The maximal length of a gap between r-graph Turán densities

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

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