Regular Partitions of Hypergraphs: Regularity Lemmas

Rödl, Vojtěch; Schacht, Mathias

doi:10.1017/s0963548307008553

Cited by 97 publications

(195 citation statements)

References 33 publications

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“…Szemerédi's regularity lemma (see, e.g., the survey by Rödl and Schacht [41]) has been extremely influential in (dense) graph property testing and there has been a great deal of work on recent extensions (see, e.g., [23,40,49]) of this lemma to hypergraphs. However, as Alon et al [2] noted, proofs of testability that avoid the regularity lemma often result in better query complexity.…”

Section: History Of Testingmentioning

confidence: 99%

Testable and untestable classes of first-order formulae

Jordan

Zeugmann

2012

Journal of Computer and System Sciences

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In property testing, the goal is to distinguish structures that have some desired property from those that are far from having the property, based on only a small, random sample of the structure. We focus on the classification of first-order sentences according to their testability. This classification was initiated by Alon et al. [2], who showed that graph properties expressible with prefix ∃ * ∀ * are testable but that there is an untestable graph property expressible with quantifier prefix ∀ * ∃ * . The main results of the present paper are as follows. We prove that all (relational) properties expressible with quantifier prefix ∃ * ∀∃ * (Ackermann's class with equality) are testable and also extend the positive result of Alon et al.[2] to relational structures using a recent result by Austin and Tao [8]. Finally, we simplify the untestable property of Alon et al. [2] and show that prefixes ∀ 3 ∃, ∀ 2 ∃∀, ∀∃∀ 2 and ∀∃∀∃ can express untestable graph properties when equality is allowed.

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Section: History Of Testingmentioning

confidence: 99%

Testable and untestable classes of first-order formulae

Jordan

Zeugmann

2012

Journal of Computer and System Sciences

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“…Sárközy gave a good bound on c(r, k) from the strong hypergraph Regularity Lemma of Rödl and Schacht [43], combined with the hypergraph Blow-up lemma of Keevash [32].…”

Section: Hypergraphsmentioning

confidence: 99%

Vertex covers by monochromatic pieces — A survey of results and problems

Gyárfás¹

2016

Discrete Mathematics

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“…Much of the notation introduced in this section was first introduced by Rödl and Skokan [34] and by Rödl and Schacht [32,33].…”

Section: Regularity and The Blow-up Lemmamentioning

confidence: 99%

“…The concept of regularity with which we shall work was first introduced in the k-uniform case by Rödl and Skokan [34], but we shall consider it in the form used by Rödl and Schacht [32,33].…”

Section: Hypergraph Regularitymentioning

confidence: 99%

Packing k-partite k-uniform hypergraphs

Mycroft

2016

Journal of Combinatorial Theory, Series A

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Abstract. Let G and H be k-graphs (k-uniform hypergraphs); then a perfect H-packing in G is a collection of vertex-disjoint copies of H in G which together cover every vertex of G. For any fixed H let δ(H, n) be the minimum δ such that any k-graph G on n vertices with minimum codegree δ(G) ≥ δ contains a perfect H-packing. The problem of determining δ(H, n) has been widely studied for graphs (i.e. 2-graphs), but little is known for k ≥ 3. Here we determine the asymptotic value of δ(H, n) for all complete k-partite k-graphs H, as well as a wide class of other k-partite k-graphs. In particular, these results provide an asymptotic solution to a question of Rödl and Ruciński on the value of δ(H, n) when H is a loose cycle. We also determine asymptotically the codegree threshold needed to guarantee an H-packing covering all but a constant number of vertices of G for any complete k-partite k-graph H.

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Regular Partitions of Hypergraphs: Regularity Lemmas

Cited by 97 publications

References 33 publications

Testable and untestable classes of first-order formulae

Testable and untestable classes of first-order formulae

Vertex covers by monochromatic pieces — A survey of results and problems

Packing k-partite k-uniform hypergraphs

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