Rational Exponents for Generalized Tur\'an Numbers

The generalized Tur\'an 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 $r\geq 1$ there exist graphs $H,F$ such that $\text{ex}(n,H,\{F\})=\Theta(n^r)$. We extend a result of Bukh and Conlon to show that for every non-empty graph $H$ on $v\geq 2$ vertices and every rational $r$ in the interval $[v-1,v]$ there exists a finite family $\mathcal{F}_r$ such that $\text{ex}(n,H,\mathcal{F}_r)=\Theta(n^r)$.