Problem

Source: Chinese TST 2007 5th quiz P2

Tags: combinatorics proposed, combinatorics



Given $ n$ points arbitrarily in the plane $ P_{1},P_{2},\ldots,P_{n},$ among them no three points are collinear. Each of $ P_{i}$ ($1\le i\le n$) is colored red or blue arbitrarily. Let $ S$ be the set of triangles having $ \{P_{1},P_{2},\ldots,P_{n}\}$ as vertices, and having the following property: for any two segments $ P_{i}P_{j}$ and $ P_{u}P_{v},$ the number of triangles having $ P_{i}P_{j}$ as side and the number of triangles having $ P_{u}P_{v}$ as side are the same in $ S.$ Find the least $ n$ such that in $ S$ there exist two triangles, the vertices of each triangle having the same color.