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Denote the infimum as $\\delta_{\\chi}(\\mathcal{F}, c)$. A fundamental result of Erd\\H{o}s, Stone and Simonovits implies that if $3\\le r+1=\\chi(\\mathcal{F})=\\min\\{\\chi (F): F\\in \\mathcal{F}\\}$, then for any $c\\le r-1$, $\\delta_{\\chi}(\\mathcal{F}, c)=1-{1 \\over r}$. 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Denote the infimum as $\\delta_{\\chi}(\\mathcal{F}, c)$. A fundamental result of Erd\\H{o}s, Stone and Simonovits implies that if $3\\le r+1=\\chi(\\mathcal{F})=\\min\\{\\chi (F): F\\in \\mathcal{F}\\}$, then for any $c\\le r-1$, $\\delta_{\\chi}(\\mathcal{F}, c)=1-{1 \\over r}$. 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