The number of congruent simplices in a point set
For 1 ≤ k ≤ d -1, let fk(d) (n) be the maximum possible number of k-simplices spanned by a set of n points in ℝd that are congruent to a given k-simplex. We prove that f2(3) = O(n5/3 2O(α2(n))), f2(4)(n) = O(n2+ε), for any ε > 0, f2(5)(n) = Θ(n7/3), and f3(4) (n) = O(n20/9+ε), for any ε > 0. We also derive a recurrence to bound fk(d)(n) for arbitrary values of k and d, and use it to derive the bound fk(d) (n) = O(nd/2+ε), for any ε > 0, for d ≤ 7 and k ≤ d - 2. Following Erdos and Purdy, we conjecture that this bound holds for larger values of d as well, and for k ≤ d - 2.
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