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

TL;DR

This paper investigates the range of possible Turán densities for r-graphs, proving that the maximal gap between such densities is bounded and confirming a conjecture about their distribution.

Contribution

It establishes bounds on the maximal length of gaps between Turán densities, advancing understanding of their distribution and confirming a related conjecture.

Findings

No Turán density can lie in the interval (0, r!/r^r).

Any other open subinterval of [0,1] avoiding Turán densities has smaller length.

Confirmed a conjecture of Grosu regarding the distribution of Turán densities.

Abstract

The Tur\'an 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. Erdos [Israel J. Math 2 (1964) 183--190] proved that no Tur\'an density can lie in the open interval $(0,r!/r^r)$. 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].