Happy ending problem

Ok! To innagurate my new "blog", here's a math problem I've been working on recently (really a lot more since school got out).

The problem is:

What's the greatest number of points you can put on a plane (with no 3 in a line) until you can be certain some subset of them make the verticies of a convex n-gon (for some n).

The guy who gave me this problem had found that certain Ramsey numbers were an upper bound, but that's not much help since we didn't think they were a very tight upper bound, and no one knows how to calculate them anyhow.

With a little research, we discovered that this is actually a well-known problem which you can read a bit more about here:
http://mathworld.wolfram.com/HappyEndProblem.html

It's bounded both above and below. There's a conjecture that the max points (I'll call it g(n)) is given by: g(n) = 2^(n-2)+1.

I was going to report some of our thoughts (they don't really count as "progress"), but it occured to me it would be much easier with pictures which I'm too busy to make right now.

The good news is that I'm almost done with some Mathematica functions that will extend the cases we can visualize and explore--so maybe when I'm done with that I'll post an update and some pictures.

In the meantime, insights are welcome!