Rational Exponents for General Graphs

A rational number $r$ is a $\mathbf{\text{realizable}}\mathbf{\text{exponent}}$ for a graph $H$ if there exists a finite family of graphs $\mathcal{F}$ such that $\mathrm{e}\mathrm{x}(n,H,\mathcal{F})=\mathrm{\Theta }(n^r)$, where $\mathrm{e}\mathrm{x}(n,H,\mathcal{F})$ denotes the maximum number of copies of $H$ that an $n$-vertex $\mathcal{F}$-free graph can have. Results for realizable exponents are currently known only when $H$ is either a star or a clique, with the full resolution of the $H=K_2$ case being a major breakthrough of Bukh and Conlon. In this paper, we establish the first set of results for realizable exponents which hold for arbitrary graphs $H$ by showing that for any graph $H$ with maximum degree $\mathrm{\Delta }\ge 1$, every rational in the interval $[v(H)-\frac{e(H)}{2{\mathrm{\Delta }}^2},v(H)]$ is realizable for $H$. We also prove a ``stability’’ result for generalized Turán numbers of trees which implies that if $T\mathrm{\ne }{K}_2$ is a tree with $\mathrm{\ell }$ leaves, then $T$ has no realizable exponents in $[0,\mathrm{\ell }]\setminus \mathbb{Z}$. Our proof of this latter result uses a new variant of the classical Helly theorem for trees, which may be of independent interest.