Intervals of hypergraph Tur\'an densities

We prove that, for every integer $r\ge 3$, the set $\Pi^{(r)}_\infty$ of Tur\'an 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, R\"odl 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 Tur\'an densities of finite families of $3$-graphs is dense in a non-degenerate interval.