Vanishing Uniform Turán Densities: Characterization and Jumps
Abstract
Uniform Turán density problems ask for the minimum density threshold that guarantees a copy of a fixed 3-uniform hypergraph F in every sufficiently large 3-uniform hypergraph whose edges are uniformly distributed. This line of research was initiated by Erdős and Sós in the 1980s, who introduced the notion of \emph{vertex-set Turán density} πvvv(F). Reiher, Rödl and Schacht later defined two related concepts, namely, the \emph{vertex-pair Turán density} πve(F) and the \emph{cherry Turán density} πee(F), which satisfy the chain of inequalities πvvv(F)≥πve(F)≥πee(F). In this paper, we investigate the problem for their vanishing densities. First, we characterize the 3-uniform hypergraphs F for which πee(F)=0, thereby answering a question posed by Schacht. We show that these are precisely the so-called \emph{max-layered} 3-uniform hypergraphs recently introduced by Lamaison. Furthermore, we answer an analogous question of Lamaison for the \emph{minimum-codegree palette density} πeepal. Together, these results imply that the following are equivalent: πee(F)=0, πeepal(F)=0, and F is a max-layered 3-uniform hypergraph. Let ⋆ ∈ {vvv, ve, ee }. We say that zero is a \emph{jump} for π⋆ if there exists c>0 such that no $3$-uniform hypergraph F satisfies π⋆(F)∈(0,c). A famous result of Reiher, Rödl and Schacht asserts that zero is a jump for πvvv. In contrast, we prove that zero is not a jump for either πve or πee. This also yields a negative answer to Schacht's well-ordering question for πve. Together with existing results, these findings complete the picture for all three notions of uniform Turán density at zero.
Related
BibTeX
@misc{lin2026vanishing,
title = {{Vanishing Uniform Turán Densities: Characterization and Jumps}},
author = {Hao Lin and Jie Ma and Wenling Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28574},
eprint = {2610.28574},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28574},
}
RIS
TY - UNPB
TI - Vanishing Uniform Turán Densities: Characterization and Jumps
AU - Hao Lin
AU - Jie Ma
AU - Wenling Zhou
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28574
UR - https://arxiv.org/abs/2610.28574
AB - Uniform Turán density problems ask for the minimum density threshold that guarantees a copy of a fixed 3-uniform hypergraph F in every sufficiently large 3-uniform hypergraph whose edges are uniformly distributed. This line of research was initiated by Erdős and Sós in the 1980s, who introduced the notion of \emph{vertex-set Turán density} π_vvv(F). Reiher, Rödl and Schacht later defined two related concepts, namely, the \emph{vertex-pair Turán density} π_ve(F) and the \emph{cherry Turán density} π_ee(F), which satisfy the chain of inequalities π_vvv(F)≥π_ve(F)≥π_ee(F). In this paper, we investigate the problem for their vanishing densities. First, we characterize the 3-uniform hypergraphs F for which π_ee(F)=0, thereby answering a question posed by Schacht. We show that these are precisely the so-called \emph{max-layered} 3-uniform hypergraphs recently introduced by Lamaison. Furthermore, we answer an analogous question of Lamaison for the \emph{minimum-codegree palette density} π_ee^pal. Together, these results imply that the following are equivalent: π_ee(F)=0, π_ee^pal(F)=0, and F is a max-layered 3-uniform hypergraph. Let ⋆ ∈ {vvv, ve, ee }. We say that zero is a \emph{jump} for π_⋆ if there exists c>0 such that no $3$-uniform hypergraph F satisfies π_⋆(F)∈(0,c). A famous result of Reiher, Rödl and Schacht asserts that zero is a jump for π_vvv. In contrast, we prove that zero is not a jump for either π_ve or π_ee. This also yields a negative answer to Schacht's well-ordering question for π_ve. Together with existing results, these findings complete the picture for all three notions of uniform Turán density at zero.
ER -