Unavoidable subhypergraphs: a-clusters

Füredi, Zoltán; Özkahya, Lale

doi:10.1016/j.jcta.2011.05.002

Cited by 16 publications

(19 citation statements)

References 19 publications

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“…It was fully developed in a series of papers including [11] and [8]. It was successfully used for starlike configurations in [8] and [14] and recently also for larger configurations (as paths and trees) in [12] and [13]. In this paper we apply the delta system method, particularly tools from [11] and [8], to determine, for all k ≥ 5 and large n, the Turán numbers of certain hypergraphs called k-uniform linear cycles.…”

Section: Introductionmentioning

confidence: 99%

Hypergraph Turán numbers of linear cycles

Füredi

Jiang

2014

Journal of Combinatorial Theory, Series A

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A k-uniform linear cycle of length ℓ, denoted by C (k) ℓ , is a cyclic list of k-sets A 1 , . . . , A ℓ such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint. For all k ≥ 5 and ℓ ≥ 3 and sufficiently large n we detemine the largest size of a k-uniform set family on [n] not containing a linear cycle of length ℓ. For odd ℓ = 2t + 1 the unique extremal family F S consists of all k-sets in [n] intersecting a fixed t-set S in [n]. For even ℓ = 2t + 2, the unique extremal family consists of F S plus all the k-sets outside S containing some fixed two elements. For k ≥ 4 and large n we also establish an exact result for so-called minimal cycles. For all k ≥ 4 our results substantially extend Erdős' result on largest k-uniform families without t + 1 pairwise disjoint members and confirm, in a stronger form, a conjecture of Mubayi and Verstraëte [23]. Our main method is the delta system method.

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Section: Introductionmentioning

confidence: 99%

Hypergraph Turán numbers of linear cycles

Füredi

Jiang

2014

Journal of Combinatorial Theory, Series A

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“…Later, Katona proposed a version of the d = 3 case, and Frankl and Füredi [3] obtained a result for n ≥ k 2 + 3k. In [6], Mubayi completely resolved the d = 3 case and proposed the conjecture for general d. This was then resolved for d = 4 and sufficiently large n by Mubayi in [7], and later for all d ≥ 3 and sufficiently large n both by Mubayi and Ramadurai in [8] and independently by Özkahya and Füredi in [4]. In [5], Keevash and Mubayi solved another case of this problem, namely where both k/n and n/2 − k are bounded away from zero.…”

Section: F| ≤mentioning

confidence: 95%

On the $d$-cluster generalization of Erdős-Ko-Rado

Currier¹

2018

Preprint

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If 2 ≤ d ≤ k and n ≥ dk/(d − 1), a d-cluster is defined to be a collection of d elements of [n] kwith empty intersection and union of size no more than 2k. Mubayi [6] conjectured that the largest size of a d-cluster-free family F ⊂ [n] k is n−1 k−1 , with equality holding only for a maximum-sized star. Here we prove two results. The first resolves Mubayi's conjecture and proves a stronger result, thus completing a new generalization of the Erdős-Ko-Rado Theorem. The second shows, by a different technique, that for a slightly more limited set of parameters only a very specific kind of d-cluster need be forbidden to achieve the same bound.

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“…The messy path is the hypergraph M = {abc, bcd, def }. Extremal results for collections of hypergraphs containing M are studied in [8], where the messy path is P 3 (1, 2) or P 3 (2, 1), and in [9], where the messy path is a (2, 1)-cluster. In [9], it is shown that for sufficiently large n, ex(n, M) = n−1…”

Section: Messy Pathmentioning

confidence: 99%

“…In [9], it is shown that for sufficiently large n, ex(n, M) = n−1 2 with the unique extremal hypergraph being a complete star. We show that this is the case for smaller n. Recall that F (a, 2) refers to the 3-uniform hypergraph with vertex set {x 1 , .…”

Section: Extremal Number Of the Messy Pathmentioning

confidence: 99%

On multicolor Ramsey numbers of triple system paths of length 3

Bohman

Zhu

2019

Preprint

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Let H be a 3-uniform hypergraph. The multicolor Ramsey number r k (H) is the smallest integer n such that every coloring of [n] 3 with k colors has a monochromatic copy of H. Let L be the loose 3-uniform path with 3 edges and M denote the messy 3-uniform path with 3 edges; that is, let L = {abc, cde, ef g} and M = {abc, bcd, def }. In this note we prove r k (L) < 1.55k and r k (M) < 1.6k for k sufficiently large. The former result improves on the bound r k (L) < 1.975k + 7 √ k, which was recently established by Łuczak and Polcyn.

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Unavoidable subhypergraphs: a-clusters

Cited by 16 publications

References 19 publications

Hypergraph Turán numbers of linear cycles

Hypergraph Turán numbers of linear cycles

On the $d$-cluster generalization of Erdős-Ko-Rado

On multicolor Ramsey numbers of triple system paths of length 3

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