Every Monotone Graph Property Has a Sharp Threshold Ehud Friedgut and Gil Kalai
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چکیده
In their seminal work which initiated random graph theory Erd os and R enyi discovered that many graph properties have sharp thresholds as the number of vertices tends to in nity We prove a conjecture of Linial that every monotone graph property has a sharp threshold This follows from the following theorem Let Vn p f g denote the Hamming space endowed with the probability measure p de ned by p n p p n k where k n Let A be a monotone subset of Vn We say that A is symmetric if there is a transitive permutation group on f ng such that A is invariant under Theorem For every symmetric monotone A if p A then q A for q p c log logn c is an absolute constant
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Given a monotone graph property P consider p P the proba bility that a random graph with edge probability p will have P The function d p P dp is the key to understanding the threshold behavior of the property P We show that if d 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 t...
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تاریخ انتشار 2009