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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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What have you tried / been taught / figured out so far? –  Eugene Shvarts Jul 5 '12 at 22:39
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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
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Wasn't this question just posted? –  Alex Becker Jul 6 '12 at 2:40

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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.

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