The Typical Structure of Maximal Triangle-Free Graphs

Balogh, József; Li, Hong; Petříčková, Šárka; Sharifzadeh, Maryam

doi:10.1017/fms.2015.22

Cited by 15 publications

(22 citation statements)

References 25 publications

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“…Combining the upper bounds in (2) and (3) with Corollary 1.5, for some C > 0, we have that almost surely [n] p is C(log log n) 4 log n , 3 -Szemerédi for p ≥ n − 1 2 +o(1) ; and for k ≥ 4 that…”

Section: Enumerating Sets With No K-term Arithmetic Progressionmentioning

confidence: 82%

The Number of Subsets of Integers with Nok-Term Arithmetic Progression

Balogh

Sharifzadeh

2016

Int Math Res Notices

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Addressing a question of Cameron and Erdős, we show that, for infinitely many values of n, the number of subsets of {1, 2, . . . , n} that do not contain a k-term arithmetic progression is at most 2 O(r k (n)) , where r k (n) is the maximum cardinality of a subset of {1, 2, . . . , n} without a k-term arithmetic progression. This bound is optimal up to a constant factor in the exponent. For all values of n, we prove a weaker bound, which is nevertheless sufficient to transfer the current best upper bound on r k (n) to the sparse random setting. To achieve these bounds, we establish a new supersaturation result, which roughly states that sets of size Θ(r k (n)) contain superlinearly many k-term arithmetic progressions.For integers r and k, Erdős asked whether there is a set of integers S with no (k+1)term arithmetic progression, but such that any r-coloring of S yields a monochromatic k-term arithmetic progression. Nešetřil and Rödl, and independently Spencer, answered this question affirmatively. We show the following density version: for every k ≥ 3 and δ > 0, there exists a reasonably dense subset of primes S with no (k+1)-term arithmetic progression, yet every U ⊆ S of size |U | ≥ δ|S| contains a k-term arithmetic progression.Our proof uses the hypergraph container method, which has proven to be a very powerful tool in extremal combinatorics. The idea behind the container method is to have a small certificate set to describe a large independent set. We give two further applications in the appendix using this idea.

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Section: Enumerating Sets With No K-term Arithmetic Progressionmentioning

confidence: 82%

The Number of Subsets of Integers with Nok-Term Arithmetic Progression

Balogh

Sharifzadeh

2016

Int Math Res Notices

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“…, n}. Following this breakthrough, Balogh, Liu, Petříčková, and Sharifzadeh [9] proved the following much stronger theorem, which states that in fact almost all maximal triangle-free graphs can be constructed in this way. Theorem 8.6.…”

Section: List Colouringmentioning

confidence: 99%

“…(a) Every set of points of S in general position is contained in some C ∈ C, (b) Each C ∈ C has size at most (1 − δ)|S|. 9 Starting with S = [m] 3 and iterating this process for O(log m) steps, we obtain the following container theorem for sets of points in general position.…”

Section: 4mentioning

confidence: 99%

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The Method of Hypergraph Containers

Balogh

Morris

Samotij

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of 'containers' for the independent sets, each of which contains few edges. We attempt to convey to the reader a general high-level overview of the method, focusing on a small number of illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry, and avoiding technical details as much as possible.

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“…One can turn the problem of counting the number of maximal L-free subsets of [n] into one of counting maximal independent sets in an auxiliary graph. Similar techniques were used in [26,2,3,14], and in the graph setting in [5,1]. To be more precise let B and S be disjoint subsets of [n] and fix a three-variable linear equation L. The link graph L S [B] of S on B has vertex set B, and an edge set consisting of the following two types of edges:…”

Section: Containers Link Hypergraphs and The Main Lemmasmentioning

confidence: 99%

On solution-free sets of integers II

Hancock

Treglown

2017

Acta Arith.

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Abstract. Given a linear equation L, a set A ⊆ [n] is L-free if A does not contain any 'non-trivial' solutions to L. We determine the precise size of the largest L-free subset of [n] for several general classes of linear equations L of the form px + qy = rz for fixed p, q, r ∈ N where p ≥ q ≥ r. Further, for all such linear equations L, we give an upper bound on the number of maximal L-free subsets of [n]. In the case when p = q ≥ 2 and r = 1 this bound is exact up to an error term in the exponent. We make use of container and removal lemmas of Green [12] to prove this result. Our results also extend to various linear equations with more than three variables.MSC2010: 11B75, 05C69.

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The Typical Structure of Maximal Triangle-Free Graphs

Cited by 15 publications

References 25 publications

The Number of Subsets of Integers with Nok-Term Arithmetic Progression

The Number of Subsets of Integers with Nok-Term Arithmetic Progression

The Method of Hypergraph Containers

On solution-free sets of integers II

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