What are interesting 3-colorings of the plane without rainbow lines?

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.

Answer

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.

Attribution
Source : Link , Question Author : domotorp , Answer Author : Chris Godsil

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