Every monotone graph property has a sharp threshold

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 V n ( p ) = { 0 , 1 } n V_n(p)= \{0,1\}^n denote the Hamming space endowed with the probability measure μ p \mu _p defined by μ p ( ϵ 1 , ϵ 2 , … , ϵ n ) = p k ⋅ ( 1 − p ) n − k \mu _p (\epsilon _1, \epsilon _2, \dots , \epsilon _n)= p^k \cdot (1-p)^{n-k} , where k = ϵ 1 + ϵ 2 + ⋯ + ϵ n k=\epsilon _1 +\epsilon _2 +\cdots +\epsilon _n . Let A A be a monotone subset of V n V_n . We say that A A is symmetric if there is a transitive permutation group Γ \Gamma on { 1 , 2 , … , n } \{1,2,\dots , n\} such that A A is invariant under Γ \Gamma . Theorem. For every symmetric monotone A A , if μ p ( A ) > ϵ \mu _p(A)>\epsilon then μ q ( A ) > 1 − ϵ \mu _q(A)>1-\epsilon for q = p + c 1 log ⁡ ( 1 / 2 ϵ ) / log ⁡ n q=p+ c_1 \log (1/2\epsilon )/\log n . ( c 1 c_1 is an absolute constant.)