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Turán density of long tight cycle minus one hyperedge

József Balogh, Haoran Luo

Abstract

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.

Turán density of long tight cycle minus one hyperedge

Abstract

Denote by the -uniform hypergraph obtained by removing one hyperedge from the tight cycle on vertices. It is conjectured that the Turán density of is . In this paper, we make progress toward this conjecture by proving that the Turán density of is , for every sufficiently large not divisible by . 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.
Paper Structure (7 sections, 21 theorems, 81 equations, 3 figures)

This paper contains 7 sections, 21 theorems, 81 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cyclic triangles in tournaments
  4. Stability result
  5. Proof of Theorem \ref{['thm::main']}
  6. Maximum number of almost similar triangles in the plane
  7. Concluding remarks

Key Result

Theorem 1.1

There is a constant $L$ such that $\pi(\mathcal{C}^-_{\ell}) = 1/4$, for every $\ell > L$ not divisible by $3$.

Figures (3)

  • Figure 1: An example of a bottle of size $8$. It has vertex set $\{1,2,\ldots, 6\}$ and hyperedges $\{1,2,3\},\{2,3,4\},\{3,4,5\},\{4,5,6\},\{5,6,2\},\{6,2,1\}$. It can be represented as $12345621$.
  • Figure 2: The tournament $D_5$.
  • Figure 3: The subsets $V'_1,V'_2,V'_3$ of $V'$.

Theorems & Definitions (74)

  • Theorem 1.1
  • Definition 1.3
  • Theorem 1.4: Theorem 1.3 in balogh2022maximum
  • Theorem 2.1: Mantel mantel1907Pro
  • Theorem 2.2: Erdős, Faudree, Pach, and Spencer, Theorem 1 in erdos1988make
  • Theorem 2.3: See Section 2 in keevash2011hypergraph
  • Theorem 2.4
  • Lemma 2.5
  • Definition 3.1
  • Definition 3.2
  • ...and 64 more