On the number of congruent simplices in a point set

We derive improved bounds on the number of k-dimensional simplices spanned by a set of n points in R^d that are congruent to a given k-simplex, for k ≤ d - 1. Let f_k^(d)(n) be the maximum 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⁾(n) = O(n^5/3 · 2^O(α(2)(n))), f₂⁽⁴⁾ (n) = O(n^2+ε), f₂⁽⁵⁾ (n) = Θ(n^7/3), and f₃⁽⁴⁾ (n) = O(n^9/4+ε). 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 d ≤ 7 and k ≤ d - 2. Following Erdös and Purdy, we conjecture that this bound holds for larger values of d as well, and for k ≤ d - 2.