论文标题
超图的正共同度密度
Positive co-degree density of hypergraphs
论文作者
论文摘要
非空的$ r $ -graph $ {h} $的\ emph {最低阳性共同学位}，表示为$Δ_{r-1}^+（{h}）$，是$ k $的最大$ k $，因此，如果$ s $是$（r-1）$，则$ s $ nintion in $ a $ a $ { {H} $。给定一个$ r $ -graph $ {f} $，我们将\ emph {proce co-degreeturán数字} $ \ mathrm {co^+ex}（n，n，{f}）$作为最大co-degree $δ__{r-n $ n $ n $ r $ r $ h $ h $ h $ r $ h $ h） subhypergraph。 在本文中，我们专注于$ \ mathrm {co^+ex}（n，{f}）$的行为，以$ 3 $ -graphs $ f $。特别是，我们确定了几种著名的混凝土$ 3 $ graphs $ f $（例如\ $ $ k_4^ - $和fano Plane）的渐近和边界。我们还表明，对于$ r $ graphs，限制\ [γ^+（f）：= \ lim_ {n \ rightArrow \ rightarrow \ infty} \ frac {\ mathrm {co^+ex}（n，{f}）} {n}）} {n} {n} \]间隔$（0,1/r）$。此外，我们表征哪个$ r $ -graphs $ f $具有$γ^+（f）= 0 $。我们的动机主要来自对（普通）共同学位的研究，其中已证明许多结果激发了我们的结果。
The \emph{minimum positive co-degree} of a non-empty $r$-graph ${H}$, denoted $δ_{r-1}^+( {H})$, is the maximum $k$ such that if $S$ is an $(r-1)$-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 \emph{positive co-degree Turán number} $\mathrm{co^+ex}(n, {F})$ as the maximum positive co-degree $δ_{r-1}^+(H)$ over all $n$-vertex $r$-graphs $H$ that do not contain $F$ as a subhypergraph. In this paper we concentrate on the behavior of $\mathrm{co^+ex}(n, {F})$ 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 \[ γ^+(F) := \lim_{n \rightarrow \infty} \frac{\mathrm{co^+ex}(n, {F})}{n} \] exists, and ``jumps'' from $0$ to $1/r$, i.e., it never takes on values in the interval $(0,1/r)$. Moreover, we characterize which $r$-graphs $F$ have $γ^+(F)=0$. 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.
