Degenerate Turán densities of sparse hypergraphs

Shangguan, Chong; Tamo, Itzhak

doi:10.1016/j.jcta.2020.105228

Cited by 13 publications

(16 citation statements)

References 40 publications

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“…A lower bound, obtained from the probabilistic construction of Brown et al [3], was explicitly calculated in [14]:…”

Section: Introductionmentioning

confidence: 99%

“…In the case when

k = (e r - v) ∕ (e - 1)

is an integer, Brown et al were able to prove

(c_{r, v, e} + o (1)) n^{k} \leq f_{r} (n, v, e) \leq (C_{r, v, e} + o (1)) n^{k} as n \to \infty .

The case

e = 2

is equivalent to the packing problem and was asymptotically solved by Rödl who proved

f_{r} (n, v, 2) = true({(\binom{r}{2 r - v})}^{- 1} + o (1) true) true(\frac{0.0pt}{n} true) as n \to \infty .

Recently, the first case with

e = 3

r \geq 3

was solved by Glock :

f_{3} (n, 5, 3) = false(\frac{1}{5} + o (1) false) n^{2}

. His result was generalized by Shangguan and Tamo to

f_{r} (n, 3 r - 4, 3) = true(\frac{1}{r^{2} - r - 1} + o (1) true) n^{2} as n \to \infty .

Shangguan and Tamo proved the upper bound

…”

Section: Introductionmentioning

confidence: 99%

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Approximate Steiner (r − 1, r, n)‐systems without three blocks on r + 2 points

Sidorenko

2019

J of Combinatorial Designs

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For a family MJX-tex-caligraphicscriptF of r‐graphs, let ex(n,MJX-tex-caligraphicscriptF) denote the maximum number of edges in an MJX-tex-caligraphicscriptF‐free r‐graph on n vertices. Let Fr(v,e) denote the family of all r‐graphs with e edges and at most v vertices. We prove that ex(n,MJX-tex-caligraphicscriptFr(r+1,2)∪MJX-tex-caligraphicscriptFr(r+2,3))=false(1r−o(1)false)true(0.0ptnr−1true).

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“…A lower bound, obtained from the probabilistic construction of Brown et al [3], was explicitly calculated in [14]:…”

Section: Introductionmentioning

confidence: 99%

“…In the case when

k = (e r - v) ∕ (e - 1)

is an integer, Brown et al were able to prove

(c_{r, v, e} + o (1)) n^{k} \leq f_{r} (n, v, e) \leq (C_{r, v, e} + o (1)) n^{k} as n \to \infty .

The case

e = 2

is equivalent to the packing problem and was asymptotically solved by Rödl who proved

f_{r} (n, v, 2) = true({(\binom{r}{2 r - v})}^{- 1} + o (1) true) true(\frac{0.0pt}{n} true) as n \to \infty .

Recently, the first case with

e = 3

r \geq 3

was solved by Glock :

f_{3} (n, 5, 3) = false(\frac{1}{5} + o (1) false) n^{2}

. His result was generalized by Shangguan and Tamo to

f_{r} (n, 3 r - 4, 3) = true(\frac{1}{r^{2} - r - 1} + o (1) true) n^{2} as n \to \infty .

Shangguan and Tamo proved the upper bound

…”

Section: Introductionmentioning

confidence: 99%

Approximate Steiner (r − 1, r, n)‐systems without three blocks on r + 2 points

Sidorenko

2019

J of Combinatorial Designs

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“…The study of strongly perfect hash matrices is relatively new [ST20b]. The thesis [Dou19] collected some recent results on a closely related topic.…”

Section: Related Workmentioning

confidence: 99%

“…We call a matrix with the property given by Theorem 1.3 a strongly t-perfect hash matrix; this can be viewed as an "error-resilient" version of perfect hash matrices. Strongly perfect hash matrices were first introduced by the third and fourth authors of this paper for t = 3, with a slightly different definition [ST20b]. Indeed, Lemma 25 of [ST20b] implies the t = 3 case of Theorem 1.3, but it breaks down at that point.…”

Section: Introductionmentioning

confidence: 98%

“…Strongly perfect hash matrices were first introduced by the third and fourth authors of this paper for t = 3, with a slightly different definition [ST20b]. Indeed, Lemma 25 of [ST20b] implies the t = 3 case of Theorem 1.3, but it breaks down at that point. We overcome this barrier, and construct strongly t-perfect hash matrices for all integers t ≥ 3.…”

Section: Introductionmentioning

confidence: 98%

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Improved List-Decodability of Reed--Solomon Codes via Tree Packings

Guo,

Li,

Shangguan

et al. 2020

Preprint

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This paper shows that there exist Reed-Solomon (RS) codes, over large finite fields, that are combinatorially list-decodable well beyond the Johnson radius, in fact almost achieving listdecoding capacity. In particular, we show that for any ε ∈ (0, 1] there exist RS codes with rate Ω( ε log(1/ε)+1 ) that are list-decodable from radius of 1 − ε. We generalize this result to obtain a similar result on list-recoverability of RS codes. Along the way we use our techniques to give a new proof of a result of Blackburn on optimal linear perfect hash matrices, and strengthen it to obtain a construction of strongly perfect hash matrices.To derive the results in this paper we show a surprising connection of the above problems to graph theory, and in particular to the tree packing theorem of Nash-Williams and Tutte. En route to our results on RS codes, we prove a generalization of the tree packing theorem to hypergraphs (and we conjecture that an even stronger generalization holds). We hope that this generalization to hypergraphs will be of independent interest.

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Sparse hypergraphs: New bounds and constructions

Shangguan

2021

Journal of Combinatorial Theory, Series B

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Degenerate Turán densities of sparse hypergraphs

Cited by 13 publications

References 40 publications

Approximate Steiner (r − 1, r, n)‐systems without three blocks on r + 2 points

Approximate Steiner (r − 1, r, n)‐systems without three blocks on r + 2 points

Improved List-Decodability of Reed--Solomon Codes via Tree Packings

Sparse hypergraphs: New bounds and constructions

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