On the Number of Graphs Without Large Cliques

Mousset, Frank; Nenadov, Rajko; Steger, Angelika

doi:10.1137/130947878

Cited by 10 publications

(16 citation statements)

References 20 publications

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“…It may well be the case that this supremum is equal to 1, though we are not prepared to state this as a conjecture. Theorem 1.1 improves a recent result of Mousset, Nenadov and Steger [20], who showed that, for the same 3 family of functions r = r(n), the number of n-vertex K r+1 -free graphs is…”

Section: Introductionsupporting

confidence: 81%

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The typical structure of graphs with no large cliques

et al. 2016

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In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed r ∈ N, almost every n-vertex K r+1 -free graph is r-partite. In this paper we extend this result to all functions r = r(n) with r (log n) 1/4 . The proof combines a new (close to sharp) supersaturation version of the Erdős-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollobás and Simonovits.

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Section: Introductionsupporting

confidence: 81%

“…3 In fact, a very slightly weaker theorem was stated in [20], but a little additional case analysis easily gives the result for all r (log n) 1/4 . 4 Similarly, we say that G is t-close to being r-partite if it is not t-far from being r-partite.…”

Section: Introductionmentioning

confidence: 97%

The typical structure of graphs with no large cliques

et al. 2016

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“…The proof of Theorem 7 uses our hypergraph containers theorem for L-structures (Theorem 6) and a powerful generalization by Aroskar and Cummings [5] of the triangle removal lemma (Theorem 10). Our proof strategies for these theorems draw on a series of enumeration results for combinatorial structures which employ the hypergraph containers theorem, namely those in [14,17,30,32,39]. We will also define generalizations of extremal graphs (Definition 17) and graph stability theorems (Definition 19).…”

Section: Summary Of Resultsmentioning

confidence: 99%

“…While our proofs are inspired by and modeled on those appearing in [14,17,30,32,39], our results are more than just straightforward generalizations of existing combinatorial theorems. We use new tools called L H -templates (see Section 3) and an application of the hyergraph containers theorem to a hypergraph whose vertices and edges correspond to certain atomic diagrams (see Theorem 11).…”

Section: Summary Of Resultsmentioning

confidence: 99%

Structure and enumeration theorems for hereditary properties in finite relational languages

Terry

2018

Annals of Pure and Applied Logic

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Given a finite relational language L, a hereditary L-property is a class of finite L-structures which is closed under isomorphism and model theoretic substructure. This notion encompasses many objects of study in extremal combinatorics, including (but not limited to) hereditary properties of graphs, hypergraphs, and oriented graphs. In this paper, we generalize certain definitions, tools, and results form the study of hereditary properties in combinatorics to the setting of hereditary L-properties, where L is any finite relational language with maximum arity at least two. In particular, the goal of this paper is to generalize how extremal results and stability theorems can be combined with standard techniques and tools to yield approximate enumeration and structure theorems. We accomplish this by generalizing the notions of extremal graphs, asymptotic density, and graph stability theorems using structures in an auxiliary language associated to a hereditary L-property. Given a hereditary L-property H, we prove an approximate asymptotic enumeration theorem for H in terms of its generalized asymptotic density. Further we prove an approximate structure theorem for H, under the assumption of that H has a stability theorem. The tools we use include a new application of the hypergraph containers theorem (Balogh-Morris-Samotij [14], ) to the setting of L-structures, a general supersaturation theorem for hereditary L-properties (also new), and a general graph removal lemma for L-structures proved by Aroskar and Cummings in [5].

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“…Below, we will make use of a version of the container theorem of [2,22], the way it was formulated by Mousset-Nenadov-Steger [18]. …”

Section: Lemma 34 Let S ⊂ V (H ) Have No Empty Columns and Contain Nmentioning

confidence: 99%

Recent Trends in Combinatorics

Beveridge¹,

Griggs²,

Hogben³

et al. 2016

The IMA Volumes in Mathematics and Its Applications

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Recently, Balogh-Morris-Samotij and Saxton-Thomason proved that hypergraphs satisfying some natural conditions have only few independent sets. Their main results already have several applications. However, the methods of proving these theorems are even more far reaching. The general idea is to describe some family of events, whose cardinality a priori could be large, only with a few certificates. Here, we show some applications of the methods, including counting C 4 -free graphs, considering the size of a maximum C 4 -free subgraph of a random graph and counting metric spaces with a given number of points. Additionally, we discuss some connections with the Szemerédi Regularity Lemma.

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On the Number of Graphs Without Large Cliques

Cited by 10 publications

References 20 publications

The typical structure of graphs with no large cliques

The typical structure of graphs with no large cliques

Structure and enumeration theorems for hereditary properties in finite relational languages

Recent Trends in Combinatorics

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