Hypergraphs with zero uniform Turán density at intermediate levels
Abstract
An r-uniform hypergraph H is uniformly dense with respect to j-graphs if, up to a small error, at least a prescribed proportion of the r-cliques of every j-uniform hypergraph on the same vertex set are edges of H. For a forbidden hypergraph F, its uniform Turán density at level j is the supremum of these density bounds over arbitrarily large F-free hypergraphs with the error tending to zero. For every r≥4 and 1≤ j≤ r−3, this paper proves that this Turán density is zero precisely when F admits a vertex ordering in which every (j+1)-set shared by two edges occupies the same positions in both edges. This generalizes the zero-density characterizations of Erdős at j=0 and of Reiher, Rödl, and Schacht at j=r−2, and establishes the latter authors' proposed extension to the remaining nontrivial levels. Consequently, at level j, every positive uniform Turán density of a possibly infinite forbidden family is at least N−N, where N=binom rj+1. The proof combines hypergraph regularity with adjacent exchanges in spreadable random hypergraphs.
Related
- Turán problems with bounded matching number in k-uniform hypergraphs
- Upper Bounds on the Turán Density of Hypergraphs Associated with the Projective Plane over a Finite Field
- The uniform Turán density of the tetrahedron
- Weighted Turán densities for hypergraphs
- A note on the Kővári--Sós--Turán theorem for stable hypergraphs
BibTeX
@misc{li2026hypergraphs,
title = {{Hypergraphs with zero uniform Turán density at intermediate levels}},
author = {Heng Li and Xizhi Liu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06964},
eprint = {2611.06964},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06964},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hypergraphs with zero uniform Turán density at intermediate levels
AU - Heng Li
AU - Xizhi Liu
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06964
UR - https://arxiv.org/abs/2611.06964
AB - An r-uniform hypergraph H is uniformly dense with respect to j-graphs if, up to a small error, at least a prescribed proportion of the r-cliques of every j-uniform hypergraph on the same vertex set are edges of H. For a forbidden hypergraph F, its uniform Turán density at level j is the supremum of these density bounds over arbitrarily large F-free hypergraphs with the error tending to zero. For every r≥4 and 1≤ j≤ r−3, this paper proves that this Turán density is zero precisely when F admits a vertex ordering in which every (j+1)-set shared by two edges occupies the same positions in both edges. This generalizes the zero-density characterizations of Erdős at j=0 and of Reiher, Rödl, and Schacht at j=r−2, and establishes the latter authors' proposed extension to the remaining nontrivial levels. Consequently, at level j, every positive uniform Turán density of a possibly infinite forbidden family is at least N^−N, where N=binom rj+1. The proof combines hypergraph regularity with adjacent exchanges in spreadable random hypergraphs.
ER -