Stable regularity for relational structures
Abstract.
We generalize the stable graph regularity lemma of Malliaris and Shelah to the case of finite structures in finite relational languages, e.g., finite hypergraphs. We show that under the modeltheoretic assumption of stability, such a structure has an equitable regularity partition of size polynomial in the reciprocal of the desired accuracy, and such that for each ary relation and tuple of parts of the partition, the density is close to either or . In addition, we provide regularity results for finite and Borel structures that satisfy a weaker notion that we call almost stability.
1.0
1. Introduction
Szemerédi’s regularity lemma for graphs is a fundamental tool in combinatorics. It can be viewed as saying that every finite graph can be approximated by one that has a small “structural skeleton” overlaid with randomness. Malliaris and Shelah [MS14] show that one can obtain more control over this approximation in the presence of a modeltheoretic tameness condition known as stability, that is essentially combinatorial in nature. In this paper, we extend the result of Malliaris and Shelah to the case of arbitrary finite structures in a finite relational language. In particular, our result yields better bounds on hypergraph regularity approximations in the presence of stability. The Szemerédi regularity lemma can be expressed more formally as saying that for any finite graph there is a partition of the vertices, known as a regularity partition, such that the partition is equitable (i.e., the sizes of the parts differ by at most ), and for all but a few pairs of (not necessarily distinct) parts of the partition, the induced subgraph on the vertices among that pair is close to a random bipartite graph (or random graph, if the parts are not distinct) having some edge density between and . The pairs for which this does not hold are called irregular. The accuracy of the approximation yielded by a regularity partition is measured both in terms of having few irregular pairs, and by the closeness of each regular pair to a random (bipartite) graph. The regularity lemma provides an upper bound on the size of a regularity partition that depends only on the desired accuracy of the approximation, and not on the particular graph being approximated. For details, see, e.g., [RS10]. While this bound on the size of the regularity partition depends only on the desired accuracy, in general one cannot guarantee a bound better than a tower of exponentials (of height that is polynomial in the reciprocal of the accuracy) [Gow97]. Further, it has long been known that if a graph contains a large halfgraph as an induced subgraph, then any regularity partition for the graph must have irregular pairs (independently observed by Lovász, Seymour, and Trotter and by Alon, Duke, Leffman, Rödl, and Yuster [ADL94]). Malliaris and Shelah [MS14] observed that the presence of a large induced halfgraph corresponds to the absence of stability, a key property from model theory that provides a sense in which a combinatorial object is highly structured, or tame (for details, see [She90]). Malliaris and Shelah [MS14] show that when a graph is stable, it admits a regularity partition with no irregular pairs, with a number of parts that is merely polynomial in the reciprocal of the accuracy, and where for each pair of (not necessarily distinct) parts, the induced bipartite graph across the parts (or induced graph on the one part) is either complete or empty. In other words, this polynomialsize partition of the vertices is such that for every pair of parts of the partition (possibly with ), the induced subgraph on can be modified by a small number of edges so that either between every pair of distinct elements, one from and the other from , there is an edge, or between every pair of distinct elements, one from and the other from , there is no edge. In this case, the graph is close in edit distance to an equitable blowup of a small finite graph (possibly with selfloops). The regularity lemma for graphs has been generalized to finite structures in a finite relational language (see, e.g., [AC14]), a key case of which are the uniform hypergraphs (see, e.g., [Tao06], [Gow07], [RS07], and [ES12]). The upper bounds on the partition size are even worse than for graphs, as Moshkovitz and Shapira have recently shown that the bounds are necessarily of Ackermanntype. The modeltheoretic notion of stability also makes sense in the context of finite relational languages. In this paper, we extend Malliaris and Shelah’s results to show that every finite stable structure in a finite relational language admits an equitable partition with polynomially many parts such that for every relation (of arity , say) and every tuple of parts (possibly with repetition), the induced substructure restricted to on can be modified by a small number of “edges” so that either every tuple of elements in forms an edge, or every tuple of elements in does not form an edge. In particular, the relational structure is close in edit distance to an equitable blowup of a small structure in the same language. This shows that in the stable case, not only is “randomness” in the edges eliminated in the approximation, but so are the “intermediate levels” that are a key complication of the general case of hypergraph regularity lemmas. Our proof closely follows the methods of [MS14]. In the case of finite relational structures that are almost stable (in a sense that we make precise), we again show that the structure is close in edit distance to an equitable blowup of a small finite structure, albeit where the few edits may not be distributed as uniformly as we can require in the stable case. Finally, we provide a similar regularity lemma for almost stable relational structures that are Borel.
1.1. Related work
Expanding on Malliaris and Shelah’s stable regularity lemma for graphs, Malliaris and Pillay [MP16] give a short proof of the stable regularity lemma for arbitrary Keisler measures. In this more general setting, they obtain most of the nice properties from the stable regularity lemma on graphs [MS14], but they do not get precise bounds on the size of the partition. Independently from our work in the present paper, Chernikov and Starchenko [CS16] prove a stable regularity lemma for Keisler measures over finite and Borel structures in a language with a single relation. In the case of finite structures, their stable regularity lemma is closely related to our main result, Theorem 4.8, restricted to languages with a single relation. However, while the partitions they obtain are definable (unlike ours), they need not be equitable. Chernikov and Starchenko also obtain two regularity lemmas for structures satisfying certain modeltheoretic conditions other than stability, one for NIP structures that generalizes a result of Lovász and Szegedy [LS10], and one for distal structures, generalizing their earlier result [CS15]. Generalizing Green’s grouptheoretic regularity lemma [Gre05], Terry and Wolf obtain a stable version for vector spaces over finite fields [TW17], and Conant, Pillay, and Terry obtain a further generalization to arbitrary finite groups [CPT17].
1.2. Road map of the proof of the main result
Before beginning our technical construction, we here provide a road map of the proof of the main result, Theorem 4.8. We will first describe how to “augment” relations and give a quick proof outline in terms of such augmented relations. Then we will provide more detail on three key aspects: obtaining excellent sets, making a partition equitable, and modifying the original structure so that it is a blowup. Let be a finite relational language, and let . Suppose that is a finite structure such that none of its relations has the socalled branching property. (In fact, a slightly weaker hypothesis will suffice.) In particular, is stable. We begin by augmenting every relation in . Each relation in can be thought of as a valued function of some arity. We replace each relation with a continuumsized family of functions (indexed by ) each of which takes values in , and further allow each argument to be either an element or a subset of . In the case where exactly one argument is a subset of , this will be done by “polling” the elements in a subset and assigning a truth value ( or ) if and only if a sufficiently large majority (namely, a ()fraction) of the elements agree on that truth value (when all other arguments are fixed), and otherwise. However, when more than one argument is a subset, the polling is more complicated. For a given order of arguments, we will define this notion of polling by induction on the number of arguments that are sets, in a way that depends on the order of arguments polled so far. These augmented relations will be used to construct collections of socalled excellent sets, that in particular are such that whenever all arguments of an augmented relation are excellent then the (function indexed by of the) augmented relation has a truth value (i.e., is assigned or ). The proof outline is as follows. Assume that is large enough (relative to ). We first find, using the augmented relations, an excellent partition of a large subset of . We then transform this into an equitable partition of into excellent sets (where depends only on ). Finally, we show that it is possible to change some fraction of the (original) relations so that an equitable partition now describes this modification of as exactly the “blowup” of a small finite structure, whose size (i.e., the number of parts of an equitable partition) is at most polynomial in , where the polynomial’s exponent depends only on and the maximum arity of .
1.2.1. excellent sets
Suppose . We now describe how to find an excellent subset of that is big in the sense that its size is among a particular collection of natural numbers determined by . We show that a witness to the nonexcellence of can be taken to consist of a relation , an order of its arguments, an index among the many arguments, an tuple of sets (satisfying a certain additional property with respect to the order) and two big disjoint subsets and , such that the truth value assigned by the augmentation of (with polling based on the given ordering) to along with in the th coordinate is different from the truth value that it assigns to along with in the th coordinate. Having found such a witness to the nonexcellence of , we then look for such witnesses to the nonexcellence of and of . We repeat this process on big disjoint subsets of and of , etc., and stop as soon as some branch can go no farther (because we have reached some big subset of that itself has no such witness), after which the resulting binary tree of subsets of is perfect. A mesa is an object of the following sort that arises from a perfect tree of such witnesses: a finite perfect binary tree, each node of which is labeled by a triple consisting of a relation symbol, an index for one of the arguments of the relation, an ordering for the arguments of the relation, and certain witnessing subsets. At least one node of a maximal mesa does not itself have witnesses; we call such a node a cap, and it turns out that the height of any maximal mesa can be bounded above in terms of . The intuitive idea is that a mesa is not too “tall”, by virtue of not being too “wide”; there can be many caps on it — by virtue of any of which it doesn’t get too “tall”. Mesas have three important properties. First, as already mentioned, each chosen subset of occurring in its tree is big (i.e., its size is in the special set of sizes). Second, also as already noted, if the mesa is maximal, then there must be at least one cap, whose corresponding subset must therefore be excellent. Third, from any mesa such that every node has the same labels for the relation, argument index, and argument order, we can extract a witness to the branching property of of the same height as the mesa. Next, by a Ramseytheoretic result, there is a function such that with the following property: whenever and is a perfect binary tree with height , each node of which is labeled by a triple consisting of a relation symbol, an index for one of the arguments of the relation, and an ordering for the arguments of the relation, there is a perfect subtree of of height such that every node of the subtree has the same label. In particular, this holds of a mesa. Hence from a bound on the branching property for we may obtain a bound on the height of any mesa arising from . Because we have bounds on how much the sets decrease in size as one proceeds down a mesa, the bound on the height of the mesa induces a bound on the size of the excellent sets. In aggregate, using the fact that no relation has the branching property, we can find a constant such that any set has an excellent set of size at least .
1.2.2. Equitable partitions
We now describe in more detail how we find an equitable partition of “most” of consisting of excellent sets. Using the method for extracting excellent subsets that have size at least a positive fraction, we repeat this procedure to get a partition of “most” of the structure where every element of the partition is excellent and the size of the partition is bounded in terms of . We then aim to modify this partition to an equitable one while only increasing the error slightly. The allowable sizes for a “big” set in fact were chosen so that their greatest common denominator is also in the set. Consider a random, equitable, refinement of the original partition where the size of each element is this greatest common divisor. Using the fact that all relations of are appropriately stable, the limiting properties of certain hypergeometric distributions imply that with high probability a random such partition is excellent provided that the structure underlying the partition is “large”. In particular, this implies that there is some such equitable refinement.
1.2.3. Modifying the original structure
We now describe how to change the truth values of each relation on just an fraction of the elements (where is the arity of the relation), so that the resulting structure is the blowup of a finite structure of size bounded by a polynomial in . This modification of the structure has two parts. First, we show that for any excellent partition of “most” of , the relations may be modified on a small portion of the elements so as to obtain a partition of the same set which is “indiscernible” (i.e., a blowup of a finite structure). Next we have to deal with the (small number of) elements of not in any part of the original partition. We show that if we add such elements to parts of the partition arbitrarily (while keeping the partition equitable), we may then modify relations on these elements (with respect to the other elements) so that in the modified structure the relations agree with the other elements within the part to which they were assigned. In aggregate these actions only require us to change the relations on a small fraction of the elements, yielding a structure that is exactly a blowup while being close to the original.
1.3. Notation
We now introduce some notation and conventions that we will use throughout the paper. All logarithms are in base 2, and we will simply write . In this paper will denote a fixed finite relational language. All formulas will be first order. Equality will be considered a logical symbol and not a member of . For any relation , we let denote the arity of . We will also need two quantities related to the arities of relations in . We let
and . We consider an element sequence of elements of to be a map of the form , and therefore is the empty sequence, and is the set of elements occurring in the sequence . We also identify a finite sequence with the tuple of its elements . For finite sequences and , we say that is an initial segment of , written
when and when for all . Given a tuple and an element , we write to denote the tuple . We now introduce two special kinds of partitions. An equitable partition is one whose pieces differ in size by at most 1, and an indivisible partition is one for which whether or not a relation holds of a tuple depends only on the parts of the partition these elements are in.
Definition 1.1.
Suppose is an structure with underlying set . We say that is a partition of if it is a partition of . We say that is equitable if for any ,
Definition 1.2.
We say that a partition of an structure is indivisible if for each relation , for all , and for any pair of sequences such that , where , we have
In other words, a partition is indivisible if we can quotient out by the equivalence relation that it induces and then assign a compatible structure relation.
Definition 1.3.
Suppose and are structures with underlying sets and respectively. A map is a full homomorphism from to if for each relation and all tuples of (distinct) elements of ,
Note that that full homomorphisms are not necessarily injective.
Definition 1.4.
We say that an structure is a blowup of a structure when there is a surjective full homomorphism . We call the witness to the blowup. If also the sets and differ in size by at most one, for all , then we say that is an equitable blowup of .
Our regularity lemmas can be seen as stating that certain types of structures are close in edit distance to a blowup of a small finite structure. The following easy lemma, whose proof we omit, makes precise the notion that an structure with an indivisible partition can be thought of as blowup of a smaller structure.
Lemma 1.5.
For an structure and a partition of the following are equivalent.

is indivisible.

There exists an structure such that is a blowup of with witness such that
Furthermore, is an equitable blowup of if and only if is equitable.
Intuitively, is a blowup of if it can be obtained by replacing each element of with an indiscernible set, while is an equitable blowup of if these indiscernible sets are all almost the same size.
1.4. Stability
We now recall some basic facts from stability theory. The notation in this section follows that of [MS14].
Definition 1.6.
Let . An formula has the order property in an structure when there exist sequences (with for all ) and (with for all ) such that for all ,
We say that has the nonorder property in when it does not have the order property in .
Note that the order property is defined for a formula along with a given partition of its free variables, not just for the formula alone. We will in fact work with a combinatorial property that holds in a structure essentially whenever the order property does.
Definition 1.7.
Let . An formula has the branching property in an structure when there exist sequences (with for all ) and (with for ) such that for all , for all , and for each , we have that
implies
We say that has the nonbranching property in when it does not have the branching property in .
We now note a connection between the nonorder property and the nonbranching property for a structure .
Lemma 1.8.
If has the nonorder property in then has the nonbranching property in , where . On the other hand, if has the nonbranching property in then has the nonorder property in , where .
Proof.
See [Hod93, Lemma 6.7.9]. ∎
We will be interested in the situation when, for each relation , the structure has on the nonproperty for some . This is equivalent to the following.
Definition 1.9.
Let be an structure. We say that has the nonbranching property (nonorder property) if for each relation and each , the formula with the partition of variables has the nonbranching property (nonorder property) in .
For the rest of this paper, fix .
2. Excellence
From now on we assume is a finite structure with underlying set . We will prove our regularity lemma by showing that, under appropriate stability assumptions, we can find a partition of any subset of that is almost a blowup. To do this, we will need a notion called excellence, which captures this idea of being almost a blowup. We begin by allowing the domain of relations to consist of both elements and sets. As such, we let where denotes the power set of . We now define how to augment a relation on to all of (according to a given tolerance).
Definition 2.1.
Let , let be an ary relation, and let be a sequence of distinct elements of . We now define, inductively on the length of , the collection of partial relations for . Such a partial relation is a function . Let and let . If , then define
Otherwise, when , we will define by induction on , as follows.
Case :
In this case, , and so .
In particular, are elements of .
Define

if , and

if .
Case :
Let be the initial subtuple of of length , and let be the last element of , so that .
Because is a tuple of distinct elements,
observe that
is a bijection. For each , define

If then define .

If then define

Otherwise .
In this definition, unless is undefined on all arguments, is the collection of indices which are sets. To understand this definition, it is worth walking through the cases where . First consider the case where . We then have , and all are elements of , and so we let agree with the relation on . Next consider the case where , with say , i.e., when there is a unique element of among the arguments . In this case we let be if, when we fix and let the th entry vary among the elements of , at least a ()fraction of the elements return a value of ; and similarly for . If this doesn’t happen, i.e., if there is no nearconsensus among the elements of , then we return signifying that its truth value is undefined. Finally consider the case when , with say . Suppose we have defined whenever . We would like to perform a similar sort of consensusgathering to determine the value of . Specifically, we would like to choose an element of from among the arguments, fix all the other arguments, of which now there exists only one set, and then take a consensus (which we can do as we have defined in this case). However, we find that we first have to make a choice as to which set, or , we want to vary. In particular the outcome of this process might be different if we first vary and then vary or if instead we first vary and then vary . The purpose of the parameter is to keep track of the order in which we are varying the parameters to calculate the truth value. As we will see, we will mainly be interested in sets which have a property called excellence, which implies that the same truth value is returned no matter which order we consider the arguments (i.e., where is independent of the order of the range of ). In order to define the notion of excellence, we first need to define a notion of goodness for each relation and each such that .
Definition 2.2.
Fix , let of arity , and let . We define the notion of goodness for any by induction on as follows.
Case :
is good if and only if .
Case :
is good if and only if is good for and for all

such that is good for every ,

, and

permutations of ,
we have
We say that is excellent if is good for all relation symbols .
Note that our definition of goodness is the same, when is a (symmetric) graph with edge relation , as the notion of goodness in [MS14], and the higher arity notions are motivated by what is needed to generalize their proof to arbitrary finite relational languages. Intuitively, we think of a set as being good provided that whenever we only fix sets for at most many coordinates in the definition of , i.e., , with the first set fixed being good (and the others being sufficiently good) then returns a truth value. Once again it will be instructive to walk through the cases when or . First, a set is good if is defined on any collection of arguments where is the only argument that is not an element of . The case of good is somewhat more complicated. Namely, is good if whenever is good then any partial relation which first varies and then varies will always return a truth value when the other arguments are in . In particular, this holds no matter which place and take in the relation. The notion of goodness generalizes this idea. A set is good if whenever we have a sequence , with , of decreasing goodness then any partial relation which first varies , then varies , , will always return a truth value (no matter what the arguments are from ). It is worth noting that if is good and then is also good. So in particular, if is excellent then is good for all . This is important because it means that if are all excellent then must have a truth value. We can further preserve goodness while weakening , leading to the following straightforward but crucial observation.
Lemma 2.3.
Let , and suppose and . If is good, then is good.
As we will see, a crucial property of excellence is that given a collection of excellent sets, all partial relations have the same truth value on that collection. In particular, the next proposition tells us that when we have a sequence of good sets, then whenever are the only set arguments of , it has a truth value, which is independent of the ordering of . In particular, creftype 2.4 will imply that for any partial relation applied to excellent sets, the augmented relation won’t depend on the ordering in which the truth value is calculated.
Proposition 2.4.
Let have arity , and let . Suppose that be good sets such that . For any two injective functions and any permutation of ,
Proof.
Without loss of generality we may assume that , as the proof of the general case is the same.
Our proof proceeds by induction.
Case :
This case is immediate, as .
Case :
As every permutation of is equal to a composition of transpositions, it suffices to prove the result when is a transposition of . Therefore, we may assume without loss of generality that and .
Let and for distinct , define the relation
Then our goal is to show that . Suppose . Then there are at most
many pairs such that . Similarly, if , then there are at most
many pairs such that . Hence if
then and cannot both hold simultaneously. A similar calculation shows that if
then and cannot both hold simultaneously. Now, , and so . Hence (as they have truth values because and are good). Therefore the result follows. ∎
From now on we will assume that . Suppose is such that exactly are good and exactly are in . Then by creftype 2.4, the value of is independent of . In this circumstance, we will refer to simply as . In particular, this holds when all arguments are excellent. In summary, we have the following corollary.
Corollary 2.5.
For any excellent elements , and any , .
The following technical lemma tells us that, for a relation and appropriately good sets, at most a small fraction of the tuples consistent with those sets disagree with the partial relation about the truth value of .
Lemma 2.6.
Let have arity and let be a permutation of . Suppose are sets such that is good for . Further suppose that . Define
Then the following hold.

If then

If then
Proof.
The proofs of (a) and (b) are essentially identical so we will only prove (a). Further we can assume without loss of generality that . To simplify notation we will will omit the superscript of the partial relation and refer to by .
For simplicity of notation let .
Note that as is good, is also good.
We will proceed in stages.
Stage 0:
Let ,
and let .
Let . Note that whenever and for , we have .
In particular, for every we have
.
Stage :
Suppose, for , that the sets , have been defined so that

whenever for and for some then , and

whenever for and for all then .
We now show how to appropriately define sets for parameters of length . Fix so that (2) holds. Let
and let
If are such that (1) holds then let .
Let (2) holds. In particular the inductive hypothesis holds.
When we reach the th inductive stage we therefore have