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 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 M.Liu and Mubayi(2022)gave the first 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 finite family of 4-uniform hypergraphs M directly such that two near-extremal M-free configurations are far from each other in edit-distance.This is the first unstable example that does not depend on 3-uniform hypergraphs.We also prove its Andrásfai-Erdős-Sós type stability theorem:Every M-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 corollary,our main result shows that the boundary of the feasible region of M has exactly two global maxima.
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