Near-optimal Turán densities of r-graphs on r+1 vertices

Abstract

Let π(H) be the Turán density of an r-uniform hypergraph H and let H_k^r denote the r-uniform hypergraph on r+1 vertices with exactly k edges, where 1≤ k≤ r+1. Sidorenko~(JCT-B, 2024) proved that π(H₃^r)≥ (1.7215−o(1))r⁻² as r→∞ and π(H_k^r)≥ (C_k+o(1))r^{−(1+1/(k−2))} for fixed k as r→∞. Clemen~later improved the first bound to π(H₃^r)≥ cr⁻²√(log r) for some constant c>0. In this article, we prove the following results. \begin{itemize} \item For any fixed ε>0, there is a constant c_ε>0 such that π(H₃^r)≥ c_ε/(r(log r)^2+ε). %π(H₃^r)≥ 1/(r(log r)^2+o(1)). Together with the known upper bound π(H₃^r)≤1/r, this implies π(H₃^r)=r^−1+o(1). \item For every 3≤ k≤ r+1, let s=min{k−2,r−k+2}. Then 0≤ (k−2)/r−π(H_k^r) ≤ 128/r(√(sloger/s)+loger/s). This estimate yields several asymptotically sharp results for π(H_k^r). For example, π(H_k^r)=(1+o(1))(k−2)/r when log(er/(k))=o(k). \end{itemize}

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BibTeX RIS

BibTeX

@misc{yang2026nearoptimal,
  title = {{Near-optimal Turán densities of $r$-graphs on $r+1$ vertices}},
  author = {Jiabao Yang and Xiutao Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.18924},
  eprint = {2608.18924},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.18924},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Near-optimal Turán densities of r-graphs on r+1 vertices
AU  - Jiabao Yang
AU  - Xiutao Zhu
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2608.18924
UR  - https://arxiv.org/abs/2608.18924
AB  - Let π(H) be the Turán density of an r-uniform hypergraph H and let H_k^r denote the r-uniform hypergraph on r+1 vertices with exactly k edges, where 1≤ k≤ r+1. Sidorenko~(JCT-B, 2024) proved that π(H_3^r)≥ (1.7215−o(1))r^−2 as r→∞ and π(H_k^r)≥ (C_k+o(1))r^−(1+1/(k−2)) for fixed k as r→∞. Clemen~later improved the first bound to π(H_3^r)≥ cr^−2√(log r) for some constant c>0. In this article, we prove the following results. \begin{itemize} \item For any fixed ε>0, there is a constant c_ε>0 such that π(H_3^r)≥ c_ε/(r(log r)^2+ε). %π(H_3^r)≥ 1/(r(log r)^2+o(1)). Together with the known upper bound π(H_3^r)≤1/r, this implies π(H_3^r)=r^−1+o(1). \item For every 3≤ k≤ r+1, let s=min{k−2,r−k+2}. Then 0≤ (k−2)/r−π(H_k^r) ≤ 128/r(√(sloger/s)+loger/s). This estimate yields several asymptotically sharp results for π(H_k^r). For example, π(H_k^r)=(1+o(1))(k−2)/r when log(er/(k))=o(k). \end{itemize}
ER  -

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