Uniform Turán density beyond 3-graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912557797736448 |
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| author | Lamaison, Ander |
| author_facet | Lamaison, Ander |
| contents | In the 1980s, Erdős and Sós first introduced an extremal problem on hypergraphs with density constraints. Given an $r$-uniform hypergraph $F$ (or $r$-graph for short), its uniform Turán density $π_u(F)$ is the smallest value of $d$ in which every hypergraph $H$ in which every linear-sized subhypergraph of $H$ has edge density at least $d$ contains $F$ as a subgraph. The first non-zero value of $π_u(F)$ was not found until 30 years later.
Progress in studying the set of values of the uniform Turán density of $r$-graphs has been uneven in terms of $r$: to this day there are infinitely many non-zero values known for $r=3$, a single non-zero value known for $r=4$ and none for $r\geq 5$. In this paper we obtain the first explicit values of $π_u$ for all uniformities, by proving that for every $r\geq 3$ there exist $r$-graphs $F$ with $π_u(F)=1/4$ and with $π_u(F)=\binom{r}{2}^{-\binom{r}{2}}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_20696 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform Turán density beyond 3-graphs Lamaison, Ander Combinatorics 05C65, 05C35 In the 1980s, Erdős and Sós first introduced an extremal problem on hypergraphs with density constraints. Given an $r$-uniform hypergraph $F$ (or $r$-graph for short), its uniform Turán density $π_u(F)$ is the smallest value of $d$ in which every hypergraph $H$ in which every linear-sized subhypergraph of $H$ has edge density at least $d$ contains $F$ as a subgraph. The first non-zero value of $π_u(F)$ was not found until 30 years later. Progress in studying the set of values of the uniform Turán density of $r$-graphs has been uneven in terms of $r$: to this day there are infinitely many non-zero values known for $r=3$, a single non-zero value known for $r=4$ and none for $r\geq 5$. In this paper we obtain the first explicit values of $π_u$ for all uniformities, by proving that for every $r\geq 3$ there exist $r$-graphs $F$ with $π_u(F)=1/4$ and with $π_u(F)=\binom{r}{2}^{-\binom{r}{2}}$. |
| title | Uniform Turán density beyond 3-graphs |
| topic | Combinatorics 05C65, 05C35 |
| url | https://arxiv.org/abs/2508.20696 |