The maximal length of a gap between \(r\)-graph Turán densities (Q888628)

Summary: The Turán density \(\pi(\mathcal F)\) of a family \(\mathcal F\) of \(r\)-graphs is the limit as \(n\to\infty\) of the maximum edge density of an \(\mathcal F\)-free \(r\)-graph on \(n\) vertices. \textit{P. Erdős} [Isr. J. Math. 2, 183--190 (1964; Zbl 0129.39905)] 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 \textit{C. Grosu} [``On the algebraic and topological structure of the set of Turán densities'', Preprint, \url{arXiv:1403.4653}].