The typical structure of graphs with no large cliques

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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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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Mathematics Subject Classification (2000)

  • 05C35
  • 05C30
  • 05C75
  • 05D40