The maximal number of edges in a homogeneous hypergraph containing no prohibited subgraphs

Sidorenko, Alexander

doi:10.1007/bf01158259

Cited by 55 publications

(64 citation statements)

References 5 publications

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“…So, Some extra work is needed to prove (2) for all n. We refer the Reader to [21] for further details. The main difficulty in proving the stability of Σ 4 via Sidorenko's argument is that the Symmetrization phase (or even just one step of it) may in principle change the hypergraph essentially.…”

Section: An Outline Of the Proofmentioning

confidence: 98%

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An exact Turán result for the generalized triangle

Pikhurko

2008

Combinatorica

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Let Σ k consist of all k-graphs with three edges D1, D2, D3 such that |D1 ∩D2| = k −1 and D1 D2 ⊆ D3. The exact value of the Turán function ex(n, Σ k ) was computed for k = 3 by Bollobás [Discrete Math. 8 (1974), 21-24] and for k = 4 by Sidorenko [Math Notes 41 (1987), 247-259]. Let the k-graph T k ∈ Σ k have edges {1, . . . , k}, {1, 2, . . . , k − 1, k + 1}, and {k, k + 1, . . . , 2k − 1}. Frankl and Füredi [J. Combin. Theory Ser. (A) 52 (1989), 129-147] conjectured that there is n0 = n0(k) such that ex(n, T k ) = ex(n, Σ k ) for all n ≥ n0 and had previously proved this for k = 3 in [Combinatorica 3 (1983), 341-349]. Here we settle the case k = 4 of the conjecture.

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Section: An Outline Of the Proofmentioning

confidence: 98%

“…Sidorenko [21] settled the case k = 4 of Bollobás' conjecture. In fact, Sidorenko proved that forbidding Σ 4 alone suffices for the upper bound, that is, (2) ex(n, T 4 ) = ex(n, Σ 4 ) = n 4 × n + 1 4 × n + 2 4 × n + 3 4 , n ≥ 4, with the complete balanced 4-partite 4-graph being the unique extremal graph.…”

Section: Introductionmentioning

confidence: 97%

An exact Turán result for the generalized triangle

Pikhurko

2008

Combinatorica

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“…a an density of this family is at least r!=r r : Bollob! a as [1] proved equality for the case r ¼ 3; and Sidorenko [10] proved equality for the case r ¼ 4:…”

Section: Introductionmentioning

confidence: 98%

On the Turán Number of Triple Systems

Mubayi¹,

Rödl

2002

Journal of Combinatorial Theory, Series A

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“…The problem of computing the Turán function ex(n, B k,m ) has been actively studied [1,5,6,[8][9][10]14,18,21,23]. Clearly, the property not containing B k,k as a subgraph is equivalent to having empty neighborhoods, so f (n, k) = ex(n, B k,k ).…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%