Proof of the Van den Berg–Kesten Conjecture

We prove the following conjecture of J. van den Berg and H. Kesten. For any events [Ascr ] and [Bscr ] in a product probability space, Prob([Ascr ]□[Bscr ]) [les ] Prob([Ascr ])Prob([Bscr ]), where [Ascr ]□[Bscr ] is the event that [Ascr ] and [Bscr ] occur ‘disjointly’.