Rational values and non-principality for ℓ-degree Turán densities of hypergraphs

@misc{tang2026rational,
  title = {{Rational values and non-principality for $\ell$-degree Turán densities of hypergraphs}},
  author = {Quanyu Tang and Wenling Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18927},
  eprint = {2609.18927},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18927},
}

TY  - UNPB
TI  - Rational values and non-principality for ℓ-degree Turán densities of hypergraphs
AU  - Quanyu Tang
AU  - Wenling Zhou
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.18927
UR  - https://arxiv.org/abs/2609.18927
AB  - Let k>ℓ≥ 1 be integers. For a family ℱ of k-uniform hypergraphs (k-graphs), the ℓ-degree Turán density γ^(k)_ℓ(ℱ) of ℱ is defined as the asymptotic maximum of the normalized minimum ℓ-degree over all ℱ-free k-graphs. In this paper, we prove that for all k>ℓ>k/2, every rational number α∈[0,1) can be realized as the ℓ-degree Turán density γ^(k)_ℓ(ℱ) for some finite family ℱ of k-graphs. Furthermore, for any k>ℓ≥ 1, we construct an explicit infinite sequence of values realized by single forbidden k-graphs: for each integer q≥2, there exists a k-graph F such that γ^(k)_ℓ(F)=1−q^ℓ−k. We also establish a strengthened non-principality property for ℓ-degree Turán densities: for all k>ℓ>1, there exist two k-graphs F_1 and F_2 satisfying 0< γ^(k)_ℓ({F_1,F_2})<min{ γ^(k)_ℓ(F_1),γ^(k)_ℓ(F_2)}.
ER  -