On the Complexity of Nondeterministically Testable Hypergraph Parameters
aa r X i v : . [ c s . D S ] M a r On the Complexity of NondeterministicallyTestable Hypergraph Parameters
Marek Karpinski ∗ Roland Markó † Abstract
The paper proves the equivalence of the notions of nondeterministic anddeterministic parameter testing for uniform dense hypergraphs of arbitraryorder. It generalizes the result previously known only for the case of simplegraphs. By a similar method we establish also the equivalence between non-deterministic and deterministic hypergraph property testing, answering theopen problem in the area. We introduce a new notion of a cut norm for hyper-graphs of higher order, and employ regularity techniques combined with theultralimit method.
Hypergraph parameters are real-valued functions defined on the space of uniform hyper-graphs of some given order invariant under relabeling the vertex set. Testing a parametervalue associated to an instance in the dense model means to produce an estimation by onlyhaving access to a small portion of the data that describes it. The test data is selected bychoosing a uniform random subset of the vertex set and exposing the induced substruc-ture of the hypergraph on this subset. A certain parameter is said to be testable if for everygiven tolerated error the estimation is within the error range of the parameter value withhigh probability, and the size of the selected random subset does only depend on the sizeof this permitted error and not on the size of the instance, precise definitions are providedbelow. Similar notions apply to testing graph properties, in that situation one also usesuniform sampling in order to separate the cases where an instance has the property or isfar from having it, where the distance is measured by the number of edge modificationsrequired. For the related notions of approximation theory and limits see [1], [3], and [4].The general reader is referred to [9], [11], and [13] for some related developments. ∗ Dept. of Computer Science and the Hausdro ff Center for Mathematics, University of Bonn. Supportedin part by DFG grants, the Hausdor ff grant EXC59-1 /
2. Research partly supported by Microsoft ResearchNew England. E-mail: [email protected] † Hausdor ff Center for Mathematics, University of Bonn. Supported in part by a Hausdor ff scholarship.E-mail: [email protected] ff ective nature.Subsequently, an explicit construction of a tester was given by Gishboliner and Shapira[8] for nondeterministically testable graph properties containing the tester of the coloredwitness property as a subroutine. They used Szemerédi’s Regularity Lemma combinedwith developments by Alon et al. [2], and provided a tower-type dependence betweenthe sample complexity of the investigated property the sample complexity of the witnessproperty.In [14], additionally the study nondeterministic testing for parameters was initiated,the definition is similar to the property testing situation. A di ff erent approach by Karpinskiand Markó [10] relying on weaker regularity methods led to an e ff ective upper bound onthe sample complexity that is a 3-fold iteration of the exponential function applied to thesample size required by the witness parameter.The previous works mentioned above dealt with graphs, it was asked in [14] if theconcept can be employed for hypergraphs. The notion of an r -uniform hypergraph (inshort, r -graph) parameter and its testability can be defined completely analogously to thegraph case, the same applies for nondeterministic testability. Naturally, first the questionarises whether or not the deterministic and the nondeterministic testability are equivalentfor higher order hypergraphs, and secondly, if the answer to the first question is positive,then what can be said about the relationship of the sample complexity of the parameterand that of its witness parameter. The statements that are analogous to the main resultsof [8], [10], and [14] do not follow immediately for uniform hypergraphs of higher orderfrom the proof for graphs, like-wise to the generalizations of the Regularity Lemma newtools and notions are required to handle these cases. In the current paper we prove theequivalence of the two testability notions for uniform hypergraphs of higher order andsettle the first question posed above. Unfortunately, we were not able to obtain an explicitupper bound for the sample complexity, this is the consequence of us applying of thelimit theory for hypergraphs developed by Elek and Szegedy [6] using methods of non-standard analysis, therefore the second problem still remains open. We also show thattesting nondeterministically testable properties is as hard as parameter testing with ourmethod in the sense that the same complexity bounds apply.The paper is organized as follows. In Section 2 we give the preliminaries requiredand formulate the precise definitions followed by our main result, Theorem 2.3. Section 3contains the testability results for r -cut norms together with a brief summary of the notionsand results regarding the ultralimit method that are needed for our purposes, Section 4comprises some auxiliary results required. Section 5 describes the proof of our main result.2n Section 6 we give an application for property testing of hypergraphs, and in Section 7we pose some questions related to possible further research. A simple r -uniform hypergraph on n vertices is a subset G of (cid:0) [ n ] r (cid:1) , the size of G is n , andthe elements of (cid:0) [ n ] r (cid:1) are r -edges. Let A G denote the symmetric { , } -valued r -array orsymmetric subset of [ n ] r that represents G , we will sometimes use also only the term G torefer to a symmetric subset of [ n ] r \ diag([ n ] r ) corresponding to the array representation.Let k be a positive integer, and let G r , kn denote the set of k-colored r -uniform hypergraphsof size n , that are partitions G = ( G α ) α ∈ [ k ] of (cid:0) [ n ] r (cid:1) into k classes, so in all what follows herecolored r -graph means a complete r -graph where to each edge e we assign exactly onecolor G ( e ) from the set [ k ]. In this sense simple r -graphs are regarded as 2-colored. In the k -colored case is also possible to speak about the array representation, A G α stands for thesymmetric { , } -valued r -array that represents the color class of α , again with slight abuseof notation we will use G α for A G α . Additionally we have to introduce the color ι and thecorresponding array A ι that always is the indicator array of the set of diagonal elementsof [ n ] r (those having repetitions in their coordinates, denoted by diag([ n ] r )). For any finiteset C the term C -colored graph is defined analogously.A k-coloring of a t -colored r -graph G = ( G α ) α ∈ [ t ] is a tk -colored r -graph ˆ G = ( G ( α,β ) ) α ∈ [ t ] ,β ∈ [ k ] with colors from the set [ t ] × [ k ], where each of the original color classes indexed by α ∈ [ t ]is retrieved by taking the union of the new classes corresponding to ( α, β ) over all β ∈ [ k ],that is G α = ∪ β ∈ [ k ] G ( α,β ) . This last operation is called k-discoloring of a [ t ] × [ k ]-colored graph,we denote it by [ ˆ G , k ] = G . We will sometimes write tk -colored for [ t ] × [ k ]-colored graphswhen it is clear from the context what we mean.Further, for a finite set S , let h( S ) denote the set of nonempty subsets of S , and h( S , m )the set of nonempty subsets of S of cardinality at most m . A real 2 r − x h( S ) denotes ( x T , . . . , x T r − ), where T , . . . , T r − is a fixed ordering of the nonempty subsetsof S with T r − = S , for a permutation π of the elements of S the vector x π (h( S )) means( x π ′ ( T ) , . . . , x π ′ ( T r − ) ), where π ′ is the action on the subsets of S induced by π .We will require some basic notation from graph limit theory, and we summarize theirrelevance outlined in previous works, Lovász [11] is a comprehensive reference for thearea.Let q ≥ G ∈ G r , kn , then G ( q , G ) denotes the random r -graph on q vertices that isobtained by uniformly picking a random subset S of [ n ] of cardinality q and taking theinduced subgraph G [ S ]. For any F ∈ G r , kq and G ∈ G r , k the F -density of G is defined as t ( F , G ) = P ( F = G ( q , G )).Let the r -kernel space W r denote the space of the bounded measurable functions W : [0 , h([ r ] , r − → R , and the subspace W r of W r symmetric r -kernels that are invariantunder coordinate permutations induced by π ∈ S r , that is W ( x h([ r ] , r − ) = W ( x π (h([ r ] , r − )for each π ∈ S r . We will refer to this invariance in the paper both for r -kernels and for3easurable subsets of [0 , h([ r ]) as satisfying the usual symmetries . Assume that the functions W ∈ W rI take their values in the interval I , for I = [0 ,
1] we call these special symmetric r -kernels r-graphons. In what follows, λ always denotes the usual Lebesgue measure in R d , where d is everywhere clear from the context.Analogously to the graph case we define the space of k-colored r-graphons W r , k whoseelements are referred to as W = ( W α ) α ∈ [ k ] with each of the W α ’s being an r -graphon. Thespecial color ι that stands for the absence of any colors in the diagonal in some sense canbe also employed in this setting, see below for the case when we represent a k -colored r -graph as a graphon. The corresponding r -graphon W ι is { , } -valued. Furthermore P α ∈ [ k ] W α ( x ) = − W ι ( x ) everywhere on [0 , h([ r ] , r − . For x ∈ [0 , h([ r ]) the expression W ( x ) denotes the color at x , we have W ( x ) = α whenever P α − i = W i ( x h([ r ] , r − ) ≤ x [ r ] ≤ P α i = W i ( x h([ r ] , r − ).Similar to the finitary case, a k-coloring of a W ∈ W r , k is a tk -colored r -graphon ˆ W = ( W ( α,β ) ) α ∈ [ t ] ,β ∈ [ k ] with colors from the set [ t ] × [ k ] so that P α ∈ [ t ] ,β ∈ [ k ] W ( α,β ) ( x ) = W α ( x ) for each x ∈ [0 , h([ r ] , r − and α ∈ [ t ]. The k -discoloring [ ˆ W , k ] of ˆ W and the term C -colored graphonis defined analogously, and simple r -graphons are treated as 2-colored.For q ≥ W ∈ W r , k the random [ k ] ∪ { ι } -colored r -graph G ( q , W ) is generatedas follows. The vertex set of G ( q , W ) is [ q ], we have to pick uniformly a random point( X S ) S ∈ h([ q ] , r − ∈ [0 , h([ q ] , r − , then conditioned on this choice we conduct independent trialsto determine the color of each edge e ∈ (cid:0) [ q ] r (cid:1) with the distribution given by P e ( G ( q , W )( e ) = α ) = W α ( X h( e , r − ) corresponding to e . Recall that ι is a special color which we want to avoidin most cases, therefore we will highlight the conditions imposed on the above randomvariables so that G ( q , W ) ∈ G r , k .For F ∈ G r , kq the F -density of W is defined as t ( F , W ) = P ( F = G ( q , W )), which can bewritten following the above description of the random graph as t ( F , W ) = Z [0 , h([ q ] , r − Y e ∈ ( [ q ] r ) W F ( e ) ( x h( e , r − )d λ ( x ) . The above notions were introduced in order to provide a concise representation forthe limit space of r -graphs in [6] and [12], in the current work we will not draw on thisdevelopment explicitly but mention their relevance here. In a nutshell, a sequence of r -graphs converges if the corresponding numerical F -density sequences converge for all r -graphs F . One of the main results of [12] for graphs and [6] in the general case is thatfor every convergent sequence of r -graphs there exists an r -graphon they converge to inthe sense that the F -densities approach the F -density of the limiting r -graphon. This waslater reproved by [15] for general r with purely combinatorial methods that are similar toconcepts employed in the current paper.We can associate to each G ∈ G r , kn an element W G ∈ W r , k by subdividing the unitcube [0 , h([ r ] , into n r small cubes the natural way and defining the function W ′ :[0 , h([ r ] , → [ k ] that takes the value G ( { i , . . . , i r } ) on [ i − n , i n ] × · · · × [ i r − n , i r n ] for distinct i , . . . , i r , and the value ι on the remaining diagonal cubes. Then set ( W G ) α ( x h([ r ] , r − ) = ( W ′ ( p h([ r ] , ( x h([ r ] , r − )) = α ) for each α ∈ [ k ] ∪ { ι } , where p h([ r ] , is the projection to thesuitable coordinates. Note that | t ( F , G ) − t ( F , W G ) | ≤ (cid:0) q (cid:1) n − (cid:0) q (cid:1) (2.1)for each F ∈ G r , kq , hence the previous representation is compatible in the sense thatlim n →∞ t ( F , G n ) = lim n →∞ t ( F , W G n ) for any sequence { G n } ∞ n = with | V ( G n ) | tending to in-finity.We proceed by providing the necessary formal definitions of the parameter testabilityin the dense hypergraph model. Definition 2.1.
An r-graph parameter f is testable if for any ε > there exists a positive integerq f ( ε ) such that for any simple r-graph G with at least q f ( ε ) nodes we have that P ( | f ( G ) − f ( G ( q f ( ε ) , G ) | > ε ) < ε. The smallest function q f satisfying the previous inequality is called the sample complexity of f .The testability of parameters of k-colored r-graphs is defined analogously. An a priori weaker characteristic than the one above, nondeterministic testability, isthe second cornerstone of the current work, and was introduced in [14].
Definition 2.2.
An r-graph parameter f is non-deterministically testable if there exist an integerk and a testable k-colored directed r-graph parameter g called witness such that for any simplegraph G the value f ( G ) = max G g ( G ) where the maximum goes over the set of k-colorings of G(regarded as an element of G r , ). Originally in [14], the witness parameter was a function of k -colored graphs, and themaximum was taken over the set of ( k , m )-colorings of the original graph in order todetermine the parameter value, meaning that the present edges are colored by elementsof [ m ], absent ones by the remaining colors from [ k ] \ [ m ]. Our modification is equivalentto that setting and is motivated by notational purposes.In the current paper we only deal with undirected structures, but similar results canbe obtained when the witness parameter is defined on the space of directed r -graphs. Inthis case, in order to obtain G from G as above after the discoloring we additionally haveto undirect the edges and neglect multiplicities created by the former operation.The maximization expression in Definition 2.2 is somewhat arbitrary and could bereplaced for example by minimization, this would however not a ff ect the testability char-acteristic of the parameter. Our main result extends the equivalence of the two testabilitynotions for arbitrary r , this was first proved by Lovász and Vesztergombi [14] for r = Theorem 2.3.
Every non-deterministically testable r-graph parameter f is testable.
Definition 2.4.
Let r ≥ and A be a real r-array of size n. Then the cut norm of A is k A k (cid:3) , r = n r max S i ⊂ [ n ] r − \ diag([ n ] r − ) i ∈ [ r ] | A ( r ; S , . . . , S r ) | , where A ( r ; S , . . . , S r ) = P ni ,..., i r = A ( i , . . . , i r ) Q rj = I S j ( i , . . . , i j − , i j + , . . . , i r ) , and the maximumgoes over sets S i that are invariant under coordinate permutations.If P = ( P i ) ti = is a partition of [ n ] r − \ diag([ n ] r − ) with symmetric classes, then the cut- P -normof A is k A k (cid:3) , r , P = n r max S i ⊂ [ n ] r − , i ∈ [ r ] t X j ,..., j r = | A ( r ; S ∩ P j , . . . , S r ∩ P j r ) | . The cut norm of an r-kernel W is k W k (cid:3) , r = sup S i ⊂ [0 , h([ r − i ∈ [ r ] | Z ∩ i ∈ [ r ] p − r ] \{ i } ( S i ) W ( x h([ r ] , r − )d λ ( x h([ r ] , r − ) | , where the supremum is taken over sets S i that satisfy the usual symmetries, and p e is the naturalprojection from [0 , h([ r ] , r − onto [0 , h( e ) . Furthermore, for a symmetric partition P = ( P i ) ti = of [0 , h([ r − the cut- P -norm of an r-kernel is defined by k W k (cid:3) , r , P = sup S i ⊂ [0 , h([ r − i ∈ [ r ] t X j ,..., j r = | Z ∩ i ∈ [ r ] p − r ] \{ i } ( S i ∩ P ji ) W ( x h([ r ] , r − )d λ ( x h([ r ] , r − ) | , where the supremum is taken over sets S i that satisfy the usual symmetries. We remark that it is also true that k W k (cid:3) , r = sup f ,..., f r ∈ [0 , h([ r − | Z [0 , h([ r ] , r − r Y i = f i ( x h([ r ] \{ i } ) ) W ( x h([ r ] , r − )d λ ( x h([ r ] , r − ) | , where the supremum goes over functions f i that satisfy the usual symmetries, and similarlyfor any symmetric partition P = ( P i ) ti = of [0 , h([ r − we have with the same conditions forthe f i ’s as above that k W k (cid:3) , r = sup f ,..., f r ∈ [0 , h([ r − t X j ,..., j r = | Z [0 , h([ r ] , r − r Y i = f i ( x h([ r ] \{ i } ) ) I P ji ( x h([ r ] \{ i } ) ) W ( x h([ r ] , r − )d λ ( x h([ r ] , r − ) | .
6n several previous works, see e.g. [1], the cut norm for r -arrays denotes a term thatis significantly di ff erent from the one in Definition 2.4 and is not suitable for our presentpurposes. The above norms give rise to a distance between r -graphons, and analogouslyfor r -graphs. Definition 2.5.
For two k-colored r-graphons U = ( U α ) α ∈ [ k ] and W = ( W α ) α ∈ [ k ] their cut distanceis defined as d (cid:3) , r ( U , W ) = k X α = k U α − W α k (cid:3) , r , and their cut- P -distance as d (cid:3) , r , P ( U , W ) = k X α = k U α − W α k (cid:3) , r , P . For two k-colored r-graphs G = ( G α ) α ∈ [ k ] and H = ( H α ) α ∈ [ k ] their corresponding distances aredefined as d (cid:3) , r ( G , H ) = d (cid:3) , r ( W G , W H ) , and d (cid:3) , r , P ( G , H ) = d (cid:3) , r , P ( W G , W H ) . Distances between an r-graph and an r-graphon, as well as for r-kernels, is analogously defined.
Note that the norms introduced above are in general smaller or equal than the 1-normof integrable functions, also d (cid:3) , r ( U , W ) ≤ d (cid:3) , r , P ( U , W ) hods for every pair. Their relevancewill be clearer in the context of the next counting lemma, we include the standard proofonly for completeness’ sake. Lemma 2.6.
Let U and W be two k-colored r-graphons with k U k ∞ , k W k ∞ ≤ . Then for every F ∈ G r , kq it holds that | t ( F , W ) − t ( F , U ) | ≤ qr ! d (cid:3) , r ( U , W ) . Proof.
Fix q and F ∈ G r , kq . Then | t ( F , W ) − t ( F , U ) | = | Z [0 , h([ q ] , r − Y e ∈ ( [ q ] r ) W F ( e ) ( x h( e , r − ) − Y e ∈ ( [ q ] r ) U F ( e ) ( x h( e , r − )d λ ( x ) |≤ X e ∈ ( [ q ] r ) | Z [0 , h([ q ] , r − [ W F ( e ) ( x h( e , r − ) − U F ( e ) ( x h( e , r − )]7 f ∈ ( [ q ] r ) , f ≺ e W F ( f ) ( x h( f , r − ) Y g ∈ ( [ q ] r ) , e ≺ g U F ( g ) ( x h( g , r − )d λ ( x ) |≤ X e ∈ ( [ q ] r ) k W F ( e ) − U F ( e ) k (cid:3) , r ≤ qr ! d (cid:3) , r ( U , W ) , where ≺ is an arbitrary total ordering of the elements of (cid:0) qr (cid:1) . (cid:3) Let d tw denote the total variation distance between probability measures on G r , [ k ] ∗ n ,where [ k ] ∗ = [ k ] ∪ { ι } for k ≥ d tw ), that is d tw ( µ, ν ) = max F ⊂G r , [ k ] ∗ n | µ ( F ) − ν ( F ) | , and let the measure µ ( q , G ), respec-tively µ ( q , W ), denote the probability measure of the random r -graph G ( q , G ), respectively G ( q , W ), taking values in G r , [ k ] ∗ q . It is a standard observation then that d tw ( µ ( q , W ) , µ ( q , U )) = X F ∈G r , [ k ] ∗ q | t ( F , W ) − t ( F , U ) | , (2.2)and that G ( q , W ) and G ( q , U ) can be coupled in form of the random r -graphs G and G ,such that d tw ( µ ( q , W ) , µ ( q , U ) = P ( G , G ) , (2.3)and further, for any coupling G ′ and G ′ it hods that d tw ( µ ( q , W ) , µ ( q , U ) ≤ P ( G ′ , G ′ ).For G ∈ G r , kn note that d tw ( µ ( q , G ) , µ ( q , W G )) ≤ q / n , (2.4)where the right hand side is a simple upper bound on the probability that if we uniformlychoose q elements of an n -element set, then we get at least two identical objects. Theinequality (2.4) follows from the fact that conditioned on the event that the independentand uniform X { i } ’s for i ∈ [ q ] fall in di ff erent intervals [ j − n , jn ] for j ∈ [ n ] the distribution of G ( q , W G ) is the same as the distribution of G ( q , G ).The next corollary is a direct consequence of Lemma 2.6. Corollary 2.7. If U and W are two k-colored r-graphons, thend tw ( µ ( q , W ) , µ ( q , U ) ≤ k q r q r r ! d (cid:3) , r ( U , W ) , and there exists a coupling in form of G and G of the random r-graphs G ( q , W ) and G ( q , U ) ,such that P ( G , G ) ≤ k q r q r r ! d (cid:3) , r ( U , W ) . r -graphs is given next. For a partition P the number of its classes isdenoted by t P . Definition 2.8.
We call an k-colored r-graphon W with r ≥ l an ( r , l ) -step function if there existpositive integers t l , t l + , . . . , t r = k, symmetric partitions P = ( P , . . . , P t l ) of [0 , h([ l ]) , and realarrays A α s : [ t s − ] h([ s ] , s − → [0 , with α ∈ [ t s ] for l ≤ s ≤ r such that P α ∈ [ t s ] A α s ( i h([ s ] , s − ) = forany choice of i h([ s ] , s − and for s ≤ r so that W α for α ∈ [ k ] is of the following form for each α ∈ [ k ] .W α ( x h([ r ]) ) = t | S | X i S = S ⊂ [ r ] , l ≤| S | A α r ( i h([ r ] , r − ) Y S ∈ ( [ r ] l ) I P iS ( x h( S ) ) Y S ⊂ [ r ] l + ≤| S | < r I ( i S − X j = A j | S | ( i h( S , | S |− ) ≤ x S ≤ i S X j = A j | S | ( i h( S , | S |− )) . We refer to the partition P as the steps of W. The most simple example is the ( r , r −
1) step function that can be written as W α ( x h([ r ]) ) = t r − X i ,..., i r = A α r ( i , . . . , i r ) r Y j = I P ij ( x h([ r ] \{ j } ) ) . r -cut norm We define a parameter of r -uniform hypergraphs that is a generalization of the groundstate energies of [5] in the case of graphs. This notion encompasses several importantquantities, therefore its testability is central to many applications. Definition 3.1.
For a set H ⊂ (cid:0) [ n ] r (cid:1) , a real r-array J of size q, and a symmetric partition P = ( P , . . . , P q ) of (cid:0) [ n ] r − (cid:1) we define the energy E P , r − ( H , J ) = n r q X i ,..., i r = J ( i , . . . , i r ) e H ( r ; P i , . . . , P i r ) , where e H ( r ; S , . . . , S r ) = |{ ( u , . . . , u r ) ∈ [ n ] r | A H ( u , . . . , u r ) = and A S j ( u , . . . , u j − , u j + , . . . , u r ) = for all j = , . . . , r }| .Let H = ( H α ) α ∈ [ k ] be a k-colored r-uniform hypergraph on the vertex set [ n ] and J α a be realq × · · · × q r-array with k J k ∞ ≤ for each α ∈ [ k ] . Then the energy for a partition P as above is E P , r − ( H , J ) = X α ∈ [ k ] E P , r − ( H α , J α ) . he maximum of the energy over all partitions P of (cid:0) [ n ] r − (cid:1) is called the ground state energy(GSE) of H with respect to J, and is denoted by E r − ( H , J ) = max P E P , r − ( H , J ) . The GSE can also be defined for r -graphons. Definition 3.2.
For an r-graphon W, a real r-array J of size q, and a symmetric partition P = ( P , . . . , P q ) of [0 , h([ r − we define the energy E P , r − ( W , J ) = X i ,..., i r ∈ [ q ] J ( i , . . . , i r ) Z ∩ j ∈ [ r ] p − r ] \{ j } ( S ij ) W ( x h([ r ] , r − )d λ ( x h([ r ] , r − ) . Let W = ( W α ) α ∈ [ k ] be a k-colored r-graphon and J α a be real q × · · · × q r-array with k J k ∞ ≤ for each α ∈ [ k ] . Then the energy for a partition P as above is E P , r − ( W , J ) = X α ∈ [ k ] E P , r − ( W α , J α ) . and the GSE of W with respect to J, and is denoted by E r − ( W , J ) = sup P E P , r − ( W , J ) , where the supremum runs over all symmetric partitions P = ( P , . . . , P q ) of [0 , h([ r − . Definitions of the above energies are analogous in the directed, and the weighted case,and also for r -kernels. The next lemma tells us about the distribution of the GSE whentaking a random sample G ( n , H ) of an H ∈ G r , k . Lemma 3.3.
The expression E r − ( G ( n , H ) , J ) is highly concentrated around its mean, that is forevery ε > it holds that P ( |E r − ( G ( n , H ) , J ) − E E r − ( G ( n , H ) , J ) | ≥ ε k J k ∞ ) ≤ − ε n r ) . Proof.
We can assume that k J k ∞ ≤
1. The random r -graph G ( n , H ) is generated by pickingrandom nodes from V ( H ) without repetition, let X i denote the i th random element of V ( H ) that has been selected. Define the martingale Y i = E [ E r − ( G ( n , H ) , J ) | X , . . . , X i ] for0 ≤ i ≤ n . It has the property that Y = E [ E r − ( G ( n , H ) , J )] and Y n = E r − ( G ( n , H ) , J ),whereas the jumps | Y i − Y i − | are bounded above by rn for each i ∈ [ n ]. The last observationis the consequence of the fact that for any partition P of (cid:0) [ n ] r − (cid:1) only at most rn r − terms inthe sum constituting E P , r − ( H , J ) are a ff ected by changing the placing of X i + in the classesof P . Applying the Azuma-Hoe ff ding inequality to the martingale verifies the statementof the lemma. (cid:3) E r − ( G ( n , W ) , J ).We will show that these hypergraph parameters are testable via the ultralimit methodand the machinery developed by Elek and Szegedy [6]. From the notational perspectiveand theoretical background this section slightly stands out from the rest of the paper. Firstwe give a brief summary of the notions that were used in [6] in order to produce a repre-sentation for the limit space of simple r -graphs. This representation led to a new analyticalproof method for several results for simple r -graphs such as the Regularity Lemma, theRemoval Lemma, or the testability assertion about hereditary r -graph properties. Subse-quently, technical results proved in [6] which are relevant here are mentioned, for moredetails and complete proofs we refer to the source paper [6].Recall that a sequence of r -graphs ( G n ) n ≥ is convergent if for every simple F thenumerical sequences t ( F , G n ) converge when n tends to infinity.We start by introducing the basic notations for ultraproduct measure spaces. Letus fix a non-principal ultrafilter ω on N , and let X , X , . . . be a sequence of finite sets ofincreasing size. We define the infinite product set ˆ X = Q ∞ i = X i and the equivalence relation ∼ between elements of ˆ X , so that p ∼ q if and only if { i | p i = q i } ∈ ω . Set X = ˆ X / ∼ , this setis called the ultraproduct of the X i ’s, and it will serve as the base set of the ultraproductprobability space. Further, let P denote the algebra of subsets of X of the form A = [ { A i } ∞ i = ],where A i ⊂ X i for each i , and [ . ] denotes the equivalence class under ∼ (for convenience, p = [ { p i } ∞ i = ] ∈ [ { A i } ∞ i = ] exactly in the case when { i | p i ∈ A i } ∈ ω ).Define a measure on the sets belonging to P through the ultralimit of the countingmeasure on the sets X i , that is, µ ( A ) = lim ω | A i || X i | , where the ultralimit of a bounded realnumerical sequence { x i } ∞ i = is denoted by x = lim ω x i , and is defined by the property thatfor every ε > { i | | x − x i | < ε } ∈ ω . One can see that the limit exists for everybounded sequence and is unique, therefore well-defined, this is a consequence of basicproperties of a non-principal ultrafilter. The set of N ⊂ X of µ -null sets is the family of sets N for those there exists an infinite sequence of supersets { A i } ∞ i = ⊂ P such that µ ( A i ) ≤ / i .Finally define the σ -algebra B on X by the σ -algebra generated by P and N , and set themeasure µ ( B ) = µ ( A ) for each B ∈ B , where A △ B ∈ N and A ∈ P . Again, everything iswell-defined, see [6], so we arrive at the ultraproduct measure space ( X , B , µ ).Let X , X , . . . and Y , Y , . . . be two increasing sequences of finite sets with ultraprod-ucts X and Y respectively, then it is true that the ultraproduct of the product sequence X × Y , X × Y , . . . is the product X × Y , but the σ -algebra B X × Y of the measure space canbe strictly larger than the σ -algebra generated by B X × B Y , and this is a crucial point whenthe aim is to construct a separable representation of the ultraproduct measure space ofproduct sets.Let r be some positive integer, and again X , X , . . . a sequence of finite sets as above.For any e ⊂ [ r ] we define the ultraproduct measure spaces ( X e , B X e , µ e ), also let P e denotethe natural projection from X [ r ] to X e . Furthermore let σ ( e ) denote the sub- σ -algebra of B X [ r ] given by P − e ( B X e ), and σ ( e ) ∗ be the sub- σ -algebra h P − f ( B X f ) | f ⊂ e , | f | < | e |i . Note thatin general σ ( e ) is strictly larger than σ ( e ) ∗ . We denote the measure µ X e simply by µ e and the σ -algebra B X e by B e . 11 efinition 3.4. Let r be a positive integer. We call a measure preserving map φ : X [ r ] → [0 , h([ r ]) a separable realization if1. for any permutation π ∈ S [ r ] of the coordinates we have for all x ∈ X [ r ] that Π ( φ ( x )) = φ ( φ ( x )) , where Π is the permutation of the power set of [ r ] induced by π , and2. for any e ⊂ h([ r ]) and any measurable A ⊂ [0 , we have that φ − e ( A ) ∈ σ ( e ) and φ − e ( A ) isindependent of σ ( e ) ∗ . We are interested in the limiting behavior of sequences of k -partitions (or edge- k -colored r -graphs on the vertex sets X , X , . . . ) of the sequence X r , X r , . . . , where conver-gence is defined in the following general way.Let G i = ( G i , . . . , G ki ) be a symmetric partition of X ri for each i ∈ N , then ( G i ) ∞ i = convergesif for every k -colored r -graph F the numerical sequences t ( F , G i ) converge, as in Section 2.The ultralimit method enables us to handle the cases where the convergence does not holdwithout going to subsequences, we describe the method next. Let us denote the size of F by m and let F ( e ) be the color of e ∈ (cid:0) [ m ] r (cid:1) , then t ( F , G i ) can be written as the measure of asubset of X mi . We show this by explicitly presenting the set denoted by T ( F , G i ), so let T ( F , G i ) = \ e ∈ ( [ m ] r ) P − e ( P s e ( G F ( e ) i )) , (3.1)where P e is the natural projection from X [ m ] i to X ei , and P s e is a bijection going from X [ r ] i to X ei induced by an arbitrary but fixed bijection s e between e and [ r ]. We define the inducedsubgraph density of the ultraproduct of k -colored r -graphs formally following (3.1), if G = ( G , . . . , G k ) is a B [ r ] -measurable k -partition of X [ r ] and F is as above then let T ( F , G ) = \ e ∈ ( [ m ] r ) P − e ( P s e ( G F ( e ) )) . (3.2)It is easy to see that λ ( T ( F , G i )) = t ( F , G i ). Forming the ultraproduct of a series of setscommutes with finite intersection, therefore lim ω T ( F , G i ) = T ( F , lim ω G i ) and lim ω t ( F , G i ) = t ( F , lim ω G i ). Observe that all of the above notation makes perfect sense and the identitieshold true for directed colored r -graphs, that is, when the adjacency arrays of the G α ’s arenot necessarily symmetric.We call a measurable subset of [0 , h([ r ]) an r -set graphon satisfying the usual symme-tries in the coordinates induced by S r permutations, we can turn it into a proper r -graphonin the sense of Section 2 by generating the marginal with respect to the coordinate cor-responding to [ r ]. Analogously a k -colored r -set graphon is a measurable partition of[0 , h([ r ]) into k classes invariant under coordinate permutations induced by permuting[ r ]. These objects can serve as representations of the ultralimits of r -graph sequences in thesense that the numerical sequences of subgraph densities converge to densities defined for r -set graphons in accordance with the notation in Section 2, we will provide the definitionnext. 12 efinition 3.5. Let F be a k-colored r-graph on m vertices, and W = ( W , . . . , W k ) be a k-coloredr-set graphon. Then T ( F , W ) ⊂ [0 , h([ m ] , r ) denotes the set of the symmetric maps g : h([ m ] , r ) → [0 , that satisfy that for each e ∈ (cid:0) [ m ] r (cid:1) it holds that ( g ( f )) f ∈ h( e ) ∈ W F ( e ) . For the Lebesgue measureof T ( F , W ) we write t ( F , W ) , this expression is referred to as the density of F in W . The reader may easily verify that the above definition of density agrees with the contentof Section 2. One of the main technical results of [6] is the following.
Theorem 3.6. [6] Let r be an arbitrary positive integer and let A be a separable sub- σ -algebra of B [ r ] . Then there exists a separable realization φ : X [ r ] → [0 , h([ r ]) such that for every A ∈ A thereexists a measurable B ⊂ [0 , h([ r ]) such that µ [ r ] ( A △ φ − ( B )) = . A lifting of a separable realization φ : X [ r ] → [0 , h([ r ]) of degree n for n ≥ r is a measurepreserving map ψ : X [ n ] → [0 , h([ n ] , r ) that satisfies p h([ r ]) ◦ ψ = φ ◦ P [ r ] , and it is equivariantunder coordinate permutations in S n , where p h([ r ]) and P [ r ] are the natural projections from[0 , h([ n ] , r ) to [0 , h([ r ]) , and from X [ n ] to X [ r ] respectively. The next lemma is central torelate the sub- r -graph densities of ultraproducts to the corresponding densities in r -setgraphons. Lemma 3.7. [6] For every separable realization φ and integer n ≥ r there exists a degree n lifting ψ . The next statement is the colored version of the homomorphism correspondence in [6](Lemma 3.3. in that paper).
Lemma 3.8.
Let φ be a separable realization and W = ( W , . . . , W k ) be a k-colored r-graphon,and let H = ( H , . . . , H k ) be a k-colored ultraproduct with µ [ r ] ( H α △ φ − ( W α )) = for each α ∈ [ k ] . Let ψ be a degree m lifting of φ and F be a k-colored r-graph on m vertices. Then µ [ m ] ( ψ − ( T ( F , W )) △ T ( F , H )) = , and consequently t ( F , W ) = t ( F , H ) for each F .Proof. By definition we have that T ( F , H ) = \ e ∈ ( [ m ] r ) P − e ( P s e ( H F ( e ) ))and T ( F , W ) = \ e ∈ ( [ m ] r ) p − r ]) ( p s e ( W F ( e ) )) . Due to the fact that ψ commutes with coordinate permutations from S n and the conditionswe imposed on the symmetric di ff erence of H α and φ − ( W α ) the statement follows. (cid:3) We turn to describe the relationship of two r -set graphons whose F -densities coincidefor each F . For this purpose we have to introduce two types of transformations and clarifytheir connection. Let us define the σ -algebras A S , A ∗ S , and B S ⊂ L [0 , h([ r ]) for each S ⊂ [ r ],the σ -algebra B S = p − S ( L [0 , h( S ) ), A S is hB T | T ⊂ S i , and A ∗ S is hB T | T ⊂ S , T , S i , where L [0 , t denotes the Lebesgue measurable subsets of the unit cube with the dimension given bythe index. 13 efinition 3.9. We say that the measurable map φ : [0 , h([ r ]) → [0 , h([ r ]) is structure preservingif it is measure preserving, for any S ⊂ [ r ] we have φ − ( A S ) ⊂ A S , for any measurable I ⊂ [0 , we have φ − ( p − S ( I )) is independent of A ∗ S , and for any π ∈ S r we have Π ◦ φ = φ ◦ Π , where Π isthe coordinate permutation action induced by π . Let L h([ r ]) denote the measure algebra of ([0 , h([ r ]) , L [0 , h([ r ]) , λ ). Definition 3.10.
We call an injective homomorphism Φ : L h([ r ]) → L h([ r ]) a structure preservingembedding if it is measure preserving, for any S ⊂ [ r ] we have Φ ( B S ) ⊂ A S , Φ ( B S ) is independentfrom A ∗ S , and for any π ∈ S r we have Π ◦ Φ = Φ ◦ Π . Another result from [6] sheds light on the build-up of structure preserving embeddings.
Lemma 3.11. [6] Suppose that Φ : L h([ r ]) → L h([ r ]) is a structure preserving embedding of ameasure algebra into itself. Then there exists a structure preserving map φ : [0 , h([ r ]) → [0 , h([ r ]) that represents Φ in the sense that for each [ U ] ∈ L h([ r ]) it holds that Φ ([ U ]) = [ φ − ( U )] , where Uis a representative of [ U ] . A random coordinate system τ is the ultraproduct function on X [ r ] of the randomsymmetric functions τ n : [ n ] r → [0 , h([ n ] , r ) that are for each n given by a uniform randompoint Z n in [0 , h([ n ] , r ) so that ( τ n ( i , . . . , i r )) e = ( Z n ) p e ( i ,..., i r ) . An important property of therandom mapping τ n is that for any r -set graphon and positive integer n it holds that( τ n ) − ( U ) = G ( n , U ), when the random sample Z n used to generate the two objects is thesame. Lemma 3.12. [6] Let U be an r-set graphon, and let H = [ { G ( n , U ) } ∞ n = ] . Then the randomcoordinate system τ = [ { τ n } ∞ n = ] is a separable realization such that with probability one we have µ [ r ] ( H △ τ − ( U )) = . A direct consequence is the statement for k -colored r -set graphons. Corollary 3.13.
Let U = ( U , . . . , U k ) be a k-colored r-set graphon, and let H = ( H , . . . , H k ) bea k-colored ultraproduct in X [ r ] , where H α = [ { G ( n , U α ) } ∞ n = ] for each α ∈ [ k ] . Then a randomseparable realization τ is such that with probability one we have µ [ r ] ( H α △ τ − ( U α )) = for each α ∈ [ k ] . The following result is a generalization of the uniqueness assertion of [6], and states thatsubgraph densities determine an r -set graphon up to structure preserving transformations. Theorem 3.14.
Let U = ( U , . . . , U k ) and V = ( V , . . . , V k ) be two k-colored r-set graphons suchthat for each k-colored r-graph F it holds that t ( F , U ) = t ( F , V ) . Then there exist two structurepreserving maps ν and ν from [0 , h([ r ]) to [0 , h([ r ]) such that µ [ r ] ( ν − ( U α ) △ ν − ( V α )) = foreach α ∈ [ k ] .Proof. The equality t ( F , U ) = t ( F , V ) for each F implies that G ( n , U ) and G ( n , V ) havethe same distribution Y n for each n . Let H = [ { Y n } ∞ n = ], then Corollary 3.13 impliesthat there exist separable realizations φ and φ such that µ [ r ] ( H α △ φ − ( U α )) = [ r ] ( H α △ φ − ( V α )) = α ∈ [ k ], therefore also µ [ r ] ( φ − ( U α ) △ φ − ( V α )) =
0. Set A = σ ( φ − ( L [0 , h([ r ]) ) , φ − ( L [0 , h([ r ]) )) that is a separable σ -algebra on X [ r ] so by Theorem 3.6there exists a separable realization φ such that for each measurable A ⊂ [0 , h([ r ]) theelement φ − i ( A ) of A can be represented by a subset of [0 , h([ r ]) denoted by ψ i ( A ). It is easyto check that the maps ψ and ψ defined this way are structure preserving embeddingsfrom L h([ r ]) → L h([ r ]) satisfying λ ( ψ ( U α ) △ ψ ( V α )) = α ∈ [ k ]. We conclude thatby Lemma 3.11 there are structure preserving ν and ν such that λ ( ν − ( U α ) △ ν − ( V α )) = α ∈ [ k ]. (cid:3) The next result is perhaps also meaningful beyond the framework of this paper andis the main contribution in the current section. Recall the definition of the ground stateenergies (GSE), Definition 3.1 and Definition 3.2.
Theorem 3.15.
For any J = ( J , . . . , J k ) with J α being a real r-array of size q for each α ∈ [ k ] theparameter of k-colored r-graphs E r − ( ., J ) is testable.Proof. We may assume that k J α k ∞ ≤ α without losing generality. We proceedby contradiction. Suppose there exist an ε > k -colored r -uniformhypergraphs { H n } ∞ n = with V ( H n ) = [ m n ] tending to infinity that are such that for each n withprobability at least ε we have that E r − ( H n , J ) + ε ≤ E r − ( G ( n , H n ) , J ). Let G n = ( G n , . . . , G kn )denote the random k -colored hypergraph G ( n , H n ) for each n with G α n = G ( n , H α n ). Theprevious event can be reformulated as stating that for each n with probability at least ε there is a partition P n = ( P n , . . . , P qn ) of (cid:0) [ n ] r − (cid:1) such that the expression1 n r k X α = q X i ,..., i r = J α ( i , . . . , i r ) e G α n ( r ; P i n , . . . , P i r n )is larger than 1 m rn k X α = q X i ,..., i r = J α ( i , . . . , i r ) e H α n ( r ; R i n , . . . , R i r n ) + ε for any partition R n = ( R n , . . . , R qn ) of (cid:0) [ m n ] r − (cid:1) .Let H denote the ultralimit of the hypergraph sequence { H n } ∞ n = that is a k -partition inthe measure space ( X [ r ]1 , B , µ ), and let σ ( S ) and σ ( S ) ∗ denote the sub- σ -algebras of B corresponding to subsets S of [ r ]. Due to Theorem 3.6 there exists a separable realization φ : X [ r ]1 → [0 , h([ r ]) such that there is a k -colored r -set graphon W = ( W , . . . , W k ) satisfying µ ( φ − ( W α ) △ H α ) = α ∈ [ k ]. Let G ( s ) stand for the point-wise ultralimit realizationof the { G n ( s ) } ∞ n = ⊂ X [ r ]2 for all s ∈ S , where ( S , S , ν ) denotes the underlying joint probabilityspace for the random hypergraphs, and ( X [ r ]2 , B , µ ) is the ultraproduct measure space inthe case of the sample sequence, σ ( S ) and σ ( S ) ∗ are the corresponding sub- σ -algebras.Note that the ultralimits G ( s ) are not k -partitions of the same ultraproduct space as H ,15oreover, it is possible that the σ -algebra generated by { G ( s ) | s ∈ S } together with µ forma non-separable measure algebra that prevents us from using Theorem 3.6 directly.Suppose that for some n we have that E E r − ( G n , J ) < E r − ( H n , J ) + / ε . This assumptionimplies by Lemma 3.3 that P ( E r − ( G n , J ) ≥ E r − ( H n , J ) + ε ) ≤ P ( E r − ( G n , J ) ≥ E E r − ( G n , J ) + ε/ ≤ − ε n r ). The last bound is strictly smaller than ε when n is chosen su ffi cientlylarge, therefore it contradicts the main assumption for large n . Therefore we can arguethat E E r − ( G n , J ) ≥ E r − ( H n , J ) + / ε for large n , throwing away a starting piece of thesequence { H n } ∞ n = we may assume that it holds for all n .A second application of Lemma 3.3 leads to a lower bound on the probability that E r − ( G n , J ) is close to E r − ( H n , J ), namely P ( E r − ( G n , J ) ≤ E r − ( H n , J ) + ε/ ≤ − ε n r ).Hence, by invoking the Borel-Cantelli Lemma, we infer that with probability one the event E r − ( G n , J ) ≤ E r − ( H n , J ) + ε/ n , let the M denote the(random) threshold for which is true that E r − ( G n , J ) > E r − ( H n , J ) + ε/ n ≥ M .It follows that lim ω E r − ( G n , J ) > lim ω E r − ( H n , J ) + ε/ G is equivalent to H in the sense that foreach k -colored r -graph F it holds that t ( F , G ) = t ( F , H ). Then, since there are countablymany test graphs F , we can conclude that the equality holds simultaneously for all F withprobability 1.We have seen above in the paragraph after (3.2) that for every fixed k -colored r -uniformhypergraph t ( F , H ) = lim ω t ( F , H n ). On the other hand the subgraph densities in randominduced subgraphs are highly concentrated around their mean, that is P ( | t ( F , G n ) − t ( F , H n ) | ≥ δ ) ≤ − δ n | V ( F ) | )for any δ >
0, this follows with basic martingale techniques, see Theorem 11 in [6] for thealmost identical statement together with a proof. The Borel-Cantelli Lemma implies thenfor every fixed F that with probability one for each δ > n ( δ ) suchthat for each n ≥ n ( δ ) it is true that | t ( F , G n ) − t ( F , H n ) | < δ/
2. Let us fix δ > F ∈ G r , k .Since the set { n | | t ( F , H n ) − t ( F , H ) | < δ/ } belongs to ω by the definition of the ultralimitfunction, it holds that { n | | t ( F , G n ) − t ( F , H ) | < δ } ∈ ω as a consequence of { n | | t ( F , G n ) − t ( F , H ) | < δ }⊃ ( { n | | t ( F , G n ) − t ( F , H n ) | < δ/ } ∩ { n | | t ( F , H n ) − t ( F , H ) | < δ/ } ) ∈ ω. Consequently, lim ω t ( F , G n ) = t ( F , H ) with probability one for each F , and the limit equationholds simultaneously for each F also with probability one, since their number is countable.Let us pick a realization { G n ( s ) } ∞ n = of { G n } ∞ n = such that it satisfies lim ω E r − ( G n ( s ) , J ) − lim ω E r − ( H n , J ) ≥ ε/ ω t ( F , G n ( s )) = t ( F , H ) for each F , the preceding discussionimplies that such a realization exists, in fact almost all of them are like this. Furthermore,let us consider the sequence of partitions P n = ( P n , . . . , P qn ) of (cid:0) [ n ] r − (cid:1) that realize E r − ( G n ( s ) , J ),and define T i , jn ⊂ [ n ] r \ diag([ n ] r ) through the inverse images of the projections A T i , jn = ( p nj ) − ( A P in ) for i ∈ [ q ], j ∈ [ r ], and n ∈ N , where p nj is the projection that maps an r -array16f size n onto an ( r − j th coordinate. Note that the T i , jn ’s are notcompletely symmetric, but are invariant under coordinate permutations from S [ r ] \{ j } forthe corresponding j ∈ [ r ]. A further property is that and T i , j n can be obtained from T i , j n swapping the coordinates corresponding to j and j .We additionally define the ultraproducts of these sets by P i = [ { P in } ∞ n = ] ⊂ X [ r − and T i , j = [ { T i , jn } ∞ n = ] ⊂ X [ r ]2 , it is clear that T i , j ∈ σ ([ r ] \ { j } ) for each pair of i and j , so ∩ ( i , j ) ∈ I T i , j ∈ σ ([ r ]) ∗ for any I ⊂ [ q ] × [ r ], and that X [ r − = ∪ i P i . The same symmetry assumptions applyfor the T i , j ’s as for the T i , jn ’s described above.We also require the fact that these ultraproduct sets defined above establish a cor-respondence between the GSE of G ( s ) and the ultralimit of the sequence of energies {E r − ( G n ( s ) , J ) } ∞ n = .This can be seen as follows: Recall that E r − ( G n ( s ) , J ) = n r k X α = q X i ,..., i r = J α ( i , . . . , i r ) | G α n ∩ ( ∩ qj = T i j , jn ) | This formula together with the identities [ { G α n ( s ) ∩ ( ∩ qj = T i j n ) } ∞ n = ] = G α ( s ) ∩ ( ∩ qj = T i j , j ), andthat the ultralimit of subgraph densities equals the subgraph density of the ultraproductimply that lim ω E r − ( G n ( s ) , J ) = k X α = q X i ,..., i r = J α ( i , . . . , i r ) µ ( G α ( s ) ∩ ( ∩ qj = T i j , j )) . Now consider the separable sub- σ -algebra A of B generated by the collection of the sets G ( s ) , . . . , G k ( s ) , T , , . . . , T q , r . Then by Theorem 3.6 there exists a separable realization φ : X [ r ]2 → [0 , h([ r ]) and measurable sets U , . . . , U k , V , , . . . , V q , r such that µ ( φ − ( U α ) △ G α ( s )) = α ∈ [ k ] and µ ( φ − ( V i , j ) △ T i , j ) = i ∈ [ q ] , j ∈ [ r ]. Additionally, wecan modify the V i , j ’s on a set of measure 0 such that each of them only depends onthe coordinates corresponding to the sets in h([ r ] \ { j } ), is invariant under coordinatepermutations induced by elements of S [ r ] that fix j , and V i , j can be obtained from V i , j by relabeling the coordinates according to the S r permutation swapping j and j . Also,( U , . . . , U k ) form a k -colored r -set graphon U when we make modifications on null sets.Most importantly, the separable realization φ is measure preserving, so we have thatlim ω E r − ( G n ( s ) , J ) = k X α = q X i ,..., i r = J α ( i , . . . , i r ) λ ( U α ∩ ( ∩ rj = V i j , j )) . (3.3)On the other hand we established that t ( F , G ( s )) = t ( F , H ) for each F ,which implies t ( F , U ) = t ( F , W ), therefore the uniqueness statement of Theorem 3.14 ensures the exis-tence of two structure preserving measurable maps ν , ν : [0 , h([ r ]) → [0 , h([ r ]) such that λ ( ν − ( W α ) △ ν − ( U α )) = α ∈ [ k ]. 17ow let us define the sets S i , j = φ − ( ν ( ν − ( V i , j ))), these satisfy exactly the same symme-try properties as the T i , j ’s above, by the measure preserving nature of the maps involvedwe have that lim ω E r − ( G n ( s ) , J ) = k X α = q X i ,..., i r = J α ( i , . . . , i r ) µ ( H α ∩ ( ∩ rj = S i j , j )) . (3.4)The properties of structure preserving maps imply that S i , j ∈ σ ([ r ] \ { j } ) for each i , j ,so ∩ ( i , j ) ∈ I S i , j ∈ σ ([ r ]) ∗ for any I ⊂ [ q ] × [ r ]. Also, the ultraproduct construction makesit possible to assert the existence of a sequence of partitions R n = ( R n , . . . , R qn ) of (cid:0) [ m n ] r − (cid:1) for ω -almost every n such that S i , j = [ { ( p m n j ) − ( R in ) } ∞ n = ]. But again by the correspondenceprinciple between ultralimits of sequences and ultraproducts in Lemma 3.8 applied to(3.3) and (3.4) we have lim ω E R n , r − ( H n , J ) = lim ω E r − ( G n ( s ) , J ) , which contradicts lim ω E r − ( G n ( s ) , J ) − lim ω E r − ( H n , J ) ≥ ε/ (cid:3) An immediate consequence is that the above theorem is also true for r -graphons. Corollary 3.16.
For any J = ( J , . . . , J k ) with J α being a real r-array of size l for each α ∈ [ k ] thereexists for any ε > a q ( ε ) integer such that for any k-colored r-set graphon W and q ≥ q ( ε ) itholds that P ( |E r − ( W , J ) − E r − ( G ( q , W ) , J ) | > ε ) < ε. Proof.
We only sketch the proof here, details are left to the reader. The main idea isto find for any fixed ε >
0, and for each k -colored r -set graphon W a G ∈ G r , k suchthat their GSE are su ffi ciently close, and further, the distributions of G ( q ( ε/ , W ) and G ( q ( ε/ , G ) are close enough in terms of ε , where q is the sample complexity of E r − ( ., J ),whose existence is ensured by Theorem 3.15. Fix ε >
0, and let W be a k -colored r -setgraphon. By measurability for any ∆ > l and a k -colored r -setgraphon U such that each U α is a union of cubes × S ∈ h([ r ]) [ z S − l , z S l ] with z ∈ Z h([ r ]) and P k α = k W α − U α k ≤ ∆ . For a fixed, but su ffi ciently small ∆ , let G be the k -colored r -graphon l vertices that is obtained by randomization form U using the independent uniform[0 , X S ) S ∈ h([ l ] , r ) \ h([ l ] , . Then by standard large deviations resultsit follows that the 1-norm of U α − W G α is highly concentrated around 0. By definition, thedeviation of the GSE’s of two r -graphons can be upper bounded by a constant multipleof their di ff erence in the 1-norm. By Corollary 2.7 the same is true for the total variationdistance of the corresponding measures for the fixed sampling depth q ( ε/ P k α = k W α − W G α k can be made arbitrarilysmall by the above discussion, which proves the result. (cid:3) We can derive a substantial property of the cut norm form the above theorem. Recallthe definition of the relevant norms, Definition 2.4.18 emma 3.17.
Let r ≥ . For any ε > and t ≥ there exists an integer l ( r , ε, t ) ≥ a such thatfor any symmetric r-kernel U that takes values in [ − , , and for any integer l ≥ l ( r , ε, t ) it holdswith probability at least − ε that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup Q , t Q ≤ t k U k (cid:3) , r , Q − sup Q , t Q ≤ t k W G ( l , U ) k (cid:3) , r , Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε, where the supremum at both places goes over symmetric partitions Q of [0 , h([ r − into at most tclasses.Proof. Let us fix ε > r , t ≥
1, and let U be arbitrary. In this lemma we deal with r -graphonsinstead of r -set graphons, Fubini’s Theorem ensures that we can apply Theorem 3.15correctly later on.Showing that there exits an l not depending on U such that for each l ≥ l it holds thatsup Q , t Q ≤ t k U k (cid:3) , r , Q − sup Q , t Q ≤ t k W G ( l , U ) k (cid:3) , r , Q ≤ ε with failure probability at most ε/ S i ) i ∈ [ r ] of symmetric subsets of [0 , h([ r − and asymmetric partition Q of [0 , h([ r − into at most t classes such thatsup Q , t Q ≤ t k U k (cid:3) , r , Q = t X j ,..., j r = | Z ∩ i ∈ [ r ] p − r ] \{ i } ( S i ∩ Q ji ) U ( x h([ r ] , r − )d λ ( x h([ r ] , r − ) | , and use Markov’s inequality. The di ffi cult part is to show that if l is large enough then foreach U it holds that sup Q , t Q ≤ t k W G ( l , U ) k (cid:3) , r , Q − sup Q , t Q ≤ t k U k (cid:3) , r , Q ≤ ε with probability at least 1 − ε/ U in order to apply the above result on k -colored r -graphs, Corollary 3.16. Therefore we split the interval [ − ,
1] into consecutive intervals I , . . . , I k of length at most ε/
4, let y j = inf I j for each j ∈ [ k ], and define the r -kernel W ( x ) = P kj = I I j ( U ( x )) y j . Then k U − W k ∞ ≤ ε/
4, so therefore (cid:12)(cid:12)(cid:12) k U k (cid:3) , r , Q − k W k (cid:3) , r , Q (cid:12)(cid:12)(cid:12) ≤ ε/ (cid:12)(cid:12)(cid:12) k W G ( l , U ) k (cid:3) , r , Q − k W G ( l , W ) k (cid:3) , r , Q (cid:12)(cid:12)(cid:12) ≤ ε/ Q and l . Thus, it su ffi ces to show the existenceof an l not depending on U or W such that for each l ≥ l we have k W G ( l , W ) k (cid:3) , r , Q − k W k (cid:3) , r , Q ≤ ε/ Q of [0 , h([ r − into at most t classes simultaneously withprobability at least 1 − ε/ Q , t Q ≤ t k W k (cid:3) , r , Q as an optimization problem, more preciselysup Q , t Q ≤ t k W k (cid:3) , r , Q (3.5) = sup Q , t Q ≤ t max A ∈ A sup T j ⊂ [0 , h([ r − j ∈ [ r ] t X i ,..., i r = A ( i , . . . , i r ) Z [0 , h([ r ] , r − W ( x h([ r ] , r − ) r Y j = I T j ∩ Q ij ( x h([ r ] \{ j } ) )d λ ( x h([ r ] , r − ) , (3.6)19here A denotes the set of all r -arrays of size t with {− , } entries, and the set andpartitions involved are symmetric.If we swap the order of the maximization operation on the right of the above formula(3.5), then it can be turned into a generalized energy for each A ∈ A . In more detail,consider W as a k -colored r -graphon with W α = I W = y α for each α ∈ [ k ], with slight abuseof notation we set W = ( W α ) α ∈ [ k ] . We also define the r -array B of size 2 r , indexed by thepower set of [ r ] so that B ( i S , . . . , i S r ) is equal to 1 if for every j ∈ [ r ] we have j ∈ S j , and isequal to 0 otherwise. Let J α A = y α ( A ⊗ B ) be the tensor product of A and B for each A ∈ A multiplied with the scalar y α with α ∈ [ k ], then J α A is an r -array of size 2 r t . It follows thatmax A ∈ A E r − ( W , J A ) = sup Q , t Q ≤ t k W k (cid:3) , r , Q . (3.7)Similarly, max A ∈ A E r − ( W G ( l , W ) , J A ) = sup Q , t Q ≤ t k W G ( l , W ) k (cid:3) , r , Q , hence sup Q , t Q ≤ t k W G ( l , W ) k (cid:3) , r , Q − sup Q , t Q ≤ t k W k (cid:3) , r , Q ≤ max A ∈ A |E r − ( W G ( l , W ) , J A ) − E r − ( W , J A ) | . The function E r − ( ., J A ) is testable by Corollary 3.16, say with sample complexity q ( ε, r , l , k ),so sup Q , t Q ≤ t k . k (cid:3) , r , Q is testable with sample complexity l ( r , ε, t ) = q ( ε/ | A | , r , m , r t ). (cid:3) In fact, we will require the version of Lemma 3.17 for k -tuples r -kernels. Lemma 3.18.
Let r , k ≥ . For any ε > and t ≥ there exists an integer q cut ( r , k , ε, t ) ≥ a such that for any k-tuple of r-kernel U , . . . , U k that take values from [ − , , and any integerq ≥ q cut ( r , k , ε, t ) it holds with probability at least − ε that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup Q , t Q ≤ t k X j = k U j k (cid:3) , r , Q − sup Q , t Q ≤ t k X j = k W G ( q , U j ) k (cid:3) , r , Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε where the supremum at both places goes over symmetric partitions Q of [0 , h([ r − into at most tclasses.Proof. We only sketch the proof as it is almost identical to that of Lemma 3.17. Let r , k , t ≥ ε > U , . . . , U k and q be arbitrary. The lower bound onsup Q , t Q ≤ t P kj = k W G ( q , U j ) k (cid:3) , r , Q can be obtained by the same argument as above using Markov’sinequality. For the upper bound we again discretize to obtain the r -kernels W , . . . , W k with common range { y i : α ∈ [ m ] } such that k U j − W j k ∞ ≤ ε k for each j ∈ [ k ], hence m = k ε W j and m -colored r -graphon W j as above and set J α A to y α ( A ⊗ B ), then max A ,..., A k ∈ A sup Q , t Q ≤ t k X j = E Q , r − ( W j , J A j ) = sup Q , t Q ≤ t k X j = k W j k (cid:3) , r , Q . Similarly, max A ,..., A k ∈ A sup Q , t Q ≤ t k X j = E Q , r − ( W G ( q , W j ) , J A j ) = sup Q , t Q ≤ t k X j = k W G ( q , W j ) k (cid:3) , r , Q . The testability of sup Q , t Q ≤ t P kj = E Q , r − ( W j , J A j ) follows from Theorem 3.15 with a slightmodification of the argument for any fixed tuple A , . . . , A k ∈ A . As the cardinality of A does not depend on W , . . . , W k the statement of the lemma follows. (cid:3) We will require the version of Szemerédi’s Regularity Lemma adapted to the Hilbert spacesetting. Let us recall this variant.
Lemma 4.1. [13] Let K , K , . . . be arbitrary subsets of a Hilbert space H . Then for every ε > and f ∈ H there is an m ≤ ε and there are f i ∈ K i and γ i ∈ R ( ≤ i ≤ m) such that for everyg ∈ K m + we have that |h g , f − m X i = γ i f i i| ≤ ε k f kk g k . We start with the following intermediate version of the regularity lemma for edge k -colored r -graphons, the partition obtained here satisfies stronger conditions than thoseimposed by the Weak Regularity Lemma [7], and weaker than by Szemerédi’s original. Lemma 4.2.
For every r ≥ , ε > , t ≥ , k ≥ and k-colored r-graphon W there exists asymmetric partition P = ( P , . . . , P m ) of [0 , h([ r − into m ≤ (2 t ) ( rk + /ε = t reg ( r , k , ε, t ) parts anda symmetric ( r , r − - step function V ∈ W r , k with steps from P , such that for any partition Q of [0 , h([ r − into at most mt classes we haved (cid:3) , r , Q ( W , V ) ≤ ε. Proof.
Our lemma is a special case of Lemma 4.1. We set H to be the space of of k -tuples ofreal measurable functions on [0 , h([ r ] , r − with the sum of the component-wise L -productsas the inner product, this space contains W r , k . Set s (1) = s ( i + = s ( i )( s ( i ) t + rk for each i ≥ K i be the set of k -tuples of indicator functions that are ( r , r −
1) stepfunctions with s ( i ) number of symmetric steps and taking values from the set {− , , } .21ote that the elements of the K i ’s are not necessarily symmetric as functions, only theirsteps are required to be such. Further, observe that s ( i ) ≤ (2 t ) ( rk + i . Now apply Lemma 4.1with the above setup for ε/ W to obtain U that satisfies all the conditions of thelemma except for symmetry, in particular k X α = k U α − W α k (cid:3) , r , P < ε. Define V with V α ( x h([ r ] , r − ) = r ! P π ∈ S r U α ( x π (h([ r ] , r − ). The symmetry of W and the triangleinequality implies that V is suitable, since k V α − W α k (cid:3) , r , P ≤ r ! X π ∈ S r k ( U α ) π − ( W α ) π k (cid:3) , r , P = k U α − W α k (cid:3) , r , P for any P and α ∈ [ k ], and U π ( x h([ r ] , r − ) = U ( x π (h([ r ] , r − ). (cid:3) The next lemma is analogous to Lemma 3.2 from [10]. It describes under what metricconditions a k -coloring of a t -colored graphon can be transfered to another one so thatthe two tk -colored graphons are close in a certain sense. For the sake if completeness wesketch the proof. Lemma 4.3.
Let ε > , U be a t-colored r-graphon that is an ( r , r − -step function with steps P = ( P , . . . , P m ) and V be a t-colored r-graphon with d (cid:3) , r , P ( U , V ) ≤ ε . For any k ≥ and ˆ U a [ t ] × [ k ] -colored r-graphon that is an ( r , r − -step function with steps from P such that [ ˆ U , k ] = U there exists a k-coloring of V denoted by ˆ V so thatd (cid:3) , r , P ( ˆ U , ˆ V ) ≤ k ε. Proof.
Fix ε >
0, and let U = ( U α ) α ∈ [ t ] , V = ( V α ) α ∈ [ t ] and ˆ U = ( U α,β ) α ∈ [ t ] ,β ∈ [ k ] as in thestatement of the lemma. Then P t α = U α = P k β = U α,β = U α for each α ∈ [ t ]. Let usdefine ˆ V = ( V α,β ) α ∈ [ t ] ,β ∈ [ k ] that is a k -coloring of V . Set V α,β = V α [ I U α = k + I U α > U α,β U α ], it iseasy to see that the factor on the right of the formula is a ( r , r − P = ( P , . . . , P m ). We estimate the deviation of each pair U α,β and V α,β from above in the r -cut norm, for this we fix the symmetric S , . . . , S r ⊂ [0 , h([ r − . Then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∩ l ∈ [ r ] p − l ( S l ) U α,β − V α,β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ t X α ,...,α r = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∩ l ∈ [ r ] p − l ( S l ∩ P α l ) U α,β − V α,β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = t X α ,...,α r = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∩ l ∈ [ r ] p − l ( S l ∩ P α l ) ( U α − V α )[ I U α = k + I U α > U α,β U α ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k U α − V α k (cid:3) , r , P . Taking the maximum over all symmetric measurable r -tuples S , . . . , S r and summing upover all choices of α and β delivers the upper bound we were after. (cid:3) Proof of the main result
The central tool in the main proof is the following lemma which can also be of independentinterest. Informally it states that every coloring of a sampled r -graph can be transferredonto the graphon from which the graph was sampled from, such that another samplingprocedure with a much smaller sample size cannot distinguish the two colored objects. Lemma 5.1.
For every r ≥ , proximity parameter δ > , palette sizes t , k ≥ , sampling depthq ≥ there exists an integer q tw = q tw ( r , δ, q , t , k ) ≥ such that for every q ≥ q tw the followingholds. Let U = ( U α ) α ∈ [ t ] be a t-colored r-graphon and let V α denote W G ( q , U α ) for each α ∈ [ t ] , also let V = ( V α ) α ∈ [ t ] , so W G ( q , U ) = V . Then with probability at least − δ there exist for every k-coloring ˆ V = ( V α,β ) α ∈ [ t ] ,β ∈ [ k ] of V a k-coloring ˆ U = ( U α,β ) α ∈ [ t ] ,β ∈ [ k ] of U = ( U α ) α ∈ [ t ] such that we have thatd tw ( µ ( q , ˆ W ) , µ ( q , ˆ U ) ≤ δ. Proof.
We proceed by induction with respect to r . The statement is not di ffi cult to verifyfor r =
1. In this case the 1-graphons U α and V α can be regarded as indicator functionsof measurable subsets B α and A α of [0 ,
1] (so for each α ∈ [ k ] we have U α = I B α and V α = I a α ) that form two partitions associated to U and V respectively. Note that ( A α ) α ∈ [ k ] isobtained from ( B α ) α ∈ [ k ] by the sampling process. A k -coloring corresponds to a refinementof these partitions with each original class being divided into k measurable parts, that is A α = ∪ ∗ β ∈ [ k ] A α,β and V α,β = I A α,β . Moreover, | t ( F , ˆ U ) − t ( F , ˆ V ) | = | Q q l = λ ( B F ( l ) ) − Q q l = λ ( A F ( l ) ) | for any of k -coloring ˆ U of U and for any [ t ] × [ k ]-colored F on q vertices. We may define asuitable coloring by partitioning each of the sets B α into parts ( B α,β ) β ∈ [ k ] so that the classessatisfy λ ( B α,β ) = λ ( B α ) λ ( A α,β ) λ ( A α ) when λ ( A α ) >
0, and λ ( B α,β ) = λ ( B α ) k otherwise for each β ∈ [ k ]. Then by setting U α,β = I B α,β and ˆ U = ( U α,β ) α ∈ [ t ] ,β ∈ [ k ] we have that d tw ( µ ( q , ˆ W ) , µ ( q , ˆ U ) = X F : | V ( F ) | = q | t ( F , ˆ U ) − t ( F , ˆ V ) | ≤ q k + α ∈ [ k ] | λ ( A α ) − λ ( B α ) | , where the sum runs over all [ t ] × [ k ]-colored 1-graphs F on q vertices.The probability that for a fixed α ∈ [ t ] the deviation | λ ( A α ) − λ ( B α ) | exceeds δ q k + is atmost 2 exp( − δ q q k + ), the union bound gives the upper bound exp(ln 2 + t − δ q q k + ) for theprobability that d tw ( µ ( q , ˆ W ) , µ ( q , ˆ U ) ≤ δ fails for our particular choice for the coloring ˆ U of U . We note that the failure probabilitycan be made arbitrary small with the right choice of q , so in particular smaller than δ ,therefore q tw (1 , δ, q , t , k ) = ( t + ln 2 − ln δ )3 q k + δ that satisfies the conditions of the lemma.Now assume that we have already verified the statement of the lemma for r − q tw . Let us proceed to the proof of the case for r -graphons, therefore let δ > t , k , q ≥ q to be determined below23nd U , V , and ˆ V as in the condition of the lemma. We start by explicitly constructinga k -coloring ˆ U for U , in the second part of the proof we verify that the construction issuitable.In a nutshell, we proceed as follows. We approximate ˆ V by the step function ˆ Z , and set Z = [ ˆ Z , k ], and also approximate U by W . Let W be the sampled version of W generatedby the same process as V . This way W and Z are close, hence we can color W usingthe coloring ˆ Z of Z to obtain ˆ W . The coloring ˆ W is then transferred onto W using theinduction hypothesis applied to the marginals of the step sets of W and W to get ˆ W with[ ˆ W , k ] = W . Finally we color U exploiting the proximity of U and W and the coloredˆ W .Our construction may fail to meet the criteria of the lemma, this can be caused at twopoints in the above outline. For one, it may happen, that W does not approximate V well enough, and secondly, when we transfer ˆ W onto W using the induction hypothesiswith r −
1, as the current lemma leaves space for probabilistic error. These two events areindependent from the particular choice of ˆ V and their probability can be made su ffi cientlysmall, we aim for to show this. We proceed now to the technical details.Let ∆ = δ r !4 k ( kt ) qr q r . Set t = t reg ( r , tk , ∆ ,
1) and t = t reg ( r , t , ∆ / , t ), and define q tw ( r , δ, q , t , k ) = max { q tw ( r − , δ/ , q , t , t ) , q cut ( r , t , ∆ / , t t ) } . Let q ≥ q tw ( r , δ, q , t , k ) be arbitrary.First we apply Lemma 4.2 with proximity parameter ∆ / t -colored r -graphon U , the lemma ensures the existence of a symmetric partition P = ( P , . . . , P t ) of [0 , h([ r − with t P ≤ t and a t -colored symmetric step function W = ( W , . . . , W t ) with steps in P that satisfies sup Q , t Q ≤ t P t d (cid:3) , r , Q ( W , U ) ≤ ∆ / , where the supremum runs over all symmetricpartitions Q of [0 , h([ r − with at most t P t classes. Applying structure preserving trans-formations to [0 , h([ r − the classes of P can be considered as piled up, meaning that foreach y ∈ [0 , h([ r − , r − the fibers { y } × [0 ,
1] are partitioned by the intersections with theclasses of P into intervals [0 , a ) , [ a , a ) , . . . , [ a t − , a t ] with { y } × [ a j − , a j ) = ( { y } × [0 , ∩ P j .We introduce the r -dimensional real arrays A , . . . , A t in order to describe the explicit formof the W α ’s. So, W α ( x h([ r ] , r − ) = t P X i ,..., i r = A α ( i , . . . , i r ) r Y l = I P il ( x h([ r ] \{ l } ) ) . Let W = ( W , . . . , W t ) denote G ( q , W ), so W α stands for G ( q , W α ) for each α ∈ [ t ], thenLemma 3.18 implies that for q ≥ q cut ( r , t , ∆ / , t t ) it holds thatsup Q , t Q ≤ t P t d (cid:3) , r , Q ( W , V ) ≤ ∆ , with probability at least 1 − ∆ /
2, since t P ≤ r . Also, W α ( x h([ r ] , r − ) = t P X i ,..., i r = A α ( i , . . . , i r ) r Y l = I P ′ il ( x h([ r ] \{ l } ) ) , α ∈ [ t ] and P ′ j = ∪ ( p ,..., p r − ) ∈ I j [ p − q , p q ] × · · · × [ p r − q , p r q ] × [0 , × · · · × [0 , I j = { ( p , . . . , p r − ) : X r [ { p ,..., p r − } ] ∈ P j } for every j ∈ [ t P ]. Note that P ′ = ( P ′ j ) j ∈ [ t P ] is asymmetric partition.We apply now Lemma 4.2 with proximity parameter ∆ in order to approximate the[ t ] × [ k ]-colored r -graph ˆ V = ( V α,β ) α ∈ [ t ] ,β ∈ [ k ] , the outcome is a [ t ] × [ k ]-colored step functionˆ Z = ( Z α,β ) α ∈ [ t ] ,β ∈ [ k ] with symmetric steps in R = ( R , . . . , R t ) of [0 , h([ r − with t R ≤ t thatsatisfies sup Q , t Q ≤ t R d (cid:3) , r , Q ( ˆ V , ˆ Z ) ≤ ∆ . We introduce the t -colored step function Z = [ ˆ Z , k ] that is the k -discoloring of ˆ Z that hassteps in R and note that sup Q , t Q ≤ t R d (cid:3) , r , Q ( V , Z ) ≤ ∆ , and therefore sup Q , t Q ≤ t R d (cid:3) , r , Q ( Z , W ) ≤ ∆ . (5.1)Define the r -arrays B , . . . , B t such that for each α ∈ [ t ] it holds that Z α ( x h([ r ] , r − ) = t R X i ,..., i r = B α ( i , . . . , i r ) r Y l = I R il ( x h([ r ] \{ l } ) ) , further define also the r -arrays ( B βα ) α ∈ [ t ] ,β ∈ [ k ] such that Z α,β ( x h([ r ] , r − ) = t R X i ,..., i r = B βα ( i , . . . , i r ) r Y l = I R il ( x h([ r ] \{ l } ) )for each α ∈ [ t ] , β ∈ [ k ]. Clearly, B α ( i , . . . , i r ) = P k β = B βα ( i , . . . , i r ).Our aim next is to find a k -coloring of W so that the new tk -colored r -graphon obtainedis close to ˆ Z . In order to produce the coloring we apply Lemma 4.3 relying on (5.1), hencewe obtain ˆ W with [ ˆ W , k ] = W . The proximity between the two tk -colored r -graphs canbe quantified by sup Q , t Q ≤ t R d (cid:3) , r , Q ( ˆ Z , ˆ W ) ≤ k ∆ . The graphon ˆ W is a symmetric step function with steps that form the coarsest partitionthat both refines P ′ and R , we denote this symmetric partition of [0 , h([ r − by S , itsnumber of classes satisfies t S = t P ′ t R ≤ t t .Let us define the t P -colored ( r − w = ( w , . . . , w t P ) that is obtained fromthe classes of the partition P by integrating along the coordinate corresponding to the25et [ r − w i ( x h([ r − , r − ) = R I P i ( x h([ r − )d x [ r − . In the same way we define the t P -colored ( r − u = ( u , . . . , u t P ) corresponding to the partition P ′ , as well as the[ t P ] × [ t R ]-colored ˆ u = ( u i , j ) i ∈ [ t P ] , j ∈ [ t R ] , where it holds that u = [ ˆ u , t R ] and ˆ u is the t R -coloringof u corresponding to the partition S . Note that w , u , and ˆ u satisfy the usual symmetries,since their origin partitions were symmetric. As the partition P ′ was constructed via thesame sampling procedure as V and W , therefore it holds that u = G ( q , w ) and u i = G ( q , w i )for each i ∈ [ t P ].We can assert that due to the induction hypothesis there exists a t R -coloring ˆ w = ( w i , j ) i ∈ [ t P ] , j ∈ [ t R ] of w that satisfies d tw ( µ ( q , w ) , µ ( q , u ) ≤ δ/ − δ/ q ≥ q tw ( r − , δ/ , q , t , t ).We construct a k -coloring for W next. Recall the proof of Lemma 4.3, therefore wehave that W α,β ( x h([ r ] , r − ) = t P X i ,..., i r = t R X j ,..., j r = A α ( i , . . . , i r ) B βα ( j , . . . , j r ) B α ( j , . . . , j r ) I B α > + k I B α = r Y m = I P ′ im ∩ R jm ( x h([ r ] \{ m } ) ) , (5.2)and set A βα (( i , j ) , . . . , ( i r , j r )) = A α ( i , . . . , i r ) (cid:20) B βα ( j ,..., j r ) B α ( j ,..., j r ) I B α > + k I B α = (cid:21) for all α ∈ [ t ], β ∈ [ k ] and(( i , j ) , . . . , ( i r , j r )) ∈ ([ t P ] × [ t R ]) r .We utilize the t R -coloring ˆ w of the ( r − w to construct a refined partitionof P that resembles S in order to enable the construction of a k -coloring of W along thesame lines as in (5.2). Let P i , j = { x ∈ [0 , h([ r − : i − X l = w l ( x h([ r − , r − ) + j − X l = w i , l ( x h([ r − , r − ) ≤ x [ r − < i − X l = w l ( x h([ r − , r − ) + j X l = w i , l ( x h([ r − , r − ) } for each i ∈ [ t P ] and j ∈ [ t R ]. Let P ′′ = ( P i , j ) i ∈ [ t P ] , j ∈ [ t R ] .Clearly, ( P i , j ) j ∈ [ t R ] is a t R -partition of the set P i , and w i , j ( x h([ r − , r − ) = R P i , j d x h([ r − . We areable now to describe the k -coloring of the W , define W α,β ( x h([ r ] , r − ) = t P X i ,..., i r = t R X j ,..., j r = A βα (( i , j ) , . . . , ( i r , j r )) r Y m = I P im , jm ( x h([ r ] \{ m } ) ) . (5.3)Note that ˆ W is a step functions with a step partition that refines P into P ′′ , but theregularity property of W allows for d (cid:3) , r , P ′′ ( U , W ) ≤ ∆ / . k -coloring ˆ U of U , so that ˆ U is [ t ] × [ k ]-colored, and d (cid:3) , r , P ′′ ( ˆ U , ˆ W ) ≤ k ∆ . It remains to show that this coloring satisfies the requirements of the current lemma for alarge enough q .In the first step of the coloring construction we employed the r -graphon version of theintermediate regularity lemma, Lemma 4.2, therefore we can assert that for each tk -colored F we have by means of Lemma 2.6 that X F ∈G r , ktq | t ( F , ˆ V ) − t ( F , ˆ Z ) | ≤ ( kt ) q r q r r ! d (cid:3) , r ( ˆ V , ˆ Z ) ≤ δ k , so we can conclude that d tw ( µ ( q , ˆ V ) , µ ( q , ˆ Z )) ≤ δ k .In the next step, as a consequence of Lemma 4.3 and Corollary 2.7, we have for ˆ W that d tw ( µ ( q , ˆ Z ) , µ ( q , ˆ W )) ≤ δ/ tk -colored random r -graph G ( q , ˆ W ), it is generated by the independentuniformly distributed [0 , { Y S : S ∈ h([ q ] , r ) } . The color ofeach edge e = { e , . . . , e } ∈ (cid:0) [ q ] r (cid:1) is decided by determining first the unique tuple (up tocoordinate permutations) (( i , j ) , . . . , ( i r , j r )) ∈ ([ t P ] × [ t R ]) r such that ( Y S ) S ∈ h( e \{ e l } ) ∈ P i l , j l , andthen check for which pair α ∈ [ t ], β ∈ [ k ] it holds that α − X l = A l (( i , j ) , . . . , ( i r , j r )) + β − X l = A l α (( i , j ) , . . . , ( i r , j r )) < Y e ≤ α − X l = A l (( i , j ) , . . . , ( i r , j r )) + β X l = A l α (( i , j ) , . . . , ( i r , j r )) , then add the color ( α, β ) to e with corresponding index. It is convenient to view thisprocess as first randomly t P ′′ -coloring an ( r − G , whoseedges are the simplices of the original edges, here we add a color ( i , j ) to an ( r − e ′ whenever ( Y S ) S ∈ h( e ′ ) ∈ P i , j , and conditioned on G we subsequently make independentchoices for each edge to determine their color based on the arrays A βα by means of therandom variables { Y S : S ∈ (cid:0) [ q ] r (cid:1) } at the top level.Let us turn to the tk -colored G ( q , ˆ W ), the above description of the random processgenerating this object remains conceptually valid also for this random graph, the r -arrays A βα are identical to the case above, only the partition P ′′ has to be altered to S . Similarlyas above, we introduce the random ( r − t P ′′ -colored hypergraph G that isgenerated as above adapted to G ( q , ˆ W ). That means that the ( r − S through the process that generates G ( q , ˆ W ), see above. The key observation here is that conditioned on G = G , one can27ouple G ( q , ˆ W ) and G ( q , ˆ W ) so that the two random graphs coincide with conditionalprobability 1. Recall that a coupling is only another name for a joint probability space forthe two random objects with the marginal distributions following µ ( q , W ) and µ ( q , W )respectively. As the conditional distributions for the choices of colors for the r -edgesare identical provided that G = G the coupling is trivial. In order to construct a goodunconditional coupling we require another coupling, now of G and G , so that P ( G , G )is negligible small for our purposes, and whose existence is exactly what the inductionhypothesis ensures, when q is large enough.As q ≥ q tw ( r − , δ/ , q , t , t ), the induction hypothesis enables us to use that there existfor any ˆ u a ˆ w so that d tw ( µ ( q , ˆ u ) , µ ( q , ˆ w )) ≤ δ/ − δ/ u simultaneously, which in turn implies that there is a coupling of the t t -coloredrandom ( r − G and G so that P ( G , G ) ≤ δ/ G ( q , ˆ W ) and G ( q , ˆ W ) such that P ( G ( q , ˆ W ) , G ( q , ˆ W )) ≤ δ/ d tw ( µ ( q , ˆ W ) , µ ( q , ˆ W )) ≤ δ/ . Since ˆ W has at most t P t steps, another application of Lemma 4.3 provides the bound d tw ( µ ( q , ˆ W ) , µ ( q , ˆ U )) ≤ δ/ , as d (cid:3) , r ( ˆ U , ˆ W ) ≤ k ∆ . Evoking the triangle inequality and summing up the upper bounds on the respectivedeviations we conclude that d tw ( µ ( q , ˆ V ) , µ ( q , ˆ U )) ≤ d tw ( µ ( q , ˆ V ) , µ ( q , ˆ Z )) + d tw ( µ ( q , ˆ Z ) , µ ( q , ˆ W )) + d tw ( µ ( q , ˆ W ) , µ ( q , ˆ W )) + d tw ( µ ( q , ˆ W ) , µ ( q , ˆ U )) ≤ (cid:18) k + + + (cid:19) δ < δ, the overall error probability is at most δ/ + ∆ /
2, which is at most δ . (cid:3) With Lemma 5.1 at hand we can overcome the di ffi culties caused by properties of the r -cut norm for r ≥ r =
2, we turn to prove the main result of thepaper.
Proof of Theorem 2.3.
We regard simple hypergraphs as 2-colored r -graphs, in the fol-lowing the term simple should be understood this way at each appearance. Let the2 k -colored witness parameter of the nondeterministically testable r -graph parameter f be denoted by g , whose sample complexity is at most q g ( ε ) for each proximity pa-rameter ε >
0. Set d ( r , ε, q , k , t ) = [ q g ( ε ) r ln( tk ) − ln( ε )][2( tk ) qr q ] ε .Let ε > q f ( ε ) = max { q tw ( r , ε/ , q g ( ε/ , k , ε q g ( ε/ d ( r , ε/ , q g ( ǫ/ , k , } . We will show that forevery q ≥ q f ( ε ) the condition P ( | f ( G ) − f ( G ( q f ( ε ) , G ) | > ε ) < ε.
28s satisfied for each G . Let q ≥ q f ( ε ) arbitrary but fixed and G be a fixed simple graph on n vertices.First we show that f ( G ( q , G )) ≥ f ( G ) − ε/ − ε/
4. For thislet us select a k -coloring G of G such that f ( G ) = g ( G ), then the random k -colored graph F = G ( q , G ) is a k -coloring of G ( q , G ), therefore f ( G ( q , G )) ≥ g ( F ), but since q ≥ q q ( ε/
4) weknow from the testability of g that g ( F ) ≥ g ( G ) − ε/ − ε/
4, whichverifies our claim.The more di ffi cult part is to show that f ( G ( q , G )) ≤ f ( G ) + ε with failure probability atmost ε/
2. Let us denote the random r -graph G ( q , G ) by F . We claim that with probabilityat least 1 − ε/ k -coloring F of F there exists a k -coloring G of G suchthat | g ( F ) − g ( G ) | ≤ ε , this su ffi ces to verify the statement of the theorem.Our proof exploits that the di ff erence of the g values between two colored r -graphs F and G can be upper bounded by | g ( F ) − g ( G ) | ≤ | g ( F ) − g ( G ( q g ( ε/ , F )) | + | g ( G ) − g ( G ( q g ( ε/ , G )) | ≤ ε/ , whenever there exists a coupling of the two random 2 k -colored r -graphs G ( q g ( ε/ , G ) and G ( q g ( ε/ , F ) appearing in the above formula such that they are equal with probabilitylarger than ε/
2. Set q = q g ( ε/ F there exists a G that satisfies the previous conditions.Recall that coupling is a probability space together with the random r -graphs G and G defined on it such that G has the same marginal distribution as G ( q , G ) and G hasthe same as G ( q , F ), their joint distribution is constructed in a way that serves our currentpurposes by maximizing the probability that they coincide. When the target spaces arefinite as in our case then a coupling that satisfies this condition can be easily constructedwhenever d tw ( µ ( q , G ) , µ ( q , F )) ≤ − ε/
2, see (2.3).By Lemma 5.1 for 3-colored r -graphs (there are 3 types of entries in the graphonrepresentation of simple r -graphs, edges, non-edges, and diagonal elements) it followsthat with probability at least 1 − ε/ F there exists a 3 k -colored U with [ U , k ] = W G such that d tw ( µ ( q , U ) , µ ( q , W F )) ≤ ε/
4. Let us condition on this event and let F be fixed.From (5.1) we know that d tw ( µ ( q , G ) , µ ( q , W G )) ≤ q / n ≤ ε/ d tw ( µ ( q , F ) , µ ( q , W F )) ≤ q / q ≤ ε/
4. Further, it follows from our condition above that there exists a 3 k -colored U thatinduces a fractional coloring of G , and d tw ( µ ( q , U ) , µ ( q , W F )) ≤ ε/
4. It remains to produce a3 k -coloring of W G from any fixed 3 k -colored U ( k of the colors of U correspond exclusivelyto diagonal cubes, so can be neglected). We do this by randomization, let ( X { i } ) i ∈ [ n ] beindependent uniform random variables distributed on [ i − n , in ], and let ( X S ) S ∈ h([ n ] , r ) \ h([ n ] , be independent uniform random variables on [0 , W G to take the color U ( X h( e ) ) on the set [ e − n , e n ] × · · · × [ e r − n , e r n ] × [0 , × · · · × [0 ,
1] for e = { e , . . . , e r } . For anyfixed H ∈ G r , kq basic martingale methods deliver P ( | t ( H , W G ) − t ( H , U ) | ≥ δ ) ≤ − δ n q )29or any δ >
0, therefore when setting δ = ε k ) qr we get that d tw ( µ ( q , U ) , µ ( q , W G )) = X H ∈G r , kq | t ( H , W G ) − t ( H , U ) | ≤ ε/ − ε/
4, since n ≥ q ≥ d ( r , ε/ , q , k , = [ q r ln(2 k ) − ln( ε/ k ) qr q ] ε .Summing up terms gives d tw ( µ ( q , F ) , µ ( q , G )) ≤ d tw ( µ ( q , F ) , µ ( q , W F )) + d tw ( µ ( q , W F ) , µ ( q , U )) + d tw ( µ ( q , U ) , µ ( q , W G )) + d tw ( µ ( q , W G ) , µ ( q , G )) ≤ ε, with failure probability at most ε/
2, this concludes our proof. (cid:3)
The concept of nondeterministic testing was originally introduced for testing propertiesby Lovász and Vesztergombi [14], and remarkable progress has been made in that context,see [8] and [14], the estimation of parameters, which is our main issue in this paper, isin close relationship to that concept. For related developments in combinatorial propertytesting using regularity methods we refer to [2].We present the definition of testability of properties in the usual and in the nondeter-ministic sense and construct a tester from the tester of the witness property with the aidof Lemma 5.1 that achieves the same sample complexity as in the parameter testing case.This result connects our contribution to previous e ff orts more directly and answers thequestion posed in [14] asking if the equivalence of the two testability notions persists foruniform hypergraphs of higher order similar to the case of graphs. Definition 6.1.
An r-graph property P is testable, if there exists another r-graph property ˆ P calledthe sample property, such that(a) P ( G ( q , G )) ∈ ˆ P ) ≥ for every G ∈ P and q ≥ , and(b) for every ε > there is an integer q P ( ε ) ≥ such that for every q ≥ q P ( ε ) and every G that is ε -far from P we have that P ( G ( q , G )) ∈ ˆ P ) ≤ .Testability for colored r-graphs is defined analogously. We remark that ε -far here means that one has to modify at least ε | V ( G ) | edges inorder to obtain an element of P . Note that and in the definition can be replaced byarbitrary constants 1 > a > b >
0, this change may alter the corresponding certificate ˆ P and the function q P , but not the characteristic of P being testable or not. Let P n denote theelements of P of size n .Next we formulate the definition of nondeterministic testability.30 efinition 6.2. An r-graph property P is nondeterministically testable, if there exists an integerk ≥ and a k-colored r-graph property Q called the witness property that is testable in the sense ofDefinition 6.1 satisfying [ Q , k ] = { [ G , k ] | G ∈ Q} = P (see Definition 2.2 above for the discoloringoperation). We formulate next the main theorem in this section.
Theorem 6.3.
Every nondeterministically testable r-graph property is testable.Proof.
Let P be a nondeterministically testable property with witness property Q of 2 k -colored r -graphs. We employ the combinatorial language with counting subgraph den-sities when referring to Q and its testability, and the probabilistic language of pickingrandom subgraphs in a uniform way when handling P in order to facilitate readability.Let ˆ Q be the corresponding sample property that certifies the testability of Q and q Q bethe sample complexity corresponding to the thresholds 1 / /
5, that is(i) if G ∈ Q , then for every and q ≥ t ( ˆ Q q , G ) ≥ /
5, and(ii) for every ε > G is ε -far from Q , then for every q ≥ q Q ( ε ) we have that t ( ˆ Q q , G ) ≤ / P together with a function q P such that they fulfill theconditions of Definition 6.1. We are free to specify the error thresholds by the remark afterDefinition 6.1, we set them to 2 / / n be a positive integer and let ε n > δ that satisfy n ≥ max { q tw ( r , / , q Q ( δ ) , , k ); 100 q Q ( δ ); d ( r , / , q Q ( δ ) , k , } from Lemma 5.1. Define foreach n the setˆ P n = { H ∈ G rn | there exists a k -coloring H of H such that t ( ˆ Q q Q ( ε n ) , H ) ≥ / } , and let ˆ P = ∪ ∞ n = ˆ P n . We set q P ( ε ) = max { q tw ( r , / , q Q ( ε ) , , k ); 100 q Q ( ε ); d ( r , / , q Q ( δ ) , k , } .We are left to check if the two conditions for testability of P hold with ˆ P and q P describedas above. Assume for the rest of the proof that n ≥ q P ( ε n ) for each n for simplicity, thegeneral case follows along the same lines with some technical di ffi culties.First let G ∈ P , we have to show that for every q ≥ G ( q , G ) ∈ ˆ P q with probability at least 3 / G ∈ P implies that there exists a a k -coloring G of G such that G ∈ Q .From the testability of Q it follows that t ( ˆ Q l , G ( l , G )) ≥ / l ≥
1. Let q ≥ F denote G ( q , G ), furthermore let F = G ( q , G ) generated by the samerandom process as F , so F is a k -coloring of F . We know by a standard sampling argumentthat P ( | t ( ˆ Q q Q ( ε q ) , G ) − t ( ˆ Q q Q ( ε q ) , F ) | ≥ / ≤ − q q Q ( ε q ) , (6.1)31nd the right hand side of (6.1) is less than 2 / ε q , since by definition q ≥ q Q ( ε q ). It follows that t ( ˆ Q q Q ( ε q ) , F ) ≥ / /
5, so by thedefinition of ˆ P we have that F ∈ ˆ P q with probability at least 3 /
5, which is what we wantedto show.To verify the second condition we proceed by contradiction. Suppose that G is ε -farfrom P , but at the same time there exists an l ≥ q P ( ε ) such that F ∈ ˆ P l with probabilitylarger than 2 /
5, where F = G ( l , G ).In this case, the latter condition implies that with probability larger than 2 / k -coloring F of F such that t ( ˆ Q q Q ( ε l ) , F ) ≥ /
5. By Lemma 5.1 and the proof ofTheorem 2.3 there exists a k -coloring G of G such that d tw ( µ ( q Q ( ε l ) , F ) , µ ( q Q ( ε l ) , G )) ≤ / /
5, in particular | t ( ˆ Q q Q ( ε l ) , F ) − t ( ˆ Q q Q ( ε l ) , G ) | ≤ , which implies that with probability at least 1 / G such that t ( ˆ Q q Q ( ε l ) , G ) > .We can drop the probabilistic assertion and can say that there exists a k -coloring G of G such that t ( ˆ Q q Q ( ε l ) , G ) > , because G and the density expression are deterministic.On the other hand, the fact that G is ε -far from P implies that any k -coloring G of G is ε -far from Q , which means that t ( ˆ Q q , G ) ≤ / k -coloring G of G and q ≥ q Q ( ε ).But we know that q Q ( ε l ) ≥ q Q ( ε ), since ε l ≤ ε which delivers the contradiction. The lastinequality is the consequence of our definitions, ε l is the infimum of the δ > l ≥ q P ( δ ), and on the other hand, l ≥ q P ( ε ). (cid:3) It would be very interesting to shed light on the explicit sample complexity bounds for thewitness parameter in Theorem 2.3. The only ingredient of our proof which is non-e ff ectiveis the part which deals with the ultralimit method in the proof of Theorem 3.15, to ourknowledge an e ff ective proof regarding this result is only known for r = Acknowledgement
We thank Gábor Elek for a number of interesting discussions.
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