On the number of graphs without large cliques

Frank Mousset*, Rajko Nenadov, Angelika Steger

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In 1976 Erdos, Kleitman, and Rothschild determined asymptotically the logarithm of the number of graphs without a clique of a fixed size ℓ. In this note we extend their result to the case of forbidden cliques of increasing size. More precisely we prove that for ℓn ≤ (log n)1/4/2 there are 2(1-1/(ℓn-1))n2/2+o(n2/ℓn)Kℓn-free graphs of order n. Our proof is based on the recent hypergraph container theorems of Saxton and Thomason and Balogh, Morris, and Samotij, in combination with a theorem of Lovász and Simonovits.

Original languageEnglish
Pages (from-to)1980-1986
Number of pages7
JournalSIAM Journal on Discrete Mathematics
Issue number4
StatePublished - 2014
Externally publishedYes


  • Asymptotic counting
  • Clique-free graphs
  • Graph theory


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