  Given a family of graphs $\mathcal{H}$, the extremal number $\textrm{ex}(n,
\mathcal{H})$ 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 $\textrm{ex}(n, \mathcal{H}_r)
= \Theta(n^r)$. This solves a longstanding problem in the area of extremal
graph theory.
