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The typical structure of graphs with no large cliques

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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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, Urbana, Illinois, 61801, USA

    József Balogh, Hong Liu & Maryam Sharifzadeh

  2. Bolyai Institute, University of Szeged, Szeged, Hungary

    József Balogh

  3. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA

    Neal Bushaw

  4. IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, Brasil

    Neal Bushaw, Maurício Collares & Robert Morris

Authors
  1. József Balogh
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  2. Neal Bushaw
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  3. Maurício Collares
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  4. Hong Liu
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  5. Robert Morris
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  6. Maryam Sharifzadeh
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Corresponding author

Correspondence to Robert Morris.

Additional information

Research supported in part by a Simons Fellowship, NSF CAREER Grant DMS-0745185, Marie Curie FP7-PEOPLE-2012-IIF 327763, Arnold O. Beckman Research Award (UIUC Campus Research Board 13039) (JB), CAPES bolsa Proex (MCN), a CNPq bolsa PDJ (NB) and a CNPq bolsa de Produtividade em Pesquisa (RM)

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Balogh, J., Bushaw, N., Collares, M. et al. The typical structure of graphs with no large cliques. Combinatorica 37, 617–632 (2017). https://doi.org/10.1007/s00493-015-3290-9

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  • DOI: https://doi.org/10.1007/s00493-015-3290-9

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