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Several decades ago, Erd\\H{o}s and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $0,1, \\frac{7}{5}, 2$, and the numbers of the form $1+\\frac{1}{m}$, $2-\\frac{1}{m}$, $2-\\frac{2}{m}$ for integers $m \\geq 1$. 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