my main problem is:

given following graph: $G=(V,E)(\text{open disks} ,R^2)$,

i'm trying the number of regions/hyperedge that can be formed by circles/open disks in 2-d dimension. I know that for $G=(V,E)(\text{rectangles},R^2)$ the number of regions/hyperedge is at most $|V|^2$, but I not able to do the same for circles/ open disks.

Let D be a finite collection of open discs in the plane. Let >0. We say that a point xR2 is -deep in D if belongs to at least D discs in D. A subset DD is called a -cover for D if every -deep point lies in the union DDD. Show that any finite collection D of discs as above always contains a -cover of cardinality O((1/)log(1/))