Let us look at ways this problem can be generalized. If we stay with three colors then we can add any additional points to the groups of two triangles and one line at will since adding a point to a convex hull will make it bigger we will still have nonzero intersection for the three sets. In particular we will have for any integer n greater than seven will have a division of n into three parts with any two parts differing in cardinality by at most one.
The next step would be to increase the number of colors and try to get a similar result. The next Tverberg result is that if we have 10 points in a plane and we divide them into four nonempty sets then the intersection of the convex hulls of the sets is nonempty. So the next step is to look at this result and try to proceed as we did in the case of three colors. I hope to do this in the next of this series of posts.