COMBINATORIAL COMPLEXITY OF SIGNED DISCS

Let C+ and C- be two collections of topological discs. The collection of discs is 'topological' in the sense that their boundaries are Jordan curves and each pair of Jordan curves intersect at most twice. We prove that the region U C+ - U C- has combinatorial complexity at most 10n - 30 where p = \C+\, q = \C-\ and n = p + q greater than or equal to 5. Moreover, this bound is achievable. We also show less precise bounds that are stated as functions of p and q.