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We say that a rational number $r$ is \\emph{realizable for $H$} if there exists a finite family $\\mathcal{F}$ such that $\\mathrm{ex}(n,H,\\mathcal{F}) = \\Theta(n^r)$. Using randomized algebraic constructions, Bukh and Conlon showed that every rational between $1$ and $2$ is realizable for $K_2$. 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