arXiv2024
The minimum positive co-degree of a nonempty $r$-graph $H$, denoted by $δ_{r-1}^+(H)$, 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 $\mathrm{co^+ex}(n,\mathcal{F})$ is the maximum of $δ_{r-1}^+(H)$ over all $n$-vertex $r$-graphs $H$ containing no member of $\mathcal{F}$. The positive co-degree density of $\mathcal{F}$ is $γ^+(\mathcal{F}) = \underset{n \rightarrow \infty}{\lim} \frac{\mathrm{co^+ex}(n,\mathcal{F})}{n}.$ While the existence of $γ^+(\mathcal{F})$ is proved for all families $\mathcal{F}$, only few positive co-degree densities are known exactly. For a fixed $r \geq 2$, we call $α\in [0,1]$ an achievable value if there exists a family of $r$-graphs $\mathcal{F}$ with $γ^+(\mathcal{F}) = α$, and call $α$ a jump if for some $δ> 0$, there is no family $\mathcal{F}$ with $γ^+(\mathcal{F}) \in (α, α+ δ)$. Halfpap, Lemons, and Palmer showed that every $α\in [0, \frac{1}{r})$ is a jump. We extend this result by showing that every $α\in [0, \frac{2}{2r -1})$ 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.