Abstract
Denote by Cℓ- the 3-uniform hypergraph obtained by removing one hyperedge from the tight cycle on ℓ vertices. It is conjectured that the Turán density of C5- is 1/4. In this paper, we make progress toward this conjecture by proving that the Turán density of Cℓ- 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) |
---|---|
Pages (from-to) | 949-976 |
Number of pages | 28 |
Journal | Combinatorica |
Volume | 44 |
Issue number | 5 |
DOIs | |
State | Published - Oct 2024 |
Keywords
- Discrete geometry
- Hypergraph
- Tight cycles
- Turán number
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics