## Sharp thresholds of graph properties, and the $k$-sat problem

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- by Ehud Friedgut and appendix by Jean Bourgain
- J. Amer. Math. Soc.
**12**(1999), 1017-1054 - DOI: https://doi.org/10.1090/S0894-0347-99-00305-7
- Published electronically: May 27, 1999
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## Abstract:

Given a monotone graph property $P$, consider $\mu _p(P)$, the probability that a random graph with edge probability $p$ will have $P$. The function $d\mu _p(P)/dp$ is the key to understanding the*threshold behavior*of the property $P$. We show that if $d\mu _p(P)/dp$ is small (corresponding to a non-sharp threshold), then there is a list of graphs of bounded size such that $P$ can be approximated by the property of having one of the graphs as a subgraph. One striking consequence of this result is that a coarse threshold for a random graph property can only happen when the value of the critical edge probability is a rational power of $n$. As an application of the main theorem we settle the question of the existence of a sharp threshold for the satisfiability of a random $k$-CNF formula. An appendix by Jean Bourgain was added after the first version of this paper was written. In this appendix some of the conjectures raised in this paper are proven, along with more general results.

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## Bibliographic Information

**Ehud Friedgut**- Affiliation: Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel
- Email: ehudf@math.huji.ac.il
**appendix by Jean Bourgain**- Affiliation: School of Mathematics, Institue of Advanced Study, Princeton, New Jersey 08540
- Received by editor(s): May 2, 1997
- Received by editor(s) in revised form: July 14, 1998
- Published electronically: May 27, 1999
- Additional Notes: This paper is part of a Ph.D. thesis prepared under the supervision of Prof. Gil Kalai.
- © Copyright 1999 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**12**(1999), 1017-1054 - MSC (1991): Primary 05C80, 28A35
- DOI: https://doi.org/10.1090/S0894-0347-99-00305-7
- MathSciNet review: 1678031