Uniform Turán densities of $k$-uniform hypergraphs
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909044080377856 |
|---|---|
| author | Lin, Hao Sun, Guowei Wang, Guanghui Zhou, Wenling |
| author_facet | Lin, Hao Sun, Guowei Wang, Guanghui Zhou, Wenling |
| contents | For $k\ge 3$, the $(k-2)$-uniform Turán density $π_{k-2}(F)$ of a $k$-graph $F$ is the supremum of $d$ for which there are arbitrarily large $F$-free $k$-graphs that are uniformly $d$-dense with respect to the $k$-vertex cliques of every $(k-2)$-graph on the same vertex set. We develop a \emph{palette framework} for this density. For every family $\mathcal F$ of $k$-graphs, we prove that $π_{k-2}(\mathcal F)$ equals the corresponding palette Turán density. We further establish palette classification tools for the existence of $k$-graphs satisfying prescribed palette colorability constraints. Those together allow us to reduce exact density computations to a palette-homomorphism framework without relying on the hypergraph regularity method.
As applications, for all $k\ge 3$ and $r\ge 2$, we establish the following values \[ \frac{r-1}{r},\quad \frac{(r-1)^2}{r^2},\quad \frac{r-1}{2r},\quad \frac{(k-1)^k}{k^k},\quad \frac{4(k-2)^{k-2}}{k^k},\quad \frac{4(k-2)^{k-2}}{3k^k} \] as $(k-2)$-uniform Turán densities of single $k$-graphs. Finally, for every $k\ge3$, we show that there exist $k$-graphs $F_1,F_2$ such that \[ π_{k-2}(\{F_1,F_2\})< \min\{π_{k-2}(F_1),π_{k-2}(F_2)\}, \] which provides the first examples of \emph{non-principal} families for this density. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15105 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform Turán densities of $k$-uniform hypergraphs Lin, Hao Sun, Guowei Wang, Guanghui Zhou, Wenling Combinatorics For $k\ge 3$, the $(k-2)$-uniform Turán density $π_{k-2}(F)$ of a $k$-graph $F$ is the supremum of $d$ for which there are arbitrarily large $F$-free $k$-graphs that are uniformly $d$-dense with respect to the $k$-vertex cliques of every $(k-2)$-graph on the same vertex set. We develop a \emph{palette framework} for this density. For every family $\mathcal F$ of $k$-graphs, we prove that $π_{k-2}(\mathcal F)$ equals the corresponding palette Turán density. We further establish palette classification tools for the existence of $k$-graphs satisfying prescribed palette colorability constraints. Those together allow us to reduce exact density computations to a palette-homomorphism framework without relying on the hypergraph regularity method. As applications, for all $k\ge 3$ and $r\ge 2$, we establish the following values \[ \frac{r-1}{r},\quad \frac{(r-1)^2}{r^2},\quad \frac{r-1}{2r},\quad \frac{(k-1)^k}{k^k},\quad \frac{4(k-2)^{k-2}}{k^k},\quad \frac{4(k-2)^{k-2}}{3k^k} \] as $(k-2)$-uniform Turán densities of single $k$-graphs. Finally, for every $k\ge3$, we show that there exist $k$-graphs $F_1,F_2$ such that \[ π_{k-2}(\{F_1,F_2\})< \min\{π_{k-2}(F_1),π_{k-2}(F_2)\}, \] which provides the first examples of \emph{non-principal} families for this density. |
| title | Uniform Turán densities of $k$-uniform hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.15105 |