Triangle-free Subgraphs of Hypergraphs

Nie, Jiaxi; Spiro, Sam; Verstraëte, Jacques

doi:10.48550/arxiv.2004.10992

Cited by 2 publications

(2 citation statements)

References 22 publications

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“…The authors and Nie [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: Counting R-graphs Of Large Girthmentioning

confidence: 59%

Counting Hypergraphs with Large Girth

Spiro

Verstraëte

2020

Preprint

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Morris and Saxton [17] used the method of containers to bound the number of n-vertex graphs with m edges containing no -cycles, and hence graphs of girth more than . We consider a generalization to r-uniform hypergraphs. The girth of a hypergraph H is the minimum such that for some F ⊆ H, there exists a bijection φ : E(C ) → E(F ) with e ⊆ φ(e) for all e ∈ E(C ). Letting N r m (n, ) denote the number of n-vertex r-uniform hypergraphs with m edges and girth larger than and defining λ = (r − 2)/( − 2) , we showwhich is tight when − 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 in random r-uniform hypergraphs.

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Section: Counting R-graphs Of Large Girthmentioning

confidence: 59%

Counting Hypergraphs with Large Girth

Spiro

Verstraëte

2020

Preprint

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“…The study of relative Turán numbers was advanced by Foucaud, Krivelevich, and Perarnau [9] and independently by Briggs and Cox [2]. Many results have been obtained for relative Turán numbers, both when H is a general host, as well as for random hosts [14,21,22,25,26].…”

Section: Introductionmentioning

confidence: 99%

Relative Turán Numbers for Hypergraph Cycles

Spiro,

Verstraete

2020

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For an r-uniform hypergraph H and a family of r-uniform hypergraphs F, the relative Turán number ex(H, F) is the maximum number of edges in an F-free subgraph of H. In this paper we give lower bounds on ex(H, F) for certain families of hypergraph cycles F such as Berge cycles and loose cycles. In particular, if C 3 denotes the set of all 3-uniform Berge -cycles and H is a 3-uniform hypergraph with maximum degree ∆, we prove ex(H, 1) e(H), and these bounds are tight up to the o(1) term.

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Triangle-free Subgraphs of Hypergraphs

Cited by 2 publications

References 22 publications

Counting Hypergraphs with Large Girth

Counting Hypergraphs with Large Girth

Relative Turán Numbers for Hypergraph Cycles

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