The maximal length of a gap between r-graph Tur\'an densities

The Tur'an density $\pi (\mathcal{F})$ of a family $\mathcal{F}$ of $r$-graphs is the limit as $n\to \mathrm{\infty }$ of the maximum edge density of an $\mathcal{F}$-free $r$-graph on $n$ vertices. Erdos [Israel J. Math 2 (1964) 183–190] proved that no Tur'an density can lie in the open interval $(0,r!\mathrm{/}{r}^r)$. Here we show that any other open subinterval of $[0,1]$ avoiding Tur'an densities has strictly smaller length. In particular, this implies a conjecture of Grosu [E-print arXiv:1403.4653v1, 2014].