Matching Random Subsets of the Cube with a Tight Control on One Coordinate

Consider a measure . on [0,1]2, and 2n points X1,.,Xn,Y1,.,Yn that are independent and distributed according to .. Consider 2n points U1,.,Un,V1,.,Vn that are independent and uniformly distributed on [0,1]. Then there exists a constant K (independent of .) such that if s..n/K, with probability close to 1 we can find a one-to-one map . from {1,.,n} to itself such that .i.n,|Ui.V.(i)|.Ks, 1n.i.n|Xi.Y.(i)|.K(sn)1/2.