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How to show that there is exist equivalence relation $\varsigma$ on ${\mathbb{R}}^2$ such that the following conditions hold:

  1. Exist only $7$ equivalence classes by $\varsigma$.
  2. For every $x,y \in {\mathbb{R}}^2$ if the distance between $x$ and $y$ is $1$, then $x,y$ are in different equivalence classes.
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1 Answer 1

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From the Wikipedia article on this problem:

The upper bound of seven on the chromatic number follows from the existence of a tessellation of the plane by regular hexagons, with diameter slightly less than one, that can be assigned seven colors in a repeating pattern to form a 7-coloring of the plane; according to Soifer (2008), this upper bound was first observed by John R. Isbell.

There's a picture of the colouring in the article, but stackexchange seems to not support svg for some reason.

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Great! Thanks!! –  17SI.34SA Nov 2 '12 at 18:06

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