Positive co-degree density of hypergraphs

The \emph{minimum positive co-degree} of a non-empty $r$-graph $H$, denoted ${\delta }_{r-1}^{+}(H)$, is the maximum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $H$, then $S$ is contained in at least $k$ distinct hyperedges of $H$. Given an $r$-graph $F$, we introduce the \emph{positive co-degree Turán number} $\mathrm{c}{\mathrm{o}}^{+}\mathrm{e}\mathrm{x}(n,F)$ as the maximum positive co-degree ${\delta }_{r-1}^{+}(H)$ over all $n$-vertex $r$-graphs $H$ that do not contain $F$ as a subhypergraph. In this paper we concentrate on the behavior of $\mathrm{c}{\mathrm{o}}^{+}\mathrm{e}\mathrm{x}(n,F)$ for $3$-graphs $F$. In particular, we determine asymptotics and bounds for several well-known concrete $3$-graphs $F$ (e.g.\ $K_4^{-}$ and the Fano plane). We also show that, for $r$-graphs, the limit $${\gamma }^{+}(F):=\underset{n\to \mathrm{\infty }}{\lim }\frac{\mathrm{c}{\mathrm{o}}^{+}\mathrm{e}\mathrm{x}(n,F)}{n}$$ exists, and ``jumps’’ from $0$ to $1\mathrm{/}r$, i.e., it never takes on values in the interval $(0,1\mathrm{/}r)$. Moreover, we characterize which $r$-graphs $F$ have ${\gamma }^{+}(F)=0$. Our motivation comes primarily from the study of (ordinary) co-degree Turán numbers where a number of results have been proved that inspire our results.