Tur\'an density of long tight cycle minus one hyperedge

TL;DR

This paper advances the understanding of Turán densities for certain 3-uniform hypergraphs by proving the conjectured density for large cycles not divisible by 3, using a novel forbidden-subhypergraph characterization.

Contribution

It proves the Turán density of long tight cycles minus one hyperedge is 1/4 for large cycles not divisible by 3, introducing a new hypergraph characterization method.

Findings

Turán density of _{\ell} is 1/4 for large _{\ell} not divisible by 3

Develops a forbidden-subhypergraph characterization linked to tournaments and cyclic triangles

Provides a human-checkable proof for an upper bound on nearly similar triangles in planar point sets

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 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.