Turán number of bipartite graphs with no 𝐾_{𝑡,𝑡}

Sudakov, Benny; Tomon, István

doi:10.1090/proc/15042

Cited by 14 publications

(3 citation statements)

References 23 publications

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“…In light of this, Conlon [5] conjectured that if we assume that a t-degenerate bipartite graph H has no K t,t then we have ex(n, H) = O(n 2−1/t−δ ) for some δ = δ(H) > 0. Lending plausibility to this conjecture, Sudakov and Tomon [25] showed that if all vertices in one of the parts of H have degree at most t but H has no K t,t then ex(n, H) = o(n 2−1/t ). For t = 2 Conlon's conjecture can be stated as: Conjecture 1.2 (Conlon [5]).…”

Section: Conjecture 11 (Constant Deficiency Besc)mentioning

confidence: 95%

A new approach for the Brown-Erdos-Sos problem

Shapira¹,

Tyomkyn²

2023

Preprint

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The celebrated Brown-Erdős-Sós conjecture states that for every fixed e, every 3-uniform hypergraph with Ω(n 2 ) edges contains e edges spanned by e + 3 vertices. Up to this date all the approaches towards resolving this problem relied on highly involved applications of the hypergraph regularity method, and yet they supplied only approximate versions of the conjecture, producing e edges spanned by e + O(log e/ log log e) vertices.In this short paper we describe a completely different approach, which reduces the problem to a variant of another well-known conjecture in extremal graph theory. A resolution of the latter would resolve the Brown-Erdős-Sós conjecture up to an absolute additive constant.

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Section: Conjecture 11 (Constant Deficiency Besc)mentioning

confidence: 95%

A new approach for the Brown-Erdos-Sos problem

Shapira¹,

Tyomkyn²

2023

Preprint

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“…Füredi proved that ex(m, n; K s,t ) ≤ (s − t + 1) 1/t nm 1−1/t + tm 2−2/t + tn for m ≥ s, n ≥ t, s ≥ t ≥ 1. Some recent results and progress on ex(m, n; H) for bipartite graphs H can be referred to [1,24,28] and references therein.…”

Section: The Bipartite Turán Extremal Problemmentioning

confidence: 99%

The bipartite Turan number and spectral extremum for linear forests

Chen¹,

Wang²,

Yuan³

et al. 2022

Preprint

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The bipartite Turán number of a graph H, denoted by ex(m, n; H), is the maximum number of edges in any bipartite graph G = (X, Y ; E) with |X| = m and |Y | = n which does not contain H as a subgraph. In this paper, we determined ex(m, n; F ℓ ) for arbitrary ℓ and appropriately large n with comparing to m and ℓ, where F ℓ is a linear forest which consists of ℓ vertex disjoint paths. Moreover, the extremal graphs have been characterized. Furthermore, these results are used to obtain the maximum spectral radius of bipartite graphs which does not contain F ℓ as a subgraph and characterize all extremal graphs which attain the maximum spectral radius.

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“…It is perhaps important to emphasize that, unlike the argument used in [8], our method does not rely on the so-called packing lemma of Haussler [11]. Instead, our approach is similar in spirit to an argument used by Sudakov and Tomon [17] in a related, but different context.…”

Section: Introductionmentioning

confidence: 99%

On the Zarankiewicz problem for graphs with bounded VC-dimension

Janzer,

Pohoata

2020

Preprint

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The problem of Zarankiewicz asks for the maximum number of edges in a bipartite graph on n vertices which does not contain the complete bipartite graph K k,k as a subgraph. A classical theorem due to Kővári, Sós, and Turán says that this number of edges is O n 2−1/k . An important variant of this problem is the analogous question in bipartite graphs with VC-dimension at most d, where d is a fixed integer such that k ≥ d ≥ 2. A remarkable result of Fox, Pach, Sheffer, Suk, and Zahl with multiple applications in incidence geometry shows that, under this additional hypothesis, the number of edges in a bipartite graph on n vertices and with no copy of K k,k as a subgraph must be O n 2−1/d . This theorem is sharp when k = d = 2, because by design any K 2,2 -free graph automatically has V C-dimension at most 2, and there are well-known examples of such graphs with Ω n 3/2 edges. However, it turns out this phenomenon no longer carries through for any larger d.We show the following improved result: the maximum number of edges in bipartite graphs with no copies of K k,k and VC-dimension at most d is o(n 2−1/d ), for every k ≥ d ≥ 3.

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Turán number of bipartite graphs with no 𝐾_{𝑡,𝑡}

Cited by 14 publications

References 23 publications

A new approach for the Brown-Erdos-Sos problem

A new approach for the Brown-Erdos-Sos problem

The bipartite Turan number and spectral extremum for linear forests

On the Zarankiewicz problem for graphs with bounded VC-dimension

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