On $3$-graphs with vanishing codegree Turán density
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909251570499584 |
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| author | Ding, Laihao Lamaison, Ander Liu, Hong Wang, Shuaichao Yang, Haotian |
| author_facet | Ding, Laihao Lamaison, Ander Liu, Hong Wang, Shuaichao Yang, Haotian |
| contents | For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree Turán density $π_{\mathrm{co}}(F)$ is the supremum over all $α$ such that there exist arbitrarily large $n$-vertex $F$-free $k$-graphs $H$ in which every $(k-1)$-subset of $V(H)$ is contained in at least $αn$ edges. Recently, it was proved that for every $3$-graph $F$, $π_{\mathrm{co}}(F)=0$ implies $π_{\therefore}(F)=0$, where $π_{\therefore}(F)$ is the uniform Turán density of $F$ and is defined as the supremum over all $d$ such that there are infinitely many $F$-free $k$-graphs $H$ satisfying that any induced linear-size subhypergraph of $H$ has edge density at least $d$.
In this paper, we introduce a layered structure for $3$-graphs which allows us to obtain the reverse implication: every layered $3$-graph $F$ with $π_{\therefore}(F)=0$ satisfies $π_{\mathrm{co}}(F)=0$. Along the way, we answer in the negative a question of Falgas-Ravry, Pikhurko, Vaughan and Volec [J. London Math. Soc., 2023] about whether $π_{\therefore}(F)\leqπ_{\mathrm{co}}(F)$ always holds. In particular, we construct counterexamples $F$ with positive but arbitrarily small $π_{\mathrm{co}}(F)$ while having $π_{\therefore}(F)\ge 4/27$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08771 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $3$-graphs with vanishing codegree Turán density Ding, Laihao Lamaison, Ander Liu, Hong Wang, Shuaichao Yang, Haotian Combinatorics 05C35, 05C65 For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree Turán density $π_{\mathrm{co}}(F)$ is the supremum over all $α$ such that there exist arbitrarily large $n$-vertex $F$-free $k$-graphs $H$ in which every $(k-1)$-subset of $V(H)$ is contained in at least $αn$ edges. Recently, it was proved that for every $3$-graph $F$, $π_{\mathrm{co}}(F)=0$ implies $π_{\therefore}(F)=0$, where $π_{\therefore}(F)$ is the uniform Turán density of $F$ and is defined as the supremum over all $d$ such that there are infinitely many $F$-free $k$-graphs $H$ satisfying that any induced linear-size subhypergraph of $H$ has edge density at least $d$. In this paper, we introduce a layered structure for $3$-graphs which allows us to obtain the reverse implication: every layered $3$-graph $F$ with $π_{\therefore}(F)=0$ satisfies $π_{\mathrm{co}}(F)=0$. Along the way, we answer in the negative a question of Falgas-Ravry, Pikhurko, Vaughan and Volec [J. London Math. Soc., 2023] about whether $π_{\therefore}(F)\leqπ_{\mathrm{co}}(F)$ always holds. In particular, we construct counterexamples $F$ with positive but arbitrarily small $π_{\mathrm{co}}(F)$ while having $π_{\therefore}(F)\ge 4/27$. |
| title | On $3$-graphs with vanishing codegree Turán density |
| topic | Combinatorics 05C35, 05C65 |
| url | https://arxiv.org/abs/2407.08771 |