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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Every monotone graph property has a sharp threshold
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by Ehud Friedgut and Gil Kalai PDF
Proc. Amer. Math. Soc. 124 (1996), 2993-3002 Request permission


In their seminal work which initiated random graph theory Erdös and Rényi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem. Let $V_n(p)= \{0,1\}^n$ denote the Hamming space endowed with the probability measure $\mu _p$ defined by $\mu _p (\epsilon _1, \epsilon _2, \dots , \epsilon _n)= p^k \cdot (1-p)^{n-k}$, where $k=\epsilon _1 +\epsilon _2 +\cdots +\epsilon _n$. Let $A$ be a monotone subset of $V_n$. We say that $A$ is symmetric if there is a transitive permutation group $\Gamma$ on $\{1,2,\dots , n\}$ such that $A$ is invariant under $\Gamma$.

Theorem. For every symmetric monotone $A$, if $\mu _p(A)>\epsilon$ then $\mu _q(A)>1-\epsilon$ for $q=p+ c_1 \log (1/2\epsilon )/\log n$. ($c_1$ is an absolute constant.)

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Additional Information
  • Ehud Friedgut
  • Affiliation: Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel
  • Email:
  • Gil Kalai
  • Affiliation: Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel and Institute for Advanced Study, Princeton, New Jersey 08540
  • MR Author ID: 195990
  • Received by editor(s): March 27, 1995
  • Additional Notes: Research supported in part by grants from the Israeli Academy of Sciences, the U.S.-Israel Binational Science Foundation, the Sloan foundation and by a grant from the state of Niedersachsen.
  • Communicated by: Jeffry N. Kahn
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 2993-3002
  • MSC (1991): Primary 05C80, 28A35, 60K35
  • DOI:
  • MathSciNet review: 1371123