A 2-Stable Family of Triple Systems

Abstract

For many well-known families of triple systems $\mathcal{M}$, there are perhaps many near-extremal $\mathcal{M}$-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 $\mathcal{M}$. Liu and Mubayi gave the first finite example that is non-stable. In this paper, we construct another finite family of triple systems $\mathcal{M}$ such that there are two near-extremal $\mathcal{M}$-free configurations that are far from each other in edit-distance. We also prove its Andrásfai-Erdős-Sós type stability theorem: Every $\mathcal{M}$-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 $\mathcal{M}$ has exactly two global maxima.