## Abstract

Denote by C_{ℓ}^{-} the 3-uniform hypergraph obtained by removing one hyperedge from the tight cycle on ℓ vertices. It is conjectured that the Turán density of C_{5}^{-} is 1/4. In this paper, we make progress toward this conjecture by proving that the Turán density of C_{ℓ}^{-} is 1/4, for every sufficiently large ℓ 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.

Original language | English (US) |
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Journal | Combinatorica |

DOIs | |

State | Accepted/In press - 2024 |

## Keywords

- Discrete geometry
- Hypergraph
- Tight cycles
- Turán number

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics