The number of congruent simplices in a point set

For 1 ≤ k ≤ d -1, let f_k^(d) (n) be the maximum possible number of k-simplices spanned by a set of n points in ℝ^d that are congruent to a given k-simplex. We prove that f₂⁽³⁾ = O(n^5/3 2^O(α2(n))), f₂⁽⁴⁾(n) = O(n^2+ε), for any ε > 0, f₂⁽⁵⁾(n) = Θ(n^7/3), and f₃⁽⁴⁾ (n) = O(n^20/9+ε), for any ε > 0. We also derive a recurrence to bound f_k^(d)(n) for arbitrary values of k and d, and use it to derive the bound f_k^(d) (n) = O(n^d/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.