The XYZ Conjecture and the FKG Inequality

Consider random variables x1,⋯,xn, independently and uniformly distributed on the unit interval. Suppose we are given partial information, Γ, about the unknown ordering of the x's; e.g., Γ = {x1 < x12, x7 < x5,⋯}. We prove the "XYZ conjecture" (originally due to Ivan Rival, Bill Sands, and extended by Peter Winkler, R. L. Graham, and other participants of the Symposium on Ordered Sets at Banff, 1981) that P(x1