On Erdős' extremal problem on matchings in hypergraphs

Łuczak, Tomasz; Mieczkowska, Katarzyna

doi:10.1016/j.jcta.2014.01.003

Cited by 58 publications

(48 citation statements)

References 7 publications

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“…For the cases k ≤ 4, it was verified asymptotically by Alon, Frankl, Huang, Rödl, Ruciński and Sudakov [1]. For k = 3, it was recently proved by Frankl [11], improving results of Frankl, Rödl and Ruciński [12], and of Luczak and Mieczkowska [21]. Bollobás, Daykin and Erdős [4] proved Conjecture 1.1 for general k whenever s < n/(2k 3 ), which extended earlier results of Erdős [8].…”

Section: Large Matchings In Hypergraphs With Many Edgessupporting

confidence: 62%

Fractional and integer matchings in uniform hypergraphs

Kühn

Osthus

Townsend

2014

European Journal of Combinatorics

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Abstract. A conjecture of Erdős from 1965 suggests the minimum number of edges in a kuniform hypergraph on n vertices which forces a matching of size t, where t ≤ n/k. Our main result verifies this conjecture asymptotically, for all t < 0.48n/k. This gives an approximate answer to a question of Huang, Loh and Sudakov, who proved the conjecture for t ≤ n/3k2 . As a consequence of our result, we extend bounds of Bollobás, Daykin and Erdős by asymptotically determining the minimum vertex degree which forces a matching of size t < 0.48n/(k − 1) in a k-uniform hypergraph on n vertices. We also obtain further results on d-degrees which force large matchings. In addition we improve bounds of Markström and Ruciński on the minimum ddegree which forces a perfect matching in a k-uniform hypergraph on n vertices. Our approach is to inductively prove fractional versions of the above results and then translate these into integer versions.

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Section: Large Matchings In Hypergraphs With Many Edgessupporting

confidence: 62%

Fractional and integer matchings in uniform hypergraphs

Kühn

Osthus

Townsend

2014

European Journal of Combinatorics

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“…For k = 2 (graphs) the value of n 0 (2, t) was determined by Erdős and Gallai [2]. The case k = 3 was recently investigated by Frankl, Rödl, and Rucinśki [10] and n 0 (3, t) was finally determined by Luczak and Mieczkowska [21] for large t, and by Frankl [6] for all t. In general, Huang, Loh, and Sudakov [16] showed n 0 (k, t) < 3tk 2 , which was slightly improved in [9] and greatly improved to n 0 (k, t) ≤ (2t + 1)k − t by Frankl [7]. Frankl [5] showed that for every n, k, t if a k-graph F on [n] has no t + 1 pairwise disjoint edges then |F| ≤ t n−1 k−1 .…”

Section: Given Setsmentioning

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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“…Luczak, Mieczkowska [119] proved (20) in the case k = 3 and s very large. In [60] (20) is proved for k = 3 and all s.…”

Section: Proposition 31 (Frankl [56]) Suppose Thatmentioning

confidence: 93%

Invitation to intersection problems for finite sets

Frankl

Tokushige

2016

Journal of Combinatorial Theory, Series A

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Abstract. Extremal set theory is dealing with families, F of subsets of an nelement set. The usual problem is to determine or estimate the maximum possible size of F, supposing that F satisfies certain constraints. To limit the scope of this survey most of the constraints considered are of the following type: any r subsets in F have at least t elements in common, all the sizes of pairwise intersections belong to a fixed set, L of natural numbers, there are no s pairwise disjoint subsets. Although many of these problems have a long history, their complete solutions remain elusive and pose a challenge to the interested reader.Most of the paper is devoted to sets, however certain extensions to other structures, in particular to vector spaces, integer sequences and permutations are mentioned as well. The last part of the paper gives a short glimpse of one of the very recent developments, the use of semidefinite programming to provide good upper bounds.

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On Erdős' extremal problem on matchings in hypergraphs

Cited by 58 publications

References 7 publications

Fractional and integer matchings in uniform hypergraphs

Fractional and integer matchings in uniform hypergraphs

Hypergraph Turán numbers of linear cycles

Invitation to intersection problems for finite sets

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