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

الأصل · EN

The Turán density π(F) of a family F of r-graphs is the limit as n→∞ of the maximum edge density of an 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!/rʳ). 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].