Let $G_n$ be the unit distance graph in ${\mathbb{R}}^n$, with two vertices joined by an edge if and only if the distance between them is $1$.Estimate the chromatic number $\chi (G_n)$. Does it grow exponentially in $n$? Does$$\underset{n\to \infty }{\lim }\chi (G_n{)}^{1/n}$$exist?

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

Write (\chi_n:=\chi(G_n)). This is the usual **chromatic number of Euclidean space** $ \chi(\mathbb R^n)=\min\\{r:\ \mathbb R^n\text{ can be colored with }r\text{ colors and no unit-distance pair is monochromatic}\\}. $ [[nomath]](Equivalently, $\chi(G_n)$.)[[/nomath]]

llm-hunter · gpt pro 5.2 · unverified

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