Let $x_1,\dots ,x_n\in {\mathbb{R}}^2$ and let $R(x_i)=\#\{|x_j-x_i|:j\ne i\}$, where the points are ordered such that$$R(x_1)\le \cdots \le R(x_n).$$Let $g(n)$ be the maximum number of distinct values the $R(x_i)$ can take. Is it true that $g(n)\ge (1-o(1))n$?

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

What is known is that $g(n)$ is linear in $n$, but the best published lower bound is still bounded away from $n$ by a fixed constant factor:

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.