# Unit distance graph in $\mathbb{R}^2$

Suppose $G$ is the simple graph with vertex set $\mathbb{R}^2$ created by connecting two points iff they are distance one from each other in the plane. How can we prove that $4\le \chi(G)\le7$? I think I have seen this problem somewhere before but can't pinpoint it. No hints please.

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I have no idea where to start and I am told it is a known result so I'd be curious to see it. :) – Aria Fitzpatrick Jul 5 '12 at 23:04
Wasn't this question just posted? – Alex Becker Jul 6 '12 at 2:40

The fact that there is a $4$-chromatic unit distance graph implies that $\chi \geq 4$, while the appropriate colouring of a hexagonal tiling shows that $\chi \leq 7$. For details, here's the Wikipedia article.