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# Positive co-degree Tur\'an number for $C_5$ and $C_5^{-}$

Dec 2022

The \emph{minimum positive co-degree} $\delta^{+}_{r-1}(H)$ of a non-empty$r$-graph $H$ is the maximum $k$ such that if $S$ is an $(r-1)$-set containedin a hyperedge of $H$, then $S$ is contained in at least $k$ hyperedges of $H$.For any $r$-graph $F$, the \emph{positive degree Tur\'an number}$\mathrm{co}^{+}\mathrm{ex}(n,F)$ is defined as the maximum value of$\delta^{+}_{r-1}(H)$ over all $n$-vertex $F$-free non-empty $r$-graphs $H$. Inthis paper, we determine the positive degree Tur\'an number for $C_5$ and$C_5^{-}$.

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