Large matchings in uniform hypergraphs and the conjectures of Erdős and Samuels

Alon, Noga; Frankl, Péter; Huang, Hao; Rödl, Vojtěch; Ruciński, Andrzej; Sudakov, Benny

doi:10.1016/j.jcta.2012.02.004

Cited by 126 publications

(262 citation statements)

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“…Despite its seeming simplicity Conjecture 1.1 is still wide open in general. 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].…”

Section: Large Matchings In Hypergraphs With Many Edgesmentioning

confidence: 83%

“…As discussed in [1], this conjecture has applications to a problem on information storage and retrieval. To prove Theorems 1.2 and 1.5, we first prove Conjecture 1.7 asymptotically for fractional matchings of any size up to 0.48n/k.…”

Section: 2mentioning

confidence: 92%

“…A key idea (already used e.g. in [1,28]) is that we can switch between considering the largest fractional matching and the smallest fractional vertex cover of a hypergraph. The determination of these quantities are dual linear programming problems, and hence by the Duality Theorem they have the same size.…”

Section: 2mentioning

confidence: 99%

“…We use Theorem 1.8, along with methods similar to those developed in [1], to convert our edge-density conditions for the existence of fractional matchings into corresponding minimum degree conditions (see Proposition 4.1). For 1 ≤ d ≤ k − 2 the following theorem asymptotically determines f s d (k, n) for fractional matchings of any size up to 0.48n…”

Section: 2mentioning

confidence: 99%

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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 Edgesmentioning

confidence: 83%

Section: 2mentioning

confidence: 92%

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

See 2 more Smart Citations

Fractional and integer matchings in uniform hypergraphs

Kühn

Osthus

Townsend

2014

European Journal of Combinatorics

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“…In particular, in recent years there has been much study of the case of a perfect matching, see e.g. [1,2,6,11,14,16,17,18,23,26,28,29,35,36]. For perfect Hpackings other than a perfect matching, results are much more sparse.…”

Section: Perfect Packings In Graphsmentioning

confidence: 99%

Packing k-partite k-uniform hypergraphs

Mycroft

2016

Journal of Combinatorial Theory, Series A

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Abstract. Let G and H be k-graphs (k-uniform hypergraphs); then a perfect H-packing in G is a collection of vertex-disjoint copies of H in G which together cover every vertex of G. For any fixed H let δ(H, n) be the minimum δ such that any k-graph G on n vertices with minimum codegree δ(G) ≥ δ contains a perfect H-packing. The problem of determining δ(H, n) has been widely studied for graphs (i.e. 2-graphs), but little is known for k ≥ 3. Here we determine the asymptotic value of δ(H, n) for all complete k-partite k-graphs H, as well as a wide class of other k-partite k-graphs. In particular, these results provide an asymptotic solution to a question of Rödl and Ruciński on the value of δ(H, n) when H is a loose cycle. We also determine asymptotically the codegree threshold needed to guarantee an H-packing covering all but a constant number of vertices of G for any complete k-partite k-graph H.

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