On the Generalized Rational Exponents Conjecture

For fixed graphs $H$ and $F$, let $\text{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. In this note, we prove the generalized rational exponents conjecture, posed by Gerbner and Palmer, showing that for every rational number $\alpha \ge 1$, there exist fixed graphs $H_{\alpha }$ and $F_{\alpha }$ such that $$\text{ex}(n,H_{\alpha },F_{\alpha })=\mathrm{\Theta }(n^{\alpha })\mathrm{.}$$ Furthermore, the counting graph $H_{\alpha }$ can always be chosen connected with diameter at most $3$. Our argument hinges on a localization–compression–shift framework, which transforms the Bukh–Conlon finite family construction for edges into a generalized Turán problem setting with a single forbidden graph.