Rational exponents in extremal graph theory

Given a family of graphs $\mathcal{H}$, the extremal number $$\text{ex}$(n, $\mathcal{H}$)$isthelargest$m$forwhichthereexistsagraphwith$n$verticesand$m$edgescontainingnographfromthefamily$$\mathcal{H}$$ as a subgraph. We show that for every rational number $r$ between $1$ and $2$, there is a family of graphs ${\mathcal{H}}_r$ such that $$\text{ex}$(n, $\mathcal{H}$_r) = \Theta(n^r)$. This solves a longstanding problem in the area of extremal graph theory.