Combinatorics (Fall 2010)/Ramsey theory

Ramsey's Theorem

The "Happy Ending" problem

Theorem (Erdős-Szekeres 1935)

For any positive integer [math]\displaystyle{ n\ge 3 }[/math], there is an [math]\displaystyle{ N(n) }[/math] such that any collection of [math]\displaystyle{ N\ge N(n) }[/math] points in the Euclidian plane, no three of which are collinear, contains [math]\displaystyle{ n }[/math] points forming a convex [math]\displaystyle{ n }[/math]-gon.