Intervals of hypergraph Turán densities
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910255371255808 |
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| author | Liu, Xizhi Pikhurko, Oleg |
| author_facet | Liu, Xizhi Pikhurko, Oleg |
| contents | We prove that, for every integer $r\ge 3$, the set $Π^{(r)}_\infty$ of Turán densities of (possibly infinite) families of $r$-graphs contains non-degenerate intervals, including an interval of the form $[1-δ_r,1]$ for some $δ_{r}>0$. This answers a question of Frankl, Peng, Rödl and Talbot from 2007. This also shows that the Hausdorff dimension of $Π^{(r)}_\infty$ has the maximum possible value 1, thus resolving a question of Grosu from 2016, whereas previously it was not even known whether it is non-zero. We also derive that the set of uniform Turán densities of finite families of $3$-graphs is dense in a non-degenerate interval. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_25914 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Intervals of hypergraph Turán densities Liu, Xizhi Pikhurko, Oleg Combinatorics We prove that, for every integer $r\ge 3$, the set $Π^{(r)}_\infty$ of Turán densities of (possibly infinite) families of $r$-graphs contains non-degenerate intervals, including an interval of the form $[1-δ_r,1]$ for some $δ_{r}>0$. This answers a question of Frankl, Peng, Rödl and Talbot from 2007. This also shows that the Hausdorff dimension of $Π^{(r)}_\infty$ has the maximum possible value 1, thus resolving a question of Grosu from 2016, whereas previously it was not even known whether it is non-zero. We also derive that the set of uniform Turán densities of finite families of $3$-graphs is dense in a non-degenerate interval. |
| title | Intervals of hypergraph Turán densities |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.25914 |