Vanishing orders and zero degree Turán densities
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908870040879104 |
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| author | Ding, Laihao Liu, Hong Yang, Haotian |
| author_facet | Ding, Laihao Liu, Hong Yang, Haotian |
| contents | For integers $1\le \ell<k$, the $\ell$-degree Turán density $π_\ell(F)$ measures the minimum $\ell$-degree threshold that forces a copy of a fixed $k$-uniform hypergraph $F$, generalizing both the classical Turán density $π_1$ and the codegree Turán density $π_{k-1}$. Motivated by Erdős' characterization of $k$-graphs with zero Turán density, we study the structural implications of vanishing $\ell$-degree Turán density.
We prove for every uniformity $k\ge 3$ that if $π_2(F)=0$, then $F$ admits a $2$-vanishing order-a global vertex ordering under which all edges align canonically. This provides a higher-degree analogue of the classical fact that $π_1(F)=0$ forces $k$-partiteness, and identifies a structural obstruction to vanishing $2$-degree Turán density. As an application, we show that, unlike $π_1$, $π_2$ accumulates at $0$. For $3\le \ell\le k-1$, we also obtain weaker necessary conditions for $π_\ell(F)=0$.
The proof combines random geometric building blocks, a design-theoretic gluing scheme, and random sparsification to reconcile positive $2$-degree with local vanishing structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05973 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vanishing orders and zero degree Turán densities Ding, Laihao Liu, Hong Yang, Haotian Combinatorics For integers $1\le \ell<k$, the $\ell$-degree Turán density $π_\ell(F)$ measures the minimum $\ell$-degree threshold that forces a copy of a fixed $k$-uniform hypergraph $F$, generalizing both the classical Turán density $π_1$ and the codegree Turán density $π_{k-1}$. Motivated by Erdős' characterization of $k$-graphs with zero Turán density, we study the structural implications of vanishing $\ell$-degree Turán density. We prove for every uniformity $k\ge 3$ that if $π_2(F)=0$, then $F$ admits a $2$-vanishing order-a global vertex ordering under which all edges align canonically. This provides a higher-degree analogue of the classical fact that $π_1(F)=0$ forces $k$-partiteness, and identifies a structural obstruction to vanishing $2$-degree Turán density. As an application, we show that, unlike $π_1$, $π_2$ accumulates at $0$. For $3\le \ell\le k-1$, we also obtain weaker necessary conditions for $π_\ell(F)=0$. The proof combines random geometric building blocks, a design-theoretic gluing scheme, and random sparsification to reconcile positive $2$-degree with local vanishing structure. |
| title | Vanishing orders and zero degree Turán densities |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.05973 |