Q:

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.

A:

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.

