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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.

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