Variant of Erdos-Szekeres theorem

Question

I'm having trouble proving the following theorem:

For all $n \in \mathbb{N},$ $\exists$ $N \in \mathbb{N}$ such that any set of $N$ points in the plane will contain either a convex $n$-gon or $n$ collinear points.

So, if the $N$ points are in general position, we can apply Erdos-Szekeres and be done. If not, it must be shown that among the $N$ points will be a line with $n$ points on it. I'm not sure how to prove the latter part.

Answer 1

Here's one approach; there are probably others.

As you've observed, there exists some $N_0$ so that, if you can find $N_0$ points, no three of which lie on a line, then you're done. If you can find $n$ points, all of which lie on a line, then you're also done. This looks a bit like Ramsey's theorem, where you want to find a large substructure of one of two types. Can you relate this to a version of Ramsey's theorem?

(The rest of the solution is contained within the following spoiler.)

Color the complete $3$-regular hypergraph on $N$ vertices red and blue, where red edges correspond to triples of points lying on a line. Supposing $N>R_2(N_0,n;3)$, there exists a structure of one of the two types by Ramsey's theorem for hypergraphs.