Positive Co-Degree Density of Hypergraphs

The minimum positive co-degree of a nonempty $r$-graph $H$, denoted ${\delta }_{r-1}^{+}\left(H\right)$, is the maximum $k$ such that if $S$ is an $\left(r-1\right)$-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 positive co-degree Turán number ${\text{co}}^{+}\text{ex}\left(n,F\right)$ as the maximum positive co-degree ${\delta }_{r-1}^{+}\left(H\right)$ over all $n$-vertex $r$-graphs $H$ that do not contain $F$ as a subhypergraph. In this paper, we concentrate on the behavior of ${\text{co}}^{+}\text{ex}\left(n,F\right)$ 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 ${\gamma }^{+}\left(F\right)≔{\lim }_{n\to \infty }\frac{{\text{co}}^{+}\text{ex}(n,F)}{n}$ exists, and “jumps” from 0 to $1/r$, that is, it never takes on values in the interval $\left(0,1/r\right)$. Moreover, we characterize which $r$-graphs $F$ have ${\gamma }^{+}\left(F\right)=0$. 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.