Can we prove or disprove this?
For a sufficiently large $n$, every set of at least $ n$ points in the plane with no three collinear has a subset that form the vertices of a convex $2012$-gon.
Gerry mentions the Happy Ending theorem but I don't see how it relates. If someone could show me the steps in the proof or disproof, that would be nice.
This follows from the theorem of Erdos and Szekeres (sometimes known as "The Happy Ending Theorem").
The statement of the theorem, with some discussion, is here. There is also a link there to the original paper.