Rational Exponents for Generalized Turán Numbers

The generalized Turán number $\text{ex}(n,H,\mathcal{F})$ denotes the maximum number of copies of $H$ in an $n$-vertex graph which contains no copies of any graph in a family $\mathcal{F}$ of graphs. The generalized rational exponents conjecture states that for every rational $r\ge 1$ there exist graphs $H,F$ such that $\text{ex}(n,H,\{F\})=\mathrm{\Theta }(n^r)$. We extend a result of Bukh and Conlon to show that for every non-empty graph $H$ on $v\ge 2$ vertices and every rational $r$ in the interval $[v-1,v]$ there exists a finite family ${\mathcal{F}}_r$ such that $\text{ex}(n,H,{\mathcal{F}}_r)=\mathrm{\Theta }(n^r)$.