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.

用\({\mathcal {C}}^-_{\ell }\)表示通过从\(\ell \)顶点上的紧循环中移除一个超边而获得的 3-一致超图。据推测\({\mathcal {C}}^-_{5}\)的 Turán 密度为 1/4。在本文中，我们通过证明对于每个足够大的\(\ell \)来说，\({\mathcal {C}}^-_{\ell }\)的图兰密度为 1/4，从而在这一猜想上取得了进展。不能被 3 整除。我们证明的主要成分之一是超图的禁止子超图表征，其中在同一顶点集上存在锦标赛，使得该锦标赛中的每个超边都是循环三角形。我们方法的一个副产品是对平面点集中几乎相似三角形的最大数量上限的人工可检查证明，最近由 Balogh、Clemen 和 Lidický 使用标志代数方法证明了这一点。