Turán Density of Long Tight Cycle Minus One Hyperedge

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.