The typical structure of graphs with no large cliques

József Balogh; Neal Bushaw; Maurício Collares; Hong Liu; Robert Morris; Maryam Sharifzadeh

doi:10.1007/s00493-015-3290-9

The typical structure of graphs with no large cliques

József Balogh, Neal Bushaw, Maurício Collares, Hong Liu, Robert Morris, Maryam Sharifzadeh

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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 ⩽ (logn)^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.

Original language	English (US)
Pages (from-to)	617-632
Number of pages	16
Journal	Combinatorica
Volume	37
Issue number	4
DOIs	https://doi.org/10.1007/s00493-015-3290-9
State	Published - Aug 1 2017

Keywords

05C30
05C35
05C75
05D40

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Computational Mathematics

Online availability

10.1007/s00493-015-3290-9

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AU - Collares, Maurício

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AU - Morris, Robert

AU - Sharifzadeh, Maryam

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N2 - In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed r∈ℕ, almost every n-vertex Kr+1-free graph is r-partite. In this paper we extend this result to all functions r = r(n) with r ⩽ (logn)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.

AB - In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed r∈ℕ, almost every n-vertex Kr+1-free graph is r-partite. In this paper we extend this result to all functions r = r(n) with r ⩽ (logn)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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The typical structure of graphs with no large cliques

Abstract

Keywords

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