Positive co-degree densities and jumps

Abstract: The minimum positive co-degree of a nonempty $r$-graph $H$, denoted by $\delta_{r-1}<sup>+(H)$, 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 Tur\'an function $\mathrm{co<sup>+ex}(n,\mathcal{F})$ is the maximum of $\delta_{r-1}<sup>+(H)$ over all $n$-vertex $r$-graphs $H$ containing no member of $\mathcal{F}$. The positive co-degree density of $\mathcal{F}$ is $$\gamma^{+(\mathcal{F})} = \underset{n \rightarrow \infty}{\lim} \frac{\mathrm{co<sup>+ex}(n,\mathcal{F})}{n}.$$ While the existence of $\gamma<sup>+(\mathcal{F})$ is proved for all families $\mathcal{F}$, only few positive co-degree densities are known exactly. For a fixed $r \geq 2$, 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 $\delta &gt; 0$, 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 $k \geq 4$ is achievable. Finally, we determine two additional achievable values for $r=3$ using flag algebra calculations.