For every positive integer t we construct a finite family of triple systems \({\cal M}_{t}\) , determine its Turán number, and show that there are t extremal \({\cal M}_{t}\) -free configurations that are far from each other in edit-distance. We also prove a strong stability theorem: every \({\cal M}_{t}\) -free triple system whose size is close to the maximum size is a subgraph of one of these t extremal configurations after removing a small proportion of vertices. This is the first stability theorem for a hypergraph problem with an arbitrary (finite) number of extremal configurations. Moreover, the extremal hypergraphs have very different shadow sizes (unlike the case of the famous Turán tetrahedron conjecture). Hence a corollary of our main result is that the boundary of the feasible region of \({\cal M}_{t}\) has exactly t global maxima.
