Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2207.05639 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911756417236992 |
|---|---|
| author | Halfpap, Anastasia Lemons, Nathan Palmer, Cory |
| author_facet | Halfpap, Anastasia Lemons, Nathan Palmer, Cory |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_05639 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Positive co-degree density of hypergraphs Halfpap, Anastasia Lemons, Nathan Palmer, Cory Combinatorics 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. |
| title | Positive co-degree density of hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2207.05639 |