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Proving the "expectation-threshold" conjecture of Kahn and Kalai [Combin. Probab. Comput. 16 (2007), pp. 495–502], we show that for any increasing property \mathcal {F} on a finite set X, \[ p_c(\mathcal {F})=O(q(\mathcal {F})\log \ell (\mathcal {F})), \] where p_c(\mathcal {F}) and q(\mathcal {F}) are the threshold and "expectation threshold" of \mathcal {F}, and \ell (\mathcal {F}) is the maximum of 2 and the maximum size of a minimal member of \mathcal {F}. [ABSTRACT FROM AUTHOR]

Resolving a conjecture of Kühn and Osthus from 2012, we show that p = 1/√n is the threshold for the random graph Gn,p to contain the square of a Hamilton cycle. [ABSTRACT FROM AUTHOR]