MathDB

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.

Problem Statement