On the union complexity of families of axis-parallel rectangles with a low packing number

Let R be a family of n axis-parallel rectangles with packing number p − 1, meaning that among any p of the rectangles, there are two with a non-empty intersection. We show that the union complexity of R is at most O(n + p²), and that the (k − 1)-level complexity of R is at most O(n + kp²). Both upper bounds are tight.