Separating Pairs of Points by Standard Boxes

Let A be a set of distinct points in ℝ^d. A 2-subset {a, b} of A is called separated if there exists a closed box with sides parallel to the axes, containing a and b but no other points of A. Let s(A) denote the number of separated 2-sets of A and put f(n, d) = max {s(A): A ⊂ ℝ^d, |A| = n}. We show that f(n, 2) = [n²/4] + n − 2 for all n≥2 and that for each fixed dimension d f(n,d)=(1−1/2 2 d−1−1)⋅n2/2+o(n2).