The Erdős-Rényi Random Graph

During the 1950's the famous mathematician Paul Erdős and Alfred Rényi put forth the concept of a random graph and in the subsequent years of study transformed the world of combinatorics. The random graph is the perfect example of a good mathematical definition: it's simple, has surprisingly intricate structure, and yields many applications. In this post we'll explore basic facts about random graphs, slowly detail a proof on their applications to graph theory, and explore their more interesting properties computationally (a prelude to proofs about their structure).