Intervals of uniform Tur\'an densities

Abstract

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