Lagrangians of hypergraphs: The Frankl-Füredi conjecture holds almost everywhere

Tyomkyn, Mykhaylo

doi:10.1112/jlms.12082

Cited by 17 publications

(19 citation statements)

References 7 publications

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“…An analogous result for r = 3 was proved by Talbot [18] and was used in [11,18,20,21] to prove that Conjecture 1.1 holds for certain ranges (see Table 1).…”

Section: Author(s)supporting

confidence: 58%

“…r Bounds on a Motzkin and Strauss [13] 2 all a Talbot [18] 3 a ≥ 2t − 3 and a ∈ {1, 2} Tang, Peng, Zhang and Zhao [19,20] 3 a ≥ 3t 2 − 5 2 and a ∈ {3, 4} Tyomkyn [21] 3 a ≥ t + δ · t 3/4 Tyomkyn [21] ≥ 4 a ≥ γ r · t r−2 Lei, Lu, Peng [11] 3 a ≥ t + ζ · t 2/3 Table 1: In this table we summarise the main progress made towards Conjecture 1.1. Here, δ and ζ are absolute constants and γ r is an absolute constant depending on r. Recall that, by definition, a ≤ t−1 r−1 .…”

Section: Author(s)mentioning

confidence: 99%

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Hypergraph Lagrangians I: The Frankl-Füredi conjecture is false

Gruslys

Letzter

Morrison

2020

Advances in Mathematics

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An old conjecture of Frankl and Füredi states that the Lagrangian of an r-uniform hypergraph on m edges is maximised by an initial segment of colex. In this paper we prove this conjecture for a wide range of sufficiently large m. In particular, we confirm the conjecture in the case r = 3 for all sufficiently large m. In addition, we find an infinite family of counterexamples for each r ≥ 4 and provide a new proof for large t of a related conjecture of Nikiforov. IntroductionThe notion of the Lagrangian of a graph was originally introduced in 1965 by Motzkin and Strauss [13] to provide a beautiful new proof of Turán's theorem. This concept was later generalised to uniform hypergraphs, where the study of Lagrangians has played an important role in the advancement of our understanding of hypergraph Turán problems. Notably, hypergraph Lagrangians were used by Frankl and Rödl [7] to disprove a conjecture of Erdős [5] on jumps of hypergraph Turán densities. See, for example, [8], [20] and the excellent survey of Keevash [9] for further applications.In many of these results, the Turán problem can be converted into the problem of determining (or finding good bounds for) the Lagrangian of a particular hypergraph. In this paper we are interested in determining the maximum value of the Lagrangian over all r-graphs (namely, r-uniform hypergraphs) with a fixed number of hyperedges.In order to introduce our main results, we require some technical definitions. For t ∈ N,for all i ∈ [t]; and v∈V w(v) = 1. Let G ⊆ [t] (r) be a hypergraph on vertex set [t]. For e ∈ G and a weighting w of [t], define w(e) := i∈e w(i), and for F ⊆ G define w(F ) := e∈F w(e). For w a weighting of [t] and G ⊆ [t] (r) , we may also say that w is a weighting of G. Define the Lagrangian of G, denoted λ(G), as follows. λ(G) := max{w(G) : w is a weighting of [t]}. Say that a weighting w of [t] is maximal for G if w(G) = λ(G). Also define Λ(m, r) := max{λ(G) : G ⊆ N (r) , |G| = m}.For a graph G, it is a simple exercise to show that λ(G) is achieved by equally distributing the weight over a largest clique in G. However, there is no easy way known for calculating the Lagrangian of a given r-graph (when r ≥ 3).Recall that the colexicographic or colex order on N (r) is the ordering in which A < B if i∈A 2 i < i∈B 2 i . Define C(m, r) to be the family containing the first m sets in the colex order on N (r) . A conjecture of Frankl and Füredi [6] from 1989 states that C(m, r) has the largest Lagrangian of any family of cardinality m. Conjecture 1.1 (Frankl and Füredi [6]). Let G ⊆ N (r) such that |G| = m. Then λ(G) ≤ λ(C(m, r)).In other words, this conjecture says that there exists some weighting w such that w(C(m, r)) = Λ(m, r). An interesting special case of this conjecture, which we (following Tyomkyn [21]) refer to as the principal case, is when m = t r , i.e. when C(m, r) is the clique [t] (r) .

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“…An analogous result for r = 3 was proved by Talbot [18] and was used in [11,18,20,21] to prove that Conjecture 1.1 holds for certain ranges (see Table 1).…”

Section: Author(s)supporting

confidence: 58%

Section: Author(s)mentioning

confidence: 99%

Hypergraph Lagrangians I: The Frankl-Füredi conjecture is false

Gruslys

Letzter

Morrison

2020

Advances in Mathematics

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“…Talbot [21] made a first breakthrough in confirming this conjecture for some cases. Subsequent progress in this conjecture were made in the papers of Tyomkyn [23], Lei-Lu-Peng [14] and Tang-Peng-Zhang-Zhao [22]. Recently, Gruslys-Letzter-Morrison [7] confirmed this conjecture for r = 3 and sufficiently large m. We focus on the Lagrangian density of an r-graph F in this paper.…”

Section: Notations and Definitionsmentioning

confidence: 87%

λ-perfect hypergraphs and Lagrangian densities of hypergraph cycles

Yan

Peng

2019

Discrete Mathematics

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“…For hypergraphs, Talbot [32] first proved the conjecture for r = 3 and 3 ≤ m ≤ 3 + −1 2 − , where > 0 is an integer. Subsequent progress in this conjecture were made in the papers of Tang, Peng, Zhang and Zhao [33,34], Tyomkyn [36], Lei, Lu and Peng [20], Nikiforov [23], Lei and Lu [19], and Lu [21]. Recently, Gruslys, Letzter and Morrison [12] confirmed this conjecture for r and r ≤ m ≤ r + −1 r−1 if is sufficiently large.…”

Section: Clearly πmentioning

confidence: 91%