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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It's not the solution, because there's (many) lines through the origin that are all blue. – Jul 02 '16 at 08:07
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Do you mean, colour a point $(x,y)$ blue if both coordinates are rational? Then every point on the line $y=\sqrt2$ will be red. – bof Jul 02 '16 at 08:15
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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.
bof
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