Hypergraph Turán numbers of linear cycles

Füredi, Zoltán; Jiang, Tao

doi:10.1016/j.jcta.2013.12.009

Cited by 58 publications

(65 citation statements)

References 20 publications

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“…For integers

r \geq 2

and

ℓ \geq 3

, an

r

‐uniform linear cycle of length

ℓ

, denoted by

C_{ℓ}^{r}

, is an

r

‐graph with edges

e_{1}, …, e_{ℓ}

such that for every

i \in [ℓ - 1], ∣ e_{i} \cap e_{i + 1} ∣ = 1, ∣ e_{ℓ} \cap e_{1} ∣ = 1

, and

e_{i} \cap e_{j} = \emptyset

for all other pairs

{i, j}, i \neq j

. Kostochka, Mubayi, and Verstraëte , and independently, Füredi and Jiang proved that for every

r, ℓ \geq 3, {ex}_{r} (n, C_{ℓ}^{r}) = normalΘ (n^{r - 1})

. Then by , we trivially have

∣ {Forb}_{r} (n, C_{ℓ}^{r}) ∣ = 2^{normalΩ (n^{r - 1})} and ∣ {Forb}_{r} (n, C_{ℓ}^{r}) ∣ = 2^{O (n^{r - 1} \log n)}

…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the number of linear hypergraphs of large girth

Balogh

2019

Journal of Graph Theory

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An r‐uniform linear cycle of length ℓ, denoted by C ℓ r, is an r‐graph with edges e 1 , … , e ℓ such that for every i ∈ [ ℓ − 1 ] ,0.33em ∣ e i ∩ e i + 1 ∣ = 1 ,0.33em ∣ e ℓ ∩ e 1 ∣ = 1, and e i ∩ e j = ∅ for all other pairs { i , j } ,0.33em i ≠ j. For every r ≥ 3 and ℓ ≥ 4, we show that there exists a constant C depending on r and ℓ such that the number of linear r‐graphs of girth ℓ is at most 2 C n 1 + 1 ∕ ⌊ ℓ ∕ 2 ⌋. Furthermore, we extend the result for ℓ = 4, proving that there exists a constant C depending on r such that the number of linear r‐graphs without C 4 r is at most 2 C n 3 ∕ 2. The idea of the proof is to reduce the hypergraph enumeration problems to some graph enumeration problems, and then apply a variant of the graph container method, which may be of independent interest. We extend a breakthrough result of Kleitman and Winston on the number of C 4‐free graphs, proving that the number of graphs containing at most n 2 ∕ 32 log 6 n0.33em C 4's is at most 2 11 n 3 ∕ 2, for sufficiently large n. We further show that for every r ≥ 3 and ℓ ≥ 2, the number of graphs such that each of its edges is contained in only O ( 1 ) cycles of length at most 2 ℓ, is bounded by 2 3 ( ℓ + 1 ) n 1 + 1 ∕ ℓ asymptotically.

show abstract

“…For integers

r \geq 2

and

ℓ \geq 3

, an

r

‐uniform linear cycle of length

ℓ

, denoted by

C_{ℓ}^{r}

, is an

r

‐graph with edges

e_{1}, …, e_{ℓ}

such that for every

i \in [ℓ - 1], ∣ e_{i} \cap e_{i + 1} ∣ = 1, ∣ e_{ℓ} \cap e_{1} ∣ = 1

, and

e_{i} \cap e_{j} = \emptyset

for all other pairs

{i, j}, i \neq j

. Kostochka, Mubayi, and Verstraëte , and independently, Füredi and Jiang proved that for every

r, ℓ \geq 3, {ex}_{r} (n, C_{ℓ}^{r}) = normalΘ (n^{r - 1})

. Then by , we trivially have

∣ {Forb}_{r} (n, C_{ℓ}^{r}) ∣ = 2^{normalΩ (n^{r - 1})} and ∣ {Forb}_{r} (n, C_{ℓ}^{r}) ∣ = 2^{O (n^{r - 1} \log n)}

…”

Section: Introductionmentioning

confidence: 99%

“…Recently, the case when H is a linear cycle received more attention. For integers r 2 ≥ and 3 ℓ ≥ , an r-uniform linear cycle of length ℓ, denoted by C r ℓ , is an r-graph with edges e e , …, [17], and independently, Füredi and Jiang [10] proved that for every r nC n , 3, ex ( , ) = Θ( )…”

mentioning

confidence: 99%

On the number of linear hypergraphs of large girth

Balogh

2019

Journal of Graph Theory

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“…If k = 2, p ≥ q = 2, the conjecture is true due to a theorem of Erdős and Gallai in [6]; if k ≥ 2, q = 2, the theorem of Frankl in [9] also confirms the conjecture; if k = 3 the theorems in [11,12] show the (p, 3)-extremal problem; and finally, if p = q and k ≥ 2 Frankl's lemma given in [8] solves the (q, q)-problem.…”

Section: Remark the Bound 2p 2 Can Be Improved Tomentioning

confidence: 97%

The (p,q)-extremal problem and the fractional chromatic number of Kneser hypergraphs

Araujo‐Pardo

Díaz-Patiño

Montejano

et al. 2016

Discrete Mathematics

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a b s t r a c tThe problem of computing the chromatic number of Kneser hypergraphs has been extensively studied over the last 40 years and the fractional version of the chromatic number of Kneser hypergraphs is only solved for particular cases. The (p, q)-extremal problem is to maximize the number of edges in a k-uniform hypergraph H with given order subject to the constraint that any p edges contain q edges that have a vertex in common. In this paper we found a link between the fractional chromatic number of Kneser hypergraphs and the (p, q)-extremal problem. We solve the (p, q)-extremal problem for graphs and with the aid of this result we calculate the fractional chromatic number of Kneser hypergraphs when they are composed with sets of cardinality 2.

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“…Füredi and Jiang [19], and Kostochka, Mubayi and Verstraëte [33] determined the Turán number of C k for all k ≥ 3, s ≥ 3 and sufficiently large n as follows.…”

Section: Preliminariesmentioning

confidence: 99%

“…It would be interesting to investigate the relation between the anti-Ramsey number and the Turán number for paths and cycles in hypergraphs. The Turán numbers of paths and cycles are extensively studied, see [12,19,20,33] or Section 2 below for details. Motivated by this, we will study the anti-Ramsey numbers of paths and cycles and compare it with the Turán numbers of paths and cycles in hypergraphs.…”

Section: Introductionmentioning

confidence: 99%

Anti-Ramsey Numbers of Paths and Cycles in Hypergraphs

Gu¹,

Li²,

Shi³

2020

SIAM J. Discrete Math.

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The anti-Ramsey problem was introduced by Erdős, Simonovits and Sós in 1970s. The anti-Ramsey number of a hypergraph H, ar(n, s, H), is the smallest integer c such that in any coloring of the edges of the s-uniform complete hypergraph on n vertices with exactly c colors, there is a copy of H whose edges have distinct colors. In this paper, we determine the anti-Ramsey numbers of linear paths and loose paths in hypergraphs for sufficiently large n, and give bounds for the anti-Ramsey numbers of Berge paths. Similar exact anti-Ramsey numbers are obtained for linear/loose cycles, and bounds are obtained for Berge cycles. Our main tools are path extension technique and stability results on hypergraph Turán problems of paths and cycles.

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Hypergraph Turán numbers of linear cycles

Cited by 58 publications

References 20 publications

On the number of linear hypergraphs of large girth

On the number of linear hypergraphs of large girth

The (p,q)-extremal problem and the fractional chromatic number of Kneser hypergraphs

Anti-Ramsey Numbers of Paths and Cycles in Hypergraphs

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