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Ehud Friedgut, Gil Kalai
2018 Proceedings of the American Mathematical Society  
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 µp( 1 , 2 , . . . , n) = p k · (1 − p) n−k , where k = 1 + 2 + · · · + n. Let A be a
more » ... 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) > then µq(A) > 1− for q = p + c 1 log(1/2 )/ log n. (c 1 is an absolute constant.)
doi:10.1090/s0002-9939-96-03732-x fatcat:qermec32eregxexboarvkbkele