PreprintCC BY 4.0arXiv
On the structure of uniform Turán densities
Abstract
Motivated by parallel developments in ordinary and ℓ-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≥3, 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≥3, the set of uniform densities omits continuum many of its limit points below the threshold and is therefore not closed.
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BibTeX
@misc{li2026structure,
title = {{On the structure of uniform Turán densities}},
author = {Heng Li and Xizhi Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07049},
eprint = {2610.07049},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07049},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the structure of uniform Turán densities
AU - Heng Li
AU - Xizhi Liu
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07049
UR - https://arxiv.org/abs/2610.07049
AB - Motivated by parallel developments in ordinary and ℓ-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≥3, 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≥3, the set of uniform densities omits continuum many of its limit points below the threshold and is therefore not closed.
ER -