Points in the plane

Prove that for every \( k \in \mathbb N \) there exists a \( n \in \mathbb N \) such that any set of \( n \) points in the plane contains:

Variant
Either \( k \) points on a line or \( k \) points in the general position.
Solution
We choose \( n = R_ {3{,}2} (k) \) according to Ramsey theorem for triplets and two colors.

We color the triplets at \( n \) points by coloring the triplets in one general position with one color and the collinear triplets with the other.

The monochromatic system of triplets at \( k \) points corresponds to either \( k \) points in the general position or \( k \) points on a common line.
Variant
Either \( k \) points on a straight line or \( k \) points in a convex position.
Solution
First we determine \( l \), which is the necessary number of points in the general position so that according to the Erdős-Szekeres theorem there is between them a \( k \)-tuple of points in the convex position.

Then, according to Ramsey theorem for triplets, we find \( n = R_3 (k, l) \) so that we have either \( k \) points on a common line or \( l \) points in general position.