This talk will cover the basics of dealing with graphs with an infinite number of verticies, focusing mostly on countably many verticies.

I will cover basic definitions of graph theory, Menger's theorem (Only 1 version, don't worry), trees, girths and chromatic numbers briefly for people not familiar with the terms. We will then jump into the sweet stuff, defining teeth and spines, local infinity, rays...

I will then present (and prove some) quite useful theorems and lemmas, including <q>de Bruijn & Erdos theorem</q> and <q>Star-Comb Lemma</q>. Finally, we will talk about Universal graphs, the Rado graph and why they are cool.