Erdős–Simonovits rational exponents conjecture

For a graph $H$, let $\mathrm{ex}(n,H)$ denote the maximum number of edges in an $n$-vertex graph containing no copy of $H$. Erdős–Simonovits rational exponents conjecture. For every rational number $r\in[1,2]$, there exists a graph $H$ such that

$$\mathrm{ex}(n,H)=\Theta(n^r).$$

The conjecture asks which rational exponents occur as exact orders of single-graph Turán numbers. The source notes that the result is known for many values of $r$, including the family $r=2-a/b$ with $a,b\in\mathbb{N}$ and $b\geq\max\{a,(a-1)^2\}$, but it is not known in full generality.