Extremal problems in uniformly dense hypergraphs and digraphs
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2026
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| _version_ | 1866914386352799744 |
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| author | Lin, Hao Wang, Guanghui Zhou, Wenling Zhou, Yiming |
| author_facet | Lin, Hao Wang, Guanghui Zhou, Wenling Zhou, Yiming |
| contents | The uniform Turán density $π_{u}(F)$ of a $3$-uniform hypergraph (or $3$-graph) $F$ is the supremum of all $d$ such that there exist infinitely many $F$-free $3$-graphs $H$ in which every induced subhypergraph on a linearly sized vertex set has edge density at least $d$. Determining $π_{u}(F)$ for a given $3$-graph $F$ was proposed by Erdős and Sós in the 1980s, yet only a few cases are known. In particular, it remains open whether $1/2$ can occur as a value of $π_{u}$.
In this paper, we establish a novel connection between Turán-type extremal problems for digraphs and uniform Turán densities of $3$-graphs. Using digraph extremal results, we give the first verifiable conditions for $3$-graphs $F$ with $π_{u}(F) = (r-1)/r$ and $π_{u}(F) = (r-1)^2/r^2$ for all $r \ge 2$, and identify the corresponding $3$-graphs. In particular, these $3$-graph classes contain some specific $3$-graphs, such as $K^{(3)-}_4$. We also present a sufficient condition ensuring $π_{u}(F)=4/27$ and construct $3$-graphs satisfying it; in particular, our examples are different from the tight $3$-uniform cycles whose uniform Turán density $4/27$ was determined in [{Trans. Amer. Math. Soc. 376 (2023), 4765-4809}]. Finally, we give a short proof of the existence of $3$-graphs $F$ with $π_{u}(F)=1/27$, originally established by Garbe, Král' and Lamaison [{Israel J. Math. 259 (2024), 701-726}] via the hypergraph regularity method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_10766 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extremal problems in uniformly dense hypergraphs and digraphs Lin, Hao Wang, Guanghui Zhou, Wenling Zhou, Yiming Combinatorics The uniform Turán density $π_{u}(F)$ of a $3$-uniform hypergraph (or $3$-graph) $F$ is the supremum of all $d$ such that there exist infinitely many $F$-free $3$-graphs $H$ in which every induced subhypergraph on a linearly sized vertex set has edge density at least $d$. Determining $π_{u}(F)$ for a given $3$-graph $F$ was proposed by Erdős and Sós in the 1980s, yet only a few cases are known. In particular, it remains open whether $1/2$ can occur as a value of $π_{u}$. In this paper, we establish a novel connection between Turán-type extremal problems for digraphs and uniform Turán densities of $3$-graphs. Using digraph extremal results, we give the first verifiable conditions for $3$-graphs $F$ with $π_{u}(F) = (r-1)/r$ and $π_{u}(F) = (r-1)^2/r^2$ for all $r \ge 2$, and identify the corresponding $3$-graphs. In particular, these $3$-graph classes contain some specific $3$-graphs, such as $K^{(3)-}_4$. We also present a sufficient condition ensuring $π_{u}(F)=4/27$ and construct $3$-graphs satisfying it; in particular, our examples are different from the tight $3$-uniform cycles whose uniform Turán density $4/27$ was determined in [{Trans. Amer. Math. Soc. 376 (2023), 4765-4809}]. Finally, we give a short proof of the existence of $3$-graphs $F$ with $π_{u}(F)=1/27$, originally established by Garbe, Král' and Lamaison [{Israel J. Math. 259 (2024), 701-726}] via the hypergraph regularity method. |
| title | Extremal problems in uniformly dense hypergraphs and digraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.10766 |