For which $n$ we may mark $n$ red and $n$ blue points on the Euclidean plane, not all on a line, so that any line which passes through two points of different colour contains another point?

For $n=1991$ this was proposed in a not-up-to-date edition of Prasolov's problem book on planimetry, but the suggested solution actually solves a different problem (in the newest edition this is fixed.)

The following example for $n=6k$ is communicated by M. Belozerov: take a regular $4k$-gon, colour its vertices alternatively and infinite points of the sides arbitrarily.