We can assign a colour, red or blue, to the points of a plane so that there isn't a segment of same colour

Question

Prove we can assign a colour, red or blue, to the points of a plane so that there isn't a segment of same colour.

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Answer 1

Colour a point $P$ red if the distance from $P$ to the origin is rational, blue otherwise. If $S$ is any straight line segment, the set of distances from points on $S$ to the origin is an interval of real numbers, so it contains both rational and irrational numbers; i.e., $S$ contains both red and blue points.

A monochromatic set for this coloring, if it is connected and has more than one point, must be a circle centered at the origin, or an arc of such a circle.