Rational exponents in extremal graph theory

Given a family of graphs H, the extremal number ex(n,H) is the largest m for which there exists a graph with n vertices and m edges containing no graph from the family H as a subgraph. We show that for every rational number r between 1 and 2, there is a family of graphs Hr such that ex(n,Hr)=Θ(nr). This solves a longstanding problem in the area of extremal graph theory.