Note: I won’t have access to my office next week, so I’ll have no blackboard. So when making a presentation you should use some kind of slides — powerpoint or (preferably, since it is optimized for math) Beamer — that you can share on the screen; or use an app that allows you to share a (virtual) blank page that you can write on in a way that everyone can see.
I’ll begin by discussing the state of the art for diagonal Ramsey numbers. Then we will go on to full-blown Ramsey’s theorem. This is Section 1.6; everyone should read this.
- Section 1.6 starts with a treatment of the case of coloring triples with 2 colors. Bailee and Sean can present on this.
- The section continues with an outline of the induction argument that allows us to extend to sets larger than triples (heading “From triples to \(p\)-tuples”), still with two colors. Casey and Nick can present on this.
- The section ends with a proof for many colors (heading “A different proof”). Colin and Alex can present on this.
Here’s a fun exercise to think about. Use Ramsey’s Theorem (you’ll need to use the more general statement about coloring \(p\)-tuples, rather than the weaker one about coloring pairs) to prove the following statement: for every \(k\) there is \(N(k)\) such that however \(N(k)\) points are put on a plane, with no three in a line, there is some \(k\) points among the \(N(k)\) that form the vertices of a convex \(k\)-sided polygon.