## Abstract

A graph is called *H-free* if it contains no copy of *H*. Denote by *f*
_{
n
}(*H*) the number of (labeled) *H*-free graphs on *n* vertices. Erdős conjectured that *f*
_{
n
}(*H*) ≤ 2^{(1+o(1))ex(n,H)}. This was first shown to be true for cliques; then, Erdős, Frankl, and Rödl proved it for all graphs *H* with *χ*(*H*)≥3. For most bipartite *H*, the question is still wide open, and even the correct order of magnitude of log_{2}
*f*
_{
n
}(*H*) is not known. We prove that *f*
_{
n
}(*K*
_{
m,m
}) ≤ 2^{O}(*n*
^{2−1/m}) for every *m*, extending the result of Kleitman and Winston and answering a question of Erdős. This bound is asymptotically sharp for *m*∈{2,3}, and possibly for all other values of *m*, for which the order of ex(*n,K*
_{
m,m
}) is conjectured to be *Θ*(*n*
^{2−1/m}). Our method also yields a bound on the number of *K*
_{
m,m
}-free graphs with fixed order and size, extending the result of Füredi. Using this bound, we prove a relaxed version of a conjecture due to Haxell, Kohayakawa, and Łuczak and show that almost all *K*
_{3,3}-free graphs of order *n* have more than 1/20·ex(*n,K*
_{3,3}) edges.

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## Additional information

This material is based upon work supported by NSF CAREER Grant DMS-0745185, UIUC Campus Research Board Grant 09072, and OTKA Grant K76099.

Research supported in part by the Trijtzinsky Fellowship and the James D. Hogan Memorial Scholarship Fund.

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### Cite this article

Balogh, J., Samotij, W. The number of *K*
_{
m,m
}-free graphs.
*Combinatorica* **31, **131 (2011). https://doi.org/10.1007/s00493-011-2610-y

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### Mathematics Subject Classification (2000)

- 05C35
- 05C30
- 05D40
- 05A16