Tur\'an density of long tight cycle minus one hyperedge
| Autor: | Balogh, József, Luo, Haoran |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 Tur\'an density of $\mathcal{C}^-_{5}$ is $1/4$. In this paper, we make progress toward this conjecture by proving that the Tur\'an 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 Lidick\'y. Comment: 23 pages, 3 figures. Some minor updates according to referees' comments |
| Databáze: | arXiv |
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