Hereditary properties of hypergraphs

Dotson, R.; Nagle, Brendan

doi:10.1016/j.jctb.2008.09.004

Cited by 15 publications

(19 citation statements)

References 25 publications

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“…For graphs, that is, ℓ = 2, this theorem was proved for p = 1/2 by Prömel [42]). For p = 1/2 and ℓ = 3 it was proved by Kohayakawa, Nagle and Rödl [36] using hypergraph regularity techniques, Dotson and Nagle [15] extending this to general ℓ.…”

Section: 4mentioning

confidence: 84%

Hypergraph containers

2015

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We develop a notion of containment for independent sets in hypergraphs. For every $r$-uniform hypergraph $G$, we find a relatively small collection $C$ of vertex subsets, such that every independent set of $G$ is contained within a member of $C$, and no member of $C$ is large; the collection, which is in various respects optimal, reveals an underlying structure to the independent sets. The containers offer a straightforward and unified approach to many combinatorial questions concerned (usually implicitly) with independence. With regard to colouring, it follows that simple $r$-uniform hypergraphs of average degree $d$ have list chromatic number at least $(1/(r-1)^2 + o(1)) \log_r d$. For $r = 2$ this improves a bound due to Alon and is tight. For $r \ge 3$, previous bounds were weak but the present inequality is close to optimal. In the context of extremal graph theory, it follows that, for each $\ell$-uniform hypergraph $H$ of order $k$, there is a collection $C$ of $\ell$-uniform hypergraphs of order $n$ each with $o(n^k)$ copies of $H$, such that every $H$-free $\ell$-uniform hypergraph of order $n$ is a subgraph of a hypergraph in $C$, and $\log |C| \le c n^{\ell-1/m(H)} \log n$ where $m(H)$ is a standard parameter (there is a similar statement for induced subgraphs). This yields simple proofs, for example, for the number of $H$-free hypergraphs, and for the sparsity theorems of Conlon-Gowers and Schacht. A slight variant yields a counting version of the K{\L}R conjecture. Likewise, for systems of linear equations the containers supply, for example, bounds on the number of solution-free sets, and the existence of solutions in sparse random subsets. Balogh, Morris and Samotij have independently obtained related results

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Section: 4mentioning

confidence: 84%

Hypergraph containers

2015

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“…Finally, we note that there has also been some important recent progress on hereditary properties of hypergraphs, by Dotson and Nagle [20] and (independently) by Ishigami [24], who (building on work of Nagle and Rödl [33] and Kohayakawa, Nagle and Rödl [28]) proved a version of the AlekseevBollobás-Thomason Theorem for k-uniform hypergraphs. Precise structural results have very recently been proved by Balogh and Mubayi [13,14] and by Person and Schacht [35] when k = 3, see Section 9.…”

Section: Alekseev-bollobás-thomasonmentioning

confidence: 82%

“…In particular, we noted the following theorem of Dotson and Nagle [20] and Ishigami [24]. In other words, it is the maximum dimension of a subspace of P n , in the product space {0, 1} ( [n] k ) .…”

Section: Questionsmentioning

confidence: 99%

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The structure of almost all graphs in a hereditary property

Alon

Balogh

Bollobás

et al. 2011

Journal of Combinatorial Theory, Series B

125

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A hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of P is the function n → |P n |, where P n denotes the graphs of order n in P. It was shown by Alekseev, and by Bollobás and Thomason, that ifP is a hereditary property of graphs thenwhere r = r(P) ∈ N is the so-called 'colouring number' of P. However, their results tell us very little about the structure of a typical graph G ∈ P.In this paper we describe the structure of almost every graph in a hereditary property of graphs, P. As a consequence, we derive essentially optimal bounds on the speed of P, improving the Alekseev-Bollobás-Thomason Theorem, and also generalising results of Balogh, Bollobás and Simonovits.

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“…By our choice of the constants in (9) and (10), and by the construction of J , for a fixed φ, any five vertex 3-graph F ⊂ J φ is a cluster 3-graph for G, and hence by the embedding lemma …”

Section: We View J As a Multiset Of Triples On [T] For Each φ : [T] mentioning

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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Hereditary properties of hypergraphs

Cited by 15 publications

References 25 publications

Hypergraph containers

Hypergraph containers

The structure of almost all graphs in a hereditary property

Almost all triangle-free triple systems are tripartite

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