Positive co-degree densities and jumps

TL;DR

This paper investigates the positive co-degree densities in hypergraphs, extending known results about jumps, and identifies new achievable values and their properties for specific hypergraph parameters.

Contribution

It extends the range of known jumps in positive co-degree densities and finds new achievable values for specific hypergraph cases, including for r=3.

Findings

Every α in [0, 2/(2r-1)) is a jump.

For r=3, infinitely many values of the form (k-2)/(2k-3) are achievable.

Two additional achievable values for r=3 are determined using flag algebra calculations.

Abstract

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 Tur\'an function $\mathrm{co^+ex}(n,\mathcal{F})$ 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 $\gamma^+(\mathcal{F}) = \underset{n \rightarrow \infty}{\lim} \frac{\mathrm{co^+ex}(n,\mathcal{F})}{n}.$ 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 $r \geq 2$, 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 $\delta > 0$, 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.