We’ll go back now to Chapter 2, on infinite Ramsey’s theorem, and its connection (via Konig’s tree lemma) to finite Ramsey’s theorem.
Everyone should read Chapter 2, through at least to the middle of page 48. There’s one substantial theorem in this span, namely Theorem 2.1 (although it’s not really too different from Theorem 1.31, finite Ramsey’s theorem, and is in fact a little easier, since one has much less “bookkeeping” to do to make the proof work. Section 2.2 through the middle of page 48 is mostly definitions (of partially ordered sets, tree orders, et cetera.).
Theorem 2.1 is proven by induction, with the base case being the obvious infinite pigeon-hole principle — if infinitely many pigeons arrange themselves among only finitely many pigeon holes, then at least one of the pigeon-holes will end up with infinitely many pigeons. I think it’s much more instructive to start with the base case \(p=2\). So: Greyson and Joe can present the proof of the following (the case [/latex]p=2[/latex] of Theorem 2.1): if the edges of an infinite complete graph are colored with finitely many colors, then the graph has an infinite complete subgraph all of whose edges have the same color. Then Luca and Sean can present the induction step that completes the proof of Theorem 2.1.
Then we’ll move onto to partial orders (to lay the groundwork for deducing finite results from infinite ones). As I said earlier, everyone should read through all the background on partial orders and tree-orders, and we can discuss and points that are confusing during the live meeting. Bailee and Anthony can first explain what it means for a partial order to be a tree (in the partial order sense), and then present Exercise 2.3 (connecting this concept to trees in the graph-theoretic sense); then Ted and Ryan can present Konig’s lemma (Theorem 2.6).