A family of 4-uniform hypergraphs with no stability

Unlike graphs, for an integer rge 3 and many classical families of r-uniform hypergraphs mathcalM there are perhaps more than one mathcalM-free r-uniform hypergraphs with n vertices and the maximum possible number of edges (such hypergraphs are called extremal configurations). Moreover, those extremal configurations are far from each other in edit-distance. Such a phenomenon is called not stable and is a fundamental barrier to determining the Turán number of mathcalM Liu and Mubayi (2022) gave the first 例 for 3-uniform hypergraphs to be not stable. A simple argument shows that for rge 4, one can get a family of r-uniform hypergraphs which is not stable through a not stable family of 3-uniform hypergraphs. In this paper, we construct a finite family of 4-uniform hypergraphs mathcalMdirectly such that two near-extremal mathcalM-free configurations are far from each other in edit-distance. This is the first unstable 例 that does not depend on 3-uniform hypergraphs. We also prove its Andrásfai-ErdHos-Sós type stability 定理: Every mathcalM-free 4-uniform hypergraph whose minimum degree is close to the average degree of extremal configurations is a subgraph of one of these two near-extremal configurations. As a 推论, our main result shows that the boundary of the feasible region of mathcalMhas exactly two global maxima.