Sets of Independent Edges of a Hypergraph

For integers n ≥ s ≥ 2 let e(n, s) denote the maximum of |F|, where F is a family of subsets of an n-element set and F contains no s pairwise disjoint members. Half a century ago, solving a conjecture of Erdős, Kleitman determined e(sm − 1, s) and e(sm, s) for all m, s ≥ 1. During the years very little progress in the general case was made.In the present paper we state a general conjecture concerning the value of e(sm − l, m) for 1 < l < s and prove its validity for s > s 0 (l, m). For l = 2 we determine the value of e(sm − 2, m) for all s ≥ 5.Some related results shedding light on the problem from a more general context are proved as well. IntroductionLet [n] := {1, 2, . . . , n} be the standard n-element set and 2[n] its power set. A subset F ⊂ 2[n] is called a family. For 0 ≤ k ≤ n we use the notationThe maximum number of pairwise disjoint members of a family F is denoted by ν(F ) and called the matching number of F . Note that ν(F ) ≤ n unless ∅ ∈ F .Two of the important classical results in extremal set theory are concerning the matching number.Definition 2. For positive integers n, k, s ≥ 2, n ≥ ks defineFor s = 2 both e(n, s) and e k (n, s) were determined by Erdős, Ko and Rado. e(n, 2) = 2 n−1 ,For m = n+1 s the family[n] ≥ m := {H ⊂ [n] : |H| ≥ m} does not contain s pairwise disjoint sets. Erdős conjectured that for n = sm − 1 one cannot do any better. Half a century ago Kleitman proved this conjecture and determined e(sm, s) as well. Kleitman [14]).e(sm, s)Note that e(sm, s) = 2e(sm − 1, s). In general, e(n + 1, s) ≥ 2e(n, s) is obvious, and, since the constructions of families that match the bounds are easy to provide, (3) follows from (4). For s = 2 both formulae give 2 n−1 , the easy-to-prove bound (1). In the case s = 3 there is just one case not covered by the Kleitman Theorem, namely n ≡ 1 ( mod 3). This was the subject of the PhD dissertation of Quinn [16]. In it a very long, tedious proof for the following equality is provided:Unfortunately, this result was never published and no further progress was made on the determination of e(n, s). Let us first make a general conjecture. Proof. Assume that P 1 , . . . , P s ∈ P(s, m, l) are pairwise disjoint. Then

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“…For n > 2k 3 s (21) was verified by Bollobás, Daykin and Erdős [1]. In the paper [8] we verify the conjecture for n ≥ (2+o (1))sk.…”

Section: Hilton-milner-type Results For Erdős Matching Conjecturesupporting

confidence: 63%

Families with no s pairwise disjoint sets

Frankl

Kupavskii

2017

J. London Math. Soc.

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“…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]. Huang, Loh and Sudakov [14] proved it for s < n/(3k 2 ).…”

Section: Large Matchings In Hypergraphs With Many Edgessupporting

confidence: 52%

“…The case d = k − 1 of Conjecture 1.4 follows easily from the determination of m s k−1 (k, n) for s close to n/k in [29]. Bollobás, Daykin and Erdős [4] determined m s 1 (k, n) for small s, i.e. whenever s < n/2k 3 .…”

Section: 2mentioning

confidence: 96%

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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“…To our knowledge, the first result relating the minimum degree and the existence of a large (though, far from perfect) matching in a k-graph was obtained by Bollobas, Daykin, and Erdos in [4]. It was further extended to perfect matchings by Daykin and Haggkvist in [7].…”

Section: A K-uniform Hypergraph or K-graph For Short Is A Pair H = mentioning

confidence: 87%

Dirac-Type Questions For Hypergraphs — A Survey (Or More Problems For Endre To Solve)

Rödl

Ruciński

2010

Bolyai Society Mathematical Studies

127

147

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In 1952 Dirac [8] proved a celebrated theorem stating that if the minimum degree 8(G) in an n-vertex graph G is at least n/2 then G contains a Hamiltonian cycle. In 1999, Katona and Kierstead initiated a new stream of research devoted to studying similar questions for hypergraphs, and subsequently, for perfect matchings. A pivotal role in achieving some of the most important results in both these areas was played by Endre Szemeredi. In this survey we present the current state-of-art and pose some open problems .

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Sets of Independent Edges of a Hypergraph

Cited by 122 publications

References 3 publications

Families with no s pairwise disjoint sets

Families with no s pairwise disjoint sets

Fractional and integer matchings in uniform hypergraphs

Dirac-Type Questions For Hypergraphs — A Survey (Or More Problems For Endre To Solve)

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