On the structure of uniform Turán densities

Heng Li; Xizhi Liu

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arXiv · 2610.07049

On the structure of uniform Turán densities

Heng Li·Xizhi Liu

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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.

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Heng Li, Xizhi Liu. 2026-10-05. On the structure of uniform Turán densities. https://arxiv.org/abs/2610.07049

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