Quasi‐random hypergraphs revisited

For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F -packing. An informal statement of a special case of our general result for 3-uniform hypergraphs is as follows. Fix an integer r ≥ 4 and 0 < p < 1. Suppose that H is an n-vertex triple system with r|n and the following two properties:• for every graph G with V (G) = V (H), at least p proportion of the triangles in G are also edges of H,• for every vertex x of H, the link graph of x is a quasirandom graph with density at least p.Then H has a perfect Kr -packing. Moreover, we show that neither hypotheses above can be weakened, so in this sense our result is tight. A similar conclusion for this special case can be proved by Keevash's hypergraph blowup lemma, with a slightly stronger hypothesis on H.

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“…• When I = [k] , then Disc (k) (I, p, µ) is closely related to the property CliqueDisc[ ] studied in [2,3,4,5,6,19,28].…”

Section: Notions Of Hypergraph Quasirandomnessmentioning

confidence: 99%

Perfect packings in quasirandom hypergraphs I

Lenz

Mubayi

2016

Journal of Combinatorial Theory, Series B

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“…Since these early papers on the subject, there have been a variety of different notions of quasirandomness defined for hypergraphs, and the relationships between these quasirandom properties are not completely understood. Chung posed the following problem. Problem (Chung ).…”

Section: Introductionmentioning

confidence: 99%

“…Chung posed the following problem. Problem (Chung ). How is a given property placed in the quasirandom hierarchy and what is the lattice structure illustrating the relationship among quasirandom properties of hypergraphs?…”

Section: Introductionmentioning

confidence: 99%

“…There are two more hypergraph quasirandom properties that have been studied. First, Chung and Graham's original property on even/odd subgraphs of the octahedron called Deviation[ℓ] and an extension of CliqueDisc p [ℓ] recently proposed by Chung . For

2 \leq ℓ \leq k

, define Deviation[ℓ] as follows:

\sum_{\begin{matrix} x_{1}, \dots, x_{k - ℓ} \in V false(H false) \\ y_{1, 0}, y_{1, 1}, \dots, y_{ℓ, 0}, y_{ℓ, 1} \in V false(H false) \end{matrix}} {(- 1)}^{| O [\vec{x}, \vec{y}] \cap E (H) |} = o (n^{k + ℓ}),

where

O [\vec{x}, \vec{y}]

is the collection of hyperedges of the squashed octahedron.…”

Section: Introductionmentioning

confidence: 99%

“…In , Chung made partial progress on Problem 1; see Table for the exact results she proved. Our main result is to determine all relationships between Expand p [π], CliqueDisc p [ℓ, s ], and Deviation[ℓ] for all

ℓ, s

, and π.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The poset of hypergraph quasirandomness

Lenz

Mubayi

2013

Random Struct Algorithms

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Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-GrahamWilson on quasirandom graphs. One feature that became apparent in the early work on k-uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them.Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990's.

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Quasi‐carousel tournaments

Coregliano

2017

Journal of Graph Theory

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A tournament is called locally transitive if the outneighborhood and the inneighborhood of every vertex are transitive. Equivalently, a tournament is locally transitive if it avoids the tournaments W4 and L4, which are the only tournaments up to isomorphism on four vertices containing a unique 3‐cycle. On the other hand, a sequence of tournaments false(Tnfalse)n∈double-struckN with Vfalse(Tnfalse)=n is called almost balanced if all but o(n) vertices of Tn have outdegree (1/2+o(1))n. In the same spirit of quasi‐random properties, we present several characterizations of tournament sequences that are both almost balanced and asymptotically locally transitive in the sense that the density of W4 and L4 in Tn goes to zero as n goes to infinity.

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Quasi‐random hypergraphs revisited

Cited by 21 publications

References 8 publications

Perfect packings in quasirandom hypergraphs I

Perfect packings in quasirandom hypergraphs I

The poset of hypergraph quasirandomness

Quasi‐carousel tournaments

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