MathNet 2010 — Part 01 Question 101

$n$ distinct points ($n \ge 3$) are marked on a plane so that no three of them lie on the same line. All points are connected with the segments. All segments are painted one of the four colors so that if in some triangle (with the vertices at the marked points) two sides have the same color, then all its sides have the same color (each of the four colors is used).
What is the largest possible value of $n$?
(S. Sobolevskii)