Erdős and Pach showed that, for $d \ge 5$ odd, there exist constants $c_1(d), c_2(d) > 0$ such that $\frac{p - 1}{2p} n^2 - c_1 n^{4/3} ≤ f_d(n) \le \frac{p - 1}{2p} n^2 + c_2 n^{4/3}$ where $p = \lfloor\frac d2\rfloor$.