Turán Density of Long Tight Cycle Minus One Hyperedge

József Balogh, Haoran Luo

Research output: Contribution to journalArticlepeer-review

Abstract

Denote by C- 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 C- 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 languageEnglish (US)
JournalCombinatorica
DOIs
StateAccepted/In press - 2024

Keywords

  • Discrete geometry
  • Hypergraph
  • Tight cycles
  • Turán number

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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