A 2-Stable Family of Triple Systems

For many well-known families of triple systems M, there are perhaps many near-extremal M-free configurations that are far from each other in edit-distance. Such a property is called non-stable and is a fundamental barrier to determining the Turán number of M. Liu and Mubayi gave the first finite example that is non-stable. In this paper, we construct another finite family of triple systems M such that there are two near-extremal M-free configurations that are far from each other in edit-distance. We also prove its Andrásfai-Erdős-Sós type stability theorem: Every M-free triple system whose minimum degree is close to the average degree of the extremal configurations is a subgraph of one of these two near-extremal configurations. As a corollary, our main result shows that the boundary of the feasible region of M has exactly two global maxima.