A fundamental barrier in extremal hypergraph theory is the presence of many near-extremal constructions with very different structures. Indeed, the classical constructions due to Kostochka imply that the notorious extremal problem for the tetrahedron exhibits this phenomenon assuming Turán’s conjecture.

Our main result is to construct a finite family of triple systems \({\cal M}\), determine its Turán number, and prove that there are two near-extremal \({\cal M}\)-free constructions that are far from each other in edit-distance. This is the first extremal result for a hypergraph family that fails to have a corresponding stability theorem.

极值超图论的一个基本障碍是存在许多具有非常不同结构的近极值结构。事实上，由于 Kostochka 的经典结构意味着臭名昭著的四面体极值问题在假设 Turán 猜想的情况下表现出这种现象。

我们的主要结果是构造一个有限的三重系统族\({\cal M}\)，确定它的 Turán 数，并证明有两个接近极值的\({\cal M}\) -free 结构是在编辑距离上彼此相距很远。这是没有相应稳定性定理的超图族的第一个极值结果。