Every monotone graph property has a sharp threshold

Ehud Friedgut*, Gil Kalai

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

243 Scopus citations

Abstract

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 Vn(p) = {0, 1}n denote the Hamming space endowed with the probability measure μp defined by μp12,...,εn) = pk · (1-p)n-k, where k = ε1 + ε2 + ⋯ + εn. Let A be a monotone subset of Vn. We say that A is symmetric if there is a transitive permutation group Γ on {1,2,...,n} such that A is invariant under Γ. Theorem. For every symmetric monotone A, if μp(A) > ε then μq(A) > 1-ε for q = p + c1 log(1/2ε)/log n. (c1 is an absolute constant.)

Original languageEnglish
Pages (from-to)2993-3002
Number of pages10
JournalProceedings of the American Mathematical Society
Volume124
Issue number10
DOIs
StatePublished - 1996

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