I just found out via the NYU seminar announcement that the empty hexagon problem has finally been solved — any sufficiently large point set in general position has six points that form the vertices of a hexagon without other points inside it. A similar result was known for pentagons, and known to be impossible for heptagons — there are infinite point sets without empty heptagons. Anyone know where to track down more details? Googling "Gerken hexagon" didn't find anything useful, just a couple more talk announcements.

ETA: Apparently the basic idea is to start with nine points in convex position (known from Erdős & Szekeres to exist in large enough point sets) and then proceed by messy case analysis. Suresh has a little more.


Dang. i just made the same post. Tobias Gerken has a web page and I just emailed him. -- Suresh