The Maximal Length of a Gap between $r$-Graph Turán Densities
The Turán density $\pi(\cal F)$ 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. Erdős [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 [arXiv:1403.4653, 2014].