Packing random rectangles

Abstract. A random rectangle is the product of two independent random intervals, each being the interval between two random points drawn independently and uniformly from [0,1]. We prove that te number Cn of items in a maximum cardinality disjoint subset of n random rectangles satisfies where K is an absolute constant. Although tight bounds for the problem generalized to d > 2 dimensions remain an open problem, we are able to show that, for some absolute constat K, Finally, for a certain distribution of random cubes we show that for some absolute constant K, the number Qn of items in a maximum cardinality disjoint subset of the cubes satisies