Abstract:We prove that, for every integer $r\ge 3$, the set $\Pi^{(r)}_\infty$ of Turán densities of (possibly infinite) families of $r$-graphs contains non-degenerate intervals, including an interval of the form $[1-\delta_r,1]$ for some $\delta_{r}>0$. This answers a question of Frankl, Peng, Rödl and Talbot from 2007. This also shows that the Hausdorff dimension of $\Pi^{(r)}_\infty$ 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.
| Comments: | 23 pages, comments are welcome |
| Subjects: | Combinatorics (math.CO) |
| Cite as: | arXiv:2605.25914 [math.CO] |
| (or arXiv:2605.25914v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25914 arXiv-issued DOI via DataCite (pending registration) |
From: Xizhi Liu [view email]
[v1]
Mon, 25 May 2026 14:50:01 UTC (30 KB)
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