aa r X i v : . [ m a t h . C O ] M a r HYPERGRAPH LIMITS: A REGULARITY APPROACH
YUFEI ZHAO
Abstract.
A sequence of k -uniform hypergraphs H , H , . . . is convergent if the sequence of homo-morphism densities t ( F, H ) , t ( F, H ) , . . . converges for every k -uniform hypergraph F . For graphs,Lov´asz and Szegedy showed that every convergent sequence has a limit in the form of a symmetricmeasurable function W : [0 , → [0 , W : [0 , k − → [0 , Introduction
One of the starting points in the theory of dense graph limits is the seminal paper by Lov´asz andSzegedy [13] where they constructed limit objects for convergent sequences of dense graphs. Thesubject has grown enormously since then with many exciting developments (see Lov´asz’s recentmonograph [12]).For any two graphs F and G , let hom( F, G ) denote the number of homomorphism from F to G ,i.e., maps V ( F ) → V ( G ) that carry every edge of F to an edge of G . The homomorphism density t ( F, G ) is defined to be the probability that a random map V ( F ) → V ( G ) is a homomorphism, i.e., t ( F, G ) := hom(
F, G ) | V ( G ) | | V ( F ) | . A sequence of graphs G , G , . . . is called convergent if the sequence t ( F, G ) , t ( F, G ) , . . . convergesfor every graph F . Convergent graph sequences were defined and studied in [4, 5]. The main resultof Lov´asz and Szegedy [13] is that for every convergent graph sequence there is a limit object in theform of a graphon , which is a symmetric measurable function W : [0 , → [0 ,
1] (here symmetric means that W ( x, y ) = W ( y, x )) such that t ( F, G n ) → t ( F, W ) as n → ∞ for all graphs F . Here t ( F, W ) is defined by t ( F, W ) := Z [0 , V ( F ) Y ij ∈ E ( F ) W ( x i , x j ) dx dx · · · dx | V ( F ) | The natural extension of these limits to hypergraphs was considered by Elek and Szegedy [7].They constructed using ultraproducts an “ultralimit hypergraph” for any sequence of hypergraphs,and established a correspondence principle which enabled them to convert statements about finitehypergraphs, such as hypergraph regularity and removal lemmas [9, 15, 16], to measure-theoreticclaims about ultralimit spaces. One of the consequences of their work is the existence of a limitobject in the form of a measurable functions W : [0 , k − → [0 ,
1] for any convergent sequence of k -uniform hypergraphs.These limit objects had actually appeared earlier in a different form, in the study of exchangeablerandom arrays, initiated by Hoover [10], Aldous [1], and Kallenberg [11] during the 1980s, buildingon the classic de Finnetti’s theorem on exchangeable random variables. This connection is explainedin the survey [3] by Austin, where he credits Tao [17] for initiating the link between exchangeable random variables and hypergraphs. These connections for graphs are also explained in the surveyby Diaconis and Janson [6] as well as Aldous’ ICM talk [2].The purpose of this paper is to provide a new proof of the existence of hypergraph limits. Ourapproach is based on weak Frieze-Kannan [8] type regularity partitions, in line with mainstreamperspectives on dense graph limits. The proof does not use any exchangeable random variables orultraproducts, and the construction of the limit is subjectively more concrete than earlier proofs.Our proof is inspired by the original approach of Lov´asz and Szegedy [13], and the paper is self-contained other than an application of the Martingale Convergence Theorem.1.1. Convergence and limit object.
For any k -uniform hypergraphs F and H , let hom( F, H )denote the number of homomorphisms from F to H , i.e., maps V ( F ) → V ( H ) that carry everyedge of F to an edge of H . Define t ( F, H ) := hom(
F, H ) / | V ( H ) | | V ( F ) | . This is the probability thata random map V ( F ) → V ( H ) is a homomorphism. Definition 1.1 (Convergence) . A sequence of k -uniform hypergraphs H , H , . . . is called conver-gent if the sequence t ( F, H ) , t ( F, H ) , . . . converges for every k -uniform hypergraph F .For any positive integer n , define [ n ] := { , , . . . , n } . For any set A , define r ( A ) to be thecollection of all nonempty subsets of A , and r < ( A ) to be collection of all nonempty proper subsetsof A . More generally, let r ( A, m ) denote the collection of all nonempty subsets of A of size at most m . So for instance, r < ([ k ]) = r ([ k ] , k − r [ k ] and r < [ k ] to mean r ([ k ]) and r < ([ k ]) respectively.Any permutation σ of a set A induces a permutation on r ( A, m ). We say that a function W : [0 , r ([ k ] ,m ) → [0 ,
1] is symmetric if it remains invariant under any permutation of the coor-dinates induced by any permutation of [ k ]. For example, W : [0 , r < [3] → [0 ,
1] being symmetricmeans that W ( x , x , x , x , x , x ) = W ( x σ , x σ , x σ , x σ σ , x σ σ , x σ σ ) (1)for any permutation σ of { , , } . Here we write x i for x { i } and x ij for x { i,j } . Definition 1.2. A k -uniform hypergraphon is a symmetric measurable function W : [0 , r < ([ k ]) → [0 , Example 1.3.
A 3-uniform hypergraphon is a measurable function W : [0 , → [0 ,
1] satisfyingthe symmetry condition (1).For any k -uniform hypergraph F and hypergraphon W , define the homomorphism density by t ( F, W ) := Z [0 , r ( V ( F ) ,k − Y A ∈ E ( F ) W ( x r < ( A ) ) d x Our convention throughout the paper is that if x = ( x A : A ∈ A ) ∈ [0 , A is a vector whosecoordinates are indexed by some set system A , and B ⊆ A is a subcollection, then we write x B = ( x B : B ∈ B ) ∈ [0 , B to mean the restriction of the vector to the coordinates indexed by B . Example 1.4. If K (3)4 = { , , , } is the complete 3-uniform hypergraph on 4 verticesand W is a 3-uniform hypergraphon, then t ( K (3)4 , W ) = Z [0 , W ( x , x , x , x , x , x ) W ( x , x , x , x , x , x ) W ( x , x , x , x , x , x ) ·· W ( x , x , x , x , x , x ) dx dx dx dx dx dx dx dx dx dx . YPERGRAPH LIMITS: A REGULARITY APPROACH 3
Every k -uniform hypergraph H can be represented as a k -uniform hypergraphon W H as follows:divide [0 ,
1] into | V ( H ) | equal-length intervals (cid:8) I , I , . . . , I | V ( H ) | (cid:9) . For each x ∈ [0 , r < [ k ] define W H ( x ) := ( x { i } ∈ I a i for i = 1 , . . . , k and { a , a , . . . , a k } is an edge of H, W H ( x ) depends only on the k coordinates of x corresponding to subsets of [ k ] of size1. It can be alternatively described as transforming the adjacency array of H into a { , } -valuedstep function and then adding 2 k − − k extra free coordinates. Observe that t ( F, H ) = t ( F, W H )for every k -uniform hypergraph F .The main purpose of this paper is to give a new proof of the following result [7, Thm. 7] on theexistence of hypergraph limits. Theorem 1.5. If H , H , . . . is a convergent sequence of k -uniform hypergraphs, then there exists a k -uniform hypergraphon W so that t ( F, H n ) → t ( F, W ) as n → ∞ for every k -uniform hypergraph F . Why are there so many coordinates?
It may initially seem somewhat strange that we need6 coordinates to describe the limit of 3-uniform hypergraphs, whereas every 3-uniform hypergraphcan be described in terms of a 3-dimensional adjacency array. These extra dimensions do not arisefor limits of graphs, but they are essential for hypergraphs. Here is a standard example illustratingwhy functions of the form [0 , → [0 ,
1] cannot capture the richness of 3-uniform hypergraphlimits. Take G n ∼ G ( n, /
2) to be a sequence of graphs on n vertices, where each edge is generatedwith probability 1 /
2, and let H n be the 3-uniform hypergraph whose edges are the triangles of G n .Then with probability one, t ( F, H n ) → −| ∂F | for every 3-uniform hypergraph F , where ∂F is thecollection of unordered pairs of vertices of F that are contained in some edge of F . The limit of H n is different from, say, the constant hypergraphon 1 /
2, which is the limit of a sequence of 3-uniformhypergraphs where every triple of vertices is taken to be an edge independently with probability1 /
2. To describe the limit of H n , we need to incorporate the limit of G n into the data, and thisis achieved by the three extra coordinates. We know that the graph sequence G n converges to theconstant graphon with value 1 /
2. To build the limit of H n , we partition each of the last threecoordinates, x , x , x into two intervals [0 , / ∪ (1 / , G n andthe limit of its complement. The limiting hypergraphon has constant value 1 on [0 , × [0 , / (as the edges of H n are supported on G n ) and 0 elsewhere. Intuitively, the first three coordinatesencode the vertex types, the last three coordinates encode the vertex-pair types. This hypergraphonis { , } -valued since it is deterministic once the vertex and vertex-pairs types are set. If we modifythe sequence H n so that each triangle of G n is included as an edge of H n with some probability p independently, then the limiting hypergraphon would be constant p on [0 , × [0 , / and 0elsewhere.For k -uniform hypergraphs, we can similarly impose some structure at each level, correspondingto j -element subsets of vertices, for every 1 ≤ j ≤ k . This is why we need a coordinate for everyproper subset of [ k ] to describe hypergraph limits.1.3. Random hypergraph model.
To further illustrate the involvement of the 2 k − W : [0 , → [0 ,
1] is a graphon, then we have the following natural random graphmodel G ( n, W ) on n vertices: choose i.i.d. uniform x , x , . . . , x n ∈ [0 , i and j with probability W ( x i , x j ) independently. It was shown [13, Cor. 2.6]using Azuma’s inequality that G ( n, W ) converges to the limit W almost surely.Similarly, a k -uniform hypergraphon W gives a natural model G ( n, W ) of a random k -uniformhypergraph on n vertices: choose a uniformly random x ∈ [0 , r ([ n ] ,k − and add the edge B = { i , . . . , i k } ⊆ [ n ] with probability W ( x r < ( B ) ) independently. Essentially the same proof for graphs YUFEI ZHAO extend over to show [7, Thm. 11] that G ( n, W ) converges to W in the sense of Theorem 1.5, as n → ∞ with probability one. Observe that the random hypergraphs H n of triangles in G ( n, / Analytic version and compactness.
It will be convenient to prove an analytic version ofTheorem 1.5. We say that a sequence of k -uniform hypergraphons W , W , . . . is convergent if thesequence t ( F, W ) , t ( F, W ) , . . . converges for every k -uniform hypergraph F . Theorem 1.6. If W , W , . . . is a convergent sequence of k -uniform hypergraphons, then thereexists a k -uniform hypergraphon f W so that t ( F, W n ) → t ( F, f W ) as n → ∞ for every k -uniformhypergraph F . In this case we say that W n converges to f W . Here is an equivalent formulation of the theorem. Theorem 1.7.
Every sequence W , W , . . . of k -uniform hypergraphons contains a subsequence thatconverges to some k -uniform hypergraphon f W . Theorem 1.7 implies Theorem 1.6 trivially since we can just take the limit f W produced byTheorem 1.7. The converse is true because [0 , N is sequentially compact, so we can restrict ( W n )to some subsequence ( W n i ) so that t ( F, W n i ) converges as i → ∞ for every F .We shall prove Theorem 1.7 with respect to another notion of convergence based on regular parti-tions, which implies the convergence of homomorphism densities. The partition-based convergencegives some structural insight into the convergence of hypergraphs.There is a neat interpretation of Theorem 1.7 in terms of compactness, discovered by Lov´aszand Szegedy [14] in the case of graphons. Let W ( k )0 denote the set of k -uniform hypergraphons.Give W ( k )0 the weakest topology for which the functions t ( F, · ) are continuous for every k -uniformhypergraph F . Identify W with W ′ if t ( F, W ) = t ( F, W ′ ) for every k -uniform hypergraph F . Callthis topology the left-convergence topology of W ( k )0 . Corollary 1.8.
The space W ( k )0 with the left-convergence topology is compact.Proof. The space is metrizable with the metric δ ( W, W ′ ) = P i ≥ − i | t ( F i , W ) − t ( F i , W ′ ) | where( F i ) is some enumeration of all isomorphism classes of k -uniform hypergraphs. We know thatcompactness is equivalent to sequential compactness in metric spaces, and Theorem 1.7 shows thatthe space is sequentially compact. (cid:4) When k = 2, Lov´asz and Szegedy [14] showed that W (2)0 is compact under the cut metrictopology, and Borgs, Chayes, Lov´asz, S´os, and Vesztergombi [4] showed that the cut metric topologyis equivalent to the left-convergence topology. Lov´asz and Szegedy interpreted the compactnesswith respect to the cut metric as an analytic form of the regularity lemma, and they showedthat the compactness of the space of graphons implies strong versions of the regularity lemma.Unfortunately, for k ≥
3, we do not know of a useful extension of the cut metric to hypergraphs(and there may be some reasons to believe that such a natural metric might be too much to askfor). This is one of the main obstacles in working with convergence of hypergraphs. It would benice to have a simple and useful description of distance between hypergraphs which agrees with thetopology induced by homomorphism densities.1.5.
Organization. In § § §
3. The proof of the main result is containedin § § § § YPERGRAPH LIMITS: A REGULARITY APPROACH 5 Limits of graphons
For any symmetric measurable function W : [0 , → R , the cut norm is defined by k W k (cid:3) := sup S,T ⊆ [0 , (cid:12)(cid:12)(cid:12)(cid:12)Z S × T W ( x, y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) , (2)where S and T range over all measurable subsets of [0 , k W k (cid:3) = sup u,v : [0 , → [0 , (cid:12)(cid:12)(cid:12)(cid:12)Z W ( x, y ) u ( x ) v ( y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) (3)where u and v range over all measurable functions [0 , → [0 , u and v , one can restrict to { , } -valued u and v , thereby reducing to (2).Recall that a graphon is a symmetric measurable function W : [0 , → [0 , φ : [0 , → [0 ,
1] and any graphon W , define W φ by W φ ( x, y ) = W ( φ ( x ) , φ ( y )).We define the cut distance between graphons by δ (cid:3) ( U, W ) = inf φ k U φ − W k (cid:3) , where the infimum is taken over all measure preserving bijections φ : [0 , → [0 , Lemma 2.1 (Counting lemma) . For any graphons U and W and any graph F , we have | t ( F, U ) − t ( F, W ) | ≤ e ( F ) k U − W k (cid:3) where e ( F ) is the number of edges of F . We illustrate the proof through the example F = K . t ( K , U ) − t ( K , W )= Z [0 , ( U ( x, y ) U ( x, z ) U ( y, z ) − W ( x, y ) W ( x, z ) W ( y, z )) dxdydz = Z [0 , ( U ( x, y ) − W ( x, y )) W ( x, z ) W ( y, z ) dxdydz + Z [0 , U ( x, y )( U ( x, z ) − W ( x, z )) W ( y, z ) dxdydz + Z [0 , U ( x, y ) U ( x, z )( U ( y, z ) − W ( y, z )) dxdydz Each of the three terms in the final sum is bounded in absolute value by k U − W k (cid:3) . For example,for the first term, for every fixed value of z , the integral has the form (3), and so it is boundedin absolute value by k U − W k (cid:3) , and the same bound holds after integrating z by the triangleinequality. It follows that | t ( K , U ) − t ( K , W ) | ≤ k U − W k (cid:3) .For any graphon W and any partition Q of [0 ,
1] into a finite collection of measurable subsets,let W Q be a graphon which is the step function obtained from W by replacing its value at ( x, y ) ∈ Q i × Q j by the average value of W on Q i × Q j , for any Q i , Q j ∈ Q , (if either Q i or Q j has measurezero, then assign value 0 on Q i × Q j ). For graphs, think of Q as a partition of the vertex set, and W Q as recording the edge densities between pairs of vertex subsets.A key tool in the construction of graph limits is the following weak regularity lemma due to Friezeand Kannan [8] (see also [14, Lem 3.1]). It can be proved by an L -energy increment argument. YUFEI ZHAO
Lemma 2.2 (Weak regularity lemma) . For every ε > and every symmetric measurable func-tion W : [0 , → [0 , , there is some partition Q of [0 , into at most /ε parts such that k W − W Q k (cid:3) ≤ ε . Lov´asz and Szegedy [14] showed that with respect to the cut metric, after identifying graphonswith cut distance zero, the space of all graphons is compact. Equivalently:
Theorem 2.3 (Lov´asz and Szegedy [14]) . Every sequence W , W , . . . of graphons contains asubsequence converging to some graphon f W in cut distance. Let us recall the idea of the proof of Theorem 2.3. Let ε >
0. We apply the weak regularitylemma to approximate every W n by some ( W n ) Q n . By replacing each W n by some W φ n n for somemeasure preserving bijection φ n , we may assume that the partition Q n divides [0 ,
1] into intervals.Take a subsequence so that the lengths of the intervals converge, and the values of ( W n ) Q n insidethe boxes induced by the partition also converge, i.e., the value inside the ( i, j )-th box of ( W n ) Q n converges to some value as n → ∞ (may be different limits for different ( i, j )). Then in thissubsequence, ( W n ) Q n converges pointwise almost everywhere to some limit e U , which is also a stepfunction.Now repeat the same procedure with a smaller ε ′ < ε . We obtain new partitions Q ′ n whichare refinements of previous partitions. Call the resulting limit e U . Note that steps of ( W n ) Q ′ n arerefinements of the steps of ( W n ) Q n , and the values of the latter can be obtained from the formerby averaging over each step. Thus a similar relation holds for e U and e U .Now we repeat this procedure for a sequence of ε k tending to zero. We obtain a sequence e U , e U , . . . of step functions so that each e U s can be obtained from e U s +1 by average over each step. Itfollows that if ( X, Y ) is a uniform random point in [0 , , then the sequence ( e U ( X, Y ) , e U ( X, Y ) , . . . )is a martingale. Since every e U s is bounded, the Martingale Convergence Theorem implies that themartingale converges with probability 1, and hence there is some f W : [0 , → [0 ,
1] which is thepointwise almost everywhere limit of e U s ’s. One then checks that f W is the desired limit.In summary, the above proof consists of two main steps:(1) For each error tolerance ε , apply a weak regularity lemma to get a finite-dimensional stepfunction approximation of each graphon. Take a subsequence so that the step functionsconverge.(2) Take a decreasing sequence of ε tending to zero, we obtain refining chains of regularitypartitions, and the corresponding subsequential limits e U s form a martingale. The existenceof the final limit graphon follows by the Martingale Convergence Theorem.3. Limits of 3-uniform hypergraphs
In this section we sketch the idea for 3-uniform hypergraph limits. To keep things simple, considera sequence H , H , . . . of 3-uniform hypergraphs (as opposed to hypergraphons).We begin with an initial attempt that does not quite work. For a 3-variable function W : [0 , → R , we might extend the cut norm (5) as follows (assume everything is measurable from now on):(bad cut norm) k W k (cid:3) = sup R,S,T ⊆ [0 , (cid:12)(cid:12)(cid:12)(cid:12)Z R × S × T W ( x, y, z ) dxdydz (cid:12)(cid:12)(cid:12)(cid:12) . (4)For each hypergraph H , one can easily extend the weak regularity lemma, Lemma 2.2, to obtaina partition Q of the vertex set of H into at most 2 /ε parts so that (cid:13)(cid:13) W H − W H Q (cid:13)(cid:13) (cid:3) ≤ ε (regard W H as a 3-variable function for now, and W H Q is derived from W by averaging over each cells The Martingale Convergence Theorem (see [18, Thm. 11.5]) says that every L -bounded martingale convergesalmost surely. Our martingales are actually bounded uniformly within [0 , YPERGRAPH LIMITS: A REGULARITY APPROACH 7 induced by Q ). Theorem 2.3 also extends with virtually no change in the proof. That is, allowingpermutations of vertices, some subsequence of H n converges with respect to the vertex-cut norm(4) to a 3-variable symmetric function f W : [0 , → [0 , H and H are close with respect to some cut norm, then t ( F, H ) and t ( F, H ) are close. If we carry through the proof of Lemma 2.1, we find that | t ( F, H ) − t ( F, H ) | ≤ e ( F ) k W H − W H k (cid:3) holds when F is a linear hypergraph , i.e., where every two edges of F intersect inat most one vertex. However, when F is not linear, say F = K (3)4 , then this claim is completely false,as t ( F, H ) and t ( F, H ) can be separated even when k W H − W H k (cid:3) is small. A counterexamplefor 3-uniform hypergraphs can be built by taking triangles of the random graph G ( n, p ), and thenkeeping each triangle as a 3-uniform edge with some probability q . With parameters ( p, q ) = (1 / , , / K (3)4 densities.Now let us scrap the vertex-cut norm (4). The proof of the counting lemma, Lemma 2.1, extendswith respect to the following modified cut norm (again we use a 3-variable W for now):(better cut norm) k W k (cid:3) = sup u,v,w : [0 , → [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [0 , W ( x, y, z ) u ( x, y ) v ( x, z ) w ( y, z ) dxdydz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5)For this cut norm, the counting lemma | t ( F, H ) − t ( F, H ) | ≤ e ( F ) k W H − W H k (cid:3) holds. However,like trying to fit a large rug in a small room, we quickly run into another issue: this norm is toostrong and we do not have the compactness result corresponding to Theorem 2.3. Indeed, takingthe sequence H n of triangles of G ( n, /
2) from § H n and H m are typicallynot close with respect to k·k (cid:3) , although they are close in homomorphism densities.Even though we do not have compactness with respect to k·k (cid:3) , we can still hope for a slightlyweaker topology that gives convergence of homomorphism densities. We can extend the weakregularity lemma, Lemma 2.2, to k·k (cid:3) , where now instead of partitioning the vertex set V = V ( H ),we partition the edges of the underlying complete graph K V = (cid:0) V (cid:1) , i.e., the collection of unorderedpairs of V . So now Q is a partition K V = G ∪ · · · ∪ G m of the edges of K V into m graphs. Thepartition Q of K V induces a partition Q ∗ on triples of vertices:( x, y, z ) ∼ Q ∗ ( x ′ , y ′ , z ′ ) ⇔ ( x, y ) ∼ Q ( x ′ , y ′ ) , ( x, z ) ∼ Q ( x ′ , z ′ ) , and ( y, z ) ∼ Q ( y ′ , z ′ ) . Being somewhat sloppy with notation for the time being, we can form W H Q by averaging W H insideeach cell of Q ∗ . Then the weak regularity lemma guarantees us a partition Q of K V into at most2 /ε parts so that k W H − W H Q k (cid:3) ≤ ε , and | t ( F, W H ) − t ( F, W H Q ) | ≤ e ( F ) ε by the counting lemma.For each hypergraph in the sequence H , H , . . . , apply the weak regularity lemma (for a uniform ε ) to obtain a partition Q n of the complete graph on V ( H n ) into m graphs: K V ( H n ) = G n, ∪ · · · ∪ G n,m , where m depends on ε but not on n .By applying Theorem 2.3 on the graph sequence ( G n, ) n ≥ , we can find a graphon e Y : [0 , → [0 ,
1] so that G n, converges to e Y as n → ∞ along some subsequence . By further restricting tosubsequences, we can find a e Y j for each 1 ≤ j ≤ m so that G n,j converges to e Y j as n → ∞ along asubsequence.For each n , { G n, , . . . , G n,m } is a partition of K V ( H n ) , so the same holds for the resulting limit ,in the sense that e Y + · · · + e Y m = 1 almost everywhere as functions [0 , → [0 , e Q of the cube [0 , = [0 , r [2] (coordinates indexed by x , x , x ) by stacking together Provided that the limits of the various graph sequences are taken in a compatible way. This is a source oftechnical/notational annoyance later on, and it is the reason for introducing branching partitions in § YUFEI ZHAO subsets whose heights are given by e Y j . More precisely, e Q = { e Q , . . . , e Q m } where e Q j = { ( x , x , x ) ∈ [0 , : ( e Y + · · · + e Y j − )( x , x ) ≤ x < ( e Y + · · · + e Y j )( x , x ) } . This is the first place where the “extra” coordinates such as x arise even though we started withhypergraphs not requiring these extra coordinates. They arise because the limit graphon e Y of asequence of graphs G n, is not always a { , } -valued function.The partition e Q of [0 , r [2] induces a partition e Q ∗ of [0 , = [0 , r < [2] :( x , x , x , x , x , x ) ∼ e Q ∗ ( x ′ , x ′ , x ′ , x ′ , x ′ , x ′ ) ⇔ ( x i , x j , x ij ) ∼ e Q ( x ′ i , x ′ j , x ′ ij ) ∀ ≤ i < j ≤ . The partition e Q ∗ should not be viewed as a regularization partition for any H n (indeed, the extracoordinates do not even appear in H n ). Instead, the partitions Q n themselves become increas-ing close to e Q . There is a correspondence of cells of Q n with those of e Q , and this induces acorrespondence between cells of Q ∗ n with those of e Q ∗ .Now we construct the first limiting hypergraphon e U as a step function [0 , → [0 ,
1] that isconstant on each part of e Q ∗ . On each part of e Q ∗ , we assign to e U the limiting value of the averageof W n on the corresponding cell of Q ∗ n , limit taken as n → ∞ along a further restricted subsequence.We have constructed e U , which plays a similar role as e U near the end of § § e U is not close in k·k (cid:3) to H n for large n . It is a limit in the following sense:we first ε -regularized H n , and then took the graph limit of the partitions, created a new partitionof [0 , using these lower order limits, and then constructed a step-function U using this limitingpartition and the limiting values on the steps. We knew from the earlier counting lemma (referredto later on as Counting Lemma I ) that (cid:12)(cid:12) t ( F, H ) − t ( F, W H Q n ) (cid:12)(cid:12) ≤ e ( F ) ε. (6)By what we will call Counting Lemma II , we have (here n → ∞ along a subsequence)lim n →∞ t ( F, W H Q n ) = t ( F, e U ) . (7)Here is some intuition why (7) holds. Both W H Q n and e U are step functions. We can split them upinto weighted sums of indicator functions, on which the claim reduces to checking homomorphismdensities for the graphons corresponding to parts of the partitions Q n and e Q . We know that thegraphs which are the parts of Q n converge to the graphons from which e Q is built. So the graphhomomorphism densities converge.This shows that e U is a O ( e ( F ) ε )-approximation to a subsequence of H n in terms of F -densities.Now, take a smaller ε ′ < ε , and build another e U , where the new partitions Q n are refinements ofthe previous ones. Continuing this process, we obtain a sequence e U , e U , . . . which is a martingaleas before. The Martingale Convergence Theorem gives a pointwise almost everywhere limit f W of e U s , s → ∞ , and f W is the desired limit.In proving 3-uniform hypergraph limits, we used the existence of graph limits. In general, weprove the existence of k -uniform hypergraph limits by induction on k . There are a few furthertechnical difficulties. For example, we need to make sure that the limit of a sequence of partitionsremains a partition, so the limit needs to be taken in a compatible way. Since we are working withmultiple partitions, we will need to deal with homomorphisms from F to a vector of hypergraphons,where the edges of F individually land in different hypergraphons. The details are addressed inthe rest of this paper. YPERGRAPH LIMITS: A REGULARITY APPROACH 9 Notation
One (not so trivial) source of difficulty in working with hypergraphs is the complexity of notation.This section collects some of the notation and conventions used in the rest of this paper. Somenotations were already introduced in § Hypergraphs. A k -uniform hypergraph F is some finite collection of k -element subsets ofsome ground set, which we denote by V ( F ). So when we talk about an element of F , we mean anedge of F , and | F | means the number of edges of F .4.2. Subsets, partitions, and hypergraphons.Definition 4.1 (Symmetric sets and partitions) . A symmetric (measurable) subset of [0 , r [ k ] is onewhich is invariant under the action of all permutations of [ k ]. A symmetric (measurable) partition of [0 , r [ k ] is a partition of [0 , r [ k ] into a finite collection of symmetric subsets.A symmetric subset P ⊆ [0 , r [ k ] is associated to a k -hypergraphon W P : [0 , r < [ k ] → [0 ,
1] byintegrating out the top coordinate: W P ( x r < [ k ] ) := Z P ( x r [ k ] ) dx [ k ] . (8)For example, for k = 3, we have P ⊆ [0 , , with coordinates indexed by r [2] = { , , } , and W P ( x , x ) = Z P ( x , x , x ) dx . This operation collapses the final coordinate in P . It will be helpful to think of P and W P asrepresenting the same object. For example, when k = 2 this means we do not care how P isplaced along the x coordinate, as we only care about how much P intersects line segments ofthe form { x } × { x } × [0 , W : [0 , → [0 , P ⊆ [0 , satisfying W P = W , e.g., any set of the form P = { ( x, y, z ) : a ( x, y ) ≤ z ≤ b ( x, y ) } where b ( x, y ) − a ( x, y ) = W ( x, y ).4.3. Homomorphism densities.
For any tuple of k -uniform hypergraphons W = ( W , . . . , W m ),any k -uniform hypergraph F , and any map α : F → [ m ], define the homomorphism density t α ( F, W ) := Z [0 , r ( V ( F ) ,k − Y e ∈ F W α ( e ) ( x r < ( e ) ) d x . Example 4.2. If k = 2, F = K = { , , } , α = (12 , , t α ( F, W ) = Z [0 , W ( x , x ) W ( x , x ) W ( x , x ) dx dx dx For any symmetric partition P = ( P , . . . , P m ) of [0 , r [ k ] , define W P := ( W P , . . . , W P m ) and t α ( F, P ) := t α ( F, W P ) . (9)4.4. Quotient and stepping operators.
Let W : [0 , r < [ k ] → [0 ,
1] be a k -uniform hypergraphonand Q a symmetric partition of [0 , r [ k − into q parts Q , Q , . . . , Q q ⊆ [0 , r [ k − . The quotient W/ Q is a 2 q k -tuple of numbers in [0 ,
1] defined by assigning to each k -tuple f = ( f , . . . , f k ) ∈ [ q ] k a pair ( v f , w f ), referred to as (volume, average), as follows: • Volume: v f equals the integral v f := Z x ∈ [0 , r< [ k ] Q f ( x r ([ k ] \{ } ) )1 Q f ( x r ([ k ] \{ } ) ) · · · Q fk ( x r ([ k ] \{ k } ) ) d x . (10) • Average: If v f = 0, then we set w f = 0. Otherwise, w f is defined to be w f := 1 v f Z x ∈ [0 , r< [ k ] W ( x r < [ k ] )1 Q f ( x r ([ k ] \{ } ) )1 Q f ( x r ([ k ] \{ } ) ) · · · Q fk ( x r ([ k ] \{ k } ) ) d x . (11)Intuitively, the partition Q induces a partition Q ∗ of [0 , r [ k ] into parts enumerated by f ∈ [ q ] k .Each cell of Q ∗ has a volume v f and an average value w f of W on the cell.If we have another k -uniform hypergraphon W ′ , and a symmetric partition Q ′ of [0 , r [ k − into q parts ( Q and Q ′ have the same number of parts) with volumes and weights ( v ′ f , w ′ f ), we define d ( W/ Q , W ′ / Q ′ ) := X f ∈ [ q ] k ( | v f − v ′ f | + | v f w f − v ′ f w ′ f | ) . (12)For any symmetric subset P ⊆ [0 , r [ k ] , we write P/ Q := W P / Q . A Q -step function U : [0 , r < [ k ] → R is a function of the form U ( x ) = X f =( f ,...,f k ) ∈ [ q ] k u f Q f ( x r ([ k ] \{ } ) )1 Q f ( x r ([ k ] \{ } ) ) · · · Q fk ( x r ([ k ] \{ k } ) ) (13)for some real values u f . Since Q is a partition, the indicator functions in (13) all have disjointsupport, which together partition the domain [0 , r < [ k ] . Usually U is a symmetric function, whichis equivalent to having an additional symmetry constraint on u f , namely that u f = u f ′ whenever f ′ is obtained from f ′ by a permutation of the coordinates.The Q -stepping operator , denoted by a subscript Q , turns a k -uniform hypergraphon W into asymmetric Q -step function W Q by averaging over each induced cell of Q ∗ . More precisely, we define W Q : [0 , r < [ k ] → [0 ,
1] to be (using v f and w f from W/ Q defined earlier) W Q ( x ) := X f =( f ,...,f k ) ∈ [ q ] k w f Q f ( x r ([ k ] \{ } ) )1 Q f ( x r ([ k ] \{ } ) ) · · · Q fk ( x r ([ k ] \{ k } ) )We can also apply the stepping operator to a tuple of hypergraphons. If W = ( W , . . . , W m ),then W Q := (( W ) Q , . . . , ( W m ) Q ) . In particular, if P = { P , . . . , P m } is a partition of [0 , r [ k ] , then we write W PQ := (( W P ) Q , . . . , ( W P m ) Q ) = ( W P Q , . . . , W P m Q )4.5. Cut norm.Definition 4.3.
For any symmetric function W : [0 , r < [ k ] → R , define k W k (cid:3) k − := sup u ,...,u k : [0 , r [ k − → [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [0 , r< [ k ] W ( x r < [ k ] ) k Y i =1 u i ( x r ([ k ] \{ i } ) ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (14)Note that by linearity of the expression inside the absolute value in (14), it suffices to considerfunctions u i ’s which are indicator functions 1 B i of symmetric subsets B i ⊆ [0 , r [ k − . The usualcut norm corresponds to the case k = 2. The following example shows k = 3. Example 4.4.
For any symmetric function W : [0 , r < [3] → R , k W k (cid:3) equals tosup u ,u ,u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [0 , W ( x , x , x , x , x , x ) u ( x , x , x ) u ( x , x , x ) u ( x , x , x ) dx dx dx dx dx dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) where u , u , u vary over all symmetric functions [0 , r [2] → [0 , YPERGRAPH LIMITS: A REGULARITY APPROACH 11 Regularity and counting lemmas
Definition 5.1.
Let W be a k -uniform hypergraphon and Q a symmetric partition of [0 , r [ k − .We say that ( W, Q ) is weakly ε -regular if k W − W Q k (cid:3) k − ≤ ε .For a symmetric subset P ⊆ [0 , r [ k ] , we say that ( P, Q ) is weakly ε -regular if ( W P , Q ) is. Lemma 5.2 (Weak regularity lemma) . Let k ≥ and ε > . Let W = ( W , . . . , W m ) be a tupleof k -uniform hypergraphons. Let Q be a symmetric partition of [0 , r [ k − . Then there exists apartition Q ′ refining Q so that every part of Q is refined into exactly ⌈ km/ε ⌉ parts (allowingempty parts) so that ( W i , Q ′ ) is weakly ε -regular for every ≤ i ≤ m .Proof. We build the partition incrementally, starting with Q . At a given stage, suppose the partitionis R . If ( W i , R ) is weakly ε -regular for every i then we stop. Otherwise there is some i with k W i − ( W i ) R k (cid:3) k − > ε , so there exists symmetric subsets B , . . . , B k ⊆ [0 , r ([ k − such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [0 , r< [ k ] ( W i − ( W i ) R )( x r < ([ k ]) ) k Y i =1 B i ( x r ([ k ] \{ i } ) ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > ε. (15)Let B : [0 , r < [ k ] → [0 ,
1] be the function (not necessarily symmetric) B ( x ) := k Y i =1 B i ( x r ([ k ] \{ i } ) ) d x . For two functions
U, U ′ : [0 , r < [ k ] → [0 , (cid:10) U, U ′ (cid:11) = Z [0 , r< [ k ] U ( x ) U ′ ( x ) d x . We will use the following easy fact: if U ′ is a Q -step function, then h U, U ′ i = h U Q , U ′ i .Now let R ′ be the the minimal partition refining R and B , . . . , B k . Since (( W i ) R ′ ) R = ( W i ) R ,applying the fact above, we obtain h ( W i ) R ′ , ( W i ) R i = h ( W i ) R , ( W i ) R i (16)Since B is an R ′ -step function, we have h ( W i ) R ′ , B i = h W i , B i . So by (15) |h ( W i ) R ′ − ( W i ) R , B i| = |h W i − ( W i ) R , B i| > ε. (17)Since k U k = h U, U i for any U , we obtain by (16), the Cauchy-Schwarz inequality, and (17) k ( W i ) R ′ k − k ( W i ) R k = k ( W i ) R ′ − ( W i ) R k ≥ |h ( W i ) R ′ − ( W i ) R , B i| > ε . (18)Furthermore, for every 1 ≤ j ≤ m , k ( W j ) R ′ k ≥ k ( W j ) R k by convexity since (( W j ) R ′ ) R = ( W j ) R .The quantity k ( W ) R k + · · · + k ( W m ) R k is at most m , and each iteration above increases thesum by at least ε . So there can be at most m/ε iterations. At the end we obtain a partition Q ′ so that ( W i , Q ′ ) is weakly ε -regular for every 1 ≤ i ≤ m . Each time we introduced at most k newsets to refine the partition, so R ′ refines each part of R into at most 2 k subparts. After at most m/ε iterations, each part of the original partition Q is refined into at most 2 km/ε parts. We canthrow in some empty parts so that each part of Q is refined into exactly ⌈ km/ε ⌉ parts. (cid:4) Lemma 5.3 (Counting lemma I) . Let U = ( U , . . . , U m ) and W = ( W , . . . , W m ) be two m -tuple of k -uniform hypergraphons and Q a symmetric partition of [0 , r ([ k − . Suppose that k W i − U i k (cid:3) k − ≤ ε for each i . Then for any k -uniform hypergraph F and any map α : F → [ m ] , we have | t α ( F, U ) − t α ( F, W ) | ≤ | F | ε. Proof.
Let V = V ( F ) and F = { e , . . . , e | F | } . Write as a telescoping sum t α ( F, U ) − t α ( F, W )= Z [0 , r ( V,k − | F | Y i =1 U α ( e i ) ( x r < ( e i ) ) − | F | Y i =1 W α ( e i ) ( x r < ( e i ) ) d x r ( V,k − = | F | X j =1 Z [0 , r ( V,k − j − Y i =1 U α ( e i ) ( x r < ( e i ) ) ! ( U α ( e j ) − W α ( e j ) )( x r < ( e j ) ) | F | Y i = j +1 W α ( e i ) ( x r < ( e i ) ) d x . The j -th term in the final sum is bounded by k U α ( e j ) − W α ( e j ) k (cid:3) k − ≤ ε . Indeed, if we fix allvariables other than x r < ( e j ) , then all the factors except for ( U α ( e j ) − W α ( e j ) )( x r < ( e j ) ) have the form u ( x r ( f ) ) for some f ( e j , where f is the intersection of e j with another edge e j ′ . So the the integralcan be bounded by the ( k − (cid:4) Lemma 5.4 (Counting lemma II) . Let U = ( U , . . . , U m ) and W = ( W , . . . , W m ) be two m -tuplesof k -uniform hypergraphons. Let Q = { Q , . . . , Q q } and R = { R , . . . , R q } be symmetric partitionsof [0 , r [ k − . Suppose that d ( U i / Q , W i / R ) ≤ δ for each i . Then for any k -uniform hypergraph F and any map α : F → [ m ] , | t α ( F, U Q ) − t α ( F, W R ) | ≤ | F | δ + X β : ∂F → [ q ] | t β ( ∂F, Q ) − t β ( ∂F, R ) | , where the sum is taken over all maps β : ∂F → [ q ] , and ∂F is the ( k − -uniform hypergraph on V ( F ) consisting of ( k − -element subsets of V ( F ) that are contained in some edge of F .Proof. We can replace each U i by ( U i ) Q as this does not change U i / Q or t α ( F, U Q ). So we mayassume that every U i is a symmetric Q -step function, i.e., U R = U . Similarly, assume that every W i is a symmetric R -step function.For each f ∈ [ q ] k , let ( v i,f , w i,f ) denote the volume and average corresponding to f in U i / Q , andlet ( v ′ i,f , w ′ i,f ) denote the same for W i / R .For each 1 ≤ i ≤ m , construct a symmetric Q -step function U ′ i from U i by changing its value onthe step corresponding to f from w i,f to w ′ i,f . So U ′ i / Q has ( v i,f , w ′ i,f ) as its volumes and averages.In other words, U i ( x r < [ k ] ) = X f =( f ,...,f q ) ∈ [ q ] k w i,f Q f ( x r ([ k ] \{ } ) ) · · · Q fk ( x r ([ k ] \{ k } ) ); (19) U ′ i ( x r < [ k ] ) = X f =( f ,...,f q ) ∈ [ q ] k w ′ i,f Q f ( x r ([ k ] \{ } ) ) · · · Q fk ( x r ([ k ] \{ k } ) ); (20) W i ( x r < [ k ] ) = X f =( f ,...,f q ) ∈ [ q ] k w ′ i,f R f ( x r ([ k ] \{ } ) ) · · · R fk ( x r ([ k ] \{ k } ) ) . (21)Write U ′ = ( U ′ , . . . , U ′ m ). We have (cid:13)(cid:13) U i − U ′ i (cid:13)(cid:13) = X f ∈ [ q ] k v i,f | w i,f − w ′ i,f | ≤ X f ∈ [ q ] k ( | v i,f w i,f − v ′ i,f w ′ i,f | + w ′ i,f | v ′ i,f − v i,f | ) ≤ X f ∈ [ q ] k ( | v i,f w i,f − v ′ i,f w ′ i,f | + | v ′ i,f − v i,f | ) = d ( U i / Q , W i / R ) ≤ δ. So k U i − U ′ i k ≤ δ for each i . It follows that (cid:12)(cid:12) t α ( F, U ) − t α ( F, U ′ ) (cid:12)(cid:12) ≤ | F | δ. (22) YPERGRAPH LIMITS: A REGULARITY APPROACH 13 (This follows from Counting Lemma I, but it’s in fact even easier.) From (20) we have t α ( F, U ′ ) = Z [0 , r ( V ( F ) ,k − Y e = { j ,...,j k }∈ F X f =( f ,...,f q ) ∈ [ q ] k w ′ α ( e ) ,f Q f ( x r ( e \{ j } ) ) · · · Q fk ( x r ( e \{ j k } ) ) d x = X β : ∂F → [ q ] Y e ∈ F w ′ α ( e ) ,β ( ∂e ) ! t ( ∂F, Q ) . (23)Here β ( ∂e ) = ( β ( e \ { j } ) , . . . , β ( e \ { j k } )) ∈ [ q ] k when e = { j , . . . , j k } . The last equality aboveneeds some pondering. Essentially we expand the product of sums in the previous line and notethat since Q is a partition, the nonzero terms in the expansion correspond to assigning an f toevery e in a compatible way: if two edges e = { j , . . . , j k } and e ′ = e ∪ { j ′ l } \ { j l } intersect in exactly k − f is assigned to e , and f ′ is assigned to e ′ , then f l = f ′ l . These assignments arein bijection with β : ∂F → [ q ], where β corresponds to the assignment assigning e to β ( ∂e ).Similar to (23) we have t α ( F, W ) = X β : ∂F → [ q ] Y e ∈ F w ′ α ( e ) ,β ( ∂e ) ! t ( ∂F, R ) . (24)Combing (23) and (24) using the triangle inequality and noting that 0 ≤ w ′ i,f ≤
1, we have (cid:12)(cid:12) t α ( F, U ′ ) − t α ( F, W ) (cid:12)(cid:12) ≤ X β : ∂F → [ q ] | t β ( ∂F, Q ) − t β ( ∂F, R ) | . (25)The lemma follows from combining (22) and (25) using the triangle inequality. (cid:4) Branching partitions
Now we are almost ready to build the limiting object. We will proceed by induction on k (for k -uniform hypergraphons). The situation is very simple when k = 1, since in this case a hypergraphonis simply a number between 0 and 1. To build the limiting hypergraphon in general, we will need torepeatedly apply the weak regularity lemma to obtain a refining chain of partitions. Since we needto apply induction on k , we need to have a stronger induction hypothesis that involves a sequenceof not just single hypergraphons, but refining chains of partitions. This motivates the followingdefinition of a branching partition, which is a special case a filtration , in the language of probability.See Figure 1. Definition 6.1. A degree p = ( p , p , . . . ) ∈ N N (symmetric) branching partition P of [0 , r [ k ] is a collection of symmetric subsets P i of [0 , r [ k ] , collected into levels , where each level P l is asymmetric partition of [0 , r [ k ] : • Level 0: P = { [0 , r [ k ] }• Level 1: P = { P , P , . . . , P p } is a symmetric partition of [0 , r [ k ] . • Level l ( l ≥ P l is a refinement of P l − , where each part of P l − gets further refined intoexactly p l parts.An index at level l is a tuple i = ( i , i , . . . , i l ) ∈ [ p ] × [ p ] × · · ·× [ p l ], which points to the symmetricsubset P i = P i ,...,i l ∈ P l at level l , where P i is the i l -th part in the refinement of the part P i ,...,i l − at level l −
1, whenever l ≥ Font convention. P is a branching partition, P is a partition, and P is a subset of [0 , r [ k ] . Example 6.2.
A symmetric subset P ⊆ [0 , r [ k ] or a k -uniform hypergraphon W (related by(8)) can be thought of as a degree (2 , , , , . . . ) branching partition: level 1 is P and P c (thecomplement of P in [0 , r [ k ] ) and all subsequent levels are trivial refinements. P Level 0: P Level 1: P Level 2: P ... [0 , r [ k ] p P p P p · · · P p p P , p · · · P ,p Figure 1.
A branching partitionWe can generalize the notion of regularity from Definition 5.1 to branching partitions as follows.
Definition 6.3.
Let P be a branching partition of [0 , r [ k ] and Q a branching partition of[0 , r [ k − . We say that ( P , Q ) is weakly ( ε , ε , . . . ) -regular if for every s ≥
1, whenever P ⊆ [0 , r [ k ] is a member of P of level at most s , and Q s is the level s partition of [0 , r [ k − in Q , thepair ( P, Q s ) is weakly ε s -regular. Lemma 6.4 (Weak regularity lemma for branching partitions) . For every k ≥ , p = ( p , p , . . . ) ∈ N N and ε = ( ε , ε , . . . ) ∈ R N > , we can find a q = ( q , q , . . . ) ∈ N N so that the following holds: forevery degree p branching partition P of [0 , r [ k ] , there exists a degree q branching partition Q of [0 , r [ k − so that ( P , Q ) is weakly ε -regular.Proof. Take q s = ⌈ k ( p + p p + ··· + p p ··· p s ) /ε s ⌉ . We build Q successively by level. To obtain the level s partition in Q , applying Lemma 5.2 with ε = ε s , W the collection of hypergraphons correspondingto all members of P of level at most s , and Q the level s − Q . (cid:4) Now we introduce two notions of convergence for branching partitions. The first notion, called left-convergence , is based on convergence of homomorphism densities. The second notion, called partitionable convergence , is based on convergence of regularity partitions. We will show, using ourcounting lemmas, that partitionable convergence implies left-convergence.
Notation.
Given degree p = ( p , p , . . . ) branching partitions P , P , . . . and f P of [0 , r [ k ] anddegree q = ( q , q , . . . ) branching partitions Q , Q , . . . and e Q of [0 , r [ k − , we use the followingnotation to refer to the partitions and parts in these branching partitions. • For each l ≥ P n,l is the level l partition in P n , and e P l is the level l partition in f P . • For each s ≥ Q n,s is the level s partition in Q n , and e Q s is the level s partition in e Q . • For each index i = ( i , i , . . . , i l ) ∈ [ p ] × · · · × [ p l ], P n,i is the index i element of P n and e P i is the index i element of f P . Definition 6.5 (Left-convergence: P n → f P ) . We say that a sequence P , P , . . . of degree p branching partitions of [0 , r [ k ] left-converges to another degree p branching partition f P of [0 , r [ k ] ,written P n → f P , if lim n →∞ t α ( F, P n,l ) = t α ( F, e P l ) for all F, l, α (26)where F ranges over all k -uniform hypergraphs, l ranges over all positive integers, and α rangesover all maps F → [ p · · · p l ]. Recall from (9) that t α ( F, P ) := t α ( F, W P ) for a partition P . Definition 6.6 (Partitionable convergence: P n f P ) . We say that a sequence P , P , . . . ofdegree p = ( p , p , . . . ) branching partitions of [0 , r [ k ] partitionably converges to another degree p branching partition f P of [0 , r [ k ] , written P n f P , if the following is satisfied (the definition isinductive on k ). YPERGRAPH LIMITS: A REGULARITY APPROACH 15
When k = 1, for every index i = ( i , . . . , i l ) ∈ [ p ] × · · · × [ p l ], we have lim n →∞ λ ( P n,i ) = λ ( e P i ),where λ is the Lebesgue measure on [0 , k ≥
2, there exists some q ∈ N N and degree q branching partitions Q , Q , . . . and e Q of[0 , r [ k − satisfying:(a) ( P n , Q n ) is weakly (1 , / , / , . . . )-regular for every n ;(b) Q n e Q as n → ∞ (defined inductively);(c) For every s ≥ i ∈ [ p ] ×· · ·× [ p l ], one has lim n →∞ d ( P n,i / Q n,s , e P i / e Q s ) = 0;(d) For every member e P ⊆ [0 , r [ k ] of f P , one has ( W e P ) e Q s → W e P pointwise almost everywhereas s → ∞ . Lemma 6.7 (Partitionable convergence implies left-convergence) . If P n f P then P n → f P .Proof. We use induction on k . When k = 1, the claim is trivial. Now assume k ≥ F, l, α . Let m = p · · · p s . Let Q n and e Q be as inDefinition 6.6, and let q = ( q , q , . . . ) be the degree of e Q .Let ε >
0. By Definition 6.6(d), W e P l e Q s converges pointwise almost everywhere in each coordinateto W e P l as s → ∞ , so lim s →∞ t α ( F, W e P l e Q s ) = t α ( F, W e P l ). We can find an s ≥ max { l, | F | /ε } so that | t α ( F, W e P l e Q s ) − t α ( F, P l ) | ≤ ε . Fix this value of s .By Definition 6.6(b) we have Q n e Q , so Q n → Q by the induction hypothesis. Thuslim n →∞ t β ( ∂F, Q n,s ) = t β ( ∂F, e Q s ) (27)for all β : ∂F → [ q q · · · q s ]. See Lemma 5.4 for the definition of ∂F . We have | t α ( F, P n,l ) − t α ( F, e P l ) |≤ | t α ( F, P n,l ) − t α ( F, W P n,l Q n,s ) | + | t α ( F, W P n,l Q n,s ) − t α ( F, W e P l e Q s ) | + | t α ( F, W e P l e Q s ) − t α ( F, e P l ) | (28)As n → ∞ , the first term on the right hand side of (28) has a limsup of at most | F | /s ≤ ε by Counting Lemma I (Lemma 5.3) since ( P, Q n,s ) is 1 /s -regular for every P ∈ P n,l by Defini-tion 6.6(a). The second term on the RHS of (28) goes to zero by Counting Lemma II (Lemma 5.4),Definition 6.6(c), and (27). The third term on the RHS of (28) is at most ε using our choice of s .It follows that lim sup n →∞ | t α ( F, P n,l ) − t α ( F, e P l ) | ≤ ε . Since ε can be made arbitarily small, weobtain lim n →∞ t α ( F, P n,l ) = t α ( F, e P l ) as desired. (cid:4) Proposition 6.8.
Let p ∈ N N . Let P , P · · · be a sequence of degree p branching partitions of [0 , r [ k ] . Then there exists another degree p branching partition f P of [0 , r [ k ] so that P n f P as n → ∞ along some infinite subsequence.Proof. We use induction on k . The claim is easy when k = 1, since we can pick a subsequence sothat for each index i , the measure λ ( P n,i ) converges to some value a i as n → ∞ , and we can takethe limit f P to be a branching partition where e P i is an interval with length a i .Now assume k ≥
2. By Lemma 6.4, there exists a q ∈ N N so that for every n we can find a degree q branching partition Q n of [0 , r [ k − so that ( P n , Q n ) is weakly (1 , / , / , . . . )-regular, therebysatisfying (a) in Definition 6.6. Applying the induction hypothesis, we can restrict to a subsequenceso that Q n e Q for some branching partition e Q of [0 , r [ k − (here and onwards in this proofwe abuse notation by only considering convergence as n → ∞ along some subsequence. We willbe repeatedly taking subsequences, and the conclusion will follow by a standard diagonalizationargument). So (b) is satisfied.By further restricting to a subsequence, we may assume that for each s ≥ i , thequotient P n,i / Q n,s converges coordinate-wise as n → ∞ . Let W n,i := W P n,i be the hypergraphon associated to P n,i . Let f W i,s : [0 , r < [ k ] → [0 ,
1] be a symmetric e Q s -step function, with valuesassigned so that d ( W n,i / Q n,s , f W i,s / e Q s ) → n → ∞ . This is possible since we previouslyassumed that P n,i / Q n,s converges coordinatewise as n → ∞ , so that are now simply putting in thelimiting values of the “average” coordinates into a template for a symmetric e Q s -step function inorder to construct f W i,s . To see that the “volume” coordinates (10) of Q n,s converge to those of e Q s , note that this amount to the claim that lim n →∞ t β ( K ( k − k , Q n,s ) = t β ( K ( k − k , e Q s ) for every β : K ( k − k → [ q ], where K ( k − k is the ( k − k − k ]. The convergence of these homomorphism densities follows from Q n → e Q which in turn follows from Q n e Q and Lemma 6.7. Claim 1. ( f W i,s +1 ) e Q s = f W i,s . Proof of Claim 1.
We have lim n →∞ d ( W n,i / Q n,s , f W i,s / e Q s ) = 0 (29)and lim n →∞ d ( W n,i / Q n,s +1 , f W i,s +1 / e Q s +1 ) = 0 (30)Since e Q s +1 is a refinement of e Q s , by merging together parts in W n,i / Q s +1 and f W i,s +1 / e Q s +1 , wededuce from (30) lim n →∞ d ( W n,i / Q n,s , f W i,s +1 / e Q s ) = 0 (31)From (29) and (31) we obtain f W i,s +1 / e Q s = f W i,s / e Q s , which implies ( f W i,s +1 ) e Q s = f W i,s since bothsides are e Q s -step functions (cid:3) It follows that f W i, , f W i, , f W i, , . . . is a martingale with respect to the filtration induced by e Q , e Q , . . . . By the Martingale Convergence Theorem, there exists some f W i , so that f W i,s → f W i pointwise almost everywhere as s → ∞ . Furthermore ( f W i ) e Q s = f W i,s . Claim 2.
Let l ≥
1, and let i = ( i , . . . , i l − ) ∈ [ p ] × · · · × [ p l − ] an index at level l −
1, whichpoints to a part in P that splits into indices { j , . . . , j p l } = i × [ p l ] at level l . Then f W j + · · · + f W j pl = f W i almost everywhere . Proof of Claim 2.
Since P n,j , . . . , P n,j ps is a partition of P n,i , we have W n,j + · · · + W n,j ps = W n,i Taking the Q n,s quotient of both sides and then take the limit as n → ∞ , we find the followingequality for these e Q s -step functions. f W j ,s + · · · + f W j s ,s = f W i,s Taking s → ∞ and using the pointwise almost everywhere convergence of f W j,s → f W j as s → ∞ for every index j , we obtain Claim 2. (cid:3) Claim 2 tells us that we can find a branching partition f P of [0 , r [ k ] so that the part e P i satisfies W e P i = f W i . Visually we can build the level s of f P by stacking together subsets of [0 , r [ k ] thatcorrespond to f W j , ranged over all indices j at level s . Then P n,i / Q n,s = W n,i / Q n,s → d f W i,s / e Q s = To be more precise, let [0 , r [ k ] be the probability space equipped with the uniform Lebesgue measure. For each s ≥ B s be the minimal σ -algebra on [0 , r [ k ] generated by functions of the form 1 Q ( x r ([ k ] \{ j } ) ) ranged over Q ∈ Q s and j ∈ [ k ]. Then f W i,s is a B s -measurable random variable, and Claim 1 implies that f W i, , f W i, , · · · is amartingale adapted to the filtration B ⊆ B ⊆ · · · YPERGRAPH LIMITS: A REGULARITY APPROACH 17 f W i / e Q s = e P i / e Q s , so (c) is satisfied. Also (d) is satisfied since W e P i e Q s = f W i,s → f W i = W e P i pointwisealmost everywhere as s → ∞ (from our application of the Martingale Convergence Theorem). (cid:4) Proof of Theorem 1.6.
Let P n be the degree (2 , , , , . . . ) branching partition built from W n as inExample 6.2. Proposition 6.8 implies that there exists a branching partition f P so that P n f P along a subsequence, and hence P n → f P along a subsequence by Lemma 6.7. Let e P be the index(1) element of f P . The associated hypergraphon f W = W e P is the desired limit of W n . By applying(26) with l = 1 and α ≡
1, we see that t ( F, W n ) → t ( F, f W n ) along the subsequence. (cid:4) We conclude the paper with a conjecture that partitionable convergence is equivalent to left-convergence, thereby proposing a converse to Lemma 6.7.
Conjecture 6.9. P n → f P if and only if P n f P . Acknowledgments
The author would like to thank Jacob Fox, L´aszl´o Lov´asz, Jennifer Chayes and Christian Borgsfor helpful conversations, and also the anonymous referees for helpful comments and for pointingout the connections between graph limits and exchangeable random arrays.
References [1] D. J. Aldous. Representations for partially exchangeable arrays of random variables.
J. Multivariate Anal. ,11(4):581–598, 1981.[2] D. J. Aldous. Exchangeability and continuum limits of discrete random structures. In
Proceedings of the Inter-national Congress of Mathematicians. Volume I , pages 141–153, New Delhi, 2010. Hindustan Book Agency.[3] T. Austin. On exchangeable random variables and the statistics of large graphs and hypergraphs.
Probab. Surv. ,5:80–145, 2008.[4] C. Borgs, J. T. Chayes, L. Lov´asz, V. T. S´os, and K. Vesztergombi. Convergent sequences of dense graphs. I.Subgraph frequencies, metric properties and testing.
Adv. Math. , 219(6):1801–1851, 2008.[5] C. Borgs, J. T. Chayes, L. Lov´asz, V. T. S´os, and K. Vesztergombi. Convergent sequences of dense graphs II.Multiway cuts and statistical physics.
Ann. of Math. (2) , 176(1):151–219, 2012.[6] P. Diaconis and S. Janson. Graph limits and exchangeable random graphs.
Rend. Mat. Appl. (7) , 28(1):33–61,2008.[7] G. Elek and B. Szegedy. A measure-theoretic approach to the theory of dense hypergraphs.
Adv. Math. , 231(3-4):1731–1772, 2012.[8] A. Frieze and R. Kannan. Quick approximation to matrices and applications.
Combinatorica , 19(2):175–220,1999.[9] W. T. Gowers. Hypergraph regularity and the multidimensional Szemer´edi theorem.
Ann. of Math. (2) ,166(3):897–946, 2007.[10] D. N. Hoover. Relations on probability spaces and arrays of random variables, 1979. Preprint, Institute forAdvanced Study, Princeton, NJ.[11] O. Kallenberg. Symmetries on random arrays and set-indexed processes.
J. Theoret. Probab. , 5(4):727–765, 1992.[12] L. Lov´asz.
Large Networks and Graph Limits , volume 60 of
Colloquium Publications . American MathematicalSociety, 2012.[13] L. Lov´asz and B. Szegedy. Limits of dense graph sequences.
J. Combin. Theory Ser. B , 96(6):933–957, 2006.[14] L. Lov´asz and B. Szegedy. Szemer´edi’s lemma for the analyst.
Geom. Funct. Anal. , 17(1):252–270, 2007.[15] B. Nagle, V. R¨odl, and M. Schacht. The counting lemma for regular k -uniform hypergraphs. Random StructuresAlgorithms , 28(2):113–179, 2006.[16] T. Tao. A variant of the hypergraph removal lemma.
J. Combin. Theory Ser. A , 113(7):1257–1280, 2006.[17] T. Tao. A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graphremoval lemma.
J. Anal. Math. , 103:1–45, 2007.[18] D. Williams.
Probability with martingales . Cambridge Mathematical Textbooks. Cambridge University Press,Cambridge, 1991.
Department of Mathematics, MIT, Cambridge, MA 02139-4307.
E-mail address ::