Regularity Lemma for k‐uniform hypergraphs

Rödl, Vojtěch; Skokan, Jozef

doi:10.1002/rsa.20017

Cited by 224 publications

(267 citation statements)

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“…First we will need a generalization of the Regularity Lemma [29] for hypergraphs. There are several generalizations of the Regularity Lemma for hypergraphs due to various authors (see [3], [6], [9], [28] and [30]). Here we will use the simplest one due to Chung [3].…”

Section: From Connected Matchings To Berge Cyclesmentioning

confidence: 99%

Monochromatic cycle partitions of edge-colored graphs

Sárközy

2010

J. Graph Theory

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The asymptotics of 2-colour Ramsey numbers of loose and tight cycles in 3-uniform hypergraphs were recently determined [16,17]. We address the same problem for Berge cycles and for 3 colours. Our main result is that the 3-colour Ramsey number of a 3-uniform Berge cycle of length n is asymptotic to 5n 4 . The result is proved with the Regularity Lemma via the existence of a monochromatic connected matching covering asymptotically 4n/5 vertices in the multicoloured 2-shadow graph induced by the colouring of K (3) n .

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Section: From Connected Matchings To Berge Cyclesmentioning

confidence: 99%

Monochromatic cycle partitions of edge-colored graphs

Sárközy

2010

J. Graph Theory

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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%

“…The Regular Approximation Lemma states that for any r-partite k-graph H there is an r-partite k-graph G on V (H) (with the same vertex partition as H) and a partition (k − 1)-complex P on V (H) such that G is ξ-close to H and the partition k-complex G[P] is ε-regular. This will suffice for our purposes as we shall avoid using any edge of G \ H whilst working with G, so any edge we do use will be an edge of H. There are other regularity lemmas for k-graphs which give information on H itself (see [9,34]) but the regular complexes yielded by these are not sufficiently dense to apply the blow-up lemma (see [13,Section 3] for further discussion of this point). The next theorem is the Regular Approximation Lemma; this is a slight restatement of a result of Rödl and Schacht (Theorem 14 of [32]).…”

Section: Hypergraph Regularitymentioning

confidence: 99%

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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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“…There are various generalisations to hypergraphs (e.g. [6,7,14] which have been instrumental in solving important problems.…”

Section: Introductionmentioning

confidence: 99%