be the value of the expression when each rooted graph is substituted with . Let be a constraint where all rooted graphs are compatible. We say that the graphon satisfies the constraint if

holds for almost every -tuple . Note that if the -tuple is chosen in such a way that a -random graph with the vertices associated with is with probability zero, then both sides of the above equality are equal to zero.

A graphon is said to be partitioned if there exist an integer , positive reals summing to one, and distinct reals , such that for each , the set of vertices in with degree has measure . The set of all vertices with degree will be referred to as a part; the size of a part is its measure and its degree is the common degree of its vertices. For example, the graphon depicted in Figure 1 has ten parts and all but have the same size. If and are parts, then we will refer to the restriction of to as to the tile . The following lemma was proven in [20, 21].

Lemma 4.

Let be positive real numbers such that and let be distinct reals. There exists a finite set of ordinary density constraints such that a graphon satisfies all constraints in if and only if it is a partitioned graphon with parts of sizes and degrees .

We next introduce a formally even stronger type of density constraint. Fix , and as in Lemma 4. A decorated graph is a graph with distinguished vertices, which are called roots and labeled with , and with each vertex (distinguished or not) assigned one of the parts, which will be called the decoration of a vertex. We allow , i.e., a decorated graph can have no roots. Suppose that are given such that each belongs to the part that the -th root is decorated with. We then define to be the probability that a -random graph is , conditioned on the -th root being associated with , and conditioned on each non-root vertex being associated with a point from the part that it is decorated with. Two decorated graphs are compatible if the subgraphs induced by their roots are isomorphic through an isomorphism preserving the labels of the roots and the decorations of all vertices. A decorated density expression is an expression that contains compatible decorated graphs only, and we define in the obvious way. A decorated density constraint is an equality between two density expressions that contain compatible decorated graphs instead of ordinary graphs; we will often say decorated constraint instead of decorated density constraint.

Before defining when a decorated constraint is satisfied, let us introduce a way of visualizing decorated graphs and decorated constraints, which was also used in [14, 16, 20]. We will draw decorated graphs as graphs with each vertex labeled by its decoration; the root vertices will be drawn as squares and non-root vertices as circles. The decorated constraint will be presented as an expression containing decorated graphs, where the roots of all the graphs appearing in the constraint will be presented in the same position in all the graphs of the constraint; this establishes the correspondence between the roots in individual decorated graphs. A solid line between two vertices will represent an edge and a dashed line a non-edge. The absence of a line between two root vertices indicates that the decorated constraint should hold for both the root graph containing this edge and the one not containing it. The absence of a line between a non-root vertex and another vertex represents the sum over all decorated graphs with and without this edge. In particular, if there are such lines missing, the drawing represents the sum of decorated graphs.

We next define when a graphon satisfies a decorated constraint; the definition is followed by an example of evaluating a decorated graph. Suppose that is a partitioned graphon and is a decorated constraint such that each decorated graph in the constraint has (the same) roots. The graphon satisfies the constraint if

for almost every -tuple such that belongs to the part that the -th root is decorated with.

To illustrate this definition, we give an example. Consider the partitioned graphon that has two parts and , each of size , and that is equal to , and on the tiles , and , respectively. The graphon is depicted in Figure 2; the three decorated graphs depicted in Figure 2 are evaluated to , and (from left to right) for any admissible choices of the roots.

Figure 2: An example of evaluating decorated constraints. The depicted partitioned graphon has two parts and , each of size , and is equal to , and on the tiles , and , respectively. The three decorated graphs that are depicted are evaluated to , and (from left to right) for any admissible choices of the roots.

In [21], it was shown that the expressive power of ordinary and decorated density constraints is the same. To be precise, the following was proven.

Lemma 5.

Let , let be positive real numbers summing to one, and let be distinct reals between zero and one. For every decorated density constraint , there exists an ordinary density constraint such that every partitioned graphon with parts of sizes and degrees satisfies if and only if it satisfies .

Lemmas 4 and 5 were used in [14, 16, 20, 21] to argue that certain partitioned graphons are finitely forcible in the following way. Suppose that is the unique partitioned graphon (up to weak isomorphism) that satisfies a finite set of decorated constraints. By Lemmas 4 and 5, the graphon is the unique graphon that satisfies a finite set of ordinary decorated constraints. It follows that the graphon is the unique graphon such that the density of each subgraph appearing in a constraint of is equal to . In particular, is finitely forcible. In this paper, we follow a similar line of arguments to show that a certain family of graphons can be forced by finitely many subgraph densities.

We next state two auxiliary lemmas, which can be found explicitly stated in [14, Lemmas 7 and 8]. The first lemma says that if a graphon is finitely forcible, then its structure may be forced on a part of a partitioned graphon. The second lemma is implicit in [34, proof of Lemma 2.7 or Lemma 3.3].

Lemma 6.

Let be an integer, positive reals summing to one, and distinct reals between zero and one. For every finitely forcible graphon and every , then there exists a finite set of decorated constraints such that any partitioned graphon with parts of sizes and degrees satisfies if and only if the tile is weakly isomorphic to , i.e., there exist measure preserving maps and such that for almost every .

Lemma 7.

Let be a measurable function, , such that

for almost every . Then, it holds that

for almost every .

3 General setup

The proof of our main theorem, Theorem 18, builds on techniques developed in relation to the finite forcibility of graphons. We find a finite set of density constraints with the property that any graphon that satisfies these constraints has most of its structure forced, except for a small part that can vary in a controlled manner. In particular, we will construct a family of graphons with structure depending on a bounding sequence, which is a notion that we define below, and the graphons in the family are indexed by elements of a subset of .

Before proceeding further, we need to introduce some definitions. We first define a linear order on finite multisets of natural numbers. Each such multiset can be associated with a vector , where is the multiplicity of the containment of in . Let be the sum of the elements of , i.e.,

If and are two multisets of natural numbers, then we will say that if either , or , or and the first entry different in and is smaller in . Let be the -th multiset in this linear order, i.e., , , , , etc. Since there exists at least one multiset with for each , we get the following observation.

Observation 8.

It holds that for every .

We call a sequence a bounding sequence if

  • each is a polynomial and for all ,

  • the coefficient of in the polynomial at the monomial satisfies that ,

  • all the partial derivatives of have values between and (inclusive) on ,

  • , and .

We remark that the second condition is needed, in particular, in order to make sure we can encode the polynomial in the graphon. We say that a bounding sequence is a strengthening of the bounding sequence if they agree on all triples except some triples where in . If is a strengthening of that agrees with on the first elements, then we say that is a -strengthening.

In Section 6, we will construct, for any bounding sequence , a family of graphons . Each of the graphons is described by a bounding sequence and a vector that satisfies for each . These families of graphons will have the following properties.

Theorem 9.

There exists a finite set of graphs and an integer such that the following holds. For any bounding sequence , there exists a polynomial of degree at most in variables such that the following two statements are equivalent for any graphon :

  • The graphon is weakly isomorphic to a graphon for some such that for all .

  • The graphon satisfies .

Furthermore, if and are weakly isomorphic, then .

We defer the precise description of the family and the proof of Theorem 9 to Section 6. For the proof of our main result, we will need additional properties of graphons in the family , which we summarize in the next two lemmas. In Section 7, we define a function for each bounding sequence and each graph . The next proposition asserts that is a totally analytic function and it describes the density of for those that satisfy .

Lemma 10.

For any finite graph and bounding sequence , the function is totally analytic. Furthermore, for every satisfying for all , it holds that .

We will also need the following proposition, which roughly says that if we change in a controlled way, then the functions do not change by much.

Lemma 11.

For any finite graph , bounding sequence , and , there exists an integer such that for any -strengthening of , the function is -close to .

Lemmas 10 and 11 will be proven in Section 7.

4 Stabilization

Our main argument involves a sequence of steps, each resulting in a strengthening of a bounding sequence that defines a family of graphons as above. The whole process is carried in a careful way so that it is still possible to vary some of the parameters without changing the densities of any given finite set of graphs. In this section, we present analytic lemmas needed to perform these steps.

We first prove the following analytic statement. The proof follows the lines of the proof of Grönwall’s inequality [22], see also [7], however, we present an entire proof here for completeness.

Lemma 12.

For every integer , reals , and , and closed interval , there exist and such that the following holds. Let be a function defined on and let . Furthermore, let be an analytic function such that for all , and each partial derivative of is -Lipschitz on , and the absolute value of the determinant of the Jacobi matrix

is at least on . For every analytic function that is -close to and for every such that , and , there exists a function such that , and for all .

Proof.

Fix a point and the functions , and for the proof. Let be the Jacobi matrix of according to , and let be the partial derivative of according to . We define and for in the analogous way. Take small enough so that the absolute value of the determinant of is at least on the whole set (note that this choice of depends only on , and ); in particular, is invertible on whole set . By the Implicit Function Theorem, if is defined for , and is invertible, then can be extended to an open neighborhood of . Let be the maximal interval containing that can be extended to in a way that and .

We next analyze the derivative of as a function of on :

Since this derivative is zero on , it follows that

Similarly, it holds that

for any in the interior of . Let and analogously . Observe that both and are analytic and for all , where is a constant depending only on , , , and . In addition, if , , , and are fixed, can be made arbitrarily close to zero by choosing sufficiently small.

Let us now consider the function

defined on . Note that . We next analyze the derivative of .

It follows that

(1)

where depends only on , , and (since each entry of the matrix is at most , and each entry of the matrix is Lipschitz on for a Lipschitz constant depending only on , and ). We now consider the function that is defined as

(2)

The derivative of is equal to

where the inequality follows from (1). This implies that for such that . Hence, we obtain using (2) that

In particular, it is possible to choose and small enough (the choice of makes small enough) that it holds that

Hence, we obtain that

for all such that .

We next show that the interval is contained in . Let be the supremum of and suppose that . Since the left limit satisfies that , we can assume that . However, this implies that and the function can be extended to a neighborhood of , contradicting the fact that is the supremum of .

The analogous argument using the function

yields that it is possible to choose and small enough that for all such that , which then yields that the interval is a subset of . This completes the proof. ∎

To state the next lemma, we need a definition. A variable system consists of

  • a sequence of non-degenerate closed intervals , , and

  • a sequence of closed sets , , with and .

We say that a variable system is -strong for if the measure of the set is at least for every . A variable system is -finite if for each .

We will need the following lemma on variable systems.

Lemma 13.

Suppose that is a -strong variable system with , and is a non-zero analytic function on , . For any , there exist subsets , , such that , with for , is a -strong variable system and is non-zero on everywhere.

Proof.

We first consider the case when . Let be the set of all such that . Since is analytic and is bounded, the set is finite. Let be the union of open -neighborhoods of the points in , where is sufficiently small so that . The set is a closed set that satisfies the statement of the lemma.

The argument for extends the one presented for but is more complex. Let . Let be the set of all such that

Since is analytic and not everywhere zero, is a closed set with measure zero.

Let be the set of points with where