Positive co-degree densities and jumps

The minimum positive co-degree of a nonempty $r$-graph $H$, denoted by ${\delta }_{r-1}^{+}(H)$, is the largest integer $k$ such that for every $(r-1)$-set $S\subset V(H)$, if $S$ is contained in a hyperedge of $H$, then $S$ is contained in at least $k$ hyperedges of $H$. Given a family $\mathcal{F}$ of $r$-graphs, the positive co-degree Turán function $\mathrm{c}{\mathrm{o}}^{+}\mathrm{e}\mathrm{x}(n,\mathcal{F})$ is the maximum of ${\delta }_{r-1}^{+}(H)$ over all $n$-vertex $r$-graphs $H$ containing no member of $\mathcal{F}$. The positive co-degree density of $\mathcal{F}$ is ${\gamma }^{+}(\mathcal{F})=\frac{\mathrm{c}{\mathrm{o}}^{+}\mathrm{e}\mathrm{x}(n,\mathcal{F})}{n}\mathrm{.}$ While the existence of ${\gamma }^{+}(\mathcal{F})$ is proved for all families $\mathcal{F}$, only few positive co-degree densities are known exactly. For a fixed $r\ge 2$, we call $\alpha \in [0,1]$ an achievable value if there exists a family of $r$-graphs $\mathcal{F}$ with ${\gamma }^{+}(\mathcal{F})=\alpha$, and call $\alpha$ a jump if for some $\delta >0$, there is no family $\mathcal{F}$ with ${\gamma }^{+}(\mathcal{F})\in (\alpha ,\alpha +\delta )$. Halfpap, Lemons, and Palmer showed that every $\alpha \in [0,\frac{1}{r})$ is a jump. We extend this result by showing that every $\alpha \in [0,\frac{2}{2r-1})$ is a jump. We also show that for $r=3$, the set of achievable values is infinite, more precisely, $\frac{k-2}{2k-3}$ for every $k\ge 4$ is achievable. Finally, we determine two additional achievable values for $r=3$ using flag algebra calculations.