Weak hypergraph regularity and linear hypergraphs

Kohayakawa, Yoshiharu; Nagle, Brendan; Rödl, Vojtěch; Schacht, Mathias

doi:10.1016/j.jctb.2009.05.005

Cited by 73 publications

(112 citation statements)

References 19 publications

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“…[5,11,32]), which is a straightforward generalization of Szemerédi's regularity lemma for graphs. This version has recently been proved [15] to be compatible (i.e., admits a counting lemma) with linear hypergraphs, i.e., those hypergraphs whose hyperedges intersect pairwise in at most one vertex. In this setting the notion of the cluster hypergraph is the same as for graphs.…”

Section: Resultsmentioning

confidence: 99%

A structural result for hypergraphs with many restricted edge colorings

Lefmann

Person

Schacht

2010

Journal of Combinatorics

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Abstract. For k-uniform hypergraphs F and H and an integer r ≥ 2, let c r,F (H) denote the number of r-colorings of the set of hyperedges of H with no monochromatic copy of F and let c r,F (n) = max H∈Hn c r,F (H), where the maximum runs over the family Hn of all k-uniform hypergraphs on n vertices. Moreover, let ex(n, F ) be the usual Turán function, i.e., the maximum number of hyperedges of an n-vertex k-uniform hypergraph which contains no copy of F .In this paper, we consider the question for determining c r,F (n) for arbitrary fixed hypergraphs F and showfor r = 2, 3. Moreover, we obtain a structural result for r = 2, 3 and any H with c r,F (H) ≥ r ex(|V (H)|,F ) under the assumption that a stability result for the k-uniform hypergraph F exists and |V (H)| is sufficiently large. We also obtain exact results for c r,F (n) when F is a 3-or 4-uniform generalized triangle and r = 2, 3, while c r,F (n) r ex(n,F ) for r ≥ 4 and n sufficiently large.

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Section: Resultsmentioning

confidence: 99%

A structural result for hypergraphs with many restricted edge colorings

Lefmann

Person

Schacht

2010

Journal of Combinatorics

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“…Hypergraphs with property DISC d were studied in [2,3,20] and a straightforward generalisation of Szemerédi's regularity lemma for this concept was observed to hold in [4,12,32] (see Theorem 23 below).…”

Section: Disc D (δ)mentioning

confidence: 99%

“…Moreover, the number of labeled, induced copies of K (3) 1,1,2 in H n is ∼ n 4 /64, while the "right" number would be 49n 4 /64 2 . However, it was shown in [20] that k-graphs having DISC d (δ) for sufficiently small δ must contain approximately the same number of copies of any fixed linear k-graph F as a genuine random k-graph of the same density. Here a linear k-graph F is defined as having no pair of edges which intersect in two or more vertices.…”

Section: Generalisation Of Theoremmentioning

confidence: 99%

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Weak quasi‐randomness for uniform hypergraphs

Conlon

Hàn

Person

et al. 2011

Random Struct Algorithms

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ABSTRACT:We study quasi-random properties of k-uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung-Graham-Wilson theorem for quasi-random graphs.Moreover, let K k be the complete graph on k vertices and M(k) the line graph of the graph of the k-dimensional hypercube. We will show that the pair of graphs (K k , M(k)) has the property that if the number of copies of both K k and M(k) in another graph G are as expected in the random graph of density d, then G is quasi-random (in the sense of the Chung-Graham-Wilson theorem) with density close to d.

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“…In addition to the regularity lemma we need the following embedding lemma recently proved by Kohayakawa-Nagle-Rödl-Schacht [17]. Recall that a hypergraph is linear if every two edges have at most one point in common.…”

Section: Outline Of the Proofmentioning

confidence: 99%

Almost all triangle-free triple systems are tripartite

Balogh

Mubayi

2012

Combinatorica

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A triangle in a triple system is a collection of three edges isomorphic to {123, 124, 345}. A triple system is triangle-free if it contains no three edges forming a triangle. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is triangle-free. We prove that almost all triangle-free triple systems with vertex set [n] are tripartite.Our proof uses the hypergraph regularity lemma of Frankl and Rödl [13], and a stability theorem for triangle-free triple systems due to Keevash and the second author [15].

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Weak hypergraph regularity and linear hypergraphs

Cited by 73 publications

References 19 publications

A structural result for hypergraphs with many restricted edge colorings

A structural result for hypergraphs with many restricted edge colorings

Weak quasi‐randomness for uniform hypergraphs

Almost all triangle-free triple systems are tripartite

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