The minimum size of 3-graphs without a 4-set spanning no or exactly three edges

Let G i be the (unique) 3-graph with 4 vertices and i edges. Razborov [A. Razborov, On 3-hypergraphs with forbidden 4-vertex configurations, SIAM J. Discrete Math. 24 (2010) 946–963] determined asymptotically the minimum size of a 3 -graph on n vertices having neither G 0 nor G 3 as an induced subgraph. Here we obtain the corresponding stability result, determine the extremal function exactly, and describe all extremal hypergraphs for n ≥ n 0 . It follows that any sequence of almost extremal hypergraphs converges, which answers in the affirmative a question posed by Razborov.