Definable Regularity Lemmas for Nip Hypergraphs

Chernikov, Artem; Starchenko, Sergei

doi:10.1093/qmath/haab011

Cited by 11 publications

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“…In the setting of graphs, a hereditary graph property is stable if and only if it has regular decompositions with no irregular pairs, see for example [76,Theorem 1.4] for a precise statement. Although it is known that in the setting of hypergraphs, stability implies an analogous result (see [1,14]), we show in [76] that it does not characterize when certain regular decomposition with no irregular triples exist. In particular, we present hereditary properties of 3-uniform hypergraphs which are unstable but still close to stable, and thus admit "stable-like" regular decompositions (see [76]).…”

Section: Introductionmentioning

confidence: 72%

Higher-order generalizations of stability and arithmetic regularity

Terry¹,

Wolf²

2021

Preprint

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We define a natural notion of higher-order stability and show that subsets of F n p that are tame in this sense can be approximately described by a union of low-complexity quadratic subvarieties up to linear error. This generalizes the arithmetic regularity lemma for stable subsets of F n p proved by the authors in [73], and subsequent refinements and generalizations [19,74], to the realm of higher-order Fourier analysis. HIGHER-ORDER GENERALIZATIONS OF STABILITY AND ARITHMETIC REGULARITY 5.2. Translation to trees 5.3. Proof of main NFOP 2 theorems 5.4. A preliminary generalized stable regularity lemma 5.5. Iterating to obtain a stronger version Appendix A. Hereditary properties A.1. Equivalent definitions A.2. Closeness A.3. An unstable property which is close to stable Appendix B. Boolean combinations of quadratic atoms are NHOP 2 Appendix C. Properties of the linear Green-Sanders example C.1. No 4-IP in GS(p, n) C.2. No 4-HOP 2 in GS(3, n) Appendix D. Density lemmas Appendix E. Counting lemmas and corollaries ReferencesThis theorem is analogous to the structural results for sets of bounded VC-dimension found in [3,22,65,73]. As an immediate corollary, we have the following.Corollary 1.19 (NIP 2 implies AQA). Let p ≥ 3 be a prime and let P be an elementary p-group property which is NIP 2 . Then P is almost quadratically atomic.We prove an analogous result in the hypergraph setting in [76] (see also [15]). We also obtain the following structural result for sets of bounded VC 2 -dimension.Corollary 1.20 (Structure of sets with bounded VC 2 -dimension). For all ǫ > 0, k ≥ 1, and rank functions ρ : N → R + , there are integers D, N such that the following holds. For all n ≥ N and A ⊆ F n p with VC 2 (A) < k, there is a quadratic factor B of complexity (ℓ, q) and rank at least ρ(ℓ+q), and a set I ⊆ At(B) such that ℓ, q ≤ D, andCorollary 1.20 is a higher-order analogue of earlier results [3,22,65,73] that state that a set of bounded VC-dimension is approximately a union of cosets of a subgroup of bounded index (in other words, a union of atoms from a purely linear factor).In fact, we believe that something stronger is true. In analogy with Definition 1.3, we say that P is almost strongly quadratically atomic if for all ǫ > 0, all functions g : N → N, and all rank functions ρ : N → R + , there are integers D, N such that for all n ≥ N, and all (G, A) ∈ P with |G| ≥ p n , the following holds. There exists (ℓ, q) satisfying ℓ + q ≤ D, and a quadratic factor B in G of complexity (ℓ, q) and rank at least ρ(ℓ + q), which is ǫ-almost ǫp −g(ℓ+q) -atomic with respect to A.Conjecture 1.21. If an epGP is NIP 2 , then it is almost strongly quadratically atomic.As was the case for linear decompositions (see Proposition 1.10), it is not difficult to show that whether or not a group property satisfies any part of Definition 1.15 depends only on its ∼-class, see Appendix A.2.Proposition 1.22. If P and P ′ are close then P is (strongly) quadratically atomic/almost (strongly) quadratically atomic if and only if P ′ is.In light of this...

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Section: Introductionmentioning

confidence: 72%

Higher-order generalizations of stability and arithmetic regularity

Terry¹,

Wolf²

2021

Preprint

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“…Consequently, it is natural to ask if restrictions on the graph being partitioned might result in a stronger form of regularity. Such restricted versions of the regularity lemma were established for example in [1,6,12,13,26,54,66,75,76]. These results establish for example a polynomial number of parts, stronger forms of regularity, the absence of exceptional pairs, etc., in restricted graph classes.…”

Section: Introductionmentioning

confidence: 92%

“…The real field (R, +, •, <, 0, 1) is an example of so-called distal structures, and the above results have then been extended to classes of graphs included in the age of a graph definable in a distal structure [12], which we call distal-defined classes.…”

Section: 3mentioning

confidence: 99%

“…Apart from real closed fields, an example of distal theories is the theory of dense linear orders without endpoints, p-adic fields with valuation, and Presburger arithmetic. We now state the graph version of [12,Theorem 5.8] in our setting.…”

Section: 3mentioning

confidence: 99%

“…Also, some of the stronger forms of the regularity lemma are based on model theoretic notions, e.g. for graphs defined in distal theories [12,75] or graphs with a stable edge relation [54].…”

Section: Introductionmentioning

confidence: 99%

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Regular partitions of gentle graphs

Jiang¹,

Nešetřil²,

Mendez³

et al. 2020

Preprint

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Szemerédi's Regularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich combinatorial context. In particular, we stress the link to the theory of (structural) sparsity, which leads to alternative proofs, refinements and solutions of open problems. It is interesting to note that many of these classes present challenging problems. Nevertheless, from the point of view of regularity lemma type statements, they appear as "gentle" classes.

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Definable convolution and idempotent Keisler measures

Chernikov

Gannon

2022

Isr. J. Math.

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We study idempotent measures and the structure of the convolution semigroups of measures over definable groups.We isolate the property of generic transitivity and demonstrate that it is sufficient (and necessary) to develop stable group theory localizing on a generically stable type, including invariant stratified ranks and connected components. We establish generic transitivity of generically stable idempotent types in important new cases, including abelian groups in arbitrary theories and arbitrary groups in rosy theories, and characterize them as generics of connected type-definable subgroups.Using tools from Keisler's randomization theory, we generalize some of these results from types to generically stable Keisler measures, and classify idempotent generically stable measures in abelian groups as (unique) translationinvariant measures on type-definable fsg subgroups. This provides a partial definable counterpart to the classical work of Rudin, Cohen and Pym for locally compact topological groups.Finally, we provide an explicit construction of a minimal left ideal in the convolution semigroup of measures for an arbitrary countable NIP group, from a minimal left ideal in the corresponding semigroup on types and a canonical measure constructed on its ideal subgroup. In order to achieve it, we in particular prove the revised Ellis group conjecture of Newelski for countable NIP groups.

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Definable Regularity Lemmas for Nip Hypergraphs

Cited by 11 publications

References 47 publications

Higher-order generalizations of stability and arithmetic regularity

Higher-order generalizations of stability and arithmetic regularity

Regular partitions of gentle graphs

Definable convolution and idempotent Keisler measures

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