On the structure of uniform Tur\'an densities

Abstract

Motivated by parallel developments in ordinary and $\ell$-degree Tur\'an problems, we study the set of $(r-2)$-uniform Tur\'an densities of possibly infinite families of $r$-uniform hypergraphs. We prove that, for all sufficiently large $r$, these densities exhibit a phase transition at $4r^{-r}$, from a countable set of algebraic values below this threshold to the full interval $[4r^{-r},1]$. For every $r\ge3$, the densities below the threshold are exactly the finite-palette Lagrangians in that range, and each is realized by a finite forbidden family. Adapting a method of Pikhurko, we show that, for every $r\ge3$, the set of uniform densities omits continuum many of its limit points below the threshold and is therefore not closed.