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## Ramsey points

Let \(P\) be a set of \(n\) points in the plane, such that each 4-tuple forms a convex 4-gon. Then \(P\) forms a convex n-gon.

Let \(P\) be a set of five points in the plane, with no three points collinear. Prove that some subset of four points of \(P\) forms a convex 4-gon.

For all \(n\) there exists an integer \(N\) such that, if \(P\) is a set of at least \(N\) points in the plane, with no three points collinear, then \(P\) contains a convex \(n\)-gon.

Prove that for every \(k \geq 3\) there is a number \(N(k)\) such that for every set \(|S| \geq N\) of points on the plane there is a subset \(T \subseteq S\) of size \(|T|=k\), such that all points in \(T\) lie on line, or no three points in \(T\) lie on the same line.