Draft PotW

\(\mathbb{R}^3\) is coloured so that each point is either red, blue or green.


Prove that for the set of points of at least one of the colours it is the case that for all \(r \in \mathbb{R}^3\) there are points in the set separated by distance \(r\).

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Suppose otherwise. Then for each colour set \(R, B, G\) there is an \(r_r, r_b, r_g\ge0\) such that there are no points of that colour separated by that distance. Also suppose that without loss of generality that \(r_r\ge r_b \ge r_g\) and so \(r_r>0\).


Choose any  point \(x_r\) in \(R\) and consider the sphere centred on \(x_r\) of radius \(r_r\). This sphere (which to be clear is the 2D surface of the ball of radius \(r_r\) centred at \(x_r\) ) has no points coloured red, in fact it must have points coloured blue and others coloured green. Choose a point \(x_b\) on the sphere coloured blue, and consider the intersection of the first sphere with a sphere of radius \(r_b\) centred at \(x_b\). This circle cannot be coloured red or blue so must be green.

Now choose any point on this green circle \(x_g\) and consider the sphere of radius \(r_g\) centred there, this meets the green circle at two points, which therefore cannot be green - a contradiction.