Closing in on Hill's Conjecture

Balogh, József; Lidický, Bernard; Salazar, Gelasio

doi:10.1137/17m1158859

Cited by 12 publications

(9 citation statements)

References 36 publications

(47 reference statements)

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“…because for each edge removed from the linkgraph L(a 1 ) the degree squared sum can go down by at most 4n. Now, we bound the sum on the right hand side of (7) ≤ n 3 1 2…”

Section: Discussion On Cliquesmentioning

confidence: 99%

“…The main power comes from the possibility of formulating a problem as a semidefinite program and using a computer to solve it. The method can be applied in various settings such as graphs [28,44], hypergraphs [3,19], oriented graphs [29,37], edge-coloured graphs [5,12], permutations [6,55], discrete geometry [7,36], or phylogenetic trees [1]. For a detailed explanation of the flag algebra method in the setting of 3-uniform hypergraphs see [22].…”

Section: Tool 2: Flag Algebrasmentioning

confidence: 99%

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Solving Turán's Tetrahedron Problem for the $\ell_2$-Norm

Balogh,

Clemen,

Lidický

2021

Preprint

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Turán's famous tetrahedron problem is to compute the Turán density of the tetrahedron K 3 4 . This is equivalent to determining the maximum 1 -norm of the codegree vector of a K 3 4free n-vertex 3-uniform hypergraph. We will introduce a new way for measuring extremality of hypergraphs and determine asymptotically the extremal function of the tetrahedron in our notion. The codegree squared sum, co 2 (G), of a 3-uniform hypergraph G is the sum of codegrees squared d(x, y) 2 over all pairs of vertices xy, or in other words, the square of the 2 -norm of the codegree vector of the pairs of vertices. Define exco 2 (n, H) to be the maximum co 2 (G) over all H-free nvertex 3-uniform hypergraphs G. We use flag algebra computations to determine asymptotically the codegree squared extremal number for K 3 4 and K 3 5 and additionally prove stability results. In particular, we prove that the extremal function for K 3 4 in 2 -norm is asymptotically the same as the one obtained from one of the conjectured extremal K 3 4 -free hypergraphs for the 1 -norm. Further, we prove several general properties about exco 2 (n, H) including the existence of a scaled limit, blow-up invariance and a supersaturation result.

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“…because for each edge removed from the linkgraph L(a 1 ) the degree squared sum can go down by at most 4n. Now, we bound the sum on the right hand side of (7) ≤ n 3 1 2…”

Section: Discussion On Cliquesmentioning

confidence: 99%

Section: Tool 2: Flag Algebrasmentioning

confidence: 99%

Solving Turán's Tetrahedron Problem for the $\ell_2$-Norm

Balogh,

Clemen,

Lidický

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The main power of the machinery comes from the possibility of formulating a problem as a semidefinite program and using a computer to solve it. The method can be applied in various settings such as graphs [28,44], hypergraphs [3,19], oriented graphs [29,37], edge-colored graphs [5,12], permutations [6,55], discrete geometry [7,36], or phylogenetic trees [1]. For a detailed explanation of the flag algebra method in the setting of 3-uniform hypergraphs, see [22].…”

Section: Tool 2: Flag Algebrasmentioning

confidence: 99%

Solving Turán's tetrahedron problem for the ℓ2$\ell _2$‐norm

Balogh

Clemen

Lidický

2022

Journal of London Math Soc

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Turán's famous tetrahedron problem is to compute the Turán density of the tetrahedron K43$K_4^3$. This is equivalent to determining the maximum ℓ1$\ell _1$‐norm of the codegree vector of a K43$K_4^3$‐free n$n$‐vertex 3‐uniform hypergraph. We introduce a new way for measuring extremality of hypergraphs and determine asymptotically the extremal function of the tetrahedron in our notion. The codegree squared sum, co2(G)$\mbox{co}_2(G)$, of a 3‐uniform hypergraph G$G$ is the sum of codegrees squared dfalse(x,yfalse)2$d(x,y)^2$ over all pairs of vertices xy$xy$, or in other words, the square of the ℓ2$\ell _2$‐norm of the codegree vector of the pairs of vertices. We define exco2(n,H)$\mbox{exco}_2(n,H)$ to be the maximum co2(G)$\mbox{co}_2(G)$ over all H$H$‐free n$n$‐vertex 3‐uniform hypergraphs G$G$. We use flag algebra computations to determine asymptotically the codegree squared extremal number for K43$K_4^3$ and K53$K_5^3$ and additionally prove stability results. In particular, we prove that the extremal K43$K_4^3$‐free hypergraphs in ℓ2$\ell _2$‐norm have approximately the same structure as one of the conjectured extremal hypergraphs for Turán's conjecture. Further, we prove several general properties about exco2(n,H)$\mbox{exco}_2(n,H)$ including the existence of a scaled limit, blow‐up invariance and a supersaturation result.

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“…In particular, proving the convex or h-convex crossing number of K n is H(n) would establish this case of the SGUBC. Flag algebras are used by Balogh et al [8] to show that lim n→∞ cr(K n )/H(n) > 0.985. Restricted to convex drawings, the same technique gets a lower bound of 0.996.…”

Section: Introductionmentioning

confidence: 99%