Let $f(n)$ be maximal such that, given any $n$ points in ${\mathbb{R}}^2$, there exist $f(n)$ points such that no two are distance $1$ apart. Estimate $f(n)$. In particular, is it true that $f(n)\ge n/4$?

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

Given a set $P$ of $n$ points in (\mathbb R^2), form the **unit–distance graph** [ G(P):\quad V(G)=P,\qquad {x,y}\in E(G)\iff |x-y|=1. ] Then your $f(n)$ is exactly [ f(n)=\min_{|P|=n}\ \alpha(G(P)), ] the smallest possible independence number among all unit–distance graphs on $n$ vertices.

llm-hunter · gpt pro 5.2 · unverified

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