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The Probabilistic Method
Tim Colmer and Erdos Renney invented the field of random graph theory from scratch. They proved amazingly precise properties about it, which I think Tim will tell us more about in a minute. The probabilistic method is an approach that lets you choose your network entirely at random. It's incredibly difficult to write down a specification that demonstrates that you really do need anything like as many people as the proof requires. But once you've seen this example, it just changes the way you think as a mathematician.