# 2012-gon- subsets of vertices.

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.

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## 1 Answer

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.

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Could you show how? –  Xuan Huang Jul 11 '12 at 7:05
Are you familiar with the theorem? Have you tried to find it and read it and see what it says? –  Gerry Myerson Jul 11 '12 at 7:28
Yes, but I don't quite see how it relates. –  Xuan Huang Jul 12 '12 at 2:08
Quoting from the paper: "Can we find for a given $n$ a number $N(n)$ such that from any set containing at least $N$ points it is possible to select $n$ points forming a convex polygon? There are two particular questions: (1) does the number $N$ exist? (2) If so, how is the least $N(n)$ determined? We give two proofs that the first question is to be answered in the affirmative. Both of them will give definite values for $N(n)$." How can you read that, and still ask how it relates to your question? It is your question! –  Gerry Myerson Jul 12 '12 at 7:18