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This question is about 3-colorings of the plane in which every line is bichromatic (or monochromatic), i.e., there are no three collinear points of different colors. Such colorings trivially exist, see e.g. http://www.cut-the-knot.org/proofs/3ColorsBichromaticLines.shtml#solution

However, there are more interesting such colorings, for example the one in Monsky's proof using valuations, see e.g. https://www.math.lsu.edu/~verrill/teaching/math7280/triangles.pdf

These colorings are useful when we want to apply Sperner's lemma (about graph-theoretic triangulations) to solve some problem about geometric triangulations (where a vertex of one triangle might fall on the side of another), that is why I wonder what different constructions there are that are not based on a simple modification of the above two examples.

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There's a definitive answer given by Hales and Straus in http://msp.org/pjm/1982/99-1/pjm-v99-n1-p03-s.pdf (but note their comments re priority in the introduction). The brief summary is that the colorings you want for Desarguesian planes correspond to valuations of the underlying field.

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  • $\begingroup$ Is there anything similar for 3-colorings of the plane such that every unit circle is bichromatic? $\endgroup$
    – domotorp
    Commented Apr 4, 2019 at 10:32
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    $\begingroup$ @domotorp: the simplest answer is that I don’t know. The answer could well depend on your choice of field. $\endgroup$
    – Chris Godsil
    Commented Apr 4, 2019 at 18:08

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