The number of congruent simplices in a point set

For 1 less than or equal to k less than or equal to 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 f(2)((3)) (n) = O (n(5/3)2(O)(alpha(2(n)))) f(2)((4)) (n) = O (n(2+epsilon)), for any epsilon > 0, f(2)((5)) (n) = Theta (n(7/3)), and f(3)((4)) (n) = O(n(20/9+epsilon)), for any epsilon > 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+epsilon)), for any epsilon > 0, for d less than or equal to 7 and k less than or equal to d -2. Following Erdos and Purdy, we conjecture that this bound holds for larger values of d as well, and for k less than or equal to d - 2.