Is there a set of $n$ points in ${\mathbb{R}}^2$ such that every subset of $4$ points determines at least $3$ distances, yet the total number of distinct distances is$$\ll \frac{n}{\sqrt{\log n}}?$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

A concrete example is a **truncated anisotropic lattice** (a “stretched grid”). Take [ P_m={(i,\sqrt2,j):0\le i,j\le m-1}\subset \mathbb Z\times \sqrt2,\mathbb Z, ] so (|P_m|=m^2). ([Erdős Problems][1])