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Every monotone graph property has a sharp threshold. (English) Zbl 0864.05078
This paper proves a conjecture of Linial about random graphs: every monotone graph property has a sharp threshold. The authors show this to be a special case of a more general result. To describe the general result, consider all \(n\)-dimensional 0-1 vectors. Endow a measure \(\mu_p\) on this set by setting each element of the vector to 1 with probability \(p\) independently of the other elements. Consider any monotone set \(A\) of 0-1 vectors such that \(\mu_p(A)>\varepsilon\). If there is a transitive permutation group on \(\{1,2,\dots,n\}\) that leaves \(A\) invariant then \(\mu_q(A)>1-\varepsilon\) where \(q=p+c\log(1/2\varepsilon)/(\log n)\). Here \(c\) is an absolute constant.
The paper describes some extensions of this result as well as applications to problems other than the one raised by Linial. Of particular interest is the connection to Talagrands work on isoperimetric inequalities.

MSC:
05C80 Random graphs (graph-theoretic aspects)
28A35 Measures and integrals in product spaces
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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