aa r X i v : . [ m a t h . C O ] M a y QUASI-RANDOM GRAPHS AND GRAPH LIMITS
SVANTE JANSON
Abstract.
We use the theory of graph limits to study several quasi-random properties, mainly dealing with various versions of hereditarysubgraph counts. The main idea is to transfer the properties of (se-quences of) graphs to properties of graphons, and to show that theresulting graphon properties only can be satisfied by constant graphons.These quasi-random properties have been studied before by other au-thors, but our approach gives proofs that we find cleaner, and whichavoid the error terms and ε in the traditional arguments using the Sze-mer´edi regularity lemma. On the other hand, other technical problemssometimes arise in analysing the graphon properties; in particular, ameasure-theoretic problem on elimination of null sets that arises in thisway is treated in an appendix. Introduction A quasi-random graph is a graph that ’looks like’ a random graph. For-mally, this is best defined for a sequence of graphs ( G n ) with | G n | → ∞ .Thomason [22, 23] and Chung, Graham and Wilson [7] showed that a num-ber of different ’random-like’ conditions on such a sequence are equivalent,and we say that ( G n ) is p -quasi-random if it satisfies these conditions. (Here p ∈ [0 ,
1] is a parameter.) We give one of these conditions, which is based onsubgraph counts, in (2.1) below. Other characterizations have been addedby various authors. The present paper studies in particular hereditarilyextended subgraph count properties found by Simonovits and S´os [19, 20],Shapira [16], Shapira and Yuster [17] and Yuster [24]; see Section 3. See alsoSections 8 and 9 for further related equivalent properties (on sizes of cuts)found by Chung, Graham and Wilson [7] and Chung and Graham [6].The theory of graph limits also concern the asymptotic behaviour of se-quences ( G n ) of graphs with | G n | → ∞ . A notion of convergence of suchsequences was introduced by Lov´asz and Szegedy [14] and further devel-oped by Borgs, Chayes, Lov´asz, S´os and Vesztergombi [4, 5]. This maybe seen as giving the space of (unlabelled) graphs a suitable metric; theconvergent sequences are the Cauchy sequences in this metric, and the com-pletion of the space of unlabelled graphs in this metric is the space of (graphsand) graph limits. The graph limits are thus defined in a rather abstract Date : May 20, 2009.2000
Mathematics Subject Classification. way, but there are also more concrete representations of them. One impor-tant representation [14; 4] uses a symmetric (Lebesgue) measurable function W : [0 , → [0 , graphon , and defines a uniquegraph limit, see Section 2 for details. Note, however, that the representationis not unique; different graphons may be equivalent in the sense of definingthe same graph limit. See further [3; 8].We write, with a minor abuse of notation, G n → W , if ( G n ) is a sequenceof graphs and W is a graphon such that ( G n ) converges to the graph limitdefined by W . It is well-known that quasi-random graphs provide the sim-plest example of this: ( G n ) is p -quasi-random if and only if G n → p , where p is the graphon that is constant p [14].A central tool to study large dense graphs is Szemer´edi’s regularity lemma,and it is not surprising that this is closely connected to the theory of graphlimits, see, e.g., [4; 15]. The Szemer´edi regularity lemma is also importantfor the study of quasi-random graphs. For example, Simonovits and S´os[18] gave a characterization of quasi-random graphs in terms of Szemer´edipartitions. Moreover, the proofs in [19; 20; 16; 17] that various propertiescharacterize quasi-random graphs (see Section 3) use the Szemer´edi regu-larity lemma. Roughly speaking, the idea is to take a Szemer´edi partitionof the graph and use the property to show that the Szemer´edi partition hasalmost constant densities.The main purpose of this paper is to point out that these, and other sim-ilar, characterizations of quasi-random graphs alternatively can be provedby replacing the Szemer´edi regularity lemma and Szemer´edi partitions bygraph limit theory. The idea is to first take a graph limit of the sequence (or,in general, of a subsequence) and a representing graphon, then the propertywe assume of the graphs is translated into a property of the graphon, andfinally it is proved that this graphon then has to be (a.e.) constant. We dothis for several different related characterizations below. Our proofs will allhave the same structure and consist of three parts, considering a sequenceof graphs ( G n ) and a graphon W with G n → W :(i) An equivalence between a condition on subgraph counts in G n anda corresponding condition for integrals of a functional Ψ of W . (Ψ isa function on [0 , m for some m , and is a polynomial in W ( x i , x j ),1 ≤ i < j ≤ m .)(ii) An equivalence between this integral condition on Ψ and a pointwisecondition on Ψ.(iii) An equivalence between this pointwise condition on Ψ and W = p .In all cases that we consider, (i) is rather straightforward, and performedin essentially the same way for all versions. Step (ii) follows from some ver-sion of the Lebesgue differentiation theorem, although some cases are morecomplicated than others. The arguments used in (iii) are similar to the ar-guments in earlier proofs that the Szemer´edi partition has almost constant UASI-RANDOM GRAPHS AND GRAPH LIMITS 3 densities (under the corresponding condition on the graphs) and the alge-braic problems that arise in some cases will be the same. However, the use ofgraph limits eliminates the many error terms and ε inherent in arguments us-ing the Szemer´edi regularity lemma, and provides at least sometimes proofsthat are simpler and cleaner. With some simplification, we can say that wesplit the proofs into three parts (i)–(iii) which are combinatorial, analyticand algebraic, respectively. This has the advantage of isolating differenttypes of technical difficulties; moreover, it allows us to reuse some stepsthat are the same for several different cases. (See for example Section 6where we prove several variants of the characterizations by modifying step(i) or (ii).) On the other hand, it has to be admitted that there can betechnical problems with the analysis of the graphons too, especially in (ii),and that our approach does not simplify the algebraic problems in (iii). (Inparticular, we have not been able to improve the results in [20], where it isthis algebraic part that has not yet been done for general graphs.) Some-what disappointingly, it seems that the graph limit method offers greatestsimplifications in the simplest cases. At the end, it is partly a matter of tasteif one prefers the finite arguments using the Szemer´edi regularity lemma orthe infinitesimal arguments using graphons; we invite the reader to makecomparisons. Acknowledgement.
This work was begun during the workshop on GraphLimits, Homomorphisms and Structures in Hraniˇcn´ı Z´ameˇcek, Czech Repub-lic, 2009. We thank Asaf Shapira, Miki Simonovits, Vera S´os and Bal´azsSzegedy for interesting discussions.2.
Preliminaries and notation
All graphs in this paper are finite, undirected and simple. The vertexand edge sets of a graph G are denoted by V ( G ) and E ( G ). We write | G | := | V ( G ) | for the number of vertices of G , and e ( G ) := | E ( G ) | for thenumber of edges. G is the complement of G . As usual, [ n ] := { , . . . , n } .2.1. Subgraph counts.
Let F and G be graphs. It is convenient to as-sume that the graphs are labelled, with V ( F ) = [ | F | ] := { , . . . , | F |} , butthe labelling does not affect our results. We define N ( F, G ) as the numberof labelled (not necessarily induced) copies of F in G ; equivalently, N ( F, G )is the number of injective maps ϕ : V ( F ) → V ( G ) that are graph homomor-phisms (i.e., if i and j are adjacent in F , then ϕ ( i ) and ϕ ( j ) are adjacent in G ). If U is a subset of V ( G ), we further define N ( F, G ; U ) as the number ofsuch copies with all vertices in U ; thus N ( F, G ; U ) = N ( F, G | U ). More gen-erally, if U , . . . , U | F | are subsets of V ( G ), we define N ( F, G ; U , . . . , U | F | )to be the number of labelled copies of F in G with the i th vertex in U i ;equivalently, N ( F, G ; U , . . . , U | F | ) is the number of injective graph homo-morphisms ϕ : F → G such that ϕ ( i ) ∈ U i for every i ∈ V ( F ). SVANTE JANSON
Quasi-random graphs.
One of the several equivalent definitions ofquasi-random graphs by Chung, Graham and Wilson [7] is: ( G n ) (with | G n | → ∞ ) is p -quasi-random if and only if, for every graph F , N ( F, G n ) = ( p e ( F ) + o (1)) | G n | | F | . (2.1)(All unspecified limits in this paper are as n → ∞ , and o (1) denotes a quan-tity that tends to 0 as n → ∞ . We will often use o (1) for quantities thatdepend on some subset(s) of a vertex set V ( G ) or of [0 , o ( a n ) for a given sequence a n similarly.)It turns out that it is not necessary to require (2.1) for all graphs F ; inparticular, it suffices to use the graphs K and C [7]. However, it is notenough to require (2.1) for just one graph F . As a substitute, Simonovitsand S´os [19] showed that a hereditary version of (2.1) for a single F issufficient; see Section 3.2.3. Graph limits.
The graph limit theory is also based on the subgraphcounts N ( F, G ) (or the asymptotically equivalent number counting not nec-essarily injective graph homomorphisms F → G , see [14; 4]). A sequence( G n ) of graphs, with | G n | → ∞ , converges , if the numbers t inj ( F, G n ) := N ( F, G n ) / ( | G n | ) | F | converge as n → ∞ , for every fixed graph F . (Here,( | G n | ) | F | denotes the falling factorial, which is the total number of injectivemaps V ( F ) → V ( G n ), so t inj ( F, G n ) is the proportion of injective maps thatare homomorphisms. Since we consider limits as | G n | → ∞ only, we could aswell instead consider t ( F, G n ), the proportion of all maps V ( F ) → V ( G n )that are homomorphisms, or the hybrid version N ( F, G n ) / | G n | | F | .) Notethat the numbers t inj ( F, G n ) ∈ [0 , G n ) of graphs with | G n | → ∞ has a convergentsubsequence. For details and several other equivalent properties, see Lov´aszand Szegedy [14] and Borgs, Chayes, Lov´asz, S´os and Vesztergombi [4, 5];see also Diaconis and Janson [8].The graph limits that arise in this way may be thought of as elementsof a completion of the space of (unlabelled) graphs with a suitable metric.One useful representation [14; 4] uses a symmetric measurable function W :[0 , → [0 , graphon , and defines a graph limitin the following way. If F is a graph and W a graphon, we defineΨ F,W ( x , . . . , x | F | ) := Y ij ∈ E ( F ) W ( x i , x j ) (2.2)and t ( F, W ) := Z [0 , | F | Ψ F,W . (2.3)(All integrals in this paper are with respect to the Lebesgue measure in oneor several dimensions, unless, in the appendix, we specify another measure.) UASI-RANDOM GRAPHS AND GRAPH LIMITS 5
A sequence ( G n ) converges to the graph limit defined by W if | G n | → ∞ and lim n →∞ t inj ( F, G n ) = t ( F, W ) (2.4)(or, equivalently, t ( F, G n ) → t ( F, W )) for every F ; as said above, in thiscase we write G n → W , although it should be remembered that the repre-sentation of the limit by a graphon W is not unique. (See [4; 3; 8; 2] fordetails on the non-uniqueness. Note that, trivially, we may change W on anull set without affecting the corresponding graph limit; moreover, we may,for example, rearrange W as in (2.11) below.)For example, the condition (2.1) can be written t inj ( F, G n ) → p e ( F ) . Sincethe constant graphon W = p has t ( F, W ) = p e ( F ) for every F by (2.2)–(2.3),this shows that, as said in Section 1, ( G n ) is p -quasi-random if and only if G n → p .2.4. Graphons from graphs. If G is a graph, we define a correspondinggraphon W G by partitioning [0 ,
1] into | G | intervals I i of equal lengths 1 / | G | ;we then define W G to be 1 on every I i × I j such that ij ∈ E ( G ), and 0otherwise. It is easily seen that if G is a graph, then N ( F, G ) = | G | | F | Z [0 , | F | Ψ F,W G + O (cid:0) | G | | F |− (cid:1) . (2.5)(The error term is because we have chosen to count injective homomorphismsonly, cf. [14; 4].) More generally, if U , . . . , U | F | are subsets of V ( G ) and U ′ , . . . , U ′| F | are the corresponding subsets of [0 ,
1] given by U ′ i := S j ∈ U i I j ,then N ( F, G ; U , . . . , U | F | ) = | G | | F | Z U ′ ×···× U ′| F | Ψ F,W G + O (cid:0) | G | | F |− (cid:1) . (2.6)2.5. Induced subgraph counts.
In analogy with Subsection 2.1 we define,for labelled graphs F and G , N ∗ ( F, G ) as the number of induced labelledcopies of F in G ; equivalently, N ∗ ( F, G ) is the number of injective maps ϕ : V ( F ) → V ( G ) such that i and j are adjacent in F ⇐⇒ ϕ ( i ) and ϕ ( j ) areadjacent in G . We further define N ∗ ( F, G ; U ) as the number of such copieswith all vertices in U and N ( F, G ; U , . . . , U | F | ) as the number of inducedlabelled copies of F in G with the i th vertex in U i . (Here U, U , . . . , U | F | ⊆ V ( G ).)For a graphon W we make the corresponding definitions, cf. Subsec-tion 2.3,Ψ ∗ F,W ( x , . . . , x | F | ) := Y ij ∈ E ( F ) W ( x i , x j ) Y ij E ( F ) (cid:0) − W ( x i , x j ) (cid:1) (2.7)and t ind ( F, W ) := Z [0 , | F | Ψ ∗ F,W . (2.8) SVANTE JANSON
Then, for any graph G , in analogy with (2.6) and using the notation there, N ∗ ( F, G ; U , . . . , U | F | ) = | G | | F | Z U ′ ×···× U ′| F | Ψ ∗ F,W G + O (cid:0) | G | | F |− (cid:1) . (2.9) Remark 2.1.
If we define t ind ( F, G ) := N ∗ ( F, G ) / ( | G | ) | F | , then the conver-gence criterion (2.4) (for every F ) is equivalent to t ind ( F, G n ) → t ind ( F, W )(for every F ) by inclusion-exclusion [14; 4].2.6. Cut norm and cut metric.
The cut norm k W k (cid:3) of W ∈ L ([0 , )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 ) d x d y (cid:12)(cid:12)(cid:12)(cid:12) . (2.10)A rearrangement of the graphon W is any graphon W ϕ defined by W ϕ ( x, y ) = W ( ϕ ( x ) , ϕ ( y )) , (2.11)where ϕ : [0 , → [0 ,
1] is a measure-preserving bijection. The cut metric δ by Borgs, Chayes, Lov´asz, S´os and Vesztergombi [4] may be defined by, fortwo graphons W , W , δ (cid:3) ( W , W ) = inf ϕ k W − W ϕ k (cid:3) , (2.12)where the infimum is over all rearrangements of W . (It makes no differenceif we rearrange W instead, or both W and W .)A major result of Borgs, Chayes, Lov´asz, S´os and Vesztergombi [4] is thatif | G n | → ∞ , then G n → W ⇐⇒ δ (cid:3) ( W G n , W ) →
0, so convergence of asequence of graphs as defined above is the same as convergence in the metric δ (cid:3) . 3. Subgraph counts in induced subgraphs
Simonovits and S´os [19] gave the following characterization of p -quasi-random graphs using the numbers of subgraphs of a given type in inducedsubgraphs. (The case F = K , when N ( K , G n ; U ) is twice the number ofedges with both endpoints in U , is one of the original quasi-random proper-ties in [7].) Theorem 3.1 (Simonovits and S´os [19]) . Suppose that ( G n ) is a sequenceof graphs with | G n | → ∞ . Let F be any fixed graph with e ( F ) > and let < p ≤ . Then ( G n ) is p -quasi-random if and only if, for all subsets U of V ( G n ) , N ( F, G n ; U ) = p e ( F ) | U | | F | + o (cid:0) | G n | | F | (cid:1) . (3.1)For our discussion of graph limit method, it is also interesting to considerthe following weaker version (with a stronger hypothesis), patterned afterTheorem 3.11 below. UASI-RANDOM GRAPHS AND GRAPH LIMITS 7
Theorem 3.2.
Suppose that ( G n ) is a sequence of graphs with | G n | → ∞ .Let F be any fixed graph with e ( F ) > and let < p ≤ . Then ( G n ) is p -quasi-random if and only if, for all subsets U , . . . , U | F | of V ( G n ) , N ( F, G n ; U , . . . , U | F | ) = p e ( F ) | F | Y i =1 | U i | + o (cid:0) | G n | | F | (cid:1) . (3.2) Remark 3.3.
Since (3.1) is the special case of (3.2) with U = · · · = U | F | ,the ’if’ direction of Theorem 3.2 is a corollary of Theorem 3.1. The ’only if’direction does not follow immediately from Theorem 3.1, but it is straight-forward to prove, either by the methods of [19] or by our methods withgraph limits, see Section 4; hence the main interest is in the ’if’ direction.(The same is true for the results below for the induced case.) Remark 3.4.
Theorems 3.1 and 3.2 obviously fail when e ( F ) = 0, sincethen (3.1) and (3.2) hold trivially and the assumptions give no informationon G n . They fail also if p = 0; for example, if F = K and G n is thecomplete bipartite graph K n,n .Shapira [16] and Shapira and Yuster [17] consider also an intermediateversion where a symmetric form of (3.2) is used, summing over all permuta-tions of ( U , . . . , U | F | ) (or, equivalently, over all labellings of F ); moreover, U , . . . , U | F | are supposed to be disjoint and of the same size. It is showndirectly in [16] that this is equivalent to (3.1). See also Subsections 6.1and 6.2.The main result of Shapira [16] is that Theorem 3.1 remains valid even ifwe only require (3.1) for U of size α | G n | with α = 1 / ( | F | + 1). (It is a simpleconsequence that any smaller positive α will also do.) This was improvedby Yuster [24], who proved this for any α ∈ (0 , Theorem 3.5 (Yuster [24]) . Let ( G n ) , F and p be as in Theorem 3.1, andlet < α < . Then ( G n ) is p -quasi-random if and only if (3.1) holds forall subsets U of V ( G n ) with | U | = ⌊ α | G n |⌋ . Theorem 3.6.
Let ( G n ) , F and p be as in Theorem 3.2, and let < α < . Then ( G n ) is p -quasi-random if and only if (3.2) holds for all subsets U , . . . , U | F | of V ( G n ) with | U i | = ⌊ α | G n |⌋ .If α < / | F | , it is enough to assume (3.2) for U , . . . , U | F | that furtherare disjoint. For F = K , Theorem 3.5 with α = 1 / α ∈ (0 ,
1) is stated in Chung and Graham [6]. Another relatedcharacterization from [6] is discussed in Section 8.Turning to induced copies of F , the situation is much more complicated, asdiscussed in Simonovits and S´os [20]. First, the expected number of induced SVANTE JANSON labelled copies of F in a random graph G ( n, p ) is β F ( p ) n | F | + o ( n | F | ), with β F ( p ) := p e ( F ) (1 − p ) e ( F ) = p e ( F ) (1 − p )( | F | ) − e ( F ) . (3.3)Hence, the condition corresponding to (3.1) for induced subgraphs is: Forall subsets U of V ( G n ), N ∗ ( F, G n ; U ) = β F ( p ) | U | | F | + o (cid:0) | G n | | F | (cid:1) . (3.4)Indeed, as observed in [19; 20], this holds for every p -quasi-random ( G n ),but the converse is generally false. One reason is that, provided F is neitherempty nor complete, then β F (0) = β F (1) = 0, and if p F := e ( F ) / (cid:0) | F | (cid:1) (the edge density in F ), then β F ( p ) increases on [0 , p F ] and decreases on[ p F , p = p F , there is another ¯ p such that β F ( p ) = β F ( p ); we call p and ¯ p conjugate . (For completeness, we let ¯ p := p when p = p F or when F is empty or complete. Note also that ¯ p depends on F as well as p .) Obviously, a ¯ p -quasi-random sequence ( G n ) also satisfies(3.4). Moreover, any combination of a p -quasi-random sequence and a ¯ p -quasi-random sequence will satisfy (3.4). Hence the best we can hope for isthe following. We say that ( G n ) is mixed ( p, ¯ p ) -quasi-random if it is p -quasi-random, ¯ p -quasi-random, or a combination of two such sequences. Definition 3.7.
Let 0 ≤ p ≤
1. We say that a graph F is hereditaryinduced-forcing ( HI ( p )) if every ( G n ) that satisfies (3.4) for all subsets U of V ( G n ) is mixed ( p, ¯ p )-quasi-random. In this case we also write F ∈ HI ( p )(thus regarding HI ( p ) as a set of graphs).We say that F is HI (and write F ∈ HI ) if F is HI ( p ) for every p ∈ (0 , p = 0 and p = 1). Remark 3.8.
The definition of mixed ( p, ¯ p )-quasi-random is perhaps betterstated in terms of graph limits. Just as ( G n ) is p -quasi-random if and onlyif G n → p , where p stands for the graphon that is constant p , ( G n ) is mixed( p, ¯ p )-quasi-random if and only if the limit points of ( G n ) are contained in { p, ¯ p } , i.e., if every convergent subsequence of ( G n ) converges to either thegraphon p or the graphon ¯ p .In general we say that a sequence ( G n ), with | G n | → ∞ as always, is mixedquasi-random if the set of limit points is contained in { p : p ∈ [0 , } , i.e.,if every convergent subsequence converges to a constant graphon. (Equiva-lently, if every convergent subsequence is quasi-random). Remark 3.9.
Just as one talks about quasi-random properties of graphs,or more properly of sequences ( G n ) of graphs, we say that a property ofgraphons W is p -quasi-random if it is satisfied only by W = p a.e., that it is quasi-random if it is p -quasi-random for some p ∈ [0 , mixedquasi-random if it is satisfied only by graphons that are a.e. constant (forsome set of accepted constants).Simonovits and S´os [20] gave a counter-example showing that the path P with 3 vertices is not HI . They also showed that every regular F (with UASI-RANDOM GRAPHS AND GRAPH LIMITS 9 | F | ≥
2) is HI , and conjectured that P and its complement P are theonly graphs not in HI . This conjecture remains open. (The methods of thepresent paper do not seem to help.) Remark 3.10.
The cases F empty or complete are exceptional and rathertrivial. If F is complete graph K m ( m ≥ N ∗ ( F, G n ; U ) = N ( F, G n ; U ),and thus (3.4) implies that ( G n ) is p -quasi-random by Theorem 3.1 (but notfor p = 0 unless m = 2, see Remark 3.4). By taking complements we seethat the same holds for for an empty graph E m ( m ≥
2) and 0 ≤ p < E m , K m ∈ HI when m ≥ HI , Shapira and Yuster [17]gave the following substitute, which is an induced version of Theorem 3.2. Theorem 3.11 (Shapira and Yuster [17]) . Suppose that ( G n ) is a sequenceof graphs with | G n | → ∞ . Let F be any fixed graph with | F | > and let < p < . Then ( G n ) is mixed ( p, ¯ p ) -quasi-random if and only if, for allsubsets U , . . . , U | F | of V ( G n ) , N ∗ ( F, G n ; U , . . . , U | F | ) = p e ( F ) (1 − p )( | F | ) − e ( F ) | F | Y i =1 | U i | + o (cid:0) | G n | | F | (cid:1) . (3.5) Moreover, it suffices that (3.5) holds for all sequences U , . . . , U | F | of disjointsubsets of V ( G n ) with the same size, | U | = · · · = | U | F | | . To show the flexibility with which our method combines different condi-tions, we also show that it suffices to consider subsets of a given size forinduced subgraph counts too, in analogy with Theorems 3.5 and 3.6.
Theorem 3.12.
In Theorem 3.11, it suffices that (3.5) holds for all se-quences U , . . . , U | F | of subsets of V ( G n ) with | U i | = ⌊ α | G n |⌋ , for any fixed α with < α < . Alternatively, if < α < / | F | , it suffices that (3.5) holds for all such sequences of disjoint U , . . . , U | F | . Theorem 3.13.
Let < α < and ≤ p ≤ , and let F be a fixed graphwith F ∈ HI ( p ) . Then every sequence ( G n ) with | G n | → ∞ such that (3.4) holds for all subsets U of V ( G n ) with | U | = ⌊ α | G n |⌋ is mixed ( p, ¯ p ) -quasi-random. Remark 3.14.
Theorems 3.6 and 3.12 fail for disjoint sets U , . . . , U | F | in thelimiting case α = 1 / | F | , at least for F = K , see Section 8 and Remark 6.4.We leave it as an open problem to investigate this case for other graphs F .4. Graph limit proof of Theorem 3.2
We give proofs of the theorems above using graph limits; the reader shouldcompare these to the combinatorial proofs in [19; 20; 16; 17; 24] using theSzemer´edi regularity lemma. In order to exhibit the main ideas clearly, webegin in this section with the simplest case and give a detailed proof ofTheorem 3.2. In the following sections we will give the minor modifications needed for the other results, treating the additional complications one byone.The first step is to recall that the space of graphs and graph limits iscompact; thus, every sequence has a convergent subsequence [4]. Hence,if ( G n ) is not p -quasi-random, we can select a subsequence (which we alsodenote by ( G n )), such that G n → W for some graphon W that is notequivalent to the constant graphon p , which simply means that W = p on aset of positive measure.Hence, in order to prove Theorem 3.2, it suffices to assume that further G n → W for some graphon W , and then prove that W = p a.e.4.1. Translating to graphons.
In this subsection we use the graph limittheory in [4] to translate the property (3.2) to graph limits.We begin with an easy consequences of Lebesgue’s differentiation theorem;for future reference we state it as a (well-known) lemma. (See Lemma 6.3below for a stronger version.) We let λ denote Lebesgue measure (in one orseveral dimensions). Lemma 4.1.
Suppose that f : [0 , m → R is an integrable function suchthat R A ×···× A m f = 0 for all sequences A , . . . , A m of disjoint measurablesubsets of [0 , . Then f = 0 a.e.Moreover, it is enough to consider A , . . . , A m with λ ( A ) = · · · = λ ( A m ) ;we may even further impose that λ ( A k ) ∈ { ε , ε , . . . } for any given sequence ε n → .Proof. For any distinct x , . . . , x m ∈ (0 ,
1) and any sufficiently small ε > A i = ( x i − ε, x i + ε ) and find(2 ε ) − m Z | y i − x i | <ε, i =1 ,...,m f ( y , . . . , y m ) = (2 ε ) − m Z A ×···× A m f = 0 . By Lebesgue’s differentiation theorem, see e.g. Stein [21, § f ( x , . . . , x m ) as ε → x , . . . , x m . (cid:3) We can now easily translate the condition (3.2) in Theorem 3.2 to acorresponding condition for the limiting graphon (which we may assumeexists, as discussed above).
Lemma 4.2.
Suppose that G n → W for some graphon W and let F be afixed graph and γ ≥ a fixed number. Then the following are equivalent: (i) For all subsets U , . . . , U | F | of V ( G n ) , N ( F, G n ; U , . . . , U | F | ) = γ | F | Y i =1 | U i | + o (cid:0) | G n | | F | (cid:1) . (4.1)(ii) For all subsets A , . . . , A | F | of [0 , , Z A ×···× A | F | Ψ F,W ( x , . . . , x | F | ) = γ | F | Y i =1 λ ( A i ) . (4.2) UASI-RANDOM GRAPHS AND GRAPH LIMITS 11 (iii) Ψ
F,W ( x , . . . , x | F | ) = γ for a.e. x , . . . , x | F | ∈ [0 , | F | .Proof. (iii) = ⇒ (ii) is trivial, and (ii) = ⇒ (iii) is immediate by Lemma 4.1applied to Ψ F,W − γ .(i) ⇐⇒ (ii). The convergence G n → W is equivalent to δ (cid:3) ( W G n , W ) →
0. By the definition of δ (cid:3) , there thus exist measure preserving bijections ϕ n : [0 , → [0 ,
1] such that if W n := W ϕ n G n , then k W n − W k (cid:3) →
0. Fix n , and let I nj (1 ≤ j ≤ n ) be the intervals of length | G n | − used to define W G n , and let as in (2.6) U ′ := S j ∈ U I nj for a subset U of V ( G n ); further,let I ′′ nj := ϕ − n ( I nj ) and U ′′ := ϕ − n ( U ′ ) = S j ∈ U I ′′ nj . Then, for any subsets U , . . . , U | F | of V ( G n ), by (2.6) and a change of variables, N ( F, G n ; U , . . . , U | F | ) = | G n | | F | Z U ′′ ×···× U ′′| F | Ψ F,W n + o (cid:0) | G n | | F | (cid:1) . Hence, (i) is equivalent to Z U ′′ ×···× U ′′| F | Ψ F,W n = γ | F | Y i =1 | U i || G n | + o (1) = γ | F | Y i =1 λ ( U ′′ i ) + o (1) , (4.3)for all subsets U ′′ i that are unions of sets I ′′ nj .We next extend (4.3) from the special sets U ′′ i (in a family that dependson n ) to arbitrary (measurable) sets. Thus, assume that (4.3) holds, andlet A , . . . , A | F | be arbitrary subsets of [0,1]. Fix n and let a ij := λ ( A i ∩ I ′′ nj ) /λ ( I ′′ nj ). Further, let B i be a random subset of [0,1] obtained by takingan independent family J ij of independent 0–1 random variables with P ( J ij =1) = a ij , and then taking B i := S j : J ij =1 I ′′ nj . Then the sets B i are of theform U ′′ i , so (4.3) applies to them, and, noting that W n is constant on everyset I ′′ ni × I ′′ nj , and hence Ψ F,W n is constant on every set I ′′ nj × · · · × I ′′ nj | F | , Z A ×···× A | F | (Ψ F,W n − γ ) = | G n | X j ,...,j | F | =1 | F | Y i =1 a ij i Z I ′′ nj ×···× I ′′ nj | F | (Ψ F,W n − γ )= E | G n | X j ,...,j | F | =1 | F | Y i =1 J ij i Z I ′′ nj ×···× I ′′ nj | F | (Ψ F,W n − γ )= E Z B ×···× B | F | (Ψ F,W n − γ ) = o (1) , (4.4)where the final estimate uses (4.3). Consequently, (4.3), for all special sets U ′′ i , is equivalent to the same estimate Z A ×···× A | F | Ψ F,W n = γ | F | Y i =1 λ ( A i ) + o (1) , (4.5) for any measurable sets A , . . . , A | F | in [0 , A , . . . , A | F | .)It is well-known that for two graphons W and W ′ , (cid:12)(cid:12)(cid:12)(cid:12)Z [0 , m (cid:0) Ψ F,W − Ψ F,W ′ (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) = O ( k W − W ′ k (cid:3) ) , see [4]; moreover, the proof in [4] (or the version of the proof in [2]) showsthat the same holds, uniformly, also if we integrate over a subset A ×· · · × A m . (In other words, extending the cut norm to functions of severalvariables as in [1], k Ψ F,W − Ψ F,W ′ k (cid:3) = O ( k W − W ′ k (cid:3) ).) Consequently, theassumption G n → W , which as said yields k W n − W k (cid:3) →
0, implies that R A ×···× A | F | Ψ F,W n = R A ×···× A | F | Ψ F,W + o (1), and thus (4.5), and hence (i),is equivalent to Z A ×···× A | F | Ψ F,W = γ | F | Y i =1 λ ( A i ) + o (1) . (4.6)Consequently, (ii) = ⇒ (i). Conversely, none of the terms in (4.6) dependson n , so if (4.6) holds, then the o (1) error term vanishes and (4.2) holds.Hence (i) = ⇒ (ii). (cid:3) An optional measure theoretic interlude.
To prove Theorem 3.2,it thus remains only to show that if W is a graphon such that Ψ F,W = p e ( F ) a.e., then W = p a.e. (In the terminology of Remark 3.9, “Ψ F,W = p e ( F ) ” isa p -quasi-random property.)We know several ways to do this. One, direct, is given in Subsection 4.4.However, as will be seen in Subsection 4.3, it is much simpler to argue ifwe can assume that Ψ F,W = p e ( F ) everywhere, and not just a.e. (The mainreason is that we then can choose x = x = · · · = x | F | .) Hence, somewhatsurprisingly, the qualification ’a.e.’ here forms a significant technical prob-lem. Usually, ’a.e.’ is just a technical formality in arguments in integrationand measure theory, but here it is an obstacle and we would like to get ridof it. We do not see any trivial way to do this, but we can do it as follows.(To say that W ′ is a version of W means that W ′ = W a.e.; this implies thatall integrals considered here are equal for W and W ′ , and thus G n → W ′ aswell.) See Subsection 4.5 and Appendix A for an alternative. Lemma 4.3.
Let F be a graph with e ( F ) > , and let W be a graphon. If Ψ F,W = γ > a.e. on [0 , | F | , then there exists a version W ′ of W suchthat Ψ F,W ′ ( x , . . . , x | F | ) = γ for all ( x , . . . , x | F | ) ∈ [0 , | F | .Proof. By symmetry, we may assume 12 ∈ E ( F ); hence Ψ F,W ( x , . . . , x | F | ),defined in (2.2), contains a factor W ( x , x ). We let x ′ := ( x , . . . , x | F | )and collect the other factors in (2.2) into a product f ( x , x ′ ) of the factors UASI-RANDOM GRAPHS AND GRAPH LIMITS 13 corresponding to edges 1 j ∈ E ( F ) with j ≥
3, and another product g ( x , x ′ )of the remaining factors. ThusΨ F,W ( x , . . . , x | F | ) = W ( x , x ) f ( x , x ′ ) g ( x , x ′ ) . By assumption, thus W ( x , x ) f ( x , x ′ ) g ( x , x ′ ) = γ (4.7)for a.e. ( x , x , x ′ ). We may thus choose x ′ (a.e. choice will do) such that(4.7) holds for a.e. ( x , x ). We fix one such x ′ and write f ( x ) := f ( x, x ′ ), g ( y ) := g ( y, x ′ ); we then have W ( x, y ) f ( x ) g ( y ) = γ for a.e. ( x, y ).We define W ( x, y ) := max (cid:0) , γ/ ( f ( x ) g ( y )) (cid:1) ; thus W = W a.e.Let | ( x , . . . , x m ) | ∞ := max | x i | . Recall that if f is an integrable functionon R m for some m (or on a subset such as [0 , m ), then a point x is a Lebesgue point of f if (2 ε ) − m R | y − x | ∞ <ε | f ( y ) − f ( x ) | d y = o (1) as ε → X εx is a random point in the cube { y : | y − x | ∞ < ε } , then f ( X εx ) L −→ f ( x ). For bounded functions, which isthe case here, this is equivalent to f ( X εx ) p −→ f ( x ) as ε →
0, which shows,for example, that if x is a Lebesgue point of both f and g , then it is also aLebesgue point of f ± g , f g , and, provided g ( x ) = 0, of f /g . It is well-known,see e.g. Stein [21, § f is integrable, then a.e. point is a Lebesguepoint of f .We can thus find a null set N ⊂ [0 ,
1] such that every x ∈ S := [0 , \ N isa Lebesgue point of both f and g . Since W ( x, y ) ≤ f ( x ) g ( y ) ≥ γ a.e., it then follows that if ( x , x ) ∈ S , then ( x , x ) is a Lebesgue point of W . This implies, by the definition (2.2), that if ( x , . . . , x | F | ) ∈ S | F | , then( x , . . . , x | F | ) is a Lebesgue point of Ψ F,W ; hence, using Ψ
F,W = Ψ F,W a.e.and Ψ
F,W = γ a.e., Ψ F,W ( x , . . . , x | F | ) = γ for ( x , . . . , x | F | ) ∈ S | F | .This would really be enough for our purposes, but to obtain the conclusionas stated, we choose x ∈ S and define ϕ : [0 , → [0 ,
1] by ϕ ( x ) = x for x ∈ S and ϕ ( x ) = x for x ∈ N ; then W ′ := W ϕ satisfies Ψ F,W ′ = γ everywhere. (cid:3) Remark 4.4.
Although we do not need it, we note that Lemma 4.3 is validfor the trivial case e ( F ) = 0 too, since then Ψ F,W = 1 for every W and thereis nothing to prove. We do not know whether Lemma 4.3 is also valid for γ = 0; consider for example F = K . (In this case it suffices to consider0/1-valued W and W ′ .)4.3. The first algebraic argument.
The proof of Theorem 3.2 is nowcompleted, by Lemmas 4.2 and 4.3 and the remarks above, by the followinglemma:
Lemma 4.5.
Let F be a graph with e ( F ) > and let W be a graphon. If p > and Ψ F,W ( x , . . . , x | F | ) = p e ( F ) for every ( x , . . . , x | F | ) ∈ [0 , | F | , then W = p . Proof.
First take x = x = · · · = x | F | = x . Then Ψ F,W ( x , . . . , x | F | ) = W ( x, x ) e ( F ) , and thus W ( x, x ) = p , for every x ∈ [0 , d of vertex 1 in F is non-zero. Let x, y ∈ [0 ,
1] and take x = x and x = · · · = x | F | = y . Then p e ( F ) = Ψ F,W ( x , . . . , x | F | ) = W ( x, y ) d W ( y, y ) e ( F ) − d = W ( x, y ) d p e ( F ) − d . Hence W ( x, y ) = p . (cid:3) This completes the first version of our graph limit proof of Theorem 3.2.4.4.
The second algebraic argument.
As said above, we can alterna-tively avoid Lemma 4.3 and instead use the following stronger version ofLemma 4.5, which together with Lemma 4.2 yields another proof of Theo-rem 3.2.
Lemma 4.6.
Let F be a graph with e ( F ) > and let W be a graphon. If Ψ F,W ( x , . . . , x | F | ) = p e ( F ) for a.e. ( x , . . . , x | F | ) ∈ [0 , | F | , then W = p a.e.Proof. We first symmetrize. If σ ∈ S | F | , the symmetric group of all permu-tations of { , . . . , | F |} , let σ ( F ) be the image of F , with edges σ ( i ) σ ( j ) for ij ∈ E ( F ), and consider Y σ ∈ S | F | Ψ σ ( F ) ,W ( x , . . . , x | F | ) = Y ≤ i Suppose that ≤ d ≤ h , and let a ( I ) be an array defined forall d -subsets I of [ h + d ] . Suppose further that for every h -subset J of [ h + d ] , X I ⊆ J a ( I ) = 0 , (4.8) summing over the (cid:0) hd (cid:1) subsets of size d . Then a ( I ) = 0 for every I .Proof. This is a form of a result by Gottlieb [11]. (It is easily proved byfixing a d -subset I and then summing (4.8) for all J with | J ∩ I | = k , for k = 0 , . . . , d ; we omit the details.) (cid:3) UASI-RANDOM GRAPHS AND GRAPH LIMITS 15 Further proofs. Instead of Lemma 4.3 we may use the weaker butmore general Lemma A.3 in Appendix A; this lemma, with Φ(( w ij ) i We next give a proof of Theorem 3.1 along the lines of Section 4. We beginwith a lemma giving an analogue of Lemma 4.1 for the case A = · · · = A m .If f is a function on [0 , m for some m , we let ˜ f denote its symmetrizationdefined by ˜ f ( x , . . . , x m ) := 1 m ! X σ ∈ S m f (cid:0) x σ (1) , . . . , x σ ( m ) (cid:1) , (5.1)where S m is the symmetric group of all m ! permutations of { , . . . , m } . Notethat for any integrable f and any subset A of [0,1], Z A m ˜ f = Z A m f. (5.2) Lemma 5.1. Suppose that f : [0 , m → R is an integrable function suchthat R A m f = 0 for all measurable subsets A of [0 , . Then ˜ f = 0 a.e. Proof. Let A , . . . , A m be disjoint subsets of [0 , ξ , . . . , ξ m ∈{ , } m , take A := S i : ξ i =1 A i . Then A = P mi =1 ξ i A i and0 = Z A m f = Z [0 , m f A m = m X i ,...,i m =1 ξ i · · · ξ i m Z A i ×···× A im f. (5.3)The monomials ξ i · · · ξ i k with i < · · · < i k , 0 ≤ k ≤ m , form a basis ofthe 2 m -dimensional space of functions on { , } m . Hence, collecting termsin (5.3), the coefficient of each such monomial vanishes. In particular, forthe coefficient of ξ · · · ξ m we obtain a contribution only when i , . . . , i m is apermutation of 1 , . . . , m , and we obtain0 = X σ ∈ S m Z A σ (1) ×···× A σ ( m ) f = m ! Z A ×···× A m ˜ f . The result follows by Lemma 4.1, applied to ˜ f . (cid:3) We can now translate the property (3.1) to graphons, cf. Lemma 4.2. Lemma 5.2. Suppose that G n → W for some graphon W and let F be afixed graph and γ ≥ a fixed number. Then the following are equivalent: (i) For all subsets U of V ( G n ) , N ( F, G n ; U ) = γ | U | | F | + o (cid:0) | G n | | F | (cid:1) . (ii) For all subsets A of [0 , , Z A | F | Ψ F,W ( x , . . . , x | F | ) = γλ ( A ) | F | . (iii) e Ψ F,W ( x , . . . , x | F | ) = γ for a.e. x , . . . , x | F | ∈ [0 , | F | .Proof. This is proved almost exactly as Lemma 4.2, with obvious notationalchanges and with Lemma 4.1 replaced by Lemma 5.1, which together with(5.2) implies (ii) ⇐⇒ (iii). The main difference is that we now use a singlerandom set B := S j : J j =1 I ′′ nj , where { J j } is a family of independent indicatorvariables. Hence, the analogue of (4.4) is not exact; we have E | F | Y i =1 J j i = | F | Y i =1 a j i (5.4)when j , . . . , j | F | are distinct, but in general not when two or more areequal. However, there are only O ( | G n | | F |− ) choices of indices with at leasttwo coinciding, and each such choice introduces an error that is at most λ ( I ′′ nj × · · · × I ′′ nj | F | ) = | G n | −| F | . Hence, we now have Z A n (Ψ F,W n − γ ) = E Z B n (Ψ F,W n − γ ) + o (1) . (5.5)The error o (1) is unimportant, and, assuming (i), the conclusion of (4.4) isvalid in the form R A n (Ψ F,W n − γ ) = o (1), which yields (ii) as in Section 4. (cid:3) UASI-RANDOM GRAPHS AND GRAPH LIMITS 17 We do not know any direct proof of the analogue of Lemma 4.3 for e Ψ F,W . (This result follows by Lemma 5.2 and Theorem 3.1 once the lat-ter is proven.) However, as in Subsection 4.5 we nevertheless can use thefollowing lemma, which is a strengthening of Lemma 4.5. Lemma 5.3. Let F be a graph with e ( F ) > and let W be a graphon. If e Ψ F,W ( x , . . . , x | F | ) = p e ( F ) for every ( x , . . . , x | F | ) ∈ [0 , | F | , then W = p .Proof. As in the proof of Lemma 4.5, first take x = · · · = x | F | = x . Then e Ψ F,W ( x , . . . , x | F | ) = Ψ F,W ( x , . . . , x | F | ) = W ( x, x ) e ( F ) , and thus W ( x, x ) = p . Using this, it is easy to see that if we take x = x and x = · · · = x | F | = y ,and d i is the degree of vertex i , then p e ( F ) = e Ψ F,W ( x , . . . , x | F | ) = 1 | F | X i ∈ V ( F ) (cid:18) W ( x, y ) p (cid:19) d i p e ( F ) . Since the right-hand side is a strictly increasing function of W ( x, y ), thisequation has only the solution W ( x, y ) = p . (cid:3) As in Section 4 there is a companion result where we allow exceptionalnull sets. Lemma 5.4. Let F be a graph with e ( F ) > and let W be a graphon. If e Ψ F,W ( x , . . . , x | F | ) = p e ( F ) for a.e. ( x , . . . , x | F | ) ∈ [0 , | F | , then W = p a.e.Proof. We have not tried to find a direct proof, since this follows directlyfrom Lemma 5.3 and Corollary A.6. (cid:3) Theorem 3.1 now follows from Lemmas 5.2 and 5.4. (Alternatively, wemay use Lemma A.3 or Theorem A.5(iii) and argue as in the proof ofLemma 5.3.) 6. Further variations Disjoint subsets. In Section 4 the sets U , . . . , U | F | of vertices werearbitrary and in Section 5 they were assumed to coincide. The oppositeextreme is to require that they are disjoint. We can translate this version tooto graphons as follows. Note that (iii) in the following lemma is that same asLemma 4.2(iii); hence the two lemmas together show that it is equivalent toassume (4.1) (or (3.2)) for disjoint U , . . . , U | F | only; this implies the generalcase. Lemma 6.1. Suppose that G n → W for some graphon W and let F be afixed graph and γ ≥ a fixed number. Then the following are equivalent: (i) For all disjoint subsets U , . . . , U | F | of V ( G n ) , N ( F, G n ; U , . . . , U | F | ) = γ | F | Y i =1 | U i | + o (cid:0) | G n | | F | (cid:1) . (ii) For all disjoint subsets A , . . . , A | F | of [0 , , Z A ×···× A | F | Ψ F,W ( x , . . . , x | F | ) = γ | F | Y i =1 λ ( A i ) . (iii) Ψ F,W ( x , . . . , x | F | ) = γ for a.e. x , . . . , x | F | ∈ [0 , | F | .Proof. Again we follow the proof of Lemma 4.2. The only difference isthat we consider only disjoint sets U , . . . , U | F | , etc. In particular, givendisjoint subsets A , . . . , A | F | of [0 , B i so that they too are disjoint. We do this by taking, for each j , the0–1 random variables J ij dependent, so that P i J ij ≤ 1. (This is possiblebecause P i a ij ≤ A , . . . , A | F | are disjoint.) The vectors ( J ij ) | F | i =1 for different j are chosen independent as before. Just as in the proof ofLemma 5.2, the dependency among the J ij means that (4.4) is not exact:in analogy with (5.4), E Q | F | i =1 J ij i = Q | F | i =1 a ij i when j , . . . , j | F | are distinct,but not in general. However, again as in the proof of Lemma 5.2, thetotal error is o (1), so the analogue of (5.5) holds, and thus the conclusion R A ×···× A | F | (Ψ F,W n − γ ) = o (1) of (4.4) holds for all disjoint sets A , . . . , A | F | .Finally, for (ii) = ⇒ (iii), note that Lemma 4.1 already is stated so that itsuffices to consider disjoint A , . . . , A | F | . (cid:3) Lemma 6.1, combined with the remainder of the proof of Theorem 3.2in Section 4, shows that in Theorem 3.2, it is sufficient to assume (3.2) fordisjoint U , . . . , U | F | .6.2. Sets of the same size. Another variation of Theorem 3.2 is to con-sider only subsets U , . . . , U | F | of the same size. (We may combine this withthe preceding variation and require that the sets are disjoint too.) This canbe translated to considering only subsets A , . . . , A | F | of the same measureby the same method as in the next subsection, when we further let the com-mon size be a given number. Since we obtain stronger results in the nextsubsection, we leave the details to the reader.6.3. Sets of a given size. Another variation of Theorem 3.2 is Theorem 3.6where we consider only subsets U , . . . , U | F | of a given size, which we assumeis a fixed fraction α of | G n | (rounded to an integer). This is translated tographons as follows. Lemma 6.2. Suppose that G n → W for some graphon W and let F be afixed graph and γ ≥ and α ∈ (0 , be fixed numbers. Then the followingare equivalent: (i) For all subsets U , . . . , U | F | of V ( G n ) with | U i | = ⌊ α | G n |⌋ , N ( F, G n ; U , . . . , U | F | ) = γ | F | Y i =1 | U i | + o (cid:0) | G n | | F | (cid:1) . (6.1) UASI-RANDOM GRAPHS AND GRAPH LIMITS 19 (ii) For all subsets A , . . . , A | F | of [0 , with λ ( A i ) = α , Z A ×···× A | F | Ψ F,W ( x , . . . , x | F | ) = γ | F | Y i =1 λ ( A i ) . (6.2)(iii) Ψ F,W ( x , . . . , x | F | ) = γ for a.e. x , . . . , x | F | ∈ [0 , | F | .If α < / | F | , we may further, as in Lemma 6.1, in (i) and (ii) add therequirement that the sets be disjoint.Proof. The equivalence (i) ⇐⇒ (ii) is proved as in the proof of Lemma 4.2,but some care has to be taken with the sizes and measures of the sets. Wenote that for any sets A , . . . , A | F | and A ′ , . . . , A ′| F | , (cid:12)(cid:12)(cid:12)(cid:12)Z A ×···× A | F | Ψ F,W n − Z A ′ ×···× A ′| F | Ψ F,W n (cid:12)(cid:12)(cid:12)(cid:12) ≤ | F | X i =1 λ ( A i △ A ′ i ) . (6.3)Hence, we can modify the sets without affecting the results as long as thedifference has measure o (1). We argue as follows.We obtain as in Section 4 that (i) is equivalent to (4.3), now for allsubsets U ′′ i of [0 , 1] that are unions of sets I ′′ nj and have measures λ ( U ′′ i ) = ⌊ α | G n |⌋ / | G n | . If (ii) holds, we may for any such U ′′ i find A i ⊇ U ′′ i with λ ( A i ) = α ; then (6.2) implies first (4.5) and then (4.3) by (6.3).Conversely, given A , . . . , A | F | with measures λ ( A i ) = α , the random sets B i constructed above (either as in Section 4 or as in Subsection 6.1 in thedisjoint case) have measures that are random but well concentrated: E λ ( B i ) = X j E J ij λ ( I ′′ nj ) = X j a ij λ ( I ′′ nj ) = λ ( A i ) = α Var λ ( B i ) = X j Var( J ij ) λ ( I ′′ nj ) ≤ | G n | − → . Hence, if δ n := | G n | − / , say, then by Chebyshev’s inequality P ( | λ ( B i ) − α | > δ n ) ≤ δ − n Var( λ ( B i )) ≤ δ n → . If | λ ( B i ) − α | ≤ δ n for all i , we adjust B i to a set U ′′ i with λ ( U ′′ i ) = ⌊ α | G n |⌋ / | G n | so that λ ( B i △ U ′′ i ) ≤ δ n + | G n | − ≤ δ n , and thus Z B ×···× B | F | Ψ F,W n = Z U ′′ ×···× U ′′| F | Ψ F,W n + O ( δ n ) . Consequently, if (4.3) holds, then R B ×···× B | F | Ψ F,W n = γα | F | + O ( δ n ) + o (1)whenever | λ ( B i ) − α | ≤ δ n for all i , and thus E Z B ×···× B | F | Ψ F,W n = γα | F | + O ( δ n ) + o (1) + O (cid:16) | F | X i =1 P (cid:0) | λ ( B i ) − α | > δ n (cid:1)(cid:17) = γα | F | + o (1) . Hence, (4.5) holds, for A , . . . , A | F | with measures λ ( A i ) = α , and thus (ii)holds by the argument in Section 4.This proves (i) ⇐⇒ (ii); we may add the requirement that the sets bedisjoint by the argument in the proof of Lemma 6.1.To see that (ii) ⇐⇒ (iii), we use the following analysis lemma. (Thisseems to be less well-known that Lemma 4.1; we guess that it is known, butwe have been unable to find a reference.) (cid:3) Lemma 6.3. Let α ∈ (0 , . Suppose that f : [0 , m → R is an integrablefunction such that R A ×···× A m f = 0 for all sequences A , . . . , A m of mea-surable subsets of [0 , such that λ ( A ) = · · · = λ ( A m ) = α . Then f = 0 a.e.Moreover, if α < m − , it is enough to consider disjoint A , . . . , A m .Proof. For f ∈ L ([0 , m ) and A , . . . , A m ⊆ [0 , f ( A , . . . , A m ) := Z A ×···× A m f, and define further the functions f A ( x , . . . , x m ) := Z A f ( x , x , . . . , x m ) d x and f A ,...,A m ( x ) := Z A ×···× A m f ( x , x , . . . , x m ) d x · · · d x m . By Fubini’s theorem, f ( A , . . . , A m ) = f A ( A , . . . , A m ) = f A ,...,A m ( A ) . (6.4)We will derive the lemma from the following claims, which we will proveby induction in m . Let B be a measurable subset of [0 , , let < α < and let f be anintegrable function on B m . (i) If α < λ ( B ) and f ( A , . . . , A m ) = 0 for all A , . . . , A m ⊂ B with λ ( A ) = · · · = λ ( A m ) = α , then f ( A , . . . , A m ) = 0 for all A , . . . , A m ⊆ B . (ii) If mα < λ ( B ) and f ( A , . . . , A m ) = 0 for all disjoint A , . . . , A m ⊂ B with λ ( A ) = · · · = λ ( A m ) = α , then f ( A , . . . , A m ) = 0 for alldisjoint A , . . . , A m ⊂ B with λ ( A ) , . . . , λ ( A m ) ≤ α . Consider first the case m = 1, in which case (i) and (ii) have the samehypotheses: α < λ ( B ) and f ( A ) = 0 if λ ( A ) = α . Suppose that A , A ⊂ B with λ ( A ) = λ ( A ) ≤ δ := ( λ ( B ) − α ). Then λ ( B \ ( A ∪ A )) ≥ λ ( B ) − δ = α , and we may thus find a set A ⊆ B \ ( A ∪ A ) with λ ( A ) = α − λ ( A ). The assumption yields f ( A ∪ A ) = 0 = f ( A ∪ A ),and thus f ( A ) = − f ( A ) = f ( A ) . (6.5)If A ⊂ B is given with λ ( A ) ≤ δ and λ ( A ) = α/N for some integer N , let A = A and choose further sets A , . . . , A N ⊂ B of the same measure α/N UASI-RANDOM GRAPHS AND GRAPH LIMITS 21 and with A , . . . , A N disjoint. By (6.5), then f ( A k ) = f ( A ) = f ( A ) forevery k ≤ N , and thus, by the assumption,0 = f (cid:16) N [ k =1 A k (cid:17) = N X k =1 f ( A k ) = N f ( A ) . Consequently, f ( A ) = 0 for every A ⊂ B with λ ( A ) ≤ δ and λ ( A ) = α/N . If x is a density point of B (i.e., a point in B that is a Lebesgue point of B ),then there is a sequence ε n → λ ( B ∩ ( x − ε n , x + ε n )) = α/n ,and thus by we just have shown, R x + ε n x − ε n f B = f ( B ∩ ( x − ε n , x + ε n )) = 0for every n . If further x is a Lebesgue point of f B , then this implies f ( x ) = f ( x ) B ( x ) = 0. Since a.e. x ∈ B satisfies these conditions, f = 0 a.e. on B , which of course is equivalent to f ( A ) = 0 for every A ⊆ B .This proves both (i) and (ii) for m = 1.For m > 1, we use, as already said, induction, and assume that the claimsare true for smaller m . To prove (i), we fix A ⊂ B with λ ( A ) = α , andsee by (6.4) that f A satisfies the assumptions of (i) on B m − . Thus, by theinduction hypothesis, f A ( A , . . . , A m ) = 0 for all A , . . . , A m ⊆ B . Fixingnow instead such A , . . . , A m , (6.4) shows that f A ,...,A m ( A ) = 0 for all λ ( A ) ⊂ B with λ ( A ) = α , and thus by the case m = 1, f A ,...,A m ( A ) = 0for all λ ( A ) ⊂ B . By (6.4) again, this proves the induction hypothesis.Thus (i) is proved in general.To prove (ii), we again fix A , and see by (6.4) that f A satisfies theassumptions of (ii) on ( B \ A ) m − , noting that ( m − α < λ ( B \ A ).Thus, by the induction hypothesis, f A ( A , . . . , A m ) = 0 for all disjoint A , . . . , A m ⊆ B \ A with λ ( A k ) ≤ α for every k . Hence, if we instead fixdisjoint sets A , . . . , A m ⊂ B with λ ( A k ) ≤ α for every k , then (6.4) showsthat f A ,...,A m ( A ) = 0 for every A ⊂ B \ ( A ∪· · ·∪ A m ) with λ ( A ) = α , andthus by the case m = 1, f A ,...,A m ( A ) = 0 for every A ⊂ B \ ( A ∪ · · · ∪ A m )with λ ( A ) ≤ α . By (6.4) again, this proves the induction hypothesis, and(ii) is proved.We have proved the claims above. We now take B = [0 , 1] and the lemmafollows immediately by Lemma 4.1. (cid:3) Remark 6.4. When α = m − , it is not enough to consider disjoint sets A , . . . , A m in Lemma 6.3. In fact, any f of the type P mi =1 g ( x i ) where R g = 0 satisfies the assumption for such A , . . . , A m . (We do not knowwhether these are the only possible f .) Taking W of this type and F = K , so that Ψ F,W = W , we get a counter-example to Lemma 6.2, and toTheorem 3.6, for disjoint sets and α = 1 / | F | ; see also Section 8 where thisexample reappears in a different formulation. We do not know whether thereare such counter-examples for other graphs F . Proof of Theorem 3.6. Theorem 3.6 follows by using Lemma 6.2 instead ofLemma 4.2 in (any version of) the proof of Theorem 3.2 in Section 4. (cid:3) A single subset of a given size. The corresponding variation ofTheorem 3.1 is Theorem 3.5 where we consider a single subset U with agiven fraction α of the vertices. Again, there is a straightforward translationto graphons. Lemma 6.5. Suppose that G n → W for some graphon W and let F be afixed graph and γ ≥ and α ∈ (0 , be fixed numbers. Then the followingare equivalent: (i) For every subset U of V ( G n ) with | U | = ⌊ α | G n |⌋ , N ( F, G n ; U ) = γ | U | | F | + o (cid:0) | G n | | F | (cid:1) . (ii) For every subset A of [0 , with λ ( A ) = α , Z A | F | Ψ F,W ( x , . . . , x | F | ) = γλ ( A ) | F | . (iii) e Ψ F,W ( x , . . . , x | F | ) = γ for a.e. x , . . . , x | F | ∈ [0 , | F | .Proof. The equivalence (i) ⇐⇒ (ii) is proved as for Lemma 6.2, using singlesets U , A and B as in the proof of Lemma 5.2.The equivalence (ii) ⇐⇒ (iii) follows by the following lemma, which stren-thens Lemma 5.1 by considering subsets of a given size only. (cid:3) Lemma 6.6. Let α ∈ (0 , . Suppose that f : [0 , m → R is an integrablefunction such that R A m f = 0 for all measurable subsets A of [0 , with λ ( A ) = α . Then ˜ f = 0 a.e.Proof. We begin by showing that the vanishing property extends to sets A with measure greater than α as follows:If A ⊆ [0 , 1] with λ ( A ) = rα for some rational r ≥ 1, then Z A m f = 0 . (6.6)(The restriction to rational r may easily be removed by continuity, but itwill suffice for us.) To see this, let N be an integer such that M := rN is an integer, and partition A into M subsets A , . . . , A M of equal measure λ ( A i ) = λ ( A ) /M = rα/M = α/N . Pick N of the sets A i at random(uniformly over all (cid:0) MN (cid:1) possibilities), and let B be their union. Thus B is a random subset of [0 , 1] with λ ( B ) = α , and thus by the assumption R B m f = 0. Taking the expectation we find0 = E Z B m f = M X i ,...,i m =1 P ( A i , . . . , A i m ⊆ B ) Z A i ×···× A im f. (6.7)If i , . . . , i m are distinct, then, letting ( N ) m denote the falling factorial, P ( A i , . . . , A i m ⊆ B ) = ( N ) m ( M ) m = (cid:18) NM (cid:19) m + O (cid:18) N (cid:19) = r − m + O (cid:18) N (cid:19) . This fails if two or more of i , . . . , i m coincide (in fact, the probabilityis ( N ) ν / ( M ) ν ≈ r − ν , where ν is the number of distinct indices among UASI-RANDOM GRAPHS AND GRAPH LIMITS 23 i , . . . , i m ), so we let U N ⊆ [0 , m be the union of all A i × · · · × A i m withat least two coinciding indices. By (6.7),( N ) m ( M ) m Z A m f = M X i ,...,i m =1 ( N ) m ( M ) m Z A i ×···× A im f = M X i ,...,i m =1 (cid:18) ( N ) m ( M ) m − P ( A i × · · · × A i m ⊆ B ) (cid:19) Z A i ×···× A im f, and thus (cid:12)(cid:12)(cid:12)(cid:12) ( N ) m ( M ) m Z A m f (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z U N | f | . (6.8)Now let N → ∞ (with rN integer). Note that λ ( U N ) ≤ (cid:0) m (cid:1) N m − ( α/N ) m ≤ (cid:0) m (cid:1) /N . Thus λ ( U N ) → f is integrable, R U N | f | → 0. Itfollows from (6.8) and ( N ) m / ( M ) m → r − m that r − m R A m f = 0, whichproves (6.6).Next, let A , . . . , A m be arbitrary disjoint subsets of [0 , 1] with equalmeasure λ ( A ) = · · · = λ ( A m ) = qα , for some rational q such that (1 + mq ) α ≤ 1. Choose A ⊆ [0 , \ S m A i with λ ( A ) = α . For any sequence ξ , . . . , ξ m ∈ { , } m , let ξ := 1 and take A := S i ≥ ξ i =1 A i . Then A = P mi =0 ξ i A i and we argue as in the proof of Lemma 5.1 with an extra set A :we have0 = Z A m f = Z [0 , m f A m = m X i ,...,i m =0 ξ i · · · ξ i m Z A i ×···× A im f. (6.9)As in the proof of Lemma 5.1, it follows that the coefficient of ξ · · · ξ m in(6.9) must vanish, and this coefficient comes from the terms where i , . . . , i m is a permutation of 1 , . . . , m . We thus obtain0 = X σ ∈ S m Z A σ (1) ×···× A σ ( m ) f = m ! Z A ×···× A m ˜ f . The result follows by Lemma 4.1 or 6.3, applied to ˜ f . (cid:3) Proof of Theorem 3.5. Theorem 3.5 follows by combining Lemma 6.5 andLemma 5.4, cf. Section 5. (cid:3) Induced subgraph counts When considering counts of induced subgraphs, we translate the condi-tions to graphons similarly as above. Lemma 7.1. Suppose that G n → W for some graphon W and let F be afixed graph and γ ≥ a fixed number. Then the following are equivalent: (i) For all subsets U , . . . , U | F | of V ( G n ) , N ∗ ( F, G n ; U , . . . , U | F | ) = γ | F | Y i =1 | U i | + o (cid:0) | G n | | F | (cid:1) . (ii) For all subsets A , . . . , A | F | of [0 , , Z A ×···× A | F | Ψ ∗ F,W ( x , . . . , x | F | ) = γ | F | Y i =1 λ ( A i ) . (iii) Ψ ∗ F,W ( x , . . . , x | F | ) = γ for a.e. x , . . . , x | F | ∈ [0 , | F | .We may further in (i) and (ii) add the conditions that, as in Lemma 6.1,the sets be disjoint, or that, as in Lemma 6.2, | U i | = ⌊ α | G n |⌋ and λ ( A i ) = α for a fixed α ∈ (0 , , or, provided α < / | F | , both.Proof. As for Lemma 4.2, using (2.9) instead of (2.6), and with the extraconditions treated as for Lemmas 6.1 and 6.2. (cid:3) Lemma 7.2. Suppose that G n → W for some graphon W and let F be afixed graph and γ ≥ a fixed number. Then the following are equivalent: (i) For all subsets U of V ( G n ) , N ∗ ( F, G n ; U ) = γ | U | | F | + o (cid:0) | G n | | F | (cid:1) . (ii) For all subsets A of [0 , , Z A | F | Ψ ∗ F,W ( x , . . . , x | F | ) = γλ ( A ) | F | . (iii) e Ψ ∗ F,W ( x , . . . , x | F | ) = γ for a.e. x , . . . , x | F | ∈ [0 , | F | .We may further in (i) and (ii) add the conditions that, as in Lemma 6.5, | U | = ⌊ α | G n |⌋ and λ ( A ) = α for a fixed α ∈ (0 , .Proof. As for Lemma 5.2, using (2.9) instead of (2.6), and with the extrasize conditions treated as for Lemma 6.5, using Lemma 6.6. (cid:3) However, it is now more complicated to do the algebraic step, i.e., to solvethe equations in (iii) in these lemmas; the reason is that Ψ ∗ F,W and e Ψ ∗ F,W are not monotone in W . For Ψ ∗ F,W , we can argue as follows. (See also thesomewhat different argument in [17].) Lemma 7.3. Let F be a graph with | F | > , let W be a graphon and let p ∈ (0 , . If Ψ ∗ F,W ( x , . . . , x | F | ) = p e ( F ) (1 − p )( | F | ) − e ( F ) for every x , . . . , x | F | ∈ [0 , | F | , then either W = p or W = ¯ p .Proof. First, take all x i equal. Recalling the definitions (2.7) and (3.3), wesee thatΨ ∗ F,W ( x, . . . , x ) = W ( x, x ) e ( F ) (1 − W ( x, x )) e ( F ) = β F ( W ( x, x )) . UASI-RANDOM GRAPHS AND GRAPH LIMITS 25 Thus, β F ( W ( x, x )) = β F ( p ), and hence, cf. Section 3, W ( x, x ) ∈ { p, ¯ p } forevery x .Next, if vertex i has degree d i and we choose x i = y and x j = x for j = i ,thenΨ ∗ F,W ( x , . . . , x | F | ) = (cid:18) W ( x, y ) W ( x, x ) (cid:19) d i (cid:18) − W ( x, y )1 − W ( x, x ) (cid:19) | F |− − d i Ψ ∗ F,W ( x, . . . , x ) , and thus (cid:18) W ( x, y ) W ( x, x ) (cid:19) d i (cid:18) − W ( x, y )1 − W ( x, x ) (cid:19) | F |− − d i = 1 , i ∈ V ( F ) . (7.1)If F is not regular, we may choose vertices i and j with d i = d j . Takinglogarithms of (7.1) and the same equation with i replaced by j , we obtain anon-singular homogeneous system of linear equations in log( W ( x, y ) /W ( x, x ))and log((1 − W ( x, y )) / (1 − W ( x, x ))), and thus these logarithms vanish, so W ( x, y ) = W ( x, x ) for every x and y in [0 , x, y ∈ [0 , W ( x, x ) = W ( x, y ) = W ( y, x ) = W ( y, y ), and it follows that W is constant,and thus either W = p or W = ¯ p .It remains to treat the case when F is regular, d i = d for all i . Note firstthat if F is a complete graph, then Ψ ∗ F,W = Ψ F,W , and the result follows byLemma 4.5. Further, if F is empty, the result follows by taking complements,replacing F by F , which is complete, W by 1 − W , and p by 1 − p . We maythus assume that 1 ≤ d ≤ | F | − i, j ∈ V ( F ) and let x i = x j = y and x k = x , k = i, j . If there is an edge ij ∈ E ( F ), thenΨ ∗ F,W ( x , . . . , x | F | ) = (cid:18) W ( x, y ) W ( x, x ) (cid:19) d − (cid:18) − W ( x, y )1 − W ( x, x ) (cid:19) | F |− − d ) × (cid:18) W ( y, y ) W ( x, x ) (cid:19) Ψ ∗ F,W ( x, . . . , x ) , and thus, using (7.1), W ( y, y ) W ( x, x ) = (cid:18) W ( x, y ) W ( x, x ) (cid:19) or W ( x, x ) W ( y, y ) = W ( x, y ) . (7.2)Choosing instead i, j ∈ V ( F ) with ij / ∈ E ( F ), we similarly obtain(1 − W ( x, x ))(1 − W ( y, y )) = (1 − W ( x, y )) . (7.3)Subtracting (7.3) from (7.2) we find W ( x, x ) + W ( y, y ) = 2 W ( x, y )and thus, also using (7.2) again, (cid:0) W ( x, x ) − W ( y, y ) (cid:1) = (cid:0) W ( x, x ) + W ( y, y ) (cid:1) − W ( x, x ) W ( y, y )= 4 W ( x, y ) − W ( x, y ) = 0 . Hence W ( x, x ) = W ( y, y ) for all x, y ∈ [0 , W is a constant, which must be p or ¯ p . (cid:3) As above, the results in the appendix imply that we can relax the as-sumption to hold only almost everywhere. Lemma 7.4. Let F be a graph with | F | > , let W be a graphon and let p ∈ (0 , . If Ψ ∗ F,W ( x , . . . , x | F | ) = p e ( F ) (1 − p )( | F | ) − e ( F ) for a.e. x , . . . , x | F | ∈ [0 , | F | , then either W = p a.e. or W = ¯ p a.e.Proof. By Corollary A.6 and Lemma 7.3, W has to be a constant c a.e.Then Ψ ∗ F,W ( x , . . . , x | F | ) = β F ( c ) a.e., and thus β F ( c ) = β F ( p ); hence c = p or c = ¯ p . (cid:3) Proof of Theorems 3.11 and 3.12. As in Section 4, we may assume that G n → W for some graphon W . By the assumption and Lemma 7.1, thenΨ ∗ F,W ( x , . . . , x | F | ) = β F ( p ) := p e ( F ) (1 − p )( | F | ) − e ( F ) for a.e. x , . . . , x | F | ∈ [0 , | F | , which by Lemma 7.4 implies either W = p a.e. or W = ¯ p a.e. (cid:3) For e Ψ ∗ F,W , the situation is even more complicated. In fact, Simonovitsand S´os [20] showed that the path P = K , and its complement P arenot HI (recall Definition 3.7). Thus, the analogue of Lemma 7.3 for e Ψ ∗ F,W cannot hold in general.We can, however, easily obtain the partial results of [20] by our methods.We note that by Theorem A.5, it suffices to study 2-type graphons; equiva-lently, it suffices to study e Ψ ∗ F,W ( x , . . . , x | F | ) for sequences x , . . . , x | F | withat most two distinct values. For any sequence x , . . . , x | F | with x i = x for k values of i , and x i = y for the | F | − k remaining values, we have e Ψ ∗ F,W ( x , . . . , x | F | ) = (cid:18) | F | k (cid:19) − Q k (cid:0) W ( x, x ) , W ( y, y ) , W ( x, y ) (cid:1) , (7.4)where Q k ( u, v, s ) is the polynomial, defined for a given graph F and k =0 , . . . , | F | , by Q k ( u, v, s ) = X A ⊆ V ( F ) | A | = k u e ( A ) (1 − u )( k ) − e ( A ) v e ( A ) (1 − v )( | F |− k ) − e ( A ) s e ( A,A ) (1 − s ) k ( | F |− k ) − e ( A,A ) , where A := V ( F ) \ A , e ( A ) is the number of edges with both endpoints in A , and e ( A, A ) is the number of edges with one endpoint in A and one in A .By symmetry, Q | F |− k ( u, v, s ) = Q k ( v, u, s ). Note that Q ( u, v, s ) = β F ( v )and Q | F | ( u, v, s ) = β F ( u ). In particular, Q ( u, v, s ) = β F ( p ) ⇐⇒ v ∈ { p, ¯ p } and Q | F | ( u, v, s ) = β F ( p ) ⇐⇒ u ∈ { p, ¯ p } UASI-RANDOM GRAPHS AND GRAPH LIMITS 27 Remark 7.5. These polynomials are essentially the same as the polynomials P ku,v ( s ) defined by Simonovits and S´os [20]. More precisely, P ku,v ( s ) := (cid:18) | F | k (cid:19) u e ( F ) (1 − u ) e ( F ) − Q k ( u, v, s ) . Hence, the condition in Theorem 7.6(iv) below is equivalent to P ku,v ( s ) = 0,with u, v ∈ { p, ¯ p } . Theorem 7.6. Let F be a graph with | F | > and let < p < . Then thefollowing are equivalent: (i) F is HI ( p ) . (ii) If Ψ ∗ F,W ( x , . . . , x | F | ) = β F ( p ) for a.e. x , . . . , x | F | ∈ [0 , | F | , theneither W = p a.e. or W = ¯ p a.e. (iii) If Ψ ∗ F,W ( x , . . . , x | F | ) = β F ( p ) for all x , . . . , x | F | ∈ [0 , | F | , theneither W = p or W = ¯ p (iv) If Q k ( u, v, s ) = (cid:0) | F | k (cid:1) β F ( p ) for k = 1 , . . . , | F | − , and u, v ∈ { p, ¯ p } ,then u = v = s .Proof. (i) ⇐⇒ (ii) follows by Lemma 7.2 and our general method.(ii) ⇐⇒ (iii) follows by Corollary A.6 (and the comment after it).(ii) ⇐⇒ (iv) follows by Theorem A.5, together with the remarks on Q and Q | F | above. (cid:3) Proof of Theorem 3.13. Again we may assume that G n → W . It then fol-lows by Lemma 7.2(i) ⇐⇒ (iii) and Theorem 7.6(i)= ⇒ (ii) that either W = p a.e. or W = ¯ p a.e. (cid:3) For F = P , it suffices by symmetry to check Q in (iv); we find Q ( u, v, s ) =2 vs (1 − s ) + (1 − v ) s , and it is easy to find solutions with u = v = p = s ,see [20] for details. On the other hand, Simonovits and S´os [20] have shownthat every regular graph (and a few others) satisfies (iv), and thus is HI ( p ).The algebraic problem of determining if there are any other cases wherethe overdetermined system in Theorem 7.6(iv) has a non-trivial root is stillunsolved. 8. Cuts Chung and Graham [6] considered also e G ( U, U ), the number of edges inthe graph G across a cut ( U, U ), where U := V ( G ) \ U . They proved thefollowing results: Theorem 8.1 (Chung and Graham [6]) . Suppose that ( G n ) is a sequenceof graphs with | G n | → ∞ and let ≤ p ≤ . Then ( G n ) is p -quasi-randomif and only if, for all subsets U of V ( G n ) , e G n ( U, U ) = p | U || U | + o (cid:0) | G n | (cid:1) . (8.1) Theorem 8.2 (Chung and Graham [6]) . Let α ∈ (0 , with α = 1 / .Suppose that ( G n ) is a sequence of graphs with | G n | → ∞ and let ≤ p ≤ .Then ( G n ) is p -quasi-random if and only if (8.1) holds for all subsets U of V ( G n ) with | U | = ⌊ α | G n |⌋ . However, as shown in [7; 6], Theorem 8.2 does not hold for α = 1 / e G ( U, U ) = N ( K , G ; U, U ) , (8.2)so these results are closely connected to Theorem 3.2 and its variants. Wemay use the methods above to show these results too, and to see why α = 1 / G n → W for some graphon W , and translate theproperties above to properties of W . We state this as a lemma in thesame style as earlier, and note that Theorems 8.1 and 8.2 are immediateconsequences. Lemma 8.3. Suppose that G n → W for some graphon W and let p ∈ [0 , .Then the following are equivalent: (i) For all subsets U of V ( G n ) , e G n ( U, U ) = p | U || U | + o (cid:0) | G n | (cid:1) . (ii) For all subsets A of [0 , , Z A × A W ( x, y ) = pλ ( A ) λ ( A ) . (8.3)(iii) W = p a.e.For any fixed α ∈ (0 , \ { } , we may further add the condition that | U | = ⌊ α | G n |⌋ in (i) and λ ( A ) = α in (ii) . (If we add these conditions with α =1 / , the equivalence (i) ⇐⇒ (ii) still holds, but these do not imply (iii) .)Proof. The equivalence (i) ⇐⇒ (ii) follows as in Lemmas 4.2 and 6.1, arguingas in Lemma 6.2 in the case of a fixed size α ∈ (0 , ⇒ (ii) is trivial, and (ii) = ⇒ (iii) follows by thefollowing lemma, applied to W − p . (cid:3) Lemma 8.4. Let α ∈ (0 , \ { } . If f : [0 , → R is a symmetric mea-surable function such that R A × ([0 , \ A ) f = 0 for every subset A of [0 , with λ ( A ) = α , then f = 0 a.e.Proof. Let f ( x ) := R f ( x, y ) d y be the marginal of f . Then0 = Z A × ([0 , \ A ) f = Z A f ( x ) d x − Z A × A f ( x, y ) d x d y = Z A × A (cid:16) α f ( x ) − f ( x, y ) (cid:17) d x d y. (8.4) UASI-RANDOM GRAPHS AND GRAPH LIMITS 29 Lemma 6.6 now shows that the symmetrization α f ( x )+ α f ( y ) − f ( x, y ) =0 a.e., i.e., f ( x, y ) = 12 α (cid:0) f ( x ) + f ( y ) (cid:1) . (8.5)Integrating (8.5) with respect to both variables we find R f = α R f , andthus, because α < R f = 0. Integrating (8.5) with respect to one variableswe then find f ( x ) = α f ( x ) a.e., and thus f ( x ) = 0 a.e. because α = 1 / f ( x, y ) = 0 a.e. (cid:3) This proof also shows what goes wrong with Theorem 8.2 when α = 1 / f ( x, y ) = g ( x )+ g ( y ) for any integrable g with R g = 0, and asa result we see that (8.1) is satisfied for all U with | U | = ⌊| G n | / ⌋ whenever G n → W where W is a graphon of the form W ( x, y ) = h ( x ) + h ( y ) with R h = p/ 2. (One such example of ( G n ), with p = 1 / h ( x ) = [ x ≥ / 2] is given in [7; 6].) Cf. Remark 6.4. Remark 8.5. The condition that f is symmetric is essential in Lemma 8.4.If f is anti-symmetric, then (8.4) implies that f satisfies the condition if andonly if R f ( x, y ) d y = 0 for a.e. x . One example is sin(2 π ( x − y )).Chung, Graham and Wilson [7] remarked that Theorem 8.2 holds in thecase α = 1 / G n ) is almost regular (see belowfor definition). We discuss and show this in the next section.9. The degree distribution If G is a graph, let D G denote the random variable defined as the degree d v of a randomly chosen vertex v (with the uniform distribution on V ( G )).Thus 0 ≤ D G ≤ | G |− 1, and we normalize D G by considering D G / | G | , whichis a random variable in [0,1]. If ( G n ) is a sequence of graphs, with | G n | → ∞ as usual, we say that ( G n ) has asymptotic (normalized) degree distribution µ if D G tends to µ in distribution. (Here µ is a distribution, i.e., a probabilitymeasure, on [0 , µ is concentrated at a point p ∈ [0 , G n ) is almost p -regular (or almost regular if we donot want to specify p ); this thus is the case if and only if D G n p −→ p , withconvergence in probability, which means that all but o ( | G n | ) vertices in G n have degrees p | G n | + o ( | G n | ). Since the random variables D G n are uniformlybounded (by 1), this is further equivalent to convergence in mean, and thusa sequence ( G n ) is almost p -regular if and only if E | D G n − p | → 0, or, moreexplicitly, cf. [7], X v ∈ V ( G ) (cid:12)(cid:12) d v − p | G n | (cid:12)(cid:12) = o ( | G n | ) . (9.1)The normalized degree distribution behaves continuously under graphlimits, and a corresponding “normalized degree distribution” may be definedfor every graph limit too. (See further [9].) For a graphon W we define the marginal w ( x ) := R W ( x, y ) d y and the random variable D W := w ( U ) = R W ( U, y ) d y , where U ∼ U [0 , 1] is uniformly distributed on [0 , Theorem 9.1. If G n are graphs with | G n | → ∞ and G n → W for somegraphon W , then D G n / | G n | d −→ D W . Hence, ( G n ) has an asymptoticallydegree distribution, and this equals the distribution of the random variable D W := R W ( U, y ) d y .Proof. It is easily seen that, for every k ≥ 1, the moment E ( D G / | G | ) k equals t ( S k , G ), where S k = K ,k is a star with k + 1 vertices, and similarly themoment E W kG = t ( S k , W ). Consequently, E ( D G n / | G n | ) k = t ( S k , G n ) → t ( S k , W ) = E D W for every k ≥ 1, and thus D G n d −→ D W by the method ofmoments. (cid:3) Corollary 9.2. Let ( G n ) be a sequence of graphs and W a graphon suchthat G n → W . Then G n is almost p -regular if and only if R W ( x, y ) d y = p for a.e. x ∈ [0 , . (cid:3) In particular, a quasi-random sequence of graphs is almost regular, butthe converse does not hold.Motivated by Corollary 9.2, we say that a graphon W is p -regular ifits marginal R W ( x, y ) d y = p a.e. This is evidently not a quasi-randomproperty of graphons, but it can be used in conjuction with the failed case α = 1 / Lemma 9.3. Let α ∈ (0 , . If f : [0 , → R is a symmetric measurablefunction such that R A × ([0 , \ A ) f = 0 for every subset A of [0 , with λ ( A ) = α , and R f ( x, y ) d y = 0 for a.e. x , then f = 0 a.e.Proof. The proof of Lemma 8.4 shows that (8.5) holds, where now by as-sumption f = 0. (cid:3) Lemma 9.4. Let p ∈ [0 , and α ∈ (0 , . Suppose that ( G n ) is an almost p -regular sequence of graphs and that G n → W for some graphon W . Thenthe following are equivalent: (i) For all subsets U of V ( G n ) with | U | = ⌊ α | G n |⌋ , e G n ( U, U ) = pα (1 − α ) | G n | + o (cid:0) | G n | (cid:1) . (9.2)(ii) For all subsets A of [0 , with λ ( A ) = α , Z A × A W ( x, y ) = pα (1 − α ) . (iii) W = p a.e.Proof. By Lemma 8.3, it remains only to show that (ii) = ⇒ (iii) in the case α = 1 / 2. However, by Corollary 9.2, W is p -regular, so (ii) = ⇒ (iii) followsby Lemma 9.3 applied to W − p . (cid:3) UASI-RANDOM GRAPHS AND GRAPH LIMITS 31 Lemma 9.4 yields, by our general machinery, immediately the followingtheorem by Chung, Graham and Wilson [7], which supplements Theorem 8.2in the case α = 1 / Theorem 9.5 (Chung, Graham and Wilson [7]) . Let ≤ p ≤ and α ∈ (0 , . Suppose that ( G n ) is a sequence of graphs with | G n | → ∞ . Then ( G n ) is p -quasi-random if and only if ( G n ) is almost p -regular and (9.2) holds forall subsets U of V ( G n ) with | U | = ⌊ α | G n |⌋ . Appendix A. A measure-theoretic lemma A multiaffine polynomial is a polynomial in several variables { x ν } ν ∈I , forsome (finite) index set I , such that each variable has degree at most 1; it canthus be written as a linear combination of the 2 |I| monomials Q ν ∈J x ν forsubsets J ⊆ I . We are interested in the case when the index set I consistsof the (cid:0) m (cid:1) pairs { i, j } with 1 ≤ i < j ≤ m , for some m ≥ 2. In this case wedefine, for any symmetric function W : [0 , → R and x , . . . , x m ∈ [0 , W ( x , . . . , x m ) := Φ (cid:0) ( W ( x i , x j )) i Suppose that Φ (cid:0) ( w ij ) i 2. Suppose further that W : [0 , → [0 , 1] is a graphon such that Φ W ( x , . . . , x m ) = γ a.e. for some γ ∈ R . Does there always exist a graphon W ′ with W ′ = W a.e. such thatΦ W ′ ( x , . . . , x m ) = γ for every x , . . . , x m ∈ [0 , E below contains the diagonal; hence we can make the equationΦ W ′ ( x , . . . , x m ) = γ hold (typically, at least) also when several, or all, x i coincide. Remark A.2. The elimination of a null set in Problem A.1 seems relatedto the infinite version of the (hypergraph) removal lemma [10], where theobjective, in a different but related context, also is to replace a null set byan empty set. Lemma A.3. Suppose that Φ (cid:0) ( w ij ) i 1] into k sets S , . . . , S k such that W is constant oneach rectangle S i × S j . Making a rearrangement, we can without loss ofgenerality assume that the sets S i are intervals. (See [13] for a study offinite-type graph limits and the corresponding sequences of graphs, whichgeneralize quasi-random graphs.) Remark A.4. In this paper, we consider for convenience only graphonsdefined on [0 , W is finite-type if itis equivalent to a graphon defined on a finite probability space. Theorem A.5. Suppose that Φ (cid:0) ( w ij ) i 1] are the intervals[0 , ] and ( , W = γ a.e. does not imply that W is a.e. constant (i.e., it is not a (mixed) quasi-random property for graphons), then there exists a counter-example that isa 2-type graphon. This generalizes one of the results for induced subgraphcounts by Simonovits and S´os [20]. Proof. (ii) ⇐⇒ (iii): A 2-type graphon W is defined by a partition ( S , S )of [0 , 1] and three numbers u, v, s ∈ [0 , 1] such that W = u on S × S , W = v on S × S , and W = s on ( S × S ) ∪ ( S × S ). It is easy to see UASI-RANDOM GRAPHS AND GRAPH LIMITS 33 that, for any S and S with λ ( S ) , λ ( S ) > 0, such a graphon W satisfiesΦ W = γ if and only if Φ(( w ij ) i 2. Since λ ( E ) = 1, the same holds for B ∩ E , and we may thus, by the regularity of theLebesgue measure, find a compact set K ⊆ B ∩ E with λ ( K ) > 0. If ( x, y ) ∈ K , then ( x, y ) ∈ E , so by (A.3), W ( x, x ) = W ( x, y ) ∈ U . Consequently, if K ′ is the projection of K onto the first coordinate, then W ( x, x ) ∈ U for x ∈ K ′ ; furthermore, K ′ is a compact, and thus measurable, subset of [0 , λ ( K ′ ) > W is not a.e. constant. Thus there exist two disjointopen intervals U and U such that W − ( U ℓ ) := { ( x, y ) : W ( x, y ) ∈ U ℓ } ⊆ [0 , has positive measure, ℓ = 1 , 2. Then also, for each ℓ = 1 , D ℓ := E ∩ W − ( U ℓ ) has positive measure, so we may pick a Lebesgue point ( x ℓ , y ℓ )in D ℓ . By what we just have shown, this implies that there exists a compactset K ℓ ⊆ [0 , 1] with λ ( K ℓ ) > W ( x, x ) ∈ U ℓ for x ∈ K ℓ .However, this means that if ( x, y ) ∈ K × K , then W ( x, x ) = W ( y, y ), andthus by (A.3), ( x, y ) / ∈ E . Hence E ∩ ( K × K ) = ∅ . Since λ ( K × K ) > λ ( E ) = 1, this is a contradiction. (cid:3) Corollary A.6. Suppose that Φ (cid:0) ( w ij ) i The assumption implies that there is no 2-type graphon W as inTheorem A.5(ii), and thus there is no graphon W as in Theorem A.5(i). (cid:3) It remains to prove Lemma A.3. In order to do this, we first prove thefollowing lemma, which is a (weak) substitute for the Lebesgue differentia-tion theorem when we consider points on the diagonal only. (The Lebesguedifferentiation theorem says nothing about such points, since the diagonal isa null set. A simple counter-example is W ( x, y ) = [ x < y ].) We introducesome further notation.If A ⊆ [0 , 1] with λ ( A ) > 0, let λ A be the normalized Lebesgue measureon A given by λ A ( B ) := λ ( A ∩ B ) /λ ( A ), B ⊆ [0 , λ A isthe distribution of a uniform random point in A .)The definition (2.10) of the cut norm generalizes to arbitrary measurespaces. In particular, if A ⊆ [0 , 1] with λ ( A ) > 0, we let k W k (cid:3) ,A denotethe cut norm on A × A with respect to the normalized measure λ A . Moregenerally, if A and B ⊆ [0 , 1] have positive measures, then k W k (cid:3) ,A × B := sup S ⊆ A, T ⊆ B Z S × T W ( x, y ) d λ A ( x ) d λ B ( y )denotes the (normalized) cut norm on A × B . Lemma A.7. For every ε > there exists δ = δ ( ε ) > such that if W :[0 , → [0 , is a symmetric and measurable function and A ⊆ [0 , with λ ( A ) > , then there exists B ⊆ A with λ ( B ) ≥ δλ ( A ) and a real number w ∈ [0 , such that k W − w k (cid:3) ,B < ε . Remark A.8. The example W ( x, y ) = [ x < y ] shows that Lemma A.7 ingeneral fails for non-symmetric functions. Remark A.9. Lemma A.7 is not true with the stronger conclusion obtainedby replacing cut norm by L norm. An example is (whp) given, for any ε < / 2, by the 0 / W corresponding to a random graph G ( n, / n .Although Lemma A.7 is a purely analytic statement, we prove it usingcombinatorial methods; in fact, the proof is an adaption of the relevantparts of the proof of one of the main theorems in Simonovits and S´os [20]to graphons (instead of graphs). Proof. By considering the restriction of W to A × A and a measure preservingbijection of ( A, λ A ) onto ([0 , , λ ), it suffices to consider the case A = [0 , r = ⌈ /ε ⌉ and let M be the Ramsey number R ( r ; r ) = R ( r, . . . , r )(with r repeated r times); in other words, every colouring of the edges ofthe complete graph K M with at most r colours contains a monochromatic K r . (See e.g. [12].) UASI-RANDOM GRAPHS AND GRAPH LIMITS 35 By the (strong) analytic Szemer´edi regularity lemma by Lov´asz and Szegedy[15, Lemma 3.2], there is an integer K = k ( ε/ (4 M )) (depending on ε only,since M is a function of ε ) and, for some k ≤ K , a partition P = { S , . . . , S k } of [0 , 1] into k sets of equal measure 1 /k with the property that for everyset R ⊆ [0 , that is a union of at most k rectangles, we have (cid:12)(cid:12)(cid:12)(cid:12)Z R ( W − W P ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε M , (A.4)where W P is the function that is constant on each set S i × S j and equal tothe average k R S i × S j W there. (I.e., W P is the conditional expectation of W given the σ -field generated by { S i × S j } ki,j =1 .) Let w ij be this average k R S i × S j W . We consider two cases separately:(i): k ≥ M . Let, for i, j = 1 , . . . , k , d ij := k W − W P k (cid:3) ,S i × S j = k W − w ij k (cid:3) ,S i × S j = max (cid:0) d + ij , d − ij (cid:1) , (A.5)where d ± ij := sup S ⊆ S i , T ⊆ S j ± k Z S × T ( W − W P ) . It follows from (A.4) that k X i,j =1 d + ij ≤ k ε M , and thus the number of pairs ( i, j ) with d + ij > ε/ k /M , andsimilarly for d − ij .Say that a pair ( i, j ) is bad if d ij > ε/ i = j , and good otherwise. By(A.5), the number of bad pairs is thus less than 2 k /M + k ≤ k /M , usingour assumption that k ≥ M and assuming, as we may, that M ≥ H on [ k ] where there is an edge ij whenever ( i, j ) is agood pair. Further, give every edge ij in H the colour c ij := max( ⌈ rw ij ⌉ , ∈ [ r ]. Since H has more than ( k − M k ) = (1 − M ) k edges, Tur´an’s the-orem shows that H contains a complete subgraph K M , and the choice of M implies that this complete subgraph contains a complete monochromaticsubgraph K r .In other words, there is a c ∈ [ r ] such that, after renumbering the sets S i in P , for all i, j ∈ [ r ] with i = j , ( i, j ) is a good pair and c ij = c .Let w := c/r ∈ [0 , ≤ i < j ≤ r , c − ≤ rw ij ≤ c , so | w ij − w | ≤ /r ≤ ε/ 3. Since ( i, j ) is good, this further implies k W − w k (cid:3) ,S i × S j ≤ d ij + | w ij − w | ≤ ε/ , ≤ i < j ≤ r. On the other hand, trivially, for every i , k W − w k (cid:3) ,S i × S i ≤ sup | W − w | ≤ . Let B := S ri =1 S i . Then λ ( B ) = r/k ≥ r/K and, recalling that the sets S i have the same measure, k W − w k (cid:3) ,B ≤ r − r X i,j =1 k W − w k (cid:3) ,S i × S j ≤ r − (cid:16) r ( r − 1) 2 ε r · (cid:17) < ε r ≤ ε. (ii): k < M . We simple take B = S and W = w . Then λ ( B ) = 1 /k > / (2 M ), and (A.4) implies k W − w k (cid:3) ,B ≤ λ ( B ) − k W − w k (cid:3) ≤ λ ( B ) − ε M < ε. This completes the proof of Lemma A.7. (cid:3) Proof of Lemma A.3. We may assume that γ = 0.For ε > η > 0, let E ε,η := n ( x, y ) ∈ (0 , : (2 ε ) − Z | x ′ − x | , | y ′ − y | <ε | W ( x ′ , y ′ ) − W ( x, y ) | d x ′ d y ′ < η o . (A.6)The Lebesgue differentiation theorem says that a.e. ( x, y ) ∈ T η S ε E ε,η ; inother words, a.e. ( x, y ) ∈ E ε,η for every η > ε > x , y and η ). For η > n ≥ 1, we can thus find ε = ε ( η, n ) ∈ (0 , /n ) such that λ (cid:0) E ε ( η,n ) ,η (cid:1) > − − n .For n ≥ 1, let δ n := δ (1 /n ) be as in Lemma A.7 with ε = 1 /n , and let η n := δ n /n , ε ( n ) := ε ( η n , n ) and E n := E ε ( n ) ,η n . Then λ (cid:0) E n (cid:1) > − − n ,so if ˜ E := S ∞ n =1 T ∞ ℓ = n E ℓ , then λ ( ˜ E ) = 1. Let E := ˜ E ∪ { ( x, x ) : x ∈ [0 , } .For x ∈ (0 , 1) and n so large that A n ( x ) := ( x − ε ( n ) , x + ε ( n )) ⊂ (0 , w n ( x ) and a set B n ( x ) ⊆ A n ( x ) with λ ( B n ( x )) ≥ δ n λ ( A n ( x )) = 2 δ n ε ( n ) such that k W − w n ( x ) k (cid:3) ,B n ( x ) ≤ /n. (A.7)If ( x, y ) ∈ E and x = y , then ( x, y ) ∈ ˜ E so for all large n , ( x, y ) ∈ E n = E ε ( n ) ,η n , and thus, by (A.6), Z B n ( x ) × B n ( y ) | W ( x ′ , y ′ ) − W ( x, y ) | d λ B n ( x ) ( x ′ ) d λ B n ( y ) ( y ′ ) ≤ (2 δ n ε ( n )) − Z A n ( x ) × A n ( y ) | W ( x ′ , y ′ ) − W ( x, y ) | d x ′ d y ′ < δ − n η n = 1 /n. (A.8) UASI-RANDOM GRAPHS AND GRAPH LIMITS 37 Let χ be a Banach limit, i.e., a multiplicative linear functional on ℓ ∞ suchthat χ (( a n ) ∞ ) = lim n →∞ a n if the limit exists. Now define W ′ ( x, y ) := ( χ (cid:0) ( w n ( x )) n (cid:1) , y = x,W ( x, y ) , y = x. (A.9)Note that W ′ is a graphon and a version of W . (Lebesgue measurability isimmediate, since the diagonal is a null set.)Assume for the rest of the proof that x , . . . , x m ∈ (0 , m with ( x i , x j ) ∈ E for all i and j . For sufficiently large n , (A.7) holds for all x i and (A.8)holds for all pairs ( x i , x j ) with x i = x j . Thus, if x i = x j , by (A.7), k W − w n ( x i ) k (cid:3) ,B n ( x i ) × B n ( x j ) ≤ /n, (A.10)and if x i = x j , by (A.8), since the cut norm is at most the L norm, k W − W ( x i , x j ) k (cid:3) ,B n ( x i ) × B n ( x j ) ≤ /n. (A.11)For notational convenience, we define the constants w ij,n := ( w n ( x i ) , x i = x j ,W ( x i , x j ) , x i = x j , (A.12)and let B ni := B n ( x i ). Thus, (A.10) and (A.11) say that for all i, j ∈ [ m ], k W − w ij,n k (cid:3) ,B ni × B nj ≤ /n. (A.13)We extend the definition of Φ W in (A.1) to families ( W ij ) ≤ i 1, say, for all i and j , andany sets B , . . . , B m ⊆ [0 , 1] with positive measures, the mapping( W ij ) Φ[( W ij ); B , . . . , B m ]:= Z B ×···× B m Φ[( W ij )]( y , . . . , y m ) d λ B ( y ) · · · d λ B m ( y m )is Lipschitz in cut norm, in each variable separately; by linearity it sufficesto consider the case when Φ is a monomial (and thus Φ W = Ψ F,W for somegraph F ), and this result then is explicit in [2, Proof of Lemma 2.2], see also[4]. Thus, by (A.13), recalling that each w ij,n here is a constant,Φ[( W ); B n , . . . , B nm ] − Φ(( w ij,n ) i 1) only. For completeness, we, trivially,may define W ′ (0 , 0) := W ′ (1 , 1) := W ′ ( , ).) (cid:3) Finally, we mention another technical problem, which might be of interestin some applications: Problem A.10. 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