Erdős Problem #1207 — Large subsets avoiding isosceles triangles

Let $P_d(n)$ be the largest integer such that every set of $n$ points in $R^d$ contains $P_d(n)$ points with no three forming an isosceles triangle. Estimate $P_d(n)$; in particular, is $P_2(n)<n^{1-c}$ for some constant $c>0$ and all sufficiently large $n$?