Rational exponents in extremal graph theory

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