Turandotdotdot

How many edges can you place in a graph on n vertices without creating a triangle? Some trial and error suggests that the best thing to do is to split your vertices into two classes of size n/2 and to connect all pairs in different classes, and indeed this is best possible. Starting from this century-old result, in this talk we'll keep asking variants of the same question and finding totally different and often surprising answers. Along the way, we'll encounter connections to algebraic geometry, number theory, probability, and topology, and we'll see a whole host of simple questions that remain wide open even today.