The Erdős-Szekeres problem on points in convex position and its proof using Ramsey theorem are well know. The problem goes like this:
For every natural number $k$ there exists a number $n(k)$ such that any $n(k)$-point set $X \subset \mathbb{R}^2$ in general position contains a $k$-point convex independent subset.
My question is, how to prove the $d$-dimensional version of the problem?
For every natural number $k$ there exists a number $n_d(k)$ such that any $n_d(k)$-point set $X \subset \mathbb{R}^d$ in general position contains a $k$-point convex independent subset.
The solution will probably use Ramsey theorem, but I don't know how.