On a plane, is it possible to arrange $6$ red points and $6$ blue points such that
- No $2$ points coincide.
- For any line containing two or more points, not all the points on that line are of the same color.
One obvious arrangement is of course for all $12$ points to be on the same line. But is there any other non-trivial arrangements? After many trials I believe that this is impossible, but cannot come up with a proof. The difficulty is that there are many possibilities to consider, and I think maybe discussing by cases is not the way to go.
I wonder whether or not this lines-and-points set up can be translated into some other mathematical objects (e.g. graphs) and the result can be proved in a clean way.
Update
As pointed out by Erick Wong this is a generalization of Sylvester's problem. This survey mentions T. Motzkin's proof in the section "colored pointsets and monochromatic linear subsets" for arbitrary numbers of red and blue points. The proof uses the duality principle of the projective plane which I cut and paste below for those who are familiar with these concepts. I would still appreciate a proof in the original point formulation.
