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 H_r such that ex (n, H_r) = Θ(n^r) . This solves a longstanding problem in the area of extremal graph theory.