A family of 4-uniform hypergraphs with no stability

Unlike graphs, for an integer r > 3 and many classical families of r -uniform hypergraphs M , there are perhaps more than one M -free r -uniform hypergraphs with n vertices and the maximum possible number of edges (such hypergraphs are called extremal con(cid:12)gurations). Moreover, those extremal con(cid:12)gurations are far from each other in edit-distance. Such a phenomenon is called not stable and is a fundamental barrier to determining the Tur(cid:19)an number of M . Liu and Mubayi (2022) gave the (cid:12)rst example for 3-uniform hypergraphs to be not stable. A simple argument shows that for r > 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 (cid:12)nite family of 4-uniform hypergraphs M directly such that two near-extremal M -free con(cid:12)gurations are far from each other in edit-distance. This is the (cid:12)rst unstable example that does not depend on 3-uniform hypergraphs. We also prove its Andr(cid:19)asfai-Erd}os-S(cid:19)os type stability theorem: Every M -free 4-uniform hypergraph whose minimum degree is close to the average degree of extremal con(cid:12)gurations is a subgraph of one of these two near-extremal con(cid:12)gurations. As a corollary, our main result shows that the boundary of the feasible region of M has exactly two global maxima.