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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_2^{(3)} (n) = O(n^{5/3} 2^{O(\alpha ^2 (n))} )$$\end{document}, 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 Erdős and Purdy, we conjecture that this bound holds for larger values of d as well, and for k ≤ d − 2.