Turán densities of hypergraph augmentations

Abstract

For a t-graph F and an integer r≥ t, the \emph{r-augmentation} F(r) is the r-graph obtained by adding the same set of r−t new vertices to every edge of F. We investigate the behaviour of the \emph{Turán density π(F(r))}, which is the asymptotically maximum edge density of a large F(r)-free r-graph, as a function of r→∞. Let H_k^r be the r-augmentation of the complete (k−1)-graph on k vertices; equivalently, H_k^r is the (unique up to isomorphism) r-graph with r+1 vertices and k edges. This paper determines the Turán density of each r-graph H_k^r within a constant factor and estimates the lower bounds on π(H_k^r) coming from the circular construction of Sidorenko. In particular, it is shown that if k/log r→∞ then π(H_k^r)=(1+o(1))k/r. Also, we consider augmentations of graphs (that is, the case t=2) and prove that π(F(r))≥(log r)^−1−o(1)/r for every 3-chromatic graph F, which comes close to the known upper bound 1/r. Note that, for all other graphs F, the function π(F(r)) is known to be either identically 0 or uniformly bounded away from 0.

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BibTeX RIS

BibTeX

@misc{pikhurko2026turán,
  title = {{Turán densities of hypergraph augmentations}},
  author = {Oleg Pikhurko and Jiabao Yang and Xiutao Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11757},
  eprint = {2610.11757},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11757},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Turán densities of hypergraph augmentations
AU  - Oleg Pikhurko
AU  - Jiabao Yang
AU  - Xiutao Zhu
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.11757
UR  - https://arxiv.org/abs/2610.11757
AB  - For a t-graph F and an integer r≥ t, the \emph{r-augmentation} F(r) is the r-graph obtained by adding the same set of r−t new vertices to every edge of F. We investigate the behaviour of the \emph{Turán density π(F(r))}, which is the asymptotically maximum edge density of a large F(r)-free r-graph, as a function of r→∞. Let H_k^r be the r-augmentation of the complete (k−1)-graph on k vertices; equivalently, H_k^r is the (unique up to isomorphism) r-graph with r+1 vertices and k edges. This paper determines the Turán density of each r-graph H_k^r within a constant factor and estimates the lower bounds on π(H_k^r) coming from the circular construction of Sidorenko. In particular, it is shown that if k/log r→∞ then π(H_k^r)=(1+o(1))k/r. Also, we consider augmentations of graphs (that is, the case t=2) and prove that π(F(r))≥(log r)^−1−o(1)/r for every 3-chromatic graph F, which comes close to the known upper bound 1/r. Note that, for all other graphs F, the function π(F(r)) is known to be either identically 0 or uniformly bounded away from 0.
ER  -

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