Ajtai–Komlós–Tusnády theorem

Statement

The Ajtai–Komlós–Tusnády theorem (also known as the AKT optimal matching theorem) is a result in probabilistic combinatorics. Given two random, distinct sets of points $X=(X_1,\dots,X_k)$ and $Y=(Y_1,\dots,Y_k)$ in the unit square $[0,1]^2$, the theorem gives upper and lower bounds for the minimal total distance needed to match the points in one set to those in the other.