**Theorem**

Show that for any integer $n ? 3$ there exists an integer $m$ such that whenever $m$ points in the plane are in general position, some $n$ of these points are the vertices of a convex n-gon.

A collection of points in the plane are in general position if no 3 of the points are collinear. A polygon with m sides, or m-gon, is convex if the line segment joining any 2 interior points is also within the m-gon.

**Proof of Theorem**

Let $m ? R^{(4)}(n,5)$ and let A be any set of $m$ points in general position. The class C of all 4-element subsets of A is partitioned into 2 subclasses, $C_1$ and $C_2$, the former being the subclass of quartets of points which determine convex quadrilaterals. Now, according to Ramsey's theorem, there exists an n-element subset, B, of A such that every 4-element subset of B belongs to $C_1$, or there exists a 5-element subset, $B'$, of $A$ such that every 4-element subset of $B'$ belongs to $C_2$. The latter alternative is impossible, by Lemma 1. The former alternative must then hold; and Lemma 2 at once gives the proof.

The proof below is based on two lemmas.

**Lemma 1**

For any 5 points in the plane in general position, some 4 of them are the vertices of a convex quadrilateral.

**Proof**

Let the smallest convex polygon that contains the 5 points be a convex n-gon; obviously, all the vertices of this n-gon belong to the given set of points. If $n = 5$ or $n = 4$, there is nothing to prove. If $n = 3$ (the only other possibility), there is a triangle formed by 3 of the 5 points (say, A, B, and C), and the other 2 points, D and E, are inside the triangle. Then the line determined by D and E will divide the triangle into 2 parts such that I of these 2 parts contains 2 vertices of the triangle (say, A and B); ABDE is the sought convex quadrilateral.

**Lemma 2**

If $n$ points are located in general position in the plane, and if every quadrilateral formed from these $n$ points is convex, then the $n$ points are the vertices of a convex n-gon.

**Proof**

Suppose the $n$ points do not form a convex n-gon. Consider the smallest convex polygon that contains the $n$ points. At least one of the $n$ points (say, the point P) is in the interior of this polygon. Let Q be one of the vertices of the polygon. Divide the polygon into triangles by drawing line segments joining Q to every vertex of the polygon. The point P then will be in the interior of one of these triangles, which contradicts the convexity hypothesis.