Sparse hypergraphs: New bounds and constructions

Ge, Gennian; Shangguan, Chong

doi:10.1016/j.jctb.2020.10.003

Cited by 7 publications

(4 citation statements)

References 24 publications

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“…A simple averaging argument shows that every (7,4)-configuration or (8,5)-configuration contains a (6,3)-configuration and, hence, f 3 (n, 7, 4), f 3 (n, 8, 5) ≥ n 2−o (1) . Similar considerations can be used to prove the same lower bound in the cases e = 7, 8 (see [10]). All other cases are open; in particular, it is not known whether f 3 (n, 9, 6) ≥ n 2−o (1) (though see [11] for results on a related problem).…”

Section: Introductionmentioning

confidence: 62%

See 1 more Smart Citation

A new bound for the Brown–Erdős–Sós problem

Conlon

Gishboliner

Levanzov

et al. 2023

Journal of Combinatorial Theory, Series B

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Section: Introductionmentioning

confidence: 62%

“…where the second equality uses Item 1 of Lemma 2.6 and that e(G ) = t • e(G), while the last equality uses our choice of t in (10). Next, note that {x 1 , .…”

Section: Proof Of Lemma 26mentioning

confidence: 99%

A new bound for the Brown–Erdős–Sós problem

Conlon

Gishboliner

Levanzov

et al. 2023

Journal of Combinatorial Theory, Series B

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“…Later in 2017, Ge and Shangguan [11] provided a construction for hypergraphs forbidding small rainbow cycles with orderoptimal edges w.r.t. Lemma V.1 (see Theorem 1.6 in [11]).…”

Section: A Tuŕan-type Problems For Sparse Hypergraphsmentioning

confidence: 99%

“…Later in 2017, Ge and Shangguan [11] provided a construction for hypergraphs forbidding small rainbow cycles with orderoptimal edges w.r.t. Lemma V.1 (see Theorem 1.6 in [11]). For general lower bound on f R (n, v, e), very recently, Shangguan and Tamo [26] proved the following result.…”

Section: A Tuŕan-type Problems For Sparse Hypergraphsmentioning

confidence: 99%

New Constructions of Optimal Locally Repairable Codes with Super-Linear Length

Kong

Wang

2020

Preprint

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As an important coding scheme in modern distributed storage systems, locally repairable codes (LRCs) have attracted a lot of attentions from perspectives of both practical applications and theoretical research. As a major topic in the research of LRCs, bounds and constructions of the corresponding optimal codes are of particular concerns.In this work, codes with (r, δ)-locality which have optimal minimal distance w.r.t. the bound given by Prakash et al.[22] are considered. Through parity check matrix approach, constructions of both optimal (r, δ)-LRCs with all symbol locality ((r, δ)a-LRCs) and optimal (r, δ)-LRCs with information locality ((r, δ)i-LRCs) are provided. As a generalization of a work of Xing and Yuan [33], these constructions are built on a connection between sparse hypergraphs and optimal (r, δ)-LRCs. With the help of constructions of large sparse hypergraphs, the length of codes constructed can be super-linear in the alphabet size. This improves upon previous constructions when the minimal distance of the code is at least 3δ + 1. As two applications, optimal H-LRCs with super-linear length and GSD codes with unbounded length are also constructed.

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Constructing dense grid-free linear $3$-graphs

Gishboliner

Shapira

2021

Proc. Amer. Math. Soc.

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Sparse hypergraphs: New bounds and constructions

Cited by 7 publications

References 24 publications

A new bound for the Brown–Erdős–Sós problem

A new bound for the Brown–Erdős–Sós problem

New Constructions of Optimal Locally Repairable Codes with Super-Linear Length

Constructing dense grid-free linear $3$-graphs

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