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The ℓ-degree Turán densities of hypergraphs: infinite families and their finite subfamilies

  • Hao Lin
  • Wenling Zhou
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Abstract

Let k>ℓ≥ 1 be integers. For a family ℱ of k-uniform hypergraphs (k-graphs), its ℓ-degree Turán density γ(k)ℓ(ℱ) is the asymptotic maximum of the normalized minimum ℓ-degree over all ℱ-free k-graphs. In this paper, we prove that for all integers k>ℓ>2 and every α∈[0,1), there exists an infinite family ℱ of k-graphs such that γ(k)ℓ(ℱ)=α, while every finite subfamily ℱ'⊂ ℱ satisfies γ(k)ℓ(ℱ')>α. This confirms the conjecture of Mubayi and Zhao on codegree Turán densities for every k > 3. As an immediate consequence, the set Γk,ℓ∞ of ℓ-degree Turán densities of nonempty k-graph families is [0,1) whenever k>ℓ>2, which answers a question of Lo and Markström from 2014.

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Cite

BibTeXRIS
BibTeX
@misc{lin2026ℓdegree,
  title = {{The $\ell$-degree Turán densities of hypergraphs: infinite families and their finite subfamilies}},
  author = {Hao Lin and Wenling Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17558},
  eprint = {2610.17558},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17558},
}
RIS
TY  - UNPB
TI  - The ℓ-degree Turán densities of hypergraphs: infinite families and their finite subfamilies
AU  - Hao Lin
AU  - Wenling Zhou
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.17558
UR  - https://arxiv.org/abs/2610.17558
AB  - Let k>ℓ≥ 1 be integers. For a family ℱ of k-uniform hypergraphs (k-graphs), its ℓ-degree Turán density γ^(k)_ℓ(ℱ) is the asymptotic maximum of the normalized minimum ℓ-degree over all ℱ-free k-graphs. In this paper, we prove that for all integers k>ℓ>2 and every α∈[0,1), there exists an infinite family ℱ of k-graphs such that γ^(k)_ℓ(ℱ)=α, while every finite subfamily ℱ'⊂ ℱ satisfies γ^(k)_ℓ(ℱ')>α. This confirms the conjecture of Mubayi and Zhao on codegree Turán densities for every k > 3. As an immediate consequence, the set Γ^k,ℓ_∞ of ℓ-degree Turán densities of nonempty k-graph families is [0,1) whenever k>ℓ>2, which answers a question of Lo and Markström from 2014.
ER  -