Games You'd Play at an Infinitely Large Birthday Party (or: How I Learned to Love Infinitary Combinatorics)

Hilbert's Hotel is back! And this time, you are throwing a birthday party. Naturally, being a social butterfly inclined towards theorizing about infinities, you decide to invite infinitely-many guests to your party at Hilbert's Hotel. However, as is common knowledge, no party is complete without a few good games. But which ones should you play with such a multitude of guests? Well, ones involving guessing hats, of course! After this ice-breaker, we will then try to see who among the guests knows the other guests; we will see that we can find an infinite subset of guests who either all know each other or who don't know anyone from the subset.

In more formal language, we will introduce a few key themes in the set-theoretic approach to infinitary combinatorics, by exploring the Axiom of Choice (hat game) and Ramsey's Theorem (guest arrangements). If time permits, we will explore König's Theorem about infinite, finitely-splitting trees and perhaps use this and Ramsey's Theorem to prove the finite Ramsey Theorem.