Year: 2026
Type: preprint
Abstract: Motivated by parallel developments in ordinary and $\ell$-degree Turán problems, we study the set of $(r-2)$-uniform Turán 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.
Source: arXiv (Cornell University)
Authors: Li, Heng, Liu, Xizhi
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220891804