Let C be a collection of n Jordan regions in the plane in general position, such that each pair of their boundaries intersect in at most s points, where s is a constant. If the boundaries of two sets in C cross exactly twice, then their intersection points are called regular vertices of the arrangement A(C). Let R(C) denote the set of regular vertices on the boundary of the union of C. We present several bounds on |R(C)|, determined by the type of the sets of C. (i) If each set of C is convex, then |R(C)| = O(n~(1.5+#epsilon#)) for any #epsilon#>0.~4 (ii) If C consists of two collections C_1 and C_2 where C_1 is a collection of n convex pseudo-disks in the plane (closed Jordan regions with the property that the boundaries of any two of them intersect at most twice), and C_2 is a collection of is tight in the worst case. (iii) If no further assumptions are made on the sets of C, then we show that there is a positive integer t that depends only on a such that |R(C)| = O(n~(2-1)/t).
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