Intervals of uniform Turán densities

Abstract

We prove that the set $Π_{\therefore,\infty}$ of uniform Turán densities of possibly infinite families of $3$-graphs contains a terminal interval: there exists $δ>0$ such that $[1-δ,1]\subseteqΠ_{\therefore,\infty}$. Consequently, $Π_{\therefore,\infty}$ has positive Lebesgue measure and Hausdorff dimension $1$.