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More precisely, we have $ex(n,H)=o(n^{k-1})$ but it exceeds $n^{k-1-c}$ for any positive $c$ for $n> n_0(k,c)$.\n  This is an extension (and simplification) of a result of Frankl and the first author from 1987 where the case $k=5$ was proven. 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There exists a $k$-uniform hypergraph $H$ (for $k\\geq 5$) without exponent, i.e., when the Tur\\'an function is not polynomial in $n$. More precisely, we have $ex(n,H)=o(n^{k-1})$ but it exceeds $n^{k-1-c}$ for any positive $c$ for $n> n_0(k,c)$.\n  This is an extension (and simplification) of a result of Frankl and the first author from 1987 where the case $k=5$ was proven. 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