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  1. ( , Notices of the American Mathematical Society)
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  2. ; ( , Journal of the American Mathematical Society)

    Proving the “expectation-threshold” conjecture of Kahn and Kalai [Combin. Probab. Comput. 16 (2007), pp. 495–502], we show that for any increasing propertyF\mathcal {F}on a finite setXX,\[pc(F)=O(q(F)log⁡<#comment/>ℓ<#comment/>(F)),p_c(\mathcal {F})=O(q(\mathcal {F})\log \ell (\mathcal {F})),\]wherepc(F)p_c(\mathcal {F})andq(F)q(\mathcal {F})are the threshold and “expectation threshold” ofF\mathcal {F}, andℓ<#comment/>(F)\ell (\mathcal {F})is the maximum of22and the maximum size of a minimal member ofF\mathcal {F}.

     
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