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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.

However, there are more interesting such colorings, for example the one in Monsky's proof using valuations, see e.g.

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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up vote 2 down vote accepted

There's a definitive answer given by Hales and Straus in (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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