Triangle-Free Subgraphs of Hypergraphs

Nie, Jiaxi; Spiro, Sam; Verstraëte, Jacques

doi:10.1007/s00373-021-02388-5

Cited by 3 publications

(3 citation statements)

References 21 publications

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“…The authors and Nie et al [19] obtained bounds for r-uniform loose triangles, 2 where for r = 3 the same essentially tight bounds as in Theorem 1.4 were obtained, but for r > 3 2 The loose triangle is the Berge triangle whose edges pairwise intersect in exactly one vertex.…”

Section: Counting R-graphs Of Large Girthmentioning

confidence: 70%

“…Due to Theorems 1.2 and 1.4, the number of linear triangle‐free

r

‐graphs with

n

vertices and

m

edges where

n^{3 ∕ 2 + o (1)} \leq m \leq ex (n, C_{MathClass-open[3 MathClass-close]}^{r}) = o (n^{2})

and

r \geq 3

N_{m}^{r} (n, 3) = N_{m}^{2} {(n, 3)}^{2 r - 3 + o (1)} = {(\frac{n^{2}}{m})}^{(2 r - 3) m + o (m)} .

The authors and Nie et al [19] obtained bounds for

r

‐uniform loose triangles, where for

r = 3

the same essentially tight bounds as in Theorem 1.4 were obtained, but for

r > 3

there remains a significant gap. In the case of subgraphs of girth larger than four, Theorem 1.2 allows us to generalize results of Morris and Saxton [17] and earlier results of Kohayakawa, Kreuter, and Steger [15] giving subgraphs of large girth in random graphs in the following way:…”

Section: Introductionmentioning

confidence: 96%

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Counting hypergraphs with large girth

Spiro

Verstraëte

2022

Journal of Graph Theory

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Morris and Saxton used the method of containers to bound the number of n‐vertex graphs with m edges containing no ℓ‐cycles, and hence graphs of girth more than ℓ. We consider a generalization to r‐uniform hypergraphs. The girth of a hypergraph H is the minimum ℓ ≥ 2 such that there exist distinct vertices v 1 , … , v ℓ and hyperedges e 1 , … , e ℓ with v i , v i + 1 ∈ e i for all 1 ≤ i ≤ ℓ. Letting N m r ( n , ℓ ) denote the number of n‐vertex r‐uniform hypergraphs with m edges and girth larger than ℓ and defining λ = ⌈ ( r − 2 ) ∕ ( ℓ − 2 ) ⌉, we show N m r ( n , ℓ ) ≤ N m 2 ( n , ℓ ) r − 1 + λ , which is tight when ℓ − 2 divides r − 2 up to a 1 + o ( 1 ) term in the exponent. This result is used to address the extremal problem for subgraphs of girth more than ℓ in random r‐uniform hypergraphs.

show abstract

Section: Counting R-graphs Of Large Girthmentioning

confidence: 70%

“…Due to Theorems 1.2 and 1.4, the number of linear triangle‐free

r

‐graphs with

n

vertices and

m

edges where

n^{3 ∕ 2 + o (1)} \leq m \leq ex (n, C_{MathClass-open[3 MathClass-close]}^{r}) = o (n^{2})

and

r \geq 3

N_{m}^{r} (n, 3) = N_{m}^{2} {(n, 3)}^{2 r - 3 + o (1)} = {(\frac{n^{2}}{m})}^{(2 r - 3) m + o (m)} .

The authors and Nie et al [19] obtained bounds for

r

‐uniform loose triangles, where for

r = 3

the same essentially tight bounds as in Theorem 1.4 were obtained, but for

r > 3

Section: Introductionmentioning

confidence: 96%

Counting hypergraphs with large girth

Spiro

Verstraëte

2022

Journal of Graph Theory

Self Cite

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show abstract

“…Very recently, Jiang and Longbrake [27] proved general upper bounds for

\text{ex}({G}_{n,p},F)

whenever

F

satisfies some mild conditions. As far as we are aware, these are the only known results concerning the random Turán problem for graphs, though there have been a number of recent results regarding the analogous problem for hypergraphs, see, for example [33, 34, 38, 39].…”

Section: Introductionmentioning

confidence: 99%