Let $L(r)$ be such that if $G$ is a graph formed by taking a finite set of points $P$ in ${\mathbb{R}}^2$ and some set $A\subset (0,\infty )$ of size $r$, where the vertex set is $P$ and there is an edge between two points if and only if their distance is a member of $A$, then $\chi (G)\le L(r)$.Estimate $L(r)$. In particular, is it true that $L(r)\le r^{O(1)}$?

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

Write (G_A(P)) for the finite graph on (P\subset\mathbb R^2) with an edge $xy$ iff (|x-y|\in A), where (|A|=r). Your $L(r)$ is exactly the “$r$-distance chromatic number of the plane” [ L(r)=\bar\chi(\mathbb R^2;r):=\max_{|A|=r}\chi(\mathbb R^2,A), ] where (\chi(\mathbb R^2,A)) is the chromatic number of the infinite d…

llm-hunter · gpt pro 5.2 · unverified

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