Turán density of long tight cycle minus one hyperedge
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914700294356992 |
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| author | Balogh, József Luo, Haoran |
| author_facet | Balogh, József Luo, Haoran |
| contents | Denote by $\mathcal{C}^-_{\ell}$ the $3$-uniform hypergraph obtained by removing one hyperedge from the tight cycle on $\ell$ vertices. It is conjectured that the Turán density of $\mathcal{C}^-_{5}$ is $1/4$. In this paper, we make progress toward this conjecture by proving that the Turán density of $\mathcal{C}^-_{\ell}$ is $1/4$, for every sufficiently large $\ell$ not divisible by $3$. One of the main ingredients of our proof is a forbidden-subhypergraph characterization of the hypergraphs, for which there exists a tournament on the same vertex set such that every hyperedge is a cyclic triangle in this tournament.
A byproduct of our method is a human-checkable proof for the upper bound on the maximum number of almost similar triangles in a planar point set, which was recently proved using the method of flag algebras by Balogh, Clemen, and Lidický. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_10530 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Turán density of long tight cycle minus one hyperedge Balogh, József Luo, Haoran Combinatorics 05C35, 05C65, 05D05 Denote by $\mathcal{C}^-_{\ell}$ the $3$-uniform hypergraph obtained by removing one hyperedge from the tight cycle on $\ell$ vertices. It is conjectured that the Turán density of $\mathcal{C}^-_{5}$ is $1/4$. In this paper, we make progress toward this conjecture by proving that the Turán density of $\mathcal{C}^-_{\ell}$ is $1/4$, for every sufficiently large $\ell$ not divisible by $3$. One of the main ingredients of our proof is a forbidden-subhypergraph characterization of the hypergraphs, for which there exists a tournament on the same vertex set such that every hyperedge is a cyclic triangle in this tournament. A byproduct of our method is a human-checkable proof for the upper bound on the maximum number of almost similar triangles in a planar point set, which was recently proved using the method of flag algebras by Balogh, Clemen, and Lidický. |
| title | Turán density of long tight cycle minus one hyperedge |
| topic | Combinatorics 05C35, 05C65, 05D05 |
| url | https://arxiv.org/abs/2303.10530 |