We show that n arbitrary circles in the plane can be cut into O(n 3/2+ε ) arcs, for any ε > 0, such that any pair of arcs intersects at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree.
We derive improved bounds on the number of fc-dimensional simplices spanned by a set of n points in R'^ that are congruent to a given fc-simplex, for fc < d -1. Let f^ '{n) be the maximum number of fc-simplices spanned by a set of n points in R"* that are congruent to a given fc-simplex. We prove that /j (n) = 0("5/3 .20(a^(n)))_ y(4)(") ^ 0{n^+n, A^\n) = 0(n^/3), and /^^'(n) = 0(TI^/**+^). We also derive a recurrence to bound fjf'^ (n) for arbitrary values of k and d, and use it to derive the bound fjfhn) = 0{n^/'^) for d < 7 and A; < d -2. Following Erdos and Purdy, we conjecture that this bound holds for laxger values of d as well, and for fc < d -2.
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